Thermal Physics (2nd Edition) - Kittel and Kroemer
502
Transcript of Thermal Physics (2nd Edition) - Kittel and Kroemer
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Library of Congress Cataloging in Publication pat;
Killcl, Clmrlcs.Thermal physics.
Bibliography:p.Includes index.
!. Statistical tiicrmodyn;miics. I. Kroe/nHerbert.1928- joiiii aullior. II. Tillc.
QC3H.5.K52 1930 536'.? 79-16677ISBN O-7167-IO8S-9
Copyright\302\260I9B0 by W. H. Freeman and Company
No pan of this bor.k may be reproduced by any
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Pcimcti in ilie United State of America
Twenty-first printing, 2000
9
About the Authors
Charles Kiitel has laught solid slate physics at the University of Californiaat Berkeley since 1951, having previously been at [he Bell Laboratories.Hisundergraduate work in physics was done at M.I.T. and at the Cavendish
Laboratory of Cambridge University. His Ph.D. researchwas in theorclicai
nuclear physics with Professor Gregory Breit at the University of Wisconsin.He has been awarded three Guggenheim fellowships, the Oliver Buckley Pme
for Solid State Physics, and, for contributions to teaching, the Oersted Medalof the American Association of Physics Teachers. He is a member of \"he
National Academy of Science and of the American Academy of Arts ;i id
Sciences. His research has been in magnetism, magnetic resonance, semio n-
ductors, and the statisticalmechanicsofsoiids.
iferbcrt Kroemcr is Professor of Electrical Engineering at the University of
California at Santa liurhara. His background mid training nre in solid state
liliy^ics. I !c received a I'lt.D. in physics in I'J52 from the University of Gulling
in Germany with a thesis on hot electron effects in lhc then new transistor.
Prom 1952through I96Stie workedin several semiconductor research labora-
laboratories in Germany and the United Stales. In I96S lie becameProfessor of
[ilixirieul Engineering at ltic University of Colorado; lie came to UCStt in
1976. His research has been in the physics and technology of semiconductors
and semiconductor devices, including high-frequency transistors, negative-
mass effects in semiconductors, injection lasers,the Gunn effect, electron-hole
drops, and semiconductor hetcrojunctions.
Preface
Thisbookgives an elementary account of thermal physics. The subject issimple,the methods are powerful, and the results have broad applica-applications. Probably no other physical theory is used more widely throughout
science and engineering.We have written for undergraduate students of physics and astronomy,
and for electrical engineering students generally. These fields for our
purposes have strong commonbonds,most notably a concern with Fermi
gases, whether in semiconductors, mcmls, stars, or ituclci. We developmethods (not original, but not easily accessible elsewhere) that are well
suited to these fields. We wrote the bookin the first place because wewere delighted by the clarity of the \"new\" methods as compared to (hose
we were taught when we were students ourselves.We have not emphasized several traditioual topics, some because they
are no longer useful and some because their reliance on classical statisn-cai mechanicswould make the course more difficult than we believe afirst course should be. Also, we have avoided the use of combinatorial
methods where they are unnecessary.Notation and units; We
generally use the SI and CGS systems in
parallel.We do not use the calorie. The kclvin temperatureT is related to
the fundamental temperature t by r =kBT,
and the conventional entropy
S is reialed lo the fundamental entropy a by 5 = ka(j. The symbol logwill denote natural logarithm throughout, simply because In is less ex-
expressive when set in type. The notafion A8) refers lo Equation A8) of
the current chapter, but C.18) refers to Equation A8) of Chapter3.Hie bookis ihe successor to course notes developed with the assisf-
ance of grants by (he University of California. Edward M. PurceSlcon-contributed many ideas to the first edition. We benefited from review of the
second editionby Seymour Geller, Paul L. Richards, and Nh.-holns
Wheeler-Help was giveii by Ibrahim Adawi, Bernard Black, G. Domo-kos, Margaret Geller, Cameron Hayne, K. A. Jackson, S. Justi, PeterKittel, Richard Kittler, Martin J. Klein, Ellen Leverenz, Bruce H. J.McKellar, F. E. O'Meara, Norman E. Phillips, B. RoswcltRussell,T. M.
Preface
Sanders. B. Stoeckly, John Verhoogen, John Wheatley, and Eyvind
Wichmann. We thank Carol Tung for the typed manuscript and Sari
Wilde for her help with the index.An
elementary treatment of the greenhouse effect in the Earth's atmo-atmosphere was added in 1994 on page 115, following an argument suggested
by Professor Richard Muiier. A page on aioinic gas experiments on theBose-Eitistein condensationwas added to page 223 in 2000.
For instructors who have adopted the course for classroom use, asolutionsmanual is available via the freeman web site (http:/Avhfreenian.
com/thermaiphysics).
Berkeley and Santa Barbara Charles K'tttel
Herbert Kroemer
Note to the Student
Forminimum coverage of the concepts presented in each chapier, the authors
recommend the following exercises.Chapter2: 1,2,3;Chapter 3: 1,2, 3,4, 8,
11; Chapter 4: 1,2,4, 5,6,8; Chapter 5: 1,3,4, 6,8; Chapter 6: 1,2,3,6,12,14,15;Chapter7:2,3,5,6,7, 11; Chapter 8: 1, 2, 3, 5, 6,7; Chapter 9: 1, 2, 3;
Chapter 10: 1,2,3; Chapter11:1,2,3;Chapter12: 3,4.5; Chapter 13: 1,2,
3,7,8,10; Chapter 14:1,3,4,5; Chapter 15: 2,3,4,6.
Contents
Guide to Fundamental Definitions xiii
GeneralReferences xv
Introduction 1
Chapter 1 States of a Model System 5
Chapter 2 Eittropy and Temperature 27
Chapter3 Boltzmann Distribution and Hdmholtz
Free Energy 55
Chapter4 Thermal Radiation and Planck Distribution 87
Chapter 5 Chemical Potential and GibbsDistribution 1 i 7
Chapter 6 Ideal Gas 151
Chapter 7 Fermi and BoseGases 181
Chapter8 tHeaZahd Work 225
Chapter 9 Gibbs Free Energy and ChemicalReactions 261
Chapter10 Phase Transformations 275
Chapter il Binary Mixtures 309
Chapter 12 Cryogenics333
Chapter13 Semiconductor Statistics 353
Chapter 14 Kinetic Theory 389
Chapter15 Propagation423
Appendix A Some Integrals Containing Exponentials 439
AppendixB TemperatureScales 445
Appendix C Poisson Distribution 453
Appendix D Pressure 459
AppendixE NegativeTemperature460
Index 465
Guide to Fundamental Definitions
Absolute activity, X ~exp(/i/t) 139
Accessible state 29
Boltzmannconstant,ka25
Boltzmann factor, exp\302\243~~ \302\243/t) 61'
Boson 183
Chemical potential, /; 119
Classicalregime,n \302\253nQ74
Ensemble of systems 31
Enthalpy, H = U + pV246
Entropy, a 40
Fermion 1S3
Gibbs factor,exp[(Nji- t)/i] 138
Gibbs free energy, G = U \342\200\224\\a + p^
Gibbs or grand sum, \342\226\240% 138
Heat capacity, C 63
Heat, Q 68,227
HelmhoHz free energy, F \342\200\224U \342\200\224xa t
Landau free energy function, FL 298
Multiplicity,g 7
Orbital 9
Partitionfunction, Z 61
Guide to Fundamental Definitions
Quantum concentration, hq=
Reversible process 64
Temperature, t 41
Thermal average 62
Thermal equilibrium 39
Work, W 227
General References
Thermodynamics
A. B. Pippard, Elements of classical thermodynamics, Cambridge University Press,1966.
M. W. Zemansfcy and R. H. DiEEman, Heat anil thermodynamics: an intermediatetextbook, 6ih ed., McGraw-Hill, 198!.
Sitttisiical Afcchanics
U. K, Agarwal and M. Eisner, Statistical mechanics, Wiicy, 1988.
T. L. Hit), Statistical mechanics:principlesand selectedapplications, Dover Pubtica-
iions, 1987, cl956.C. Kittct, Elementary statistical physics, Wiicy, 1958. Parts 2 and 3 treat applications
to noise and to elemeniary transport Eheory. Part 1 has been expanded ioEo thepresent Eext.
R. Kubo, Statistical mechanics, North-Holland, 1990,cI965.R, Kubo, M. Toda, N. Hashitsume, Statistical physics !! (NanequHibrium), Springer,
1985.
L D. Landau and E. M. Lifshitz, Statistical physics, 3rd cd. by K. M. Lifshitz and L. P.
Piiaevskii, Pcrgamon, 1980,part 1.
Shang-Keng Ma, Statistical mechanics. World Scientific, 1985.
M. Toda, R. Kubo, N. Saito,Statisticalphysics ! (Equilibrium), Springer, 1933,
Mathematical tables
H. B. Dwight, Tables of integrals and other mathematical data, 4ih ed., MacmUton,
1961. A widely useful smati collection.
Applications
AsirophysicsR. J. Taylor, The stars: their structure and evotitiioit. Springer, 1972.
S. Weinbcrg, The first three minutes: a modern v:\\-w of the origin of the universe, new
ed., Bainam Cooks, 1984.
Biophysics and macromolccules
T. L. Hill, Cooperathity theory in biochemistry: steady stale and equilibrium systems,
Springer, 1985.
General Refer,
Cryogenicsand low lempcrature physics
G. K. White, Experimental techniques in low-temperature physics, 3rd ed., Oxford
University Press, 1987, ct979 . . pa.J. Wilks and D. S. Betis, An introduction to liquid helium, 2nd ed , Oxford Univesity
Press, 1987.
Irreversible thermodynamicsJ. A. McLennan,Introduction to non-equilibrium statistical mechanics, Prentice-Hall,
1989.I. Prigogine and I. Stcngers, Order out of chaos: man's new dialog with nature.
Random House, 1934.
Kjnclic theory and transport phenomena
S. G. Brush, The kind of motion we call heal, North-Holland, 1986, cI976.H. Smith and H. H. Jensen, Transport phenomena,Oxford University Press, 19S9.
Plasma physics
I...Spitzer, Jr., Physical processes in the interstellar medium, Wiley, 197S.
Phase transitions . .
P. PfeiHy and G. Touiouse, Introduction to the renormalizat'ton group and to criticalphenomena, Wiley, 1977.
H. E. Stanley, Introduction to phase transitions and critical [ihenomena, Oxford Uni-University Press, 1987.
Metais and affoys
P. Haasen, Physical metallurgy, 2nd ed., CambridgeUniversity Press, 1986. Superb.
Boundary value problems
H.S-Carslaw and J. C. Jaeger, Conduction of heat in solids, 2nd ed., Oxford Univer-sily Press,19S6,ci959.
Semiconductor devices
R. Datven, Introduction to appliedsolidstate physics, Plenum, t990.
K. Seeger, Semiconductorphysics:an introduction, 5th ed., Springer, 1991,S. M. Sze,Physics of semiconductor devices, 2nd ed., Wiley, t981.
Solid state physics
C. Kittel, Introduction to solid state physics, 6th ed., Wiley, 1986. Referred to ssISSR
Thermal
Introduction
Ourapproachto thermal physics differs from the tradition followed in beginning
physics courses. Therefore we provide this introduction 10setoul what we are
going to do in the chapters that follow. We show the main lines of the logicalstructure:in this subject all the physics comes from the logic. In order of lhcir
appearance,the leadingcltaractersin our story are the entropy, the temporaiure,the Boltzmann factor, the chemical potential, the Gibbs factor, and the disiribu-
tion functions.
The entropy measures the number of quantum states accessible to a system.A closed system might be in any of these quantum states and (weassume)with
equal probability. The fundamental statistical element, ihe fundamental logical
assumption, is that quantum states are eitheraccessibleor inaccessible to the
system, and the system is cquaiiy likely to be in any one accessible slate as in
any olher accessible slate. Given g accessiblestates, the cniropy is defined as
a = logg. The entropy lhtis defined will be a function of ihe energy U, lhe
number of particles N, and the volume V of the system, because theseparam-parameters ciilcr the dctcrminaiion of y; other parametersmay enter as wirii. The
use of the logarithm is a mathematical convenience: it is easier to write 1010than expA020),and it is morenatural for two systems to speak of a-y + o, lhan
ofg,3j.When two systems, each of specified energy, are brought into thermal coniact
ttiey may transfer energy; their total energy remainsconstant,but the comlraints
on their individual energies are lifted. A transfer of energy in one direction, or
perhaps in the other, may increase the product g^g, that measures the tiumber of
accessiblestates of thecombinedsystems. The fundamental assumption biases
the outcome in favor of that allocation of the total energy that maximizes the
number of accessible states: more is better, and more likely. This statement isthe kernel of the law of increase of entropy, which is the general expression of
the second law of thermodynamics.
We have brought two systemsinto thermalcontactso that they may transfer
energy. What is the most probableoutcomeofthe encounter? One system will
gain energy at lhe expense of the other, and meanwhilethe lotal entropy of the
two systems will increase. Eventually the entropy will reach a maximum for
the given total energy. It is not difficult to show (Chapter 2) that the maximum
is atiained when ihe value o((ca/cU}K_y for one system is equal to the value ofihesamequantity for the Second system. This equality property for Iwo systems
in ihermai coniaa isjust the property we expect of the icmperat lire. Accordingly,we define the fundamental lemperaiure i by the relation
1
CUJU)
The use of 1/rassuresthat energy will flow from high x to low r; no more com-
complicated rclaiion is needed. it will foilow'that ihe Kelvin temperature T is
directly proportional to t, with i = Ayr, where kB is the Boltzmannconstant.The conventional entropy 5 is given by .S\" - kRa.
Now consider a very MiiipJe csampli? uf ibe Dolt/nvnm factor treau-d in
Uiajner 3. i.ti a Miiall sysicm wiih uiilyiwn sinies, i.fie :it cnorj>y tl :iml une :n
energy c, be pjaeed in thermal coniaci with u large system Uial we eaH thereservoir.The loia! energy of the combined systems is UQ; when the smallsystem is in ihe stale of energy 0. the reservoir has energy Uo and wiil have
g{U0) states accessibleto it. When the small sysiem is in the slate of energy e, thereservoirwill have energy Uo \342\200\224\302\243and wiii have g{U0 \342\200\224
e) slates accessible lo
it. By the fundamental assumpiion, the ratio of tbe probability of finding [hesmali system with energy s to the probability of finding it with energy 0 is
, - c)^@) gWo) exp[<*(f o)]
The reservoir entropy a may be expanded in a Taylor series:
g{U0-e)
~a{Ua)
~\302\243{ca/tV0)
= c(t/0)- e/r ,
B)
C)
by the definition A) of the temperature. Higher order termsin the expansion
may be dropped. Cancellation of the term exp[_a{U0)], which occurs in the
numerator and denominator of B) after the substitution of C), leaves us with
This is Boitzmantt's result. To show its use, we calculate the thermal average
energy (e) of the two state system in thermal contact with a reservoir at tem-
temperature t;
E)
where we have imposed the normalization conditionon the sum of the prob-probabilities:
P@) + P{e) - 1. F)
Theargument can be generalized immediately to find the average energy of a
harmonic oscillator at temperature r, and we do this in Chapter 4 as the first
step in the derivation of the Planck radiationlaw.
The most important extension of the theory is to systems that can transfer
particles ns well as energy with ttie reservoir. For two systems in diffusive andthermalcontact,the entropy will be a maximum with respect to the transfer
of particles as well as to the transfer of energy. Not only must {ca/cU)\\- %\342\226\240be
eqtuti for Hie two systems, btil (f it/jW)^,. must also be equal, where N refers to
the mmilKr of particles of ;i {uwn spevk's.Tin-* ih'w cttii:tiity coiutiiion is Hio
For two systems in thermal and diffusive contact, r, <=r2 and jt1 =
\\it. The
sign in G) is chosen to ensure thai the direction of particle flow as equilibriumis approachedIs from high chemicai potential to low chemical potential.
The Gibbsfactorof Chapter 5 is an extension of Ihe Bolizmann factor and
ailows us to treat systems ilia! can transfer particles.Thesimplestcx:unpieis a
system with two states, one witli 0 panicles and 0 energy, and one with 1 parltcie
and energy e. The system is in contact with a reservoir at temperature r andchemicalpotential;i. We extend C) for the reservoir entropy:
tf(f/0-
t;HQ- 1)=
0(r/o;A'o)- z{ca/dUo) - l-{ca/dNo)
~o(U0;NQ)
~c/r + ^/r. (S)
By anaiogy with D), we have
P{U)/P@,Q) =\302\273cxp[{/(
-\302\243)/r] , (9)
for the ratio ofihe probability thesystem
is occupied bv J particle at energy z
to the probability the system ts unoccupied, with energy 0. The result (9)after
normalization is readily expressed as
A0)
This particular result is known as the Fermi-Dirac distribution function and
is used particularly in the theory of metals to describe the electron gas at low
temperature and high concentration (Chapter 7).The classicaldistribution function used in the derivation of the idealgas law-
is just the limit of A0) when the occupancyPA,e)is much lessthan 1:
The properties of the ideal gas aredevelopedfrom this result in Chapter 6.The HeSmholtzfree energy F= V \342\200\224to appears as an important computa-
computational function, because the relation {5F/cx)s v= \342\200\224a offers the easiest method
for finding the entropy, once we have found out how to calculate F from the
energy eigenvalues (Chapter 3). Other powerful tools for the calculation of
thcnnodyiumic functions arc developed in the text. Most of the reiftaiiuier of
the tcxl concerns applications that are useful in their own right and that illumi-illuminate the meaning and utility of the principal thermodynamic functions.
Thermalphysics connects the world of every iky objects, of astronomic;*!objects,and of chemical and biological processes with the world of moiecular.
atomic, and electronic systems. It unites the two parts of our world,the micro-
microscopic and the macroscopic.
Chapter 1
States of a Model System
BINARY MODEL SYSTEMS 10
Enumeration of States and the Multiplicity Function UBinary Alloy System 16
Sharpness of the Multiplicity Function 18
AVERAGE VALUES :\342\226\240 22
Energy of ihe Binary Magnetic System' ' --. -3
Example: Multiplicity Function for Harmonic Oscillators 24
SUMMAItY 26
Chapter !: States of a ModelSystei.
Bul although, as a matter of history, statisticalmechanicsowes its origin to
investigations in thermodynamics, it seems eminently worthy of an independent
development, both on account of the elegance and simplicity of its principles,and becauseit yields new results and places old truths in a new light in
departments quite outside of thermodynamics,J. W. Cibbs
A theory is the more impressive the greater the simplicity of its premises, themore different kinds of things it relates, and the more extendedits areaofapplicability. Therefore the deep impression that classical thermodynamics made
upon me. It is ihu only physical theory of universal content which I am convinced
mil never be overthrown, within the framework of applicability of its basicconcepts.
A. Einstein
Chapter 1: States of a ModelSyster,
Thermal physics is the fruit of the union ofslatistica! and mechanical principles.Mechanicstellsus the meaning of work; thermal physics tells us the meaning of
heat. There are three new quantities in thermal physics that do not appear in
ordinary mechanics:entropy, temperature, and free energy. We shall motivatetheir definitions in the first three chapters and deduce their consequencesthereafter.
Our point of departure for the developmentofthermal physics is the concept
of the stationary quantum statesofa system of particles. When we can countthe quantum states accessibleto a system, we know the entropy of the system,for the entropy is defined as the logarithm of the number of states (Chapter 2).The dependenceof the entropy on the energy of the system defines the tempera-temperature.From rhe entropy, the temperature, and the free energy we find the pressure,the chemicalpotential,and all other therm odynamic properties of tlie system.
Fora system in a stationary quantum state, all observable physicalpropertiessuch as the energy and the number of particlesare independentofthe time.Forbrevity we usually omit the word stationary; the quantum slates that we treatare stationary exceptwhen we discuss transport processes in Chapters 14-55.
The systems we discuss may be composedof a single particle or, more ofi^n,
of many particles. The theory is developed to handle genera! systems of inter-
interacting particles, but powerful simplilicarions can be madein specialproblemsfor which the interactions may be neglected.
Each quantum slate has a definite energy. Stales with identical energies arc
said to belong to the same energy level. The multiplicity or degeneracy of an
energy level is the number of quantum states wiih very nearly the some energy,it is the number of quantum slates that is important in thermal physics, not
tlte number of energy levels. We shall frequently deal fti'th sums over all quantumstates. Two states at the same energy must always be counted as two slates,
not as one level.
Let us look antic qiu.Umusuresam! eitcryy lewis uOveia! alomic s>^u-ms.The simpler is hydrogen, with oik electron and one proron. The !u.\\-I>iiigenergy levels of hydrogen are shown in Figure !.!. The zero of energy in the
figure is taken at the state of lowest energy. Tlte number of quantum stales
belongingto the sameenergy level is in parentheses, in lite figure we overlookthat the prolon has a spin of \\h and has two independent orientations, parallel
Chapter 1; States of a Model SyMm
Hydrogen
Low-tying ei
Lithium Boron
lergy levelsof atomic hydrogen, lithium, ant! boron. Theenergiesare given in electron votis, with t eV = 1.602 x tO\0211 erg. The numbers \302\243
parentheses give ihe number of quantum stales having the same energy, with no ac
lakenofthe spinoflhe nucleus.The zero of energy in the figure is taken forcouvei
ai die lowest energy slaie of each aiom.
or antiparaliel to the direction of an arbitrary externalaxis,such as the direction
of a magnetic field.To takeaccount of the two orientations we should doublethe values of the multiplicities shown for atomic hydrogen,
An atom of lithium has three electrons which move about the nucleus. Each
electron interacts with the nucleus,and eachelectronalsointeracts with ail the
Chapter I: Statesofa Model Systet
30
25
20
i 15
5 to
Mill lip liciiy
J
-
3
_
IUdiq
4
4
4
3
43
3
23
2
1
presc
3
2
2
3
1
2
2
2
1
1
'
1
1
2
i
2
1
\\
1
Figure 1.2 Energy levels, multiplicities, and quanlum numbers
\302\253_,,jjj,, n, ofa particle confined to a cube.
other electrons.Theenergiesofthe levels of lithium shown in the figure are the
collective energies of the entire system. The energy leveis shown for boron, whichhas five electrons, are also the energies of the entire system.
The energy ofa system is the total energy ofal! particles, kinetic plus potential,with account taken of interactions between particles. A quantum state of the
system is a state ofall particles.Quantum states ofa one-pariicie system arecalled orbitals.The low-lying energy levels ofa single particle of mass M con-confined to a cube of stde \302\243are shown in Figure 1.2. We shall find in Chapter 3
Chapter 1: State*ofa Model System
that an orbital of a free particlecanbe characterized by three positive integral
quantum numbers nf, nyin,. The energy is
The multiplicities of the levelsare indicated tn the figure. The three orbitals
wiih(HJl^>.fi.)equ.iltoD,l,l),(l,4,]),and{l,l,4)ai!l!aveii/ + nr2 + n.1 = 18;the corresponding energy levelhas the multiplicity 3.
To describe the statistical properties of a system of A' particles, it is essemial
to know these!ofvalues of the energy \302\243S(N),where c is theenergy of the quantumsiaie soTthejV particle system. Indices such as s may be assigned lo the quantumstates in any convenient arbitrary way, bul two different states should not beassignedthe same index.
It is a good idea to siart our program by studying the properties of simplemodelsystems for which the energies Ej{ A') can be calculated e.vacily. We chooseas a modela simplebinary system because the genera! statisiical propeniesfound for the model system are believed to appiy equally well to any realistic
physical system. This assumptionleadsto predictions that always agree with
experiment. What general statistical properties are of concern will become clear
as we go along.
BINARY MODEL SYSTEMS
Thebinary model syslern is illustrated in Figure 1.3. We assume there are N
separate and distinct sites fixed in space, shown for convenience on a line-Attached to each site is an elementary magnet titat can point only up or down,
corresponding to magneticmoments \302\261n>.To understand tlie system means to
count the slates.This requitesno knowledge of magnelism: an element of the
system can be any site capable of two states, labeled as yes or no, red or blue,
occupied or unoccupied,zeroor one,plus one or minus one. The sites arenumbered,and sties with dtSFercni numbers are supposed not to overlap in
physical space. You might even tltink of the sites as numbered parking spaces in
a car parking lot, as in Figiire 1A Cacti parking spuce has two states, vacant or
occupied by one car.Whatever llic milure of otlr objects, we may desigreiic the two slates by
arrows that can only point straight up or straight down. If (he magnet pointsup, we say thai ilie magnetic moinenr is -Hii.If the magnet points down, the
magnetic moment is -m.
Binary Model Syster
123456789 10
Number of the site
Ffgure 1.3 Mode! system composedof 10elementary
magnets at fixed sites on a line,each having magncric
moment \302\261m.The numbers shown arc aflachcd to ihc silcs;each sire has ils own magnet We assume there are no
magne'ic field.Each magnetic moment may be oriented in
two ways, up or down, so ihai there are 210disiinclarrangements of the 10 magnetic moments shown in the
figure. If ihe arrangements arc selectedin a random process,(he probability of finding tile particular arrangement shownis 1/210.
Figure 1.4 State ofa parking lot with 10 numbered parking
spaces. TiseO's denotespaces occupied by a car; theO'sdenote vacant spaces. This particular state is equivalent to that
shown in Figure (.3.
Now consider N different sites, each of which bears n moment thai may
assume the values +\302\253i. Each moment may be oriented in two ways with a
probability independent of the orientation ofa!! other moments.The total
number of arrangements nf the N moments is 2 x 2 x 2 x \342\226\240\342\200\242\342\226\2402 *= 2\\ A
state of the system h sjveitiedUy yiviiig the orient at ion of the moment oilc:k!isite;there are 2'v states. We may use ilio followingsimplettotation for a single
state of the system of N sites:
nuimrr- B)
\342\226\240rl:States of a Model Syst
Figure 1,5 The four diflercnl Malesof as> stem of two elements numbered | and 2,vs here ench clctnetit can hsvc two conditions
The element is a magnel which can be in
condition f orcondiiion [.
The sites themselves are assumed to bearrangedin a definite order. We maynumbcr4hem in sequence ftom left to right, as we did in Figure1.3.According
to this convention the state {2}also can be wriitcn as
C)
Both sets of symbols B) and {3}denotethe same state of the system, the slatein which the magnetic moment on site 1 is +m; on site2, the moment is -t-m;
on site 3, the moment is -m; and so forth.
It is not hard to convince yourself that every distinct state of the system iscontainedin a symbolic product of N factors:
U)(U D)
Themultiplication ruie is defined by
(Tt + liXti + li) ti + till + UU + E)
The function D) on muitipltcationgeneratesa sumof2*v terms, one for each of
the 2'v possible states. Each term is a product of N individual magnetic moment
symbols, with one symbol for each elementary magnet on the line.Eachtermdenotesan independent state of the system and is a simpleproductof the form
T1T3I3''\"
t\\i f\302\260rexample.
For a system of two elementary magnets,we multiply (}x + li)by(t2 + |j)to obtain the four possible states of Figure 1.5:
(Ti + I1KT2+ ii) Till itTa + I1I2. F)
The sum is not a state but is a way of listingthe four possible states of the system.The producton the left-hand side of the equation is calleda generatingfunction:
it generates the states of the system. . \342\226\240 \342\226\240\342\226\240
.
Binary Model Systems
The generating function for the slates of a system of three magnetsis
(Ti + li)(?2+ IjHTj + U)-
This expressionon multiplicationgenerates21 = S different states:
Three magnets up: T1T1T3
Two magnets up: T1T2I3 T1I2T3 lihti
Onemagnctup: tihli IJ2I3 lilif3None up: lilils-
The totat magnetic moment of our model system of N magnets each of
magnetic moment m will be denoted by Mt which we will relate to the energyin a magnetic field. The value of M varies from Nm to -- Nm. The set of possiblevalues is given by
M \302\253Nm, {N - 2)m, (N -
4>n, (N- 6)m,
\342\200\242\342\200\242\342\200\242,-A'\302\273i- G)
The set of possible values of M isobtainedif we start with the state for whichallmagnets are up (M = Nm) and reverseoneat a time. We may reverse iV magnets
to obtain llie ultimate state for which al! magnets are down (A/= - Nm).
There are N + ] possible values of the total moment, whereas thereare2sstates.When N \302\273!, we have 2N \302\273N + 1. There are many more states than
values of [he total moment. !fW = 10,thereare210= ! 024 states distributed
among 11 different values of the total magnetic moment. For large N manydifferent states of the
system may have the same value of the total moment ft/.
We will calculate in the next section how many states have a given value of M.
Only one state of a system has the moment M = Nm;that state is
TTTT--
-TTTT- (S)
There arc N ways to form a slate with one magnet down:
mt \342\200\242\342\226\240\342\226\240mt AJ)
is one sue! 1 state; another is
tin \342\226\240\342\226\240.\342\226\240tin,
\"
do)
Chapter 1: Slates of a Mode!System
and the other slates with one magnet down are formedfrom (S) by reversing
any single magnet. The states(9) and \302\243!0)have lot.il moment M = Nw - 2w.
Enumeration of Stales and the Multiplicity Function
We use the word spin as a shorthand for elementary magnet. It is convenient loassume that N is an even number. We need a mathematical expressionfor the
number of states with W, ={W + s magnets up and Nl = jN \342\200\224s magnets
down, where sis an integer. When we turn one magnet from Ihe up to [hedown
orientation, {.V + 5 goes to jW + s - I and ?N- s goes to jiV
~ 5 + I.
The difference (number up \342\200\224number down) changes from 2s to 2a \342\200\2242. The
difference
W, -/V,
= 25 (ID
is called ihc spin excess. The spin excess of the 4 states in Figure 1.5 is 2,0, 0, \342\200\2242,
from left to right. The facior of 2 in (I!} appears to be a nuisance at this stage,but it will prove to be convenient.
The productin D) may be written symbolically as
\342\226\240.(T + if-
We may drop the site labels{thesubscripts)from D) when we are interested
only in how many of the magnets in a state are up or down, and not in which
particular sites have magnets up or down, ft we drop the labels and neglectthe order in which the arrows appear in a given product, then E) becomes
(t-
II;
further,
(t + I)' = Itt + 3ItJ + 3IJJ + jjj.
We find (I + |)v for arbitrary iV by the binomial expansion
A2)
Enumeration of Stales and the Multiplicity Function
We may write the exponents of x and y in a slightly different, but equivatem,form by replacing t with \\N
\342\200\224s:
With ihis result ihe symbolic expression{| + |)'v becomes
4- ivv =* y tj-v+j M*V\"JA4)
The coefficient of the term in fiA-+* M*\"' is the number of stittes having
W,= $N + 5 magnets up and N, = i.V - s magnets down. This class of
states lias spin excess JV, \342\200\224JVj
= 2s and net raagneiic moment 2sm.Let us
denote the number of states tn this class by g{N,s), for a system of N magnets:
,n\342\202\254>T
Thus A4) is written as
(IS)
(I + i)'v= I stJMT^l1\"\" A6)
We shall call g(N,s) ihemultiplicity function; it is ihe number of slates having
lliesamevalue of 5. The reason for our deltnttion emerges when a magnetic
field is appliedto the spin system: in a magnetic field, stales of different values of
s have different values of the energy,so that \302\260ur9 is equal to the multiplicityof an energy level tn a magnetic field. Until we introduce a magneticfield, all
states of the model system have the sameenergy, which may be taken as zero.
Note from A6) that the total number of states is given by
'
L g{Nts) =A + l)-v = 2-v (H)
Examples related to g{h',s)for A' ~ \\Q are given in Figures 1.6 and 1.7. For a
coin, \"heads\" could stand for \"magnet upland \"tails\" could stattd for \"magnet
down.\"
Chapter t: Slatesofa Model System
Figtorc 1.6 Number of distinct arrangements
of 5 -f j- spins up and 5 ~ 5 spins down.Values of yf Npi) are for N - 10, when: 2.v h
tUc spin oixss N \\ - K I. Tlic toul numtwt of
stales is
TTicvalues of the 9's arc taken frothe binomial coefficients.
I -8 I -4 j 0 2 4 6-10 -6 -2
Spin excess 2s
Binary Alloy System
To illustrate that the exact nature of the two states on each site is irrelevant tothe result,we consider an alternate system\342\200\224an alloy crystal with N distinct
sites, numbered from 1 through 12 in Figure 1.8. Each siteisoccupiedby either
an atom of chemical species A or an atom of chemical species B,with no provi-
provision for vacant sites. In brass, A could be copper and B zinc. In analogy to C),a single state of the alloy system can be written as
- A8)
nry Allay Sya,m
\342\226\240=o
S 3 20
01 23456789 10Number of heads
Figure 1,7 An experiment Was done in which 10 pennieswere thrown NX) times. The number of heads in each
throw was recorded.
0\302\25100I 2 3 A
\302\251 0
0 0
Fijutc 1.8 A binary alloy syslcm of two
chemical componenls A and 1!,whoseatoms
5 6 7 S
10 II 12
Every distinct state of a binary alloy system on N sites is contained in the
symbolic product of N factors:
(A, + B1)(A3 4- B2)(A3 + Bj)---(A.V +BN) , A9)
in analogy to D). The Liverage composition of a binary alioy is specified con-
conventionally by the chemical formula A1_1B1, which means thai out of a tola!of N atoms, the number of A atoms is NA ~
A ~x)N and the number of Batoms is NB
~ .\\JV. Here .v lies between Oand 1.Thesymbolicexpression
is analogous to the result A2). The coefficient of the term in A'v\"' B' gives thenumber g{i\\\\f) of possible arrangements or states of N ~
\302\243atoms A and /
atoms B on N sites:
which is identical to the result A5) for the spin model system, except for notation.
Sharpnessof ihe MultiplicityFunction
We know from common experience that systemsheldat constanttemperatureusually have well-defined properties; this stability of physicalpropertiesis amajorprediction of thermal physics. The stability follows as a consequenceofthe exceedingly bharp peak in the multiplicity function and of the steep variation
of that function away from the peak. We can show explicitlythai for a very
large system, the function <j(Af,s) defined by A5) is peaked very sharply aboutthe value 5 = 0. We look for an approximation that allows us to examine the
form of g(S,s) versus 5 when jV \302\273i and js| \302\253N. We cannot (ook up these
values in tables: common tables of factorials do not goabove N = 100, and we
may be interested in Af =^ 10'\302\260,of the order of the number of atoms in a soiid
specimen big enoueli to be seenand felt. An approximation is clearly needed,and a goodoneis available.
It is convenient to work with log*/, f-xccpl where .-ilierwise specified, till
logarithms are understood to be loy base*?,written here as log. The international
standard usage is In for log base c, but it is clearer to write log when there is no
ambiguity whatever. When you confront a very, very large number such ;is
Sharpnessof the Multiplicity Function
2iV, where iV ~ lG:o,it is a simplification to look at the logarithmoft he number.
We take the logarithm of both sidesofA5) lo obtain
\\ogcj(N,s) = logN! - logfrV + 5)!- log&N- s)l , B2)
by virtue of ihe characteristicproperlyofthe logarithm of a product:
logxy = logx + logy; log(.\\-/_v)= log.v ~ logy. {23)
With the notation
jV, =\\N 4- 5; A', =\302\253
\\N~ s B4)
for the numberof magnetsupand down, B2) appears as
logg(N,s) - logW! - logiV,!- logW,!. B5)
We evaluate the logarithm of N\\ in B5) by use of die Stirling approximation,accordingtowhich 1 ZjTi'Go
s-t\\Q\\ \\
.V! a BrtN)ti:iNNcxpl~N + IJ[12N) + \342\226\240\342\200\242\342\226\240], B6)
for N \302\2731. This result is derived in Appendix A. For sufficiently large N, the
terms 1,A2iV) + \" \342\226\240in the argument may be neglected in comparison with N.
We take the logarithm of both sidesofB6)toobtain
logN! S I log 2k + (N + |) log N ~ N. B7)
Similarly
!ogjVt! s I log2?r + (JV, + JJlogN, - jV,; BPJ
logN,! s iiog2* + (,Vi + J)log.V,- Nu B9)
After rearrangement of B7),
log.Y! S Ito\302\243B!r/A') '.- f.V, + \\ + -V, + i)logJV- (.V,+ Nj) , CUf
wlK'i'ciwcliaVcuscilW = .V, I- .Vt. \\VL-siil>ii;iLl{2S);iiKM2'J)froin{.!U)loobl:iiii
for B5):
tog</ S ilog(l/2;r,V) -(.V, + iJlogty./.Y)
- (,V, -f i)Uui(.V,/.V). C1)
This may be simplified because
logOV,/iV) = log^l 4- 2s/jV)= -iog2 + iog(i -f- 2s/S)
~ -iog2 + Bs/,V) - I2s2/N2) C2)
by virtue of the expansion logfl + x) = .v- jx2 + \342\200\242\342\226\240
-, valid for x \302\253I.
Similarly,
!og(W,/A') = logld - 2s/N)= -log2 -B5/N)
- Qs2/,\\'!). C3)
On subsliiuiion in C!) we obtain
iogg s |logB/;E/V) + Wiog2-
2s!/N. C4)
We write this result as
C5)
WnH)\"*?1. C6)
Such a distribution of values of sis caiicda Gaussian distribution. The integral*
of C5) over the range \342\200\224co to + co for s givesthe correctvalue 2*for the total
number of states. Several useful integrals are treated in Appendix A.
The exact value of y(N,0) is given by A5) with s = 0;
C7\302\273
\342\200\242The replacement of a slim by an iniegrai, such as \302\243{. - \342\226\240)by f{_. .)Js, usually does
significanl errors. For example, ihc rai io of '
X s =--i{N2 + N) to Ts./s = IN2
\302\273^o
is equal io 1 + A/N), which approaches i as N approaches co.
10
X.\" 6
4
0 y
1
1
n
\\
\\
\\
\342\200\224
Figure 1.9 Tlie Gaussian approsimaiion (o(he binomial cocfficicnis g{!OO,s) plotted on a
iinearscaie. On this scale ii is not possible iodistinguish on ihc drawilig the approximation
plotted The entire range of s is from - 50 to
4- SO. The dashed linesai\302\253drawn from the
points at t/e or (he maximum vaiuc or y.
-20 -10 0
For N = 50, the value of sE0,0) is 1.264 x 101*,from C7).The approximate
vylue from C6) is 1.270 x 1014.Thedistribution plottedin Figure t.9 is centered
in a maximum at s = 0. When s1 = {N, the value of g is reducedtoe~' ofthe
maximum value. That is, when
s/N = A/2NI'2 , CS).
the value of g ise\021of g(N,0).Thequantity A/2NI'1 is Ihus a reasonable mea-measure of the fractional width of the distribution. For N =s 1022, the fractional
width is of the order of 10\"u. When N is very large, ihc distribution is exceed-exceedingly sharply defined, in a relative sense. It is this sharppeakand the continued
sharp variation of the multiplicity function far from the peak that will lead to a
prediction that the physical properties of systems in thermal equilibriumare
well defined. We now consider one suchproperty,
the mean value of s1.
ChapterI: Stales of a Modal System
AVERAGE VALUES
The averagevalue, or mean vaiue, of a function f(s) taken overa probabilitydistribution function P(s) is defined as
</> = Z/<*m*), 09)
provided that Uic distribution function is normalized to unity;
\302\243?(*)-!. D0)
The binomial distribution A5) has liteproperty A7) that
Zs(iV,5)\302\2532W , D1)
and is nol normalized lo unity. If all states are equally probable, ihen P(s) ~g{N,s)/2s, and we have
\302\243J'($)
- t. Tfic average of/(s) over this distribution
will be
D2)
Consider tile function f{s) = s2.In the approximation that led to C5) and
C6),we replace in D2) the sum\302\243
over s by an integral J\342\200\242\342\226\240\342\226\240
ds between ~- co
and + co.Then
- [2/nNI'3(jV/2)J'JJ^Jxx^\"'*
= {2/7r,V)\"!{.V/2):i'J (jt/4)\022 ,
whence
<i-J>= iiV; <B,y> = ,V. D3)
The quantity <{2i);> is the mean square spin excess.The root mean square
spin excess is
<B5}2>';J =>yfN t-
D4)
<uy Magnetic Sys
and the fractional fluctuation in 2s is defined as
D5)
The larger N is, the smaller is the fractional fluctuation. This means that the
central peak of the distribution function becomes relatively more sharplydefined as the size of the system increase;, the size being measured by ihenumber of sitesN. For10:oparticles, $F - 10\0210,which is very small.
Energy of the Binary Magnetic System
The thermal properties of the model system become physically relevant when
the elementary magnetsarc placedin a magnetic ticid, for ihen the energiesofthe different states ate no longer all equal. If the energyof the system is specified,
ihen only the suites having this energy tn;ty occur. The energy \302\273fink-net ion
ofa single magnetic moment m with a fixed external magnetic field B is
V \302\273-m-B. H6)
This is the potential energy of the magnet m in the field B.
For the model system of Ar elementary mngncts, each with two allowedorientations in a uniform magnetic field 1J,the tola!potentialenergy
U is
m,- =Q. 2smB}=- MB , D7)
using ihe expression M for the toial magnetic moment 2s/n.In this example lite
spcclnttn of values of ihe encrcy U is discrete. We shall sec later (hat a con-continuous or quast-conttnuous spectrum will creafe no difficulty. Furthermore,ihe spacingbetween adjacent energy levels of (his model is constant,as in
Figure t.tO. Constant spacing is a special feature of the particular tnodcl, but
this feature will not restrict the generality of (he argument that is developedm
ihe Following sections.
The value ^f (he energy for moments ihat inferact only with the exicnial
magnetic field ?.; ^ornrfcicly ridennined by the value of s. This funcitotutl
dependence is, i..ucaka by wrtitng U{s). Reversing a single moment lowers
2s by -2, lowers the total magneticmoment by ~2m, and raises the energy
by 2mB.The energy difference bsiween adjacent levels is dcnotcJ by Ac, where
Ae = DS)
Chopset
+ 2 \342\226\240
+ 3
+ 4
+ 5
UM./mB
+ 10+ S
'\342\226\240
+ 6
+ 4
+2
0
__2
-4
-6
-8
-10
s(.)1
10
45
120
210
252
210
120
45
10
1
log g(
0
2.30
3.81
4.79
5.355.53S.3S
4.79
3.81
2.30
0
magnetic nmmersis m m a magnetic field S. The levelsare labeledby their s values, where 2s is ihe spin excess
and \\N + s ==\342\226\2405 + i isihe number of tip spins. The
energies UD ititd muliiplictlics g(>) ^fe showti. Tor this
Example: Multiplicity function for harmonic oscillators. The problem of tlic binary model
system is the simplest problem for which an exact solution for the multiplicity function is
known. Another exactly solvable problem is the harmonic oscillator, for which the solutionwas originally given by Max Planck. The original derivation is often felt to be not entirely
simple. The beginning Sludent need nol worry about this derivation. The modern way to
do the problem is given in Chapter 4 and is simple.The quantum slates ofa harmonic oscillator have the energy eigenvalues
es = sho) , D9)
where the quantum number s is a positive integer or zero, and to is the angular frequency of
the oscillator. The number of states is infinite, and the multiplicity of each is one. Now
consider a system of N such oscillators,all of the same frequency. We want to find the
number of ways in which a given total excitation energy
can be distributed among the oscillators. That is, we want the multiplicity function g{N,n)for tlie Af oscillators. The oscillator multiplicity function is not the same as the spin mufti-
pitcitv function fount! e^rher.
We begin the analysis by going back to tlie multiplicity function for a single oscillator,
forwm'chff(i,\302\253)= 1 for ail values of the quantum numbers, here identical to m. To sojve the
problem of E3) below, we need a function to represent or generate ihe scries
E1)
AS!Y,fl!!1 from ^ S\302\260co- ^CfC ' 's Jusl a temporary tool that will help us find the result
(S3),but t docs not appear in the final result. The answer is
(S2)
provided we assume\\i\\
< |. For the problem of JV oscillators, the generating function is
E3)
becausetlie number of w;iys n term i\" can ;\\\\i\\Kai in the N-fold pftiJuct is picciscly ihe
number of onSctedwuys in which the integer n c;m be foiuicJ as the sum of iV non-iicg.nive
We observe that
tj{N,n)
Thus for the system of oscillators,
2) - \342\200\242\342\226\240(W + n - 1). E4)
ES)
This result will be needed in solving a problem in the next chapter.
Chapter 1: States of a Mode!System
SUMMARY
1. The multiplicity function for a syslemof N magnets with spin excess 2s =
N, -N't is
In ihe limit s/N \302\2531, with A' \302\2731, we have the Gaussian approximation
g[N,s) * {2/rlN)m2xexp{~2s2/\\').
2. Ifal!states of the mode! spin system are equally likely, the average value of2
52> =j''^JsstgtN,s) p
in the Gaussian approximation.
3. The fractional fluctuation of s2 is defined as (s2yll2/N and is equal to
S/2N\022.
4. The energy of the modelspin syslem in a siaie of spin excess2sis
where in is the magnetic moment of one spin and B is the magnetic field.
Chapter 2
Entropy and Temperature
FUNDAMENTAL ASSUMPTION 11
PROBABILITY 3'\\
Example: Construction of ;in Ensemble 3-Most ProbableConfiguration 33
Example: Two Spin Systems in Thermal Contact 3?
THERMALEQUILIBRIUM 39
TEMPERATURE
ENTROPY -\342\226\240!
Example: Entropy Increase On Heat Floiv 41Law of increase of Entropy 45
LAWS OF THERMODYNAMICS -iS
Entropy as a Logarithm 50
Example: Perpetual Motionof the Second Kind 50
SUMMARY 51
PROBLEMS 5:
1. Entropy and Temperature 52
2. Paramagnetism 52
3. Quantum HarmonicOscillator 52
4. The Meaning of \"Never\" 535. Additivity of the Entropy for Two Spin Systems 536. IntegratedDeviation 54
Note on problems: The melhoJ of fhis chapter c
we Jo iitil cinplusi^e problem soKing dl lliis siu
Chapter 2; Entropy and Temperatui
One shouldno! imagine thai two gases in a 0.1 liter container,initially unmixed,
will mix, then again after afew days separate, then mix again, and so forth. On
the contrary, one finds .., ilia!not until a time enormously long compared to
W10\302\260
years will there by any noticeable unmixing of the gases. One mayrecognize that this is practically equivalent to never. . . .
ff we wish to find in rational mechanics an a priori foundation for the principles
of thermodynamics, we must seek mechanicaldefinitions of temperature and
entropy.
J. W. Gibbs
The genera} connection between energy and temperaturemay only be established
by probability considerations. {Twosystems]are in statistical equilibrium when
a transfer of energy does not increase the probability.
M. Planck
We slart this chapter with a definition of probability that enables us todefine the average value of a physical property of a system.We then consider
systems in thermal equilibrium, the definition of entropy, and the definition of
temperature. The second law of thermodynamicswill appear as the taw ofincrease of entropy.This chapteris perhapsthe most abstract in the book. Thechapters th;it follow wilt apply the concepts to physical problems.
FUNDAMENTALASSUMPTION' \" \342\226\240-
The fundamental assumption of thermal physicsis ttt;tt a closed system is equallylikely to be in any of the quantum states accessible to it. All accessible quantum
states arc assumed to be equally probable\342\200\224there is no reason to prefer someaccessiblestates over other accessible states.
A closed system will have constant energy, a constant numberof particles,constant volume, and constant values of all external parameters that may
influence the system, including gravitational, electric, and magneticfields.A quantum state is accessible if its properties arc compatiblewith the physical
specification of the system: the energy of the stale must be in the range within
which the energy of the system is specified,and the number of particles must bein the range within which the number of parlictcsis specified.Wtlh large systems
we can never know either of theseexactly,but it will suffice to have.SU/l/ \302\2531
tmd&N/N \302\253I.
Unusual properties of a system may sometimes make it impossible for
certain states to beaccessibleduring the time the system is under observation.Fof example,the states of the crystalline form of SiO2 are inaccessibleat low
temperatures in any observation that starts with the glassy or amorphousform: fused silica will not convert to quartz in our lifetime in a low-tcmpcraturcexperiment. You will recognize many exclusions of this type by commonsense.We treat all quantum states as accessible unless they are excluded by the
specification of the system(Figure2.1)and the time scale of the measurement
process.Statesthat are not accessible are said to have zero probability.Ofcourse,it is possible to specify the configuration of a closedsystem to a
point that its statistical propertiesas such are of no interest. If we specifythat ihe
Chapter 2; Enxropy and Temperature
I imtt of spcMftcation of ihe sjstcn
V\\uil- 2, t A iwdy symbolic Ji:iKr;ihi: L-:idi solid s|x'
represents an accessible quantum slate of a closed sysn
fundymema! assumption of statistical pliysics is tliat a
system is equally likely to be in any of tlic quantum si;accessible to it. \"Die empty circles represent some of thi
that are not accessible because their properties do nc
the specification of the system. IlovG vjfju -1 ,-/<_
system is exactly in a stationary quaniurn state s, no statistical aspect is left in
the problem.
PROBABILITY
Suppose we have a closed sysiem that we know is equally likely to be tn any
of they accessiblequantumstates.Let s be a genera] state label (a\302\273dnot one-half
ihe spin excess). The probabtHty P(s)of finding ihe sysiem in this slate is
P(s) =\\fg (t)
if the state 5 is accessible and P[s)= 0 otherwise, consistent with ihe fun-
fundamental assumption. We shall be concerned taier with systems that are not
dosed, for which the energy V and panicle number N may vary. For thesesystems P(s) wtH not be a constant as in A), but wilt have a functional dependenceon [/and on A'.
Probubitiiy
The sum\302\243P(s) of the probability over alt states is always equal to unity,
because the total probability that the sysiem is in some state is unity:
_, . . B)
The probabilities defined by (I) tead to ihe definitionof the averagevalue of
any physical properly. Suppose iliat the physical property X has the value
X{s) when the system is in the state s. Here X might denote magnetic moment,energy, square of the energy, charge density near a point r, or any property that
can be observed wlien the system is in a quaniumstate.Then the average of the
observations of the quantity X taken overa system described by the proba-probabilities I'{s) is
This equation defines the average value of X. HereP(s] is the probability that
the sysiem is in the state s. The angular brackets <-\342\226\240\342\226\240>are used to denote
average value.For a dosed system, the average value of A' is /\"i>-'-f >':
' !'i -
D]
because now alt g accessible slates are equally likely, with P(s) =\\jg. The
average in D) is an elementary exampleof what may be called an ensemble
average; we imagineg simitarsyslems,one in each accessible quantum stale.
Such a group of systems constructed alike ts catted an ensemble of syslems.Theaverageof any properly over the group is catted ihc ensembleaverageof that
property. _1_ '\342\200\242\342\226\240>-\342\226\240',' .\342\226\240>.\342\226\240'
An ensemble of systems is composed of many systems, all consfrueled alike.
Each system in the ensemble is a replica of the actual system in one of the
quanium states accessible to the system. If there are g accessible stales, then
therewilt be g systems in the ensemble, one systemfor each stale Each system
in the ensemble is equivalentfor all practical purposes to the actual sysiem.Each sysiem satisfies all external requirements placed on the original system
and in this sense is \"jus! as good\"as the actual system. Every quantum stale
\"U.:'j\\ \342\200\242\342\200\242Tit Y t : t t
'[!\".;\"*'\342\226\240\342\231\246* -rrrt \302\273'.\"*
t
'';\"(:!' t t:\302\273\":;'\302\273''\342\226\240
t : t';t ;i
'[in :y t: r;jt t i t
/[r;i t t 'nrt yi t
sT'\"\302\273 t i rn; t -t t t
/.:Y:t I t fit Y \\ [\\ Y.
-\342\200\242Titt t rft t t t t
i'YY t t :t:;Y \\ ;m I
l;ij;ure 2.2 Tliis cuscmbJc \302\253iJirotijiJij icpresciiis ;l
of lOspiiw wiih etwujy -Sui/Jimd spin excess 2.v
Tlic n!Litij]>]iciiy !/{.\\'.n) is yUU,4)= |0, so tliut ilic
rcpfcscutiiiive cl]sv-]I]!iIc iiium ^inlnin Hi syML-ins,order in which Uic vuriou^ sy>(cnis Mi tlie cusciubl
listed has no sighificance.
accessible lo the actualsystem is represented in the ensemble by one system in a
slaiionary quantum stale, as in Figure 2.2. We assume that the ensemble
representsthe real system\342\200\224this is implied in the fundameniat assumption.
Example; Construction of an ensemble- We conslruci in Figure 2.3 an ensctnMe to
represent a dosedsystem of five spins, each system with spin excess 2s = {.The energy ofeach in a magnetic field is -mB. |Do not confuse !he use of s in spin excess with our
frequent use of s as a stale index or label.) Each system represents one of the muliiptes of
t tl'OutibiC K.O[ij\\j}U\\iilI0
i\\ tj jTIT: f\\ t'T
Figure 2.3 The enscmbie represents a system with N = 5 spins and spin excess 2i - i.
\342\200\236 R\302\253ute2.4 With A' = 5 and 2s - 5,a singie
| f \342\226\240\342\200\242!t .! I ! t Sy^m may represent ti1Ca\302\273embio.ThiS is .o!ii ii
i states at tin's enetcy.Tlic number ofsuch slatesisgiven by tlic multiplicity function
Tile 10systems shown m Figure 23 make up ttic cuwnibk.
If the energy in llie tiiLiitueliefieid weic siidi lli^t 2.\\- \342\226\240=5r ltn-'ii n sinylc sysitm tomjiriscs
5 systems have 2s = 3; |Q sysiems have 7s = I: 10 systems have 2s = -1; 5 systems
have 2s - -3; and t system has 2j = - 5.
Most ProbableConfiguration
Let two systems 5, and\302\243t
be brought into contact so that energy can betransferred freely from one to tttc other. This is called (hermnl contact (Figure2.5).The two systems in contact form a larger closed system & ~ Sx + \302\243z
with constant energy U =. Ut + U2.Whai determines whether there will be a
net flow ofenergy from one system to another? Theanswerleadsto the concept
of temperature. The direction of energyflow js not simply a matter ofwhethef
the energy of one system is greater than the energy of the other, because the
r 2: Entropy anil Temperature
Two closed
in eomacl
The systems ure in
thermal contact
u\\ +\342\226\240f/; s= ul + ut
sulaliiin Thermal conduclor allowsexchange of
energy
Figure 2.5 Establishment of Micrmal contact between [wo systems &, and
systemscan be dificrcin in size and constitution. A constant lotal energy can besbared in many ways between two systems.
The most probabledivision of the t6tal energy is that for which the combinedsystem has the maximum number of accessible states. We shall enumerate the
accessible slates of two model systems and then study what characterizes the
systemswhen in thermal contact. We first solve in detail the problem of thermalcontactbetween two spin systems, 1 and 2, in a niagoetic fieldwhich isintroducedin order to define the energy. The numbersofspins N u N2 maybe different, and
the values of the spin excess2s,,2sz may be different rOr the two systems. All
spins have magnetic moment m, Tlie actual exchangeof energy might take place
via some weak (residual)couplingbetween the spins near the interface between
the two systems. We assume that the quantum states of the total system & can
be represented accurately by a combination of any state of 3, with any state of
S2. We keep N,, N2 constant, but the values of the spin excessare allowedtochange.Thespin excess of a state of ilie combinedsystem will be denoted by 2s,where s =
sx + sz. The energy of ihe combinedsystem is dirccily proportional
to the total spin excess:
U{s) s2)= ~2inBs. E)
The tola] number ofparticlesisN = A'( + .Vj.
Most ProbableConfiguration
We assume that the energy splittings betweenadjacentenergy leveh are equalto 7n\\B in both systems, so that the magneticenergy given up by system I whenone spin is reversedcan bo taken itp by the reversal of one spin of systctn 2 in
the opposite sense.Any large physical system will have enough diverse modesofenergy storage so that energy exchange with another system is always possible.The value of s = st + s2 is constantbecausethe total energy is constant, but
when the two systems are brought into thermal contact a redistributionispermittedin ihe separate values of s,, s3 and thus in the energies I/,, Uz.
The multiplicity function g{N,s) of the combined system & is related to the
product of the multiplicity functions of the individual systems 5[ and 5Z bythe relation;
H2(Nj,s - s,) , F)
where the multiplicity functions gx, g2 are given by expressionsof the form of
A.15). The range of s, in the summation is from -fiY, to %Nt, if Nl < N2.To see how F) comesabout,considerfirst that configuration of the combined
system for which the first system has spin excess2sj and the second systetn has
spin excess2s2.A configuration is defined as the set of all stateswith specified
values of s( and sz. The first system hasg^N^s,) accessible states, each ofvvhich
may occur togeiher with any of the g1(N2,Si) accessible stales of ihe secondsystem. The total number of states in one configurationof thecombinedsystem
is given by the product gl{N1,sl)g2{N1,Si) of the multiplicity functions of Sk
and &2. Because s2 ~ s ~ sit the product of the g's may be written as
This product forms one term of the sum F).Different configurations of Uie combined system are characterized by different
values of j{. We sum over a!!possiblevalues ofs( to obtain the total number ofsuitesof all ihe configurations with fixed s or fixed energy. We thus obi a in F),
where y{N,s) is the numberofaccessiblestates of the combined system. In thesum we hold s, N,, and N2 constant, as part of the specification of therma!
contact.The result F) is a sum of products of the form G). Such a product\\sill be a
maximum for some value of s,, say sL, to be read us \"st hat\" or \"si caret\".
The configuration for which glg1 is a maximum is called the most probab!ecmtfigurntiun; ilie number of states in it is
gxiN1.Sl)g2[Nl.s- 5,). . (8)
Chapter 2: Entropy am! Temper
A*. 0
Thermal equilibrium
Figure 2.6 Sclicmalic repressaiion of Ihe dependence of the
coiifiguralioji multiplicity on [he division of ihe tola!energy
belwcen two syslems.-S, and Sj.
!f ihe systems are large, the maximum with respect to changes in Sj will be
extremelysharp,as in Figure 2.6. A relatively small number of configurationswill dominate the statistical properties of the combined system. The mostprobable configuration alone will describe many of theseproperties.
Such a sharp maximum is a property of every realistictype of large system
for which exact solutionsareavailable; we postulate that it is a genera! propertyof all largesystems- From the sharpness properly it follows that'fluctuations
about the most probable configuration are small, in a sense ihat we will define
The imporlani result follows l hat the values of the average physicalpropertiesof a large system in thermal contact with another large system are accuratelydescribed by the properties nf the most probable configuration, the configura-configurationfor which the number of accessible st3tes is a maximum.Such average
values (used in either of these two senses)arecalledthermal equilibrium values.
Because of the sharp maximum, we may replace the average of a physicalquantity over all accessibleconfigurations F) by an average over only ihemostprobable configuration (8). In the example below we estimateiheerrorinvolved
in such a replacement and find the error lo be negligible.
Most Prabahte ConfiSui
Example: Tn-o sprn systems in thermal contact. We investigate for lUc modd spin syslemtfiL- sharpness of the produci G) near the maximum (8) as follows. We form the product ofIlie muliiplicity functions for(j,(W,.s,) and g^i-^J- ^olh ofiho form of|U5):
white ?i(Q)denotesjj,(:V,.G) and ^@} denotes ffsf.Vj.Q). We replace s, by s - *,:
--^--
-l^iiLj. (jo)
This product* gives the number of statesaccessibleto the combined system when the spin
excess of the combined system is 2s, and the spin excess of ihc first system is 2s,.We find the maximum value of(tO}as a function of s, when the total spin esccss 25 ishd.l
constant; that is, when the energy of the combined systems is constant, it is convenient
to use the properly that the maximum of fog.m) occurs at the same \\atue of x as the
maximum of >'{.y). The calculation can be done eilherway. From f !0|,
)~^T-~
^~-. (II)
uy be a maximum! a minimuni, or z point of inflection. The e\\tr\302\243nuim is a
if the second derivative of the function is negative, so that the curve bends
downward.Ai the cxt
where Nt, N2. and s are held constant as s, is varied. The secondderivative c1/csi1o(
Equation A1) U
* The product fund ion of two Gaussian fund ions is always a Gaussian.
and is negative, so thai the extremum is a maximum. Tims the most probable configurationof the combined system is thai for which A2) is satisfied:
t
The two systems arc in equilibrium with respect to interchange ofenergy when the fractional
spin excessof system 1 is equal to the fractional spin excess of system 2.
We prove itiat nearly alt the accessible stales of ttic combined systems satisfy or very
nearly satisfy (U). if s, and 52denote the values of s, and sj at the maximum, then A3}
o find iiic number of slates in the most probjble configuration, we insert A4} in {9} to
To investigate the sharpness of i
such that
(-25I/N). A5)
of gigl at a given value of s, introduce &
- s, + <5; sj= s2
- 5. A6)
Hcre^ measures the deviarion of su st from their values Su Sx at the maximum o(g,gi.
Square j,,ij to fonn
which we substitute in {9}and A5) to obtain the number of sialcs
/ 45,^ 2Sl 4s2S 2d2
We know from (H) thai s,/jVj\302\253V.vi.so !ii;it the number ofslatesin a configuration of
deviation ii from equilibrium is.'.\\'u so that the number of slates in a configuration ot
\\
l(N2J2 - 3) = te.ffiU.e ~ ~
;v~
As a numerical example in wliidi the fraciionat devbiion from equilibrium is very smalt,
let Ni \302\253,V. = !0:i and 5 = 10'2; ilia!is, <5/A'i= !O\0210. Tlien,2ii!,'iVi
= ^00, and the
Thermal Equilibrium
product g,^i is reduced lo g\"\02100 = lO\0217* of its maximum value.This is an extremely
laigc reduction, so that g,g, is truly a very sharply peaked function of st. Tiic probability
that tlie fractional deviation will be 10\"'\302\260or larger is found by integrating {17} from
& = I0u out lo a value of (heorder of s or of N, thereby including ihc area under the wings
of the probability distribution. Tiits is the subject of Problem 6. An upper limit to the
integrated probability is given by N x tO\"t14 = 10~i!2,si ill very smalt. When two
systems are in ihermal contact, the values ofsi, Sj thai occur most often will be very close tothe values off,, S] for w hich ihe product g]g1 is a maximum. It is e.Uremely rare io (ind
systems with values of*,, s, perceptiblydifferent from ?\342\200\236st.
What does it mean to say that the probability of ftitding the system wjlh a fractional
deviation larger limn 5 .V, = \\Q~10 is Only \\0'!ii of the probability of finding the systemill equilibrium? We mean that the system wilt never be found with a deviation as much as
1 part in tO10, smalt as this deualion seems. We would have to sample!0IJ2similar systems
lo have a reasonable chance ofsuccessin such an experiment. If we satnptc one system every
10'1 s, which is prctlv Usi sunk, we would iuive to sample for 101J\"s. The age of the
universe is only 1O'Bs.Thereforewe say with great surely that lite deviation described wilt
never be observed. The estimate is rough, but the message is correct. The quotation from
Uoli/iitaun given at iUi beginning of this chapter is relevum here.We may expect to ob>mc substantial fractional deviations only in the properties of it
imiili system in thermal contact with a targe system or reservoir. The energy of a small
system, say a system of 10 spins, in thermal contact with a large reservoir may undergofluclualions lhat are largi! in a fractional sense, as (lavebeenobservedin experiments on ihc
Brown tan motion of small particles in suspension in liquids. The average energy of a small
system ill contact with a targe system can altvays be determined accurately by observationsa! one time on a tatgc number of identical smart systems or by observations on one sronlt
sysicm over a long period of lime.
THERMAL EQUILIBRIUM
The result for ihe number of accessible stales of two model spin systems in
thermal contact may be generalized to any two systems in thermal contact, with
constant toial energy U - Ul +\342\226\240U2. By direct extension of the earlier argu-argument, the multiplicity g(S, V) of the combiited system is:
summed over all values of Us < V.Wzxz g X{N itU %) is the number of accessible
states of system 1 at Jiicrgy Ux. A coniigiiratioit of the combined system is
specified by ihe value of (.\342\226\240',.together with the constants U,Nt, -V2.The numberofaccesstblcstatesin a configuration is lite product gt{N],Ui)()i(Nz,U
~ Ut).The sum o\\er :nl configuraiioits gi
The largest term in the sum in (!S) governs the propertiesof the total system
ill thermal equilibrium. For an extremuni tt is necessary that ihe differential* of
g{N,U)be zero for an infinitesimal exchange of energy:
dg = (-\342\200\224] g2t!Ul +9i(^~\\d
lV, + dU2 - 0. A9}
We divide by glgl and use the result dV2= ~tiUx to obtain the thermal
equilibriumconciiiion:
Lflfs
which we may write as
B0b)
We defitie the quanttiy a, calledIhe entropy, by
a{N,U) s \\
where a is the Greek letter sigm:i.We now write B0) in the iimii form
B1)
means lhai Ns is held consiam in ihe differttiljalion o(g,{Nt,U,) with tespeel lo U,. Thai is, ili=
panial dcrivalivc wiih respect to 0, is defined as
Teutperutute
This is the condition for thermal equilibrium for two systems in thermal
contact. Here Afi ant! /V2 may symbolize not only the numbers ofparticles,but
ail constraints on the systems.
TEMPERATURE
The last equality B2) leads us immediately to the concept of temperature.We
know the everyday rule: in thermal equilibrium the temperatures of the two
systems are equal:
r, = t2. B3)
Thisrulemust be equivalent to B2), so that T must be a function of (ro/f V)s.
If T denotes ttie ahsohite temperature in ketvin, this function is simply iSie
inverse relationship
B4)
The proportionality constant ku is ;i universal constant called the ilott/iiumu
L-on^mnt. As determined experimentally,
kn\302\273U81 x 10~\"joulcs/ke1vin
\302\2531.3X1 x 10\"Ulcrgs/kelvin. B5)
We defer the discussionto Appendix B becausewe prefer 10 use a more natural
temperature scale:we define the fundamental temperature r by
B6)
This temperaturediffers from ihc Kelvin temperature by the scale factor, kB:
r - kBT. | B7)
Becausea isapurenumber,thefundamental temperature t has the dimensions
of energy.We can use as a iemperature scale the energyscale,in whatever unit
C/iapier 2; Entropy and Temperature
may be employed for the latter\342\200\224joule or erg. This procedure is much simplerthan the introduction of the Kelvin scale in which the unit of temperature is
arbitrarily selected so that the triple point of wmer isexactly273.16K.Thctriple
point of water is the unique tcmperaiurc at which water, ice, and water vaporcoexist.
Historically,the conventional scale dates from an age in which it was possibleto build accurate thermometerseven though the relation of temperature to
quantum stateswas as yet not understood. Even at present, it is still possible to
measure temperatures with thermometers calibrated in kelvin to a higherprecisionthan the accuracy wit h which tlie conversion factor kB iiself is known\342\200\224
about 32 parts per million. Questions of praciical thermomeiryarediscussed in
Appendix B.
permissible to take the reciprocal of both sides to
B8)
The iwo expressions B6) and B8} have a slightly diOcrcu
was given as a function of the independent variables Udetermined from B6) has the same indcpendeiu variab]differentiation of U with respect lo a with N consiant
V). The definition of Eemr^caiufe is die same in both cases, but it is expressed
function of different independent variables. The qvariables?\"arises frequently in lhermal phy.some variabtcs, and in oilier ej.pcrinicnts we
leaning, in B6). the entropy <r
id M as a = a{U,X).Henceir \342\204\242t(U,N). In B8). however,
mpties V = V(a,N), so that t =
in \"What are ihc independentbecause in some experiments we coniroi
itrot other variables.
ENTROPY
Tile quantity a s loggwas introduced in B1) as ihe entropy of the system:the
entropy ts defined as the logarithm of the number of slates accessible to the
system. As defined, theentropy is a pure number. In classical thermodynamics
the entropy i'is defined by
129)
Entropy
Figure 2.7 if the temperature r, is higherthan t2, the transfer of a positive amount of
energy & U from system t to system 2 v.itt
increase the total entropy a, -f a2 of ihecombinedsystemsover ihe initiat value
<j,(initia!) + ^(initiat). In oiher words, the
finat system wiii bt: in a more probablecondition if energy flows from the wanner bto the cooler body when lhermat contact isestablished.This is an otamptc of the Saw of
increasing cm ropy.
Energy transfer
o,(final) + cMXinai) > o^iniiia!) + oa(iniiiai)
As a consequenceof B4),we see that 5 and o are connected by a scale factor:
U2+8U
130)
We wilt call S the conventional entropy.The more statesthat are accessible, the greater the entropy, fn the definition
of a{N,U) we have indicated a functional dependence of the entropy on thenumber of particlesin the system and on the energy of the system.Theentropy
may depend on additional independent variables: the entropy of a gas(Chapter3) depends on (he volume.
in Uicearly history of thermal physics the physical significance of the cm ropywas not known. Thus the author of the article on thermodynamics in the
Encyclopaedia Britaiiuka, t lih ed. A905), wrote: \"The utility of ihe conception
of entropy ... is limited by the fact that it does not correspond directly to anydirectly measurablephysical property, but is merely a mathematical function
of the definitionofabsolutetemperature.\"We now know v.hat absolute physical
properly the entropy measures,An example of the comparison of the experi-experimental determination and theorctic.il calcutiition of the entropy is discussed in
Chapter 6.
Consider the tot;il entropy changeAa when we remove a positive amount of
energy All from 1 and add the same amount of energy to 2, ;is in Figure 2.7.
Clmptcr 2: Entropy urn! Tehtjter
The tolnl entropy change is
When t, > r2 ihe quantiiy in p;ireniheseson ihe righ>hand side is positive,
so that the Loialchange of entropy is positive when the direction of energy flow
is from the system with the higher temperature to the system with the lower
temperature.
Example: Entropy increase on heiitjlow. This example makesuse of ihc rentier's previousfansitiariiy with heal and specific heat.
(aj Let a 10-g specimen of copper at a temperatureof 350 K be placed in thermal contact
with an identical specimenat a temporal tire of 290 K. Lei us find the quantiiy of energytransferred when ihe iwo specimens arc ptaced m contact and come Lo equilibrium at ihe
final temperature Tf, Ttie specifichc;ii of meiaitic copper over ihe LempcraLure range 15:Cto t007C is approxiinaieiy O.3S9Jg~! K~l, according Lo a standard handbook.
The energy increase of the second specimen i> cmiai Lo the energy loss of ihc first) ihus
the energy increase of ihz second specimen is, in joules,
AV = C.89J K-'HTV- 290K) - C.89JK-')C5OK~
Tf) ,
where ihe tempcraiures are in kcMn. Ttie linat temperature after contact is
Tj\302\253
|C5O + 290JK = 32OK.
Thus
At/, \302\253C.89JK~!)(~3OK)= -11.7 J ,
and
At/3 = -At/, = U.7J.
(b) What is the change of entropy of the two specimens when a transfer pf 0.1J hastaken place,almost immediately after initial con'act? Notice that this transfer is a smallfraction of ihe final energy transfer as calculated above.Becausethe energy transfer con-
considered is small, we may suppose the specimens are approximately at their initial tempera-temperaturesof 350 and 290 K. The entropy of the firsi body is decreased by
Lan of Imrrrmc of Enlnpy
Tile cnlropy of iiic second hody is increased by
S2=
,7~= 3.45 x
Tile iotal enlropy increases by
AS, + AS, = (-2.S6 + 3.45)x 10-4JK-' = 0.59x 1CTJJK~
In fundamental, units the increase of entropy *s
where tjisihe Botl/.ma\302\253n constant. This resuil mcaaS thai (lie number of accessible st;it
of the two systems increases by (he factor exp{M - [email protected] 10l9>.
Law oflncrease of Eniropy
We can show thai ihc loial ctttrapy always increases when two systems ;trc
broughi into thermal contact. We have jusi demonstrated this in a special case.
If (he total energy V ~ V% + U2 is consiant, the lotal multiplicity after the
systems arc in thermal contact is
ff(t/)\302\273= ^0,A/^A/- t/,) , C3)
by A8). This expressioncontainsthe term gi(EAo)i/i(^~~
^to^ ^or l^e i-XiiiVd^
niuitiplicuy before contact and many osher terms besides.Here ViQis the
initial energy ofsystem 1 and V ~~l/lois the initial energy ofsyslem 2.Because
all termsin C3) are positive numbers, ihe muitipliciiy is always increased by
establishmenl of ihermal conlaci bclween two systems. This is a proof of Ihetaw of increase of entropy for a weli-definedoperalion.
Thesignificant effect of conlact, the effect that slands out evenafter lakingthe
logarithm of ihe multiplicity, is not just that Ihe number of tcims in iiic summa-
summationis large, but that the largest singleterm in the summationmay be very, very
much larger than the initial muilipiicity. That is,
(Mi).,, = 9i(O,)gJiU - 0.) C4)
Chapter 2: Entropy and Teniperatur
with Ut= 0 anJ U, * U.Exchange of energy takes piao; between h
parts and presently the syucm will be found in or dose 10 tlie most
probable con figuration. Hie cm ropy increases as the jysicm attains
conftgL rat ions ofincreasirtg muhiplicily or probabilhy. The entropyeventually reaches ihe entropy a{U) oflhe moit probable configuratic
may be very, very much larger than ihe initial term
9iiVtMU'~ Vl0). C5)
Here 0s denotesthe value of Vl for which ihe product g^x is a maximum.The essential effect is lhat the syslems after contacl cvoive from iheir initial
configurations lo their final configurations. The fundamental assumption
implies thai evoluiion in this operation will always lake place, with ali accessible
final states eqtutlly probable.
The statement
fffjnil C6)
is a statement of the law of increaseofcnlropy:the entropy of u closed system[ends to remain constant or to iucreasewhen a constraint iniernul to she sjstemis removed.The operationofebtabitsliiiig thermal contact is equivalent to iheremoval of the constraint that Vu V2 each be constant; after contact only
Ux + U, need be constant.Theevolutionof the combined system lowards ihe final thermal equilibrium
configuration takes a certain time. If we separate ihe two systems before they
Add energy
Decompose molecules
Let a linear polymer curl up
Figure 2.9 Operations thai lend io increase the entropy ofa syslcm.
reach (his cotifiguraiion, we will obtain an intermediate configuration with
intermediate energies and an intermediate entropy. Ii is ihesefore meaningful io
view the entropy as a function of the lime tli.i' lias elapsed sinceremovalof the
constraint, called (he lime of evolution in Figure 2.8.
Processes that lend lo increase the eniropy ofa system are shown in Figure
2.9; the arguments in support of each process will be developed in the chaptersthat follow.
Chapter 2: Entropy and Temperature /
For a largesysiem* (in thermal coniaci with another large sysiem)iherewill
never occur spontaneously significant differences belween the actual value of the
entropy and ihc value of the entropyof the most probableconfiguration of the
system. We showed ihis for ilie model spin sysiem in the argument following A7);we used \"never\" in ihe sense of not once in ilie entire age of the universe. 10's s.
We can only find a significant difference beiwcen Ihc actual entropy asid ihe
entropy of ihe most probable configuration of the macroscopic system veryshortly afier we have changed the nature of ihe contactbetween two systems,
which implies that we had prepared the system initially in some special way.
Special preparation couldconsistof lining up all the spins in one system parallel
to one another or collectingall the molecules in the air of the room into the
system formed by a small volume m one corner of the room. Such extreme
situations never arise naturally m systems left undisturbed, but arise fromartificial operations informed on the system.
Consider ihc gas in a room: the gas in one half of the room might be prepared
initially wjiti a low value of the average energy per molecule, while the gas in ihc
other half of the room might be prepared initially with a higher value of the
average energy per molecule. If the gas in the two halves is now allowed tointeract by removal of a partition, the gas moleculeswill come very quickly'
to a most probable configuration m which ihe molecules m both halves of theroom have the same average energy. Nothing else will ever be observed to
happen. We will never observe the sysiem to leave ihe most probableconfigura-
configurationand reappear later in the initial specially prepared configuration.This is true
even ihough the equations of motion of physics are reversiblein time and do not
distinguish past and failure.
LAWS OF THERMODYNAMICS
When thermodynamics is studiedas a nonslatisticaisubject,four posluiales
are introduced. Tiicse postulates are caiied ihe laws of thermodynamics, in
essence, these laws are containedwithin our statistical formuiaiion of thermal
physics, bul it is useful to exhibit Ihem as separateslatemenis.Zeroth law. If two sysleins are in thermal equilibriumwith a third sysiem,
Ihey must be in thermal equilibrium with each olher. This iaw isa consequence
1The calculation of Ihe lime required for Ihe process is largely a problem in hydrodyna
Laws of Thtentotiynamks
of the condition B0b)for equilibrium in thermal comact:
(e\\oSgt\\ feioSg3\\ /cloggA /cloggA{~\342\204\242rk
*
{-furl: {imX= {~7urk
in oilierwords, r,= t3 and r3 = t3 imply rj
= r2.
First law. Heat is a form of energy. This law is no more than a slaicment of
liie principle of conservationofenergy.Chapter 8 discusses wliat form of energyheat is.
Secondlaw. There are many equivalent statements of ihe second law. We
shali use the statisiicai statement, which we have called ilie law of increase ofentropy, applicablewhen a constraim iniernal lo a dosed systemis removed. The
commonly used statement of the law of increase of entropy is: \"Ifaclosedsystem
is ill :l configuration tluit is not (he equilibrium cotiliyitnilton.ilicmosi |>rubnble
consequence will be lhat ihc enlropy of the systemwill increase monoiotiic;ilty
in successive instants of time.\"Tins is a looser siaicineiil ilian I he one we gavewilh Eq. C6} above.
The traditionalthermodynamic statement is the Kelvin-Pianck formulationof second iaw of thermodynamics; \"it is impossible for any cyclic process to
occur whose soie effect is the extraction of heal from a reservoir and the per-
performance of an equivalent amount of work.\"An engine that vioiaies lhe secondiaw by extractingthe energy of one heat reservoir is said to be performing
perpeiual motion ofthe second kind. We will see in Chapter 8 that the Kelvin-Pianck formulationis a consequenceoflhe statistical statement.
Third iaw. The entropy of a system approachesa constant value as the
temperature approaches zero.Theear!test statement of this law, due loNemst, isthat at ihe absolute zero the entropy difference disappears betweenall thoseconfigurations of a system which are in internal thermal equilibrium. The third
iaw follows from the statistical definition of the entropy, provided ihat thegroundstateof the system has a weii-defined multiplicity. If lhe ground stalemultiplicity is g@), the corresponding entropy is o{0)= iogy@) as t -* 0.
From a quantum point of view, the law does not appear to say much that is
not implicit in the definition of enlropy, provided, however, that the system is
in its lowest se! of quantum states at absolute zero. Except for glasses, therewould not be any objection to affirming that (j{0) is a small numberand c{0)
is essentially zero. Glasses have a frozen-in disorder, and for them o{0) can besubstantial,of the order of [he number of atoms N. What the third law tells us
in real life is that curves of many reasonable physical quantities plotted against x
must come in flat as r approaches 0.
Chapter2: Entropy and Temperature 1
Entropy as a Logarithm
Severaluseful properties follow from ihe definition of (he cmropyas the ioga-
rithm of the number of accessiblestates,hut cad of as the number of accessiblestales itself.First, the entropy of two independent systems is liie sum of lheseparateentropies.
Second,the entropy is entirely insensitive\342\200\224for ali practical purposes\342\200\224-to
the precision 6U with wiiich the energy of a closedsystem is defined. We have
never meant to imply that tite system energy is known exactly, a circumstancethat for a discrete spectrum of energy eigenvalues would make the number of
accessible stales depend erraticallyon ttie energy. We have simply not paidmuch attention io lhe precision,wlicthcr ii be determined by the uncertainlyprinciple <5U 5{time)
- h, or determined otherwise. Define<0{U)as the numberof accessiblesrates per unit energy range; O{U) can be a suilablesmoothedaverage centered at V. Then y{U) ~ i>{UNU is lhe number of accessible
stales in the range SU al V. The cmropy is
a{U) logO(t/) -t- \\ C7)
Typieally, as for the system of N spins, the total number of states will be of ihc
order of 2V. If lhe tola! energy is of the order of N times same average one-
particle energy A, then C(l') - 2;7jVA. Thus
. . a{V) =*Nlog2 --'logNA + log^C.
Let N ~ IO20;A = IO\021* erg; and 5U \302\27310^' erg.
a(U) *\302\2730.69 x 10:o - 13-82- 2.3.
C8)
C9)
We see from this exam pie that the value of the entropy is dominated overwhelm-
overwhelminglyby the value of N; lhe precision dU is without perceptible effect on the
result. 1 n the problem of A' Tree particles in a box, the numberofstalesis propor-
proportional io something like U*dU, whence a - JVIogf + \\ogSU. Again lhe
term in N is dominant, a conclusion independentof even the system of units
used for lhe energy.
Example: Perpetual iiuxion o
more energy in.in ii absorbs.
Early in our study of physics we came tolion machine, a machine ilui wjli gi\\c forth
Equally impossible is a perpetual motion machine of the second kind, as it is called, in
which heat is exiracicd from part of the environment and deliveredto another part of ihc
environment the difference in temperature ihus established being usedto power a heat
engine that delivers mechanicalwork available for any purpose at no cost to us.In brief, we
cannot propel a ship by cooling tlic surrounding ocean lo extract the cnefgy necessary to
propel ihe ship. The Spontaneous transfer of energy from ihe low temperature ocean 10 a
higher temperature boileron the ship would decrease ihe total entropy of the combined
systems and would thus be in violation of the law of increase of entropy.
SUMMARY
1. The fundamental assumption is that a cioscd system is equally likely to be
in any of lite quantum slatesaccessible!Oit.2, ifPfi) is the probability that a system is in the stale s, the average value ofa
quanliiy X is
3. An ensemble of systems is composedof very many systems, all constructed
alike.
4. The numberofaccessiblestates of the combined systems 1 and 2 is ,^ ^
*^'
where s, + s2 = s.
5. The entropy a(N,U) = to$g{N,U). Tiie relation S =kBa connects the
conventional entropy S with the fundamental entropy o.
6. The fundamental temperature r is defined by
1/t s [ca;cV)sx.
The relation r = kgTconnectsthe fundamental temperature and the con-
conventional temperature.
7. The law of increase of entropy states that the entropy ofa closed sysicnitends to remain constantor lo increase when a constraint intorn.il lo llic
systemis removed.
Chapter 2: Entropy and Temperature j
8. The thermal equilibriumvalues of the physical properties of a system aredefined as averages over all states accessible when the systemis in contact
with a large system or reservoir.If the first system also is large, the thermalequilibriumpropertiesare given accurately by consideration of the states inthe most probableconfiguration alone.
PROBLEMS
1. Entropy and temperature. SupposegW)= CUiNn, wiiere C is ;t constant
and N is tile number of particles,{a}Show thai U *={Nt. (b) Show that
{c^a/rU2)* is negative.Tliisform of y( U) ;iciii;tlly applies hi an ideal gits.
2. Ptirtitttagiwtisni. Find the equilibrium value ui ictn|KT;tit:rc i of ilie (Vac*lional mngntrlizalion
M \\'m = 2<s>/N
of the sysiem of h' spinseachofmagnetic momem in in a magnelic field B.Thespin excess is 2s. Take the eniropy as lhe iogarlhilhmof lhc muliipliciiy g(N,s)
as given in A.35):
c(s) =logg(Ar.O)- 2s2/N , D0)
for |s| \302\253iV. Him: Show that in this approximation
g{U) = c0- U2/2m2B2N, D1)
wtih oQ- 1og(/{N,0). Further, show thai \\jx
~ -U/)\302\2732B2N, where U denotes
<t/>, the lhermai averageenergy.
3. Quantum hartitonic oscillator, (a) Find the enlropy of a selofN oscillators
of frequency m as a fund ionof the loiai quanlum number n. Use the muiiipiicily
funciion A.55) and make lhe Sliding approximalion iogiV!= iViogiV\342\200\224iV.
Replace JV \342\200\2241 by iV. (b) Let U denote lhc tola!energy \302\273/ituof the oscillators.
Express the entropy as a(U,N).Show that the total energy at temperature x is
exp{/itu/r)\342\200\2241
,42)
This !S the PSanck result; it is derived again tn Chapter 4 by a powerful method
that does not require us to find ihemuilipiicity function.
where we have used !ogl0 44 = 1.64345.
{b} Show that the probability that a monkey-Hamlet will be typed in the ageof the universe is approximately 10\"'64316. The probability of Hamlet isthereforezeroin any operational sense of an event, so that the original statement
at the beginning of this problemis nonsense: one book, mudi less a library,will never occur in the total literary produclion of the monkeys.5. Addith-ity of entropy far two spin systems. Given two systems of JV, =s
A', = 1012spins with multiplicity functions g^x^i) and g2{N2.s - ss), the
product gig2 as a functtonofsj is relatively sharply peakedat s( = s,.ForSj=ss + I012, the product #sj72 is reduced by jO\02174 from its peak value. Use theGaussian approximationto the multiplicity function: the form A7) may beuseful.
(a) Computegigz/{9-i9z)m3X for s,= s, + lOn and s = 0.
(b) For s = 1O10, by what factor must you multiply (gijh),,,a!l fo make it
equal to Yai9'it^i*5i)9i{^i's~
5i)\"> \302\247've l^e ^ctor to the nearest order ofmagnitude.
' j. Jeans, htysteriota utirerst, Cambridge Universily Press, 1930, p. 4. The slalenuill is attributed
to Huxley.' For a rctaicd malhematfco-iherary study, sec'The Libtary of Babel,\" by ihe fascinating Argentine
Clarke in 2001. We arc gralcfut to the Population Reference Bureau and to Dr. Rosier Revcttc for
explanations of the evidence. The cumulative number of man-secondsis 2 x iO10. if we take iheaverage trretrme as 2 x 10 s 3nd tl^c number of lives as I ^ 10 , i he cumu*ai[\\c numoc^ oi
man-seconds ii njocti ksi than the'numbcr of monkey \342\200\242sccondi(t0\"> iaken in the problem.
Cliapter2; Entropy and Tcmperatur
(c) How iarge is the fractional error in the entropy when you ignore thisfactor?
6, Integrateddeviation. For the example lhat gave ihe result A7), calculateapproximatelythe probability that the fractional deviation from equilibrium\342\226\2405/JVjis 10\"l0 or larger. Take iVj
= JV2 = IO2Z. You will find it convenient to
use an asymptotic expansion for the complementary error function. When
x \302\273i,
xp(x2) (\"\"e x 1 + small terms.
Chapter 3
Boltzmann Distribution andHelmholtz Free Energy
BOLTZMANN FACTOR 58
Partition Function 61
Example: Energy and HeatCapacityof a Two State System 62
Definition; Reversible Process 64
PRESSURE 6-1
Tlicrmodynaimc identity ;>7
HELMHOLTZ KttEE ENERGY 6S
Example: Minimum Property of the Free Energy of ;iParamagneticSystem 69
Differential Relations 70
Maxwell Relation 7!Calculationof f from 2 71
IDEAL GAS: A FIRST LOOK 72
One Atom in a Box 72
Example; Af Atoms in a Box 74
Energy 76Example:Equipartition of Energy 77
Example: Entropy of Mixing 7$
SUMMARY SO
PROBLEMS 81
1. Free Energy of a Two State Sysiem85
2. Magnetic Susceptibility Si
3. Free Energy of a Harmonic O.ollaior S3
4. EnergyFluctuations S3
5. Overhauser Effect S4
6. Rotation of DiatomicMolecules 84
7. Zipper Problem S5
S. Quantum Concentration S3
ChapfcrS: Bol auJ lleliiiiioiiz Free Energy
9. Partition Function for Two Systems10. Elasticity of Polymers
11. One-Dimensional Gas
Chapter3; BolRinattn Distribution and lleimkoitz Free Energy
The laws of thermodynamics may easily be obtainedfrom the principles of
statistical mechanics, of which they are the incomplete expression.Gibbi
We are able to distinguish in mechanical terms the thermal action of one systemon anotherfrom lhai which we call mechanical in the narrower sense . , . so asto specify cases of thermal action and cases ofmechanicalaction.
Glbbf
In this chafer we develop the principles that permit us to calculate the valuesof the physical properties of a system as a ftinciion of ihe temperature. We
assume that the sysiem & of interest io us is in thermal equilibrium wiih a veryiarge sysiem (ft, called [he reservoir. The system and the reservoirwill have a
common temperature r because ihcy are in thermal contact.The iota!system (ft + & is a closed sysiem, insuiaicd from u!! external
influences, as in Figure 3.1. The total energy Uo =* U^ + \302\243/jis constant. In
particular, if the sysiem is in a staleof energy Ea, tltcn Uo -r,x is ihe energy of
the reservoir.
Toial sysiem
. Constant energy Vo
J.I RcprcsirOtiiUhcn
illation of a cioscd loiat systa! coniaci with a sjsieni S.
n decomposed iisioa
BOLTZMANN FACTOR
A central problem of thermal physics is 10 find the probability iltai ihe system5 \302\253iilbe in a specific quantum siaiu s of energy t,. This probability is propor-
proportional to the Boitzmann factor.
When we specify that S should be in ihe state s, ihe number of accessible
Slates of the louil sysiem is reduced10the number of accessible states of ihe
reservoir (H, ai ihe appropriate energy. That is, ihe number g\302\256tj of siaies
Figure 3.2 The change of cnlropy when ihensetvoit u&nsfcssenergy t la (he system. Thefractionaleffec! of itie transfer on ihe reservoir
laigc TCMivoii vil\\ luua high ci\302\253iop>.
Energy of the reservoir -
accessibleto (ft -
A)
because for our present purposes we have specified the state of .S ,If the system energy is \302\243\342\200\236the reservoir energy is Uo \342\200\224
\302\243,.The number of
stales accessible to the reservoir in this condition is^f/o-
\302\243,},as in Figure 3.2.
The ratio of the probability that the system is in quantum stale I al energy
\302\243,to Ihe probability that the system is in quantum state 2 at energy Ei is Ihe
ratio of Ihe two multiplicities:
Multiplicity of <R at energy Uo-
r.t _ \302\243a(^o
- ei)
Multiplicity of (ft at energy [/q \342\200\224\302\2432 91s(^'o
\342\200\224e2)
B)
This result ts a direct consequence of what we have called the fundamental
assumption. The two situations are shown in Figure 3.3. Although questionsabout the system depend on the constitution of the reservoir, we shall seeihat
the dependence is only ow the temperature of the reservoir.
if the reservoirsare very large, the multiplicities arc very, very l.-irgcnumbers.
We write B) in terms of the entropy of the reservoir:
\302\273-
E.)-
ff\302\253(t/0-
f C)
Chapters: Bohzmnnn Distribution and Helmholtz Free Energy'
E
Oi
ergy (/\342\200\236
yo- ',)
-',stale
Sta
Ene
8
tc I
gy\302\253>
En
01
\302\253gy (/\342\200\236
;\302\260-
'='
-\302\253,
slates
i
State 2
Energy c.
Fi\302\273urc3.3 The system in (a), (b) is in quantum slate t, 2. The reservoirhas a,,(U0- t,},(,\302\273(()\342\200\236
- c,) acccisiblc quanluin slates, in (a) and (b)id
the probability ratio for the two states 1, 2 of the syslem is simply
Let us expand the entropies in D) in a Taylor series expansion about
The Taylor seriesexpansionoff[x)about /(x0)is
- t)
D)
E)
0)
where 1/t=
(S^/cCV^ gives the temperature. The partial derivative is taken
al energy Uo. The higher order termsin liie expansion vanish in the limit of
an infinitely large reservoir.*Therefore
Acr^ defined by D) becomes
Affffl= -(\302\243, -\302\2432)/T. (8)
The final result of E) and C) is
P{ez) expft/r)''
A term of the form exp( \342\200\224e/t)is known ,as 3 Boltzmann factor. This result isofvust utility. It gives the ratio of Hie probability of finding the system in a
single: quantum state I to ttie probability of finding ihc system in :i single
quantum state 2.
Partition Function
h is helpfui to consider the function
Z(r) = 5>p(~Ei/T) , A0}
ealied the partition function. The summation is over the Boltzmann factor
exp(-e,/t) for a!! states 5 of the system. The partition function is the pro-
proportionality factor between the probability p[Et) and (he Boltzmann factor
We see that\302\243?(\302\243,)
= ZjZ = 1: the sum of all probabilitiesis unity.
The result (II) is one of the most useful results of statistical physics. The
average energyof the system is V = (e) = X^fo). or
U =,
Zh^Zh!A= T^logZ/ct).
'A2)
\342\200\242We expand oft'j, ~\342\202\254)andnolg({/B
~ e) because [he cupansion of ihc tatter quanliiyimmcdiatctygives convergence difficulties.
ChapterS: Battvnanti Distribution and Helmholtz Free Energy
0.5
0.4
Energy and heat capacity of aystcm as functions ofthe temperaturergy is plotteJ in units ol t.
0.1
0
A
J V
u
J-\342\200\224\342\200\224
\342\226\240
The average energy refers to those statesof a system that can exchange energywith a reservoir. The notation <\342\226\240
\342\226\240) denotes such an average value and is
called the thermal average or ensembie average. In A2) the symbol U is usedfor (e) in conformity with common practice; U will now refer to the systemand not, as earlier,to the system -f reservoir.
Exwiiplc? Energy uiid licut cupticiiy of a two itatt systexn. We trc^l a sy^lcm ofonc pitrti-
clc wftli two slates, one of energy 0 and one of energy e. 11w partide is in iltcmut contact
wiiti a reservoir at temjKrature t. We want lo find the energy and ttie heat cupaciiy of ihe
system as a function of the temperature i. The pat lit ion function for tin iwo stales of
Z = exp(-G/t) + exp(~\302\243/r)= i -f expf-s/t). A3)
\"fhe average energy is
This function is plotted in Figure 3.4.If we shift iliczcro of energy and lak^: W.c energies of the two states as -\\e and +\\e,
instead ofas 0 and e, the results appear different ty. We have
\302\253p(-\302\243/2r)= 2cosh(fi/2t) , A5)
Partition Function
and
= -jctanh(c/2c). A6)
The heat capacityCv of a syslem at conslant volume is defined as
Cv s x(iajdx)r , A7a)
which by the ihermodynamic identity C4a) derived below is equivalent lo the allcmate
definition
Cr s (SU/dt)y. A7b)
We hold V conslant because the values of the energy arc calculaied for a syslem at a specifiedvolume. From A4) and (!?b},
C - a ' -fEY eXp(E/t)A8a)
The same resull follows from A6).In conventional unils C, is defined as VfSIST), or gU/dT),. ivliencc
dimensions of energy per kcivin. Tim siicctfic heat is drfmed us itio deal capacity per unil
ri^i-immp in ]Ilc pio! of Iicii]cnpiiciiy versus icmpcjiiiurc \\\\\\ t'iiHirc3 \"\342\226\240is Ciiiic^l ^ jLtiOiiKy
anomaly. For i \302\273i, the heat capacuy A8a) becomes
Cr^(E/2iJ. A9)
N'oiicc ihat Cy cc r\"\": in ihis high tempcraiufe limit. Iii ilie low teraperaiuie limil ihc
lemperaiuie is small in comparison whh llic energy level spacinge. For i \302\253e we have
Cy ^ {c/lJ^Xpl-E/lX CO}
The exponen'.ial factor exp[ \342\200\224r,'t} rcJuces Cv rapidly as r decreases, because exp[ ~ 1/v) -> 0
as a- - 0.
Chapter}: Boltzmann Distribution and Hdmhoki Free Energy
Definition: Reversible process, A process is reversible if carried out in such away ihai ihe system is always infmiiesimally close to the equilibriumcondition.Forexample, if the entropy is a function of the volume, any change of volume
must be carried out so slowly ihat the entropy at any volume V is closely equallo the equilibrium entropy a{V), Thus, the entropy is well defined at every
stage of a reversible process, and by reversing the direction of the change thesystem will be returned to its initial condition. In reversible processes, thecondition of the system is welt defined at ail times, in contrast to irreversibleprocesses,where usually we wilt not know what is going on during the process.
We cannot apply the mathematical methods of thermal physics 10 systems
whose condition is undefined.A volume change that leaves the system in the same quantum state is an
exampleofan isentropicreversible process. If the system always remains in the
same state the entropy change will be zero between any two stages of the pro-process, because the number of states in an ensemble (p. 31) of similar systemsdoesnot change. Any process in which the entropy change vanishes is an isentropic
reversible process. But reversible processesare not limited to isentropic pro-
processes, and we shaHhave a special interest also in isothermal reversible processes.
PRESSURE
Consider a system in the quantum state s of energy e,. We assume e, to be a
function of the volume of the syslem.The volume is decreased slowly from V
to V - AV by applicalron of an external force. Let the volumechangetakeplace sufTkrenliy slowly that the system remains in the same quantum state sthroughout the compression. The \"same\" stale may be characterized by its
quantum numbers (Figure 3.5) or by the number of zeros in the wavefunclion.
The energy of the state s after the reversible volume change ts
t\302\243V- &V)~ eJ^V) -(dtJdV)bV + \342\226\240\342\226\240-. B1)
Consider a pressure p, applied normal to all facesof a cube.The mechanical
work done on the system by the pressure in a contraction (Figure 3.6)of thecubevolume from Kto V \342\200\224
AKappears as the change of energy of the system:
U(V- Af) - U(V) .=* At/
--(deJdVyLV. . B2)
\342\226\240\342\200\224-\342\200\224_
O.S i.O t.S
Volume, relative scale
Figure 3.5 Dependence of energy on volume, for the energy levels
particle- connned 10a cube. Ttie curves arc labeled oy m ^\342\200\224i^ -j- .
as in Figure t.2. The niuhtpltctties g are also given. The volume ch;
^s isolropier a cub1\" remains & cube, i he criercy ranfie oc of ]tic stat1
represented in an ensemble of systems will increase in a revcrsibk
of a free
t -%- n_\\
nge here
of ilie energy ranee iiself is of no practical impottance. ti is )he cha
llu: av cragc energy lhal is impotlaiU.
Tigure 3.6 Volume cliangc - AI' i
compression of a cube.
Chapter i; Batiziriann Distribution and Ihtmhottz Free Energy
Here V denotes the energy of the system.Let A be the area of one face of ihecube;then
A{Ax + Ay + Az) = AV , B3)
if ail increments A V and Ax =sAy
- Az are taken as positive in the compres-
compression.The wotk done in the compression is
AV = psA(&x + Ay + Az) = PiAV , B4)
so that, on comparisonwith B2),
P, = -thjdv B5)
is the pressureon asystem in the stale s.
We average B5) overallstates of the ensemble to obtain the averagepressure<\302\243>,usually written as p:
B6)
where U m <t>. The entropy a is held constant in ihc tlcrivaiivi; because ihc
number of states in the ensemble is uuclianged in the reversible compression
we have described. We hove a collection of systems, each in some stme, and
each remains in fhis _st:ile in ihe compression.The result B6)correspondsto our mechanic! picture of the pressure on a
system lhat is maintained in some specific state. Appendix D discusses the
result moredeeply.Forapplicationswe shall need also the later result E0) for
the pressure on a system maintained at constant temperature.We look for other expressions for the pressure. The numberofstatesand thus
the entropy depend only on U and on V, for a fixed number of particles, sothat only the two variables U and V describe the system. The differential of
the entropy is
da[U.V)Uu
B7)
This gives the differential change of the entropy Tor arbitrary independent
differential ch;i:v:_\342\226\240\342\226\240.JU and dV. Assume now that we select dV and dV inter-
dependency, in such a way th;it the two terms on the right-hand sideof B7}
Thermodyaamie Identity
cancel. The overall entropy change da will be zero. If we denote these inter-interdependent values of dll and dV by {W)a and {&V}at the entropy change will
be zero:
B8)
B9))
Bui \302\273heratio (*5[/)\342\200\236/(<)F), is the partial Jerivative of U wilh respectto V at
constant o:
(iV)J(SV), s ldU/dV),. C0)
With this and the dcfiniiion l/i s (So/5!/),.,Eq.B9)becomes
\302\273\342\226\240-<\302\243).-
By B6) the kft-h;ll)d side of C1) is equal to -ft whence
Therniodynaniic Identity
Consideragain the differential B7) of the entropy; substitute the new result for
ihe pressure and the definition of i to obtain
Ttla = dU -f pdV.
Chapter3: Bohynaaa Distribution and Helmbohi Free Energy
This useful relation will be called the thcrtnodjnsmk identity, The form with
variable will appear in E.38). A simple transposition gives
dU = TVS - pdV. C4b)
If the actual process of changeof stateof the system is reversible, we can
identify xda as the healaddedto the system and -pdV as the work done On
the system. The increase of energy is causedin part by mechanical work andin part by (he transfer of heal. Heat is defined as the transfer of energy be!weentwo systems brought into thermal contact (Chapter 8).
HELMHOLTZ FREE ENERGY
The function
C5)
is called the Hclmhoitz free energy. This function plays the part in thermal
physicsat constant temperature that the energy V plays in ordinary mechanical
processes, which arc always understoodto be at constant entropy, because no
internal changes of state are allowedThe free energy tells us how to bulancetiic conflicting demands of a system for minimum energy and maximum en-
entropy. The Helnihoitz free energy will be a minimum for a system S in thermalcontactwith a reservoir (R, if the volume of the systemis constant.
We first show that F is an cxtrcmum in equilibrium ;it constant r and V.
By definition, for infinitesimal reversible transfer from 01 to &,
at constaot temperature. But 1/t h {das/cUi)yiso that dUL
stant volume. Therefore C6) becomes
dFt *= 0 ,
C6)
C7)
which is the condition for F to be an extremum with respect to all variations
at constant volume and temperature.We like F because we can calculate it from
the energy eigenvalues e, of the system(seep. 72).
tfehuitoh: Free Energy
Comment. We can show that the extiemum is a minimum. The total energy is V ==
i',R + Us \342\226\240Then ihe tola! eniropy is
* e^U) - UtfffJtVrivji + ffsiUj). C3)
We know that
(\302\253*A'^),,v* i/t , C9)
so thai C8) becomes
a^a^iV) - FJx , D0)
where Fj \302\273L/j- w^ is ihc free energy of the system.-Now <rlk(Lr) is consiant; and we
rewll thai a = o, + ^ in equilibrium is a maximum with respect to Us . It follows from
D0) lha) Ft mus) be a minimum vviiii respect lo Vs when the system is in ihc mosi probableconfiguraiion. The free energy of the system at constant r, V will increase for any deparluce
hxti'iiplc: Slltthnuiii property vf the free energy tifa parcmti^nctic iyswin, Coilsklef tliC
model sy^tcfn of tliaptcr I. \\viUi N't spins up *uid Ni spins down. Let N = A1? -+- N^',
the spin excess is 2s = A't- W(. Tlie entropy in the SUHiiig approximmion is found
with tlicliclpofan approximate form of A.31}:
,4,,
The energy in a magnetic field Bis.- 2\\tttfl, w licrc m is ilic magnetic woniciilofan elemen-
elementarymagnet. Tile fice cncigy funaion (to be called ldc lapdau function in Chapter tO) is
FJjaB) & V{s,m- io(s),or
D2)
Al ihc minimum of Ft{r,*.B) with respeel io s, this function becomes cquat lo the equilibrium
free energy F[i.B). Thai is, f Jt.<5>,BJ- F(r,B), because <s> is a function of z and B. Theminimum of FL with respect io the spin excess occurs when
= 0 = -JmB + tlog^4-|- D3)
Thus in the magnetic field B lhe thermal equilibrium value ofihe spin excess Is i
or, on dividing numerator and denominator by exp(/\302\273B/r),
<2s>\302\273
Ntanh(mi?/r). D5)
The magnetization ,U is the magnetic momeni per unit volume. If n is the number ofspinsper unii volume, the magnetizalion in thermal equilibrium in the magnetic field is
M = <2s>m/K=Hi\302\273wnh{mB/r). D6)
The free energy of ihe system in equilibrium can be obtained by substituting D5) in D2).It is easier, however, to obtain F directly from the partition function for one magnet:
Z =exp(\302\273iB/r) + expt-niB/r)
\302\2532cosh(mB/t). D7)
Now use the relation F =\342\200\224ilogZas derived below. Multiply by N to obtain the result
for N magneis. (The magnetization is derived more simply by the method of Problem 2.)
Differential Relations
The differential off is
dF = dU - xda - oiit,
or, with use of the thermodynamic identity C4a),
dF = -adt \342\200\224fdV\342\226\240
for which
D9)
These relations are widely used.The free energy F in the result p =\302\273-(rf/^K), acts as the eilecttvc energy
for an isvihcnwil change of volume; contrast this result witii B6). The result
Calculation of Ffrom Z
may be written as
' --($.
\342\231\246
<D,\342\226\240
by use of F = U ~ to. The two terms on the right-hand side of E0) representwhat we may call the energy pressure and the entropy pressure.The energy
pressure ~-(cU/dV)f is dominant in most solids and the entropy pressure
x(?a/3V)t is dominant in gases and in elastic polymerssuelias rubber {Problem
10). The enlropy contribution is testimony of the importanceof the entropy:
Ihe natve feeling from simple mechanics that -JUjdV must tell everything
about the pressure is seriouslyincompletefor a process at constant temperature,because the entropy can changein response to she volume change even if the
energy is independent of volume, as for an ideal gas at constant temperature.
Maxwell relation* We can now derive one of a group of useful tliermodynamic relationscalled Maxwell relations.Form the cross-derivauvirscV/^l\" Hx and ^FfcxcV,which must
be equat 10 each other. It follows from D9) that
(ca/dV)t *{ep;eT)y , E!)
a relation that is not at all obvious. Other Maxwell relations will be derived later at
appropriate points, by similar arguments. The methodology of obtaining thermody-namic relationsis discussed by R. Gtlmore, L Chem. Phys. 75, 5964 A981).
Calculation of F from Z
Because F = U - %a and a = -{cFjdi)Vt we have the differential equation
F = U + i{SF/dt)Vt or ~x2c(F/z)Jcx\302\253U. E2S
Wo show thul this equation is satisfied, by
F/x - -log^ . E.1)
where Z is the partition fund ton. On substitution.
Chapter3:Bollzittnnn Distribution and Hchniipliz Free Energy
by A2). This proves that
F= -riogZ E5)
satisfies the required differential equafion E2).It would appear possible for Fjx to contain an additive constant a such that
F = -riogZ + ar. However, the entropy must reduce to\\oggQ when the
temperature is so low that only the cj0 coincident states at the lowest energy \302\2430
are occupied. In that limit logZ -* logg0~W*. s0 thai a = -cFjcr \342\200\224
g{t log Z)fct = logg0 only if a ~ 0.
We may write the result as
Z - cxp{-f/t); E6)
and the Boftzmann factor (II) for the occupancy probability of a quantum
state s becomes
E7)
IDEAL GAS: A FIRST LOOK
One atom in a box. We calculate the partition function Zx of one atom ofmassM free to move in a cubical box of volume V ~ L1. The orbitals of thefree particle wave equation -{/iJ/2.U)V:^ ==
ei/i are
\\p{x,y,z) = Asia(nxitx L)sm{iiyity}L)sin{n.iiz}L) , E8)
where ir,, nytn. are any positive integers, as in Chapter 1. Negative integers do
not give independent orbitals,and a zero does not give a solution. The energy
values are
We neglect the spin and all otherstructure of the atom, so that a state of the
system is entirely specified by the values of nlt nft n:.
Ideal Cm: A First Look
The partiiion function is ihe sum over the statesE9);
Provided the spacing of adjacent energy values is small in comparisonwiih t,
we may replace the summations by integrations:
2] = I t!nx I dnf (/ji,cxp[\342\200\224
*2{nx2 + ny2 + 'u2)]. F1)
The notation a\" & Azji2/2A/12t is introduced for convenience. The exponentialmay
be written as ihe product of Ihrec factors
F2)
in ierms of ihe concentration ji ~ XjV.
Here
F3!
is catted the quantum concentration. It is the conccnlration associated with oneatom in a cube of side equal to die thermalaverage lie Broglie wavelength,
which is a length roughly equal to /i/M<i\302\273>- h'{Mi)ul. Here <r> is a thermal
averagevelocity. This concentration will keep turning up in ihe thermal physicsofgases,in semiconductor theory, and in the theory of chemical reactions.
For helium at atmospheric pressure at room lemperature, n. s= 2.5 x
lO^cnr3 and uQ = 0.8 x I02scrrT3.Thus, ii/iiq*3x I(T6, which is very
small compared 10 unity, so ihat helium is very dilute under normal conditions.Whenever n/nQ \302\2531 we say ihat the gas is in the ctassieai
regime. An ideal gasis defined us a gas of noninteractingatoms in the classical regime.
The thermal average energyof the alom in the box is, as in (]2),
F4)
becauseZj\"' exp{- eJx) is (he probability the system is in the state it. From F2),
log^i = \342\200\224jlog(I/i) -f- terms independent of t ,
so that for an ideal gas of oneatom
F5)
If t =*kaT, where kB is :Iic Boltzmann constant, then U = lkHT, the well-
known result for the energy per atom of an ideal gas.The thermal average occupancy of'a free particle orbital satisfies the in-
equiility
which sets an upper limit of 4 x !0\026 for the occupancy of an orbital by ahelium atom at standard concentration and temperature. For the classicalregimeto apply, this occupancy must be \302\2531. We note that \302\243\342\200\236as defined by E9)
is always positive for a free atom.
temporarily tiiitil ac develop in Chapter 6 a powerfulmeihod 10 deal with the problem of
many noniulcmaing identical atoms i\302\273a box. We iitsi ireai an idealgas of .V aioms in a
box, all atoms of different species or different isotopes. This is a simple extension of the
one utoni result. We then discuss the major correction factor that arises when ail atoms are
identic!!, of ik b.unc isotope of ihe same specks.
IdealGas;A First Look
\342\226\241
I 2 1
!\342\226\241
Figure 3.7 An .V particle system of free particles with one panicle in eachof.V bo\\es.Theenergy is N limesthat for one particlein one box.
*o Figure 3.8 Atoms of different species in a
If we have cfne atom in each of iV distinct boxes (Figure 3.7), the partition funciion is fhe
product of ihc separateone aiom partition funclioiis:
ZXha*\302\273= Zt{l)Z1B)---Zl(N) , F6)
because ihe product on !he right-hand sido includesevery independent state of llic N
':,(!) f- r.^2) +\342\200\242\342\200\242\342\200\242r^N)
, F7)
where 3, fi.... Cdenotethe orbital indices oUlotm in ihc suewssive boxes. The result F6)
also gives lite pjiiiiioit function of iV itottmiontciJng aiuiits all ordilTcrctii speciesin a
single box (Figure 3.S):
iliis bang the same problem bseauic the energy eigenvalues are iliesameas for F7). If i!icmusses of all thsic dilTercni atoms hapjienedlo be lite same, ihe lotal partiiion funciion
would be Z,-\\ whsrc Zl is given by F2).
When we consider ihe more common pfublem of N identical panicles in one box, we
have to correct Zts~ because it overcounts the disiincl siaics of the ,V idutitic.tl parliclc
lion numbers. For mo hbded particles \302\251and - in a single bos, the siate ^(O) + M*land ihe Male \302\243,(\342\200\242)+ cfrtO|;irc tlisiinct Mates, and both combiti.tttons tmibi be counted i\302\253
lite pmtiuon function. Bui for hs'o tdetiitccil pitritclcs lHc st.iie of energy c, + \302\243*Ij Hie
idcmicai siate as c^ + \342\202\254\342\200\236and only one cnlry is to be made in tfte si.ite sum in the p.irtiiioti
futiciion.
Chapter 3: BoHzmann Distribution ami Hdmhohz Free Energy
If tlicorbiial indices are alt diffrfem, each cniry wilt occur S< limes iti Zts, whereas llic
entry slsould occur onlj once if ihe panicles are identical. Thus, Z,v ovcrcounisthe Stales by
a faclor of ,V!, ;md the correct partition function for N identical particles is
\342\226\240mz'\"-*]<\302\273>**\" |
F8)
in ihe classical regime.HerenQ
= {\\(zi2nk3)yl from F3).There isaslep in ihe argitmcni whcii we assume iliat ail jV occupied orbiuis arc always
diticrcnt oEotla's- It i\302\243no Simple rnallei lo cvaltsaieo^rccily {lieerror tmrouuecci by ih^S
^pproxitnaiiorij but laicr %sfg v,'il\\ cotifj^ni by another rnctliod the validiiy of FSJ in (lie
tlassicai regime n <i na. Tile ft'! fjcior changesihe result for the entropy of !hcideal gas.
The enlropy is an cxpaimemalty measurablequaniily, and i! lias been confirmed ihat the
jV! facior is corrcci in this low concentration irmit.
Energy, The energy of {he ideal gas follows from {he N particle partitionfunciion by use of{l2):
l/ = T3^l0gZ.v/(}T}=^T, F9)
consistent with {65} for one panicle. The free energy is
F = -ilogZy - -i]ogZi'v + ilogN!. G0)
With {he earlier result Zt = naV = (Mr/2n/!2K'i2KandtheStirlingappfoxima-tionlogN! c= NlogN
\342\200\224N, we have
F = -tNIog[{.Ui/2n/i2K'2K] + rNtogN -xN. {71}
From the free energy we can calculateihe entropy and the pressure of the ideal
gasof N aioms. The pressure follows from {49}:
p - -BF/3K),= NxjV , {72}
IPK
= Nr, G3)
IdtalGas: A first Look
which is called the iJcal gaslaw.In conventional units,
1>V = NkuT. G-1)
The entropy follows from D9};
a = -{CF/?z)r =W!og[{Afr/2\302\253/r}3';F] + \\N
- NlogN + S , G5}
G6)
with (he concentration n \342\200\224NjV, This result is known as the Sackur-Tctrodc
equation for the entropy of a monatomtc ideal gas.It agreeswith experiment.
The result involves h through the term hq, so even for the classical ideal gasthe entropy involves a quantum concept. We shall derive theseresultsagain tn
Chapter 6 by a direct method that does not explicitly involve the N\\ or identical
particle argument. The energy {69}also follows from U *= F + xa; with use of
{7!}and {76}we have V = \\Nr.
Example! Equipanlihn of energy. The energy U =|j\\'r from F9} is ascribed to a contri-
contribution i* from each \"degree of freedom\" of each panicle, where the number of degrees offreedom is tlic number of dimensions of the spsce in which ttie slonis move: 3 in Itus
example. In ihe classical focm of siatisiicat mechanics, the partition function contains the
kinetic energy of the particles in an iniegral over the momentum components p,. pt, p,.For one free particle
Jjjexp[~( pI1)/2Mx']dpIdpfilp. , {77}
a result similar to Ft). The limits of integration are \302\261aofat each component. The thermal
average energy may be calculated by use of A2) and is equalwfr.The result ts generalized in the classical theory. Whenever the Kamiltonian of the system
is homogeneous of degree2 in a canonical momentum component, the classical limit of the
thermal averagekinetic energy associated with thai momentum wilt be \302\243r.Further, if the
hamiltonian is homogeneous of degree 2 in a posilion coordinate component, theihermalaverage potential energy associated with that coordinate will also be Jr. The resull thus
3
2
0
-
T,\302\273s
I
^\342\200\224\302\273
lion
!
Vibt
25 50 75100 250 500 1000 2500 5000Temperature.
K
Figure 3.9 Heat capacity at constant volume of one molecule of Hj in
the gas phase. The vertical scale is in fundamental units; !o obtain a value
in conventional units, multiply by kB. The contribution from the three
transnational degrees of freedom is j; the contribution at high temperaturefrom the two rotational degrees of freedom is 1;and the contribution
from ihe potential and kineiic energy of the vibrattonal motion in the
high icinpciaturc limit is I. The classicallimits are attained wlicni \302\273relevant energy level separations.-.
applies to ihe harmonic oscillator in the classical limit. The quantum results for the har-harmonic oscillator and for the diatomic roiator are derived in Problems 3 and 6, respectively.At high temperatures the dais teal limits are attained, as in Figure 3.9,
Example: Entropy ofttuxia*. In Chapter 1 we calculated the number of possible arrange-arrangementsof A and 8 in a solid made up of ,V - ( atoms A arid t atoms B. We found in A.20)
for the number of arrangements:
A/iG3)
The crtiopy associated with these arrangements is
c{A',0 = logflMf) - bg,V!-!og{iV
-r)! ~ logti , G9)
and is piotied in Fi^me 3.10for jV~ 20. This contribution to the total entropy of ;tn alloy
itfcalGas; A First lj>ak
vj7
\\
i
4 4\342\200\224-~
0 0.2 0-4 0.6 0.S 1.0
Alloy composition At_x Br
Figure 3.10 Nftxing entropy ora random binary alloy as a function of
ihe proportions or the constituent atoms A and B. The curve plotted
was calculated for a total of 20atoms.We see that this entropy is a
maximum when A and B are present in equal proportions (x \302\2730.5),
anci trie entropy is zero For pure A or pure H.
system iscalled the entropy of mining. The result G9) may be put in a more convenient u
by use of the Sliding approximation:
a(N,i) =* NlagN - N -(N
- i)\\ag{N- i) + N - t ~ rlogt + r
= NlogN - (N -!)\\og(N
- t)~~ l\\agl
= -(,V - t)Iog(l- //N)- l Iog(f/jV) ,
.viih x = t/N,
- -v)Iog(I --v) .vlog.v].
This result gives the entropy of mi\\iug of an n'loy A^^B, treated as a random (homo-
(homogeneous} soiici solmx^i. TIic problem is J^dopjd in detail in Chapter 11.
We ask i Is the homogeneous solid solution ihe equilibrium condition of a mixture of A
and B atoms, or is the equilibrium a two-phase system, sudi as a miMure ofcrysi2l!::i*s of
pure A and crystallitesof pure B?The complete answer is the basis of much of the science
of metallurgy: the answer will depend on the temperature and on the imcniioiTiicii;;;r-action energies t/M, f/EB,and UAa. In the special case iluit tltc iiue faction energies bciw.vn
A A., BB, and AB neighbor pairs are all equal, ihe homogeneous solid solution will have a
lower free energy than Hie corresponding mb-lure ofc/yitatlilci of thepure elements. The
free energy of the solid solution A [ -^B, is
F = Fo -tct(.v)
\302\273Fo + A't[A
- .\\)Iog{l- a) + xlogx] , (81)
which we must compare with
F =A
-x)F0 + xF0 - Fo (82)
Tor the mixture of A and B crystals in the proportion A-
.vj to x. The entropy of mixing
is always positive\342\200\224all entropies arc positive\342\200\224so that the solid soiution has ihe lowerfree energy in this special case.
There is a tendency for at least a very small proportion of any element B to dissolve in
any other element A, evert if a strong repulsive etlergy exists between a B atom and ihesutroundiug A atoms. Let this repulsive energy be denoted by U, a positive quantity. If a
very small proportion -\\ \302\253t of B aioms is present, the Iota! repulsive energy is xi\\U, wlk-re
xN is ihe number of B atoms.Tlic mixing eniropy (SO) is approximately
a = ~.\\N\\ogx (83)
Fix)- N(sU + txiog.v) , (84)
which has a minimum when
BFfBx \302\273N(U + xiog-v + t) = 0 , {85}
x =exp(~!}exp(-t//T}. (86)
This shows there is a natural impurity content in all crystals.
SUMMARY
1. The factor
?{\302\243j)= expf-e./tJ/Z
. is the probabiHty of finding a system in a slate s of energyc,when the system
is in thermal comaei with a large reservoir ai temperature r. The numberofparticlesin the system is assumed constant.
2. The partition function is
3. The pressure is given by
p = ~{cUldV)a = x$gIc\\')v.
4. The Hejmltoltz free energy is defined as F ^ U ~ xa. H is a mittimum in
equilibrium for a system held at constant i, V.
5. a - -i?F/dT)y; p \302\273~{8F/dV)r
'
6. }\342\226\240ss -tlogZ.Tltisrcsulitsvcry ttscru! incilcuhitiansorf and ofquuniiiicssuchas p and o derived from F.
7. For an idealmonatomicgasof N atoms of spin zero,
ZH = (nQVf/N\\ ,
if u ~ N/V \302\253\302\273a.The quantum concentration nQ s (A/r/2n/t2K/I. Further,
pV- JVt; o =
W[log(iic/H)+ 5]; Q =
fW.
8. A process is reversible if the sysiem remainsinfinitesimally close to ihe
equilibrium state at all timesduring iheprocess.
PROBLEMS
/. Free energy of a two state systenu (a) Find an expression for the free
energy as a function of t of a system with two states, one at energy 0 and onein energy c. (b) From ihe fro; ettcrgy, find expressions for the energy itnd entropy
of the system. Tttc entropy is plotted in Figure 3.11.
2. Magnate susceptibility. 00 Use ihe pitriiiion function io find nn exaci
expression for ihe tnagiteliAUiou M end tltu susceptibility x s ilM/tlB as afunction of temperature and magnetic field for the model system of magnetic
moments in a magnetic field. The result for the magnetization is M ~
mutanli^jiB/i}, as derived in {46}by another method. Here n is the particle
Chapter Si Bolizmann Distribution and tjelinholt; Free Energy
OR 2
0.6
0.4
0.2
ft
-
/
/
A-
/ ]
0 0.5 1.0 1.5 2.0
Figure 3.12 Plot of ihe loui macaeiicmoment as a function
ofmfl'r. Notice thai 31 lou mB/x the momeni is a U.icarfunciio
of HiiJ/r, bul Ai high hiS r the manicnl Sends 10salurale.
contciiiraitot!. Tlie result is plottcJ in Hgure 3.12. (b) Find ihe free energy and
express the result as a function only ofr and the parameicr.v s M/tuu, (c) Show
thul the sust;eptibiltty is x \342\200\224j\302\253\302\273Vin the ttniit /\302\273B\302\253r.
i. /Vcc energy of a harmonic oscillator. A one-dt'mcnsian;il liurmonic oscil-oscillator has an infinite series of equally spacedenergy states, with \302\243,
= sho, where
Figure 3.13 Enin
osciliuSor of frequency
s is a positive imegeror zero, and to is tlic classical frequency of she oscillaior.Wu have chosen the zero of energy at the state s = 0. (;s) Show that for a
harmonic oscillator the free energy is
(S7)
Notethat at high temperatures such ihal x \302\273fitu we may expand the argumentof the logarithm to obtainF^i !og(/itu/t). (b) From (87) show thai the entropy
($8}
The entropy is shown in Figure 3.13 and the heat capacity in Figure 3.1-i.
4. Energy fluctuations. Consider a systemof fixed volume ta thermal contact
wiiis a reservoir- Show that the mean square fluctuation ia she esiemyof lhcsyslcin is
Here U is the conventtooai symbol for <e). Hint: Use the partition fitnctwn Z
to relate c-U/ci to ihc mean square fluctuation. Also, multiply out the :crm
(\342\226\240\342\226\240
J. Note: The temperature t of a system is a quantity that by delirsiiion does
Chapter 3: Bohzmnn Distrihut mJ IIcIihUoUz Free Energy
S\302\273re3.S4Heal capacity Versus u-mperatufe
r harmonic oscillator of frequency u>. The
ifiiontal scale is in units of i/fiw, whkh isunncal wilh T flE. \302\253herc0, is calkd rhc
mlcln Icniperamte. \\n the high temperatureiiit Cv
- kB.ot 1 in run<iamciiia! miiis. This
iluc is k^own 'as \\\\\\t c^issic^I V'^Iv:;. i\\i low
mpcratures C(- decreases csponeniially.y
^
0.5 1.0
t r
not fluctuate v,\\ value \\shcn the system is in thermal contact with a reservoir.
Any otltcr attiiude would be inconsibletit with our definition of the temperatureof a system.The energy of such a system may fiuetijiitc, but the temperaturedo^s ltot. Sorr.c workers do not acllterc to a rigorous definition of temperature.
Thus Landau arid Ltfshttz etve Hie result
(90)
equal to- r=C,.,
\302\253AtJ>=> t2/Cv ,
but this should be viewed as just another form of (89) with At
AV;CV. We know that AU = Cy At, whence (90) becomes <(Akwhich is our result (89).
5. Ovcrhaussr effect. Suppose that by a suitable external mechanicalorelectrical arrangement one can add ae to the energy of 'he heat reservoir
whenever the reservoir passes to the system the quantum of energy e.The net
increase of er.eray of the reservoir is(a \342\200\224l}e. Here a ts some numerical factor,
positive or negative. Show that the effective Boltzmantifactor for this abnormal
system is given by
This reasoninggives the statistical basts of the Ovcrhattsereffect whereby the
nuclear polarization in a magnetic field can be enhanced above the thermalequilibrium polarization. Such a condition requires the active supply of energyto the system from an external source. The system is not in equilibrium, but ts
said to be in a steady state. Cf. A. W. Overhauser, Phys. Rev. 92, 411 A953).
6. Rotation of diatomic molecules. In our first look at the ideal gas we con-considered only the translationa! energy of the particles.But molecules can rotate,
with kinetic energy. The rotational motion is quantised; and the energy levels
of a diatomic molecule arc or the form
ft;} = jlj + l)\302\243o (92}
where; is any posilive integer includingzero:/ = 0,I,2,.... The multiplicity
of e;ich rotational level is </(;) - 2; + I. (a) Find the partition function Zk(t)for the rotational states of one molecule. Remeniber that Z is a sum over allstiitcs,not over all le\\cls\342\200\224this makes a difference, jb) Evaluate ZH(x) approxi-
approximately for r \302\273e0, by converting the sum to an integral,(c)Do the same for
r \302\253sOl by truncating the sum after the second term,(d) Give expressions for
the energy V and the heat capacity C, as functions oft, tn both limits. Observethat the rotational contribution to the heat capacity of a diatomicmoleculeapproaches 1 {or, in conventional units, A-,,)\\Uv?n r \302\273r.0.lc) Sketch the behaviorof L'(t) nnd C{x),showing the limiting behaviors for r -\302\273ro and r -* 0.
7. Zipperpruhh'W. A zipper h:is N links; each link has n state tti which it is
closed with citefgy 0 and u siate in which ii is open with energy t:. We require,
however, that the zipper can only im/jp from the left end, and that site link
mmiher s can only open if all links to jhe left {1,2,.. .,s - 1}are already open,
(a) Sliow ihat the parution fuiicliotl can be summed tn the form
193}
{b) In the iirnii e \302\273i, find the average number of open links.The modelis a
very simplified model of lhe unwinding ofnvo-siraiided DNA molecules\342\200\224see
C. Kittel, Arner. J. Physics 37, 917A969).
5, Quantum concentration. Consider one particle confined to a cubeof sideL;lhe concern rai ion in effect is n = 1/L3. Find the kineiic energyof i he particle
when in the ground orbtial. There will be a value ofthceoticentraiioii for whichlhis zero-poiniquantum kineiic energy is equal to the temperature r. (At this
concern ration the occupancy of the lowestorbitalisof the order of unity; ihe
lowest orbital always hasa higheroccupancyilian any oilier orbiial.) Show thai
the concentration nQ thus defined is equal to the quantum eoticcntraitonnQ
defined by {63), wtthtti a facior of ihe orderof utitiy.
9. Partition function for two systems. Show thai the partition functionZ(l 4-2)of two independent sysiems I and 2 in thermal contaci at a common
temperature t is equal10ihe produci of ihe paniUon funciions of the separatesystems:
Z(I + 2)= Z(I)ZB). (94)
ChaptcrS: Boltzntann Distribution and Heimhaltz free Energy
10. Elasticity of potythevs. The lhermodyiiamic identity for a One-dimensional
system is
zda = dU ~ fdi (95)
when / is the external forceexertedon the line and ill is the extension of theline. By analogy witSi C2) we form the derivative to find
.-Ha-
The direction of the force is opposite to the conventional direction of thepressure.
We consider a polymeric chain of N links each of length />, with each link
equally likely to be directedto the right and to the left, (a)Show ihat the numberof arrangementsthat give a head-ioiail length of / =
2js|p is
q(N,-s) + q(N,s) =-T-
\342\200\224~'
(97)(kN 4- 5)! (\302\243N
- s)\\
(b) For\\s\\
\302\253JV show that
(98)
(c) Show that the force at extension / is
/ = h/Np1. (99)
The force is proportional to the temperature. The forcearises because the
polymer wants to cur! up: the entropy is higher in a random coil than iti ati
uncoiled configuration. Warming a rubber band makesit contract; warming a
steel wire makes tt expand. The theory of rubber elasticity is discussedby
H. M. James and E. Guilt, Journal of ChemicalPhysics II, 455 A943); Journalof Polymer Science4, 153A949);seealsoL. R. G. Treloar, Physics of rubber
elasticity, Oxford,1955.
II. Ouc'dimcmionaigas.Consideran idea! gas of A' particles, each of massM, confined to a one-dimcsisional line oflcnath L. Find the entropy at tempera-
temperaturer. The pti5tides have spin zero.
Chapter 4
Thermal Radiation andPlanck Distribution
PLANCK DISTRIBUTION FUNCTION
PLANCK LAW AND STEFAN-EOLTZMANN LAW
Emission and Absorption: KirchhorT LawEstimation of SurfaceTemperatureExample: Cosmic Black Body Background Radiation
ELECTRICAL NOISE
PHONONSIN SOLIDS:DEBYE THEORY
Numberof Phonon Modes
Number of Thermal PhotonsSurface Temperature of Ike Sun
Average Temperature of the interiorof the Sun
Age of the SunSurface Temperature of tlte Earth
Pressure of Thermal RadiationFree Energyofa Photon Gas
Heat Shields
Photon Ga:Heat CapaiHeatCapa'HeatCapaiEnergy Flu>
HeatCapai
Angular D:
in One Dimensionty ol\" intergalactic Space
ty of Solids in High Temperature Limit
;ty of Photons and Phononstuations in a Solid at Low Temperaturesty of Liquid 4He at Low Temperaturestribution of Radiant Energy Flux
image of a Radiant ObjectEntropy and Occupancy
Isentropic Expansion of Photon GasReflective Heat Shield and Kirchhoff's Law
969793
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112113113113113113114
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SUPPLEMENT: GREENHOUSE EFFECT
Chapter's: Thermal Radio
[We consider] the distribution of the energy U among N oscillators offrequencyv, // U is viewed as divisiblewithout limit, then an infinite number ofdistributionsarepossible.We consider however\342\200\224wUl this is the essential pointof the whole calculation\342\200\224Vas made up of an entirely determined nimiber of
finite equal parts, and we make useof the natural constant h = 6.55 x fO~21
erg-sec.This constant when multiplied by the common frequency v of the
oscillators gives the element of energye in ergs ....
M. Planck
Planck Distribution Fu
PLANCK DISTRIBUTION FUNCTION
The Planck distributiondescribesthe spectrum of the electromagnetic radiationin thermal equilibrium within a cavity. Approximately, it describes the emission
spectrum of the Sun or ofmeta!heated by a welding torch. The Planck distribu-distributionwas the first appHcation of quantum thermal physics. Thermalelectro-electromagnetic radiation is often caHed black body radiation.ThePlanckdistribution
also describes the thermal energy spectrum of lattice vibrationsm an clastic
solid.
The word \"mode\"characterizesa particularoscillationamplitude pattern in
the cavity or in the solid. We shall always refer low = 2nf as the frequency of
the radiation. The characteristic feature of the radiationproblemisthat a mode
of oscillation of frequency w may be excited only in units of the quantum of
energy hio. The energy \302\243,of the state with s quanta in the mode is
e, U)
where s is zeroor any positive integer (Figure 4.!). We omit the zero pointenergy \\hai.
These energies are the same as the energiesofa quantum harmonic oscillator
of frequency to, but there is a differencebetween the concepts.A harmonic
Figure 4.1 Stales of an oscillator that
represents a mode of frequency tu of an
to s photons in the mode.
(-haptcr 4- Thtftjml Radiation and Planck Distribution
modes a and b, of frequency &>\342\200\236and \302\253>t.The amplitude
magnetic field is suggested in the figures for one pliotoroccupancy ofeach mode.
oscillator is a iocasizedosciiiator.whereasthe electric and magnetic energy ofan electromagneticcavity mode is distributed throughout the: interior of the
cavity (Figure 4.2). For both problems the energy eigenvaluesare integral
multiples of ho, and this is the reason for the similarity in the thermal physics ofthe two problems. The language used to describe an excitationis different; s for
the oscillator is called the quantum number, and s for the quantized electro-
electromagnetic mode is called the number of photons in the mode.We first calculate the thermal average of the number of photonsin a mode,
when these photo ns are in thermal equilibriumwith a reservoir at a temperaturet. The partitionfunction C.10) is the sum over the states A):
}{~shia/r}. B)
Thissum is of ihc form TV, with .\\- s exp{-/i<y/0- Because .\\- is smaller than 1,
the infinite scries may be summed and has the value 1/A -- x),whence
1 -exp(-tou/T)C)
Planck Lax and Slefan-BohimannLt*
The probability that the system is in the state s of energy slim is given by the
Bohzntann factor:
P(s)
Thethermalaveragevalue of s is
exp( \342\200\224shci/x)
E)
With >' = ftej/r, the summation on the rmht-hand side has the form:
From C) and E) we find
-cxpt-v)/
I -exp(->.)'
F)
This is the Planck distribution function for the thermal average number of
photons(Figure4.3) in a singie mode of frequency w. Equally, it is the average
number of phonons in the mode.The result applies lo any kind of wave field
with energy in the form of (!).
PLANCK LAW AND STEFAN-BOLTZMANN LAW
The thermal average energy in the modeis
)\342\200\2241*
G)
Chapter 4: Thermal Radiation and Planck Disiribttlio
Figure 4.3 Planck distribution as a function
ofihe reduced temperature i./rw.Here <s(w)>
is Hie thermal average of the rmniber of
photons in the mods: of frequency en. A plot of
O(o)> + i is also given, where $ is the effective7ciopoint occupancy of ihc mode; the dashed
line is i!ie classical asymptote.Noie that we
0.5
A
/
*(\302\253)+
/
0.5
The high temperature limit t \302\273ha) is often called the classical limit. Hereexp{frfcj/t) may be approximated as 1 4- lna/r 4- \342\200\242*
\342\226\240,whence the classical
average energy is
<\302\243>̂ T. (8)
There is an infinite number of electromagnetic modes within any cavity. Each
mode n has its own frequency wn. For radiation confined within a perfectlyconducting cavity in the form of a cube of edgeL, there is a set of modes of theform
Ex= ExOitn wtcos(fiJji.v/L)sin(iiyjij'/L)sin(fi.Jiz/L) , (9a)
ID sin(fl=Tiz/L) , (9b)
Et = E-0 (9c)
Here Ex,Er and Ex are the three electric field components, and \302\243lQ,Ey0and
\302\243;0are the corresponding amplitudes. The three components are not indepen-
independent,because the field must be divergence-free:
A0)
Planck Law and Slcfan-Bolwtann Law
When we insert {9}into A0} and drop ail common factors, we find the condition
\302\243,0\", + E^nr 4- E:Qn: -. Eo \342\226\240n \302\2730- A1)
This states rhar the field vectors must be perpendicular to the vector n with
the components nx, nyand >l, so that the electromagnetic field in the cavity is a
transversely polarized field.Thepolarizationdirection is defined as the directionof Eo.
For a given triplet n,, nft n. we can choose two mutually perpendicular
polarization directions, so rhat rhere are mo distinct modes for each iriplei
On substitution of (9)in the wave equation
V'-v1 i:y trJJ. '. .(V
wilh c the velocity of light, we find
cWnJ 4-Hya
+ n;2) = w3L2. A3)
This determinesthe frequency \302\253)of the mode in terms of Hie triplet of integers>h< \"y \";\342\226\240if we define
\342\200\236=(rtj[2 + ,,/ + rta\302\273)\302\273'i, A4)
then the frequencies are of the form
w.= mrc/L. A5)
The total energy of the photons in the cavity is, from G),
The sum is over the triplet of integers nx, ny,n.. Positive integers alone will
describe all independent modes of tlic form (9). We replace the sum over nx,
ny, iij by an integral over the volumeclementditx dny dnx in the space of the modeindices.That is, we set
Chapte
where the factor | =h(\302\243Karises because only the positive octant of the Spaceis
involved. We now multiply the sura or integral bya factor of 2 because Ihere are
two independent polamations of the electromagnetic field (two independentsetsofcavity modes). Thus
, ft1 = jiJo
hu)n
(nVic/L) r dnn* \342\200\224
\302\273\302\260 exA8)
with A5) for \302\253\342\200\236.Standard practice is to transform the definite integralto oneover a dimensionless variable. We set x = nhcn/LT, and A8) becomes
A'J)
The definite integral has the value z*/l5; it is found in good standard tablessuch as Dwight (cited in the general references}. Tlie energy per unit volume is
\\ShsB0)
with the volume V = L1. The result that the radiant energy density is propor-lional !olhe fourth power oflhe lemperalure is known as theStefan-Boltzmannlaw of radiation.
For many applications of this theory we decompose B0) into the spectraldensity of the radiation.Thespectraldensity is defined as the energy per unil
volume per unit frequency range, and is denoted as \302\253\342\200\236,.We can find \302\273ufrom
(IS) resvritlen in terms of w:
U/V B1)
so that the spectral density is
B2)
Planck Law andSufan-Boltzmam Law
A
1.2
1.0
0.6
0.4
0.2
0
1
/
/
/
/
/
/
\\
\342\200\224 \342\200\224
\\
\\
\342\200\224
Figure-1.4 Ploiof.vJ/(c* ~ l)willi.v = bttf/t. T\\\\h
runciion h involved in the Planck radiation law for llic
spectral density uw. flic temperature of a black body may
be found from ilie frequency tjmil ai which the radiant
energy density is a maximum, per unit ffequency range.
This frequency is directly proportional so ihe tempera sure.
This result is the Planck radiation law; It gives the frequency distribution of
thermal radialion (Figure4.4).Quantum theory began here.
The entropy of the thermal photonscanbe found from the relation A34a)
at constant volume:da ~ dUfr,whence from B0),
Thus the entropy is
B3)
The constantofintegrationis zero, from C.55) and the relation belsvecnF and a.
Chapter 4: Thermal Radiation and Planck Distribution
gy y v gy p
The flux density is of the order of the energy contained in a column of unit
area and length equal to thevelocity
of light times the unit of time. Thus,
The geometrical factor isequalto \302\243;the derivation is (he subject of Problem 15.Thefinal result for the radiant energy flux is
by use of B0) for the energy density U/V. The result is often written as
Jv \302\253 B6)
aB s= b2V/60AV B6a)
has the vahie 5.670 x 10~8 W m~2 K~* or 5.670x 10\"*erg cm\022 s~' K~\".
(Here cts is not the entropy.) A body that radiates at this rate is said to radiateasa black body. A small hole in a cavity whose walls are in thermal equilibrium
at temperature T will radiate as a blackbody at the rate given in B6). The rate
is independent of the physicalconstitutionof the walls of the cavity and de-
depends only on the temperature.
Emission and Absorption;Ktrchhoff Law
The ability of a surface to emit radiationis proportionalto the ability of the
surface to absorb radiation. We demonstrate this relation, first for a black bodyor biacksurface and, second, for a surface with arbitrary properties. An object
is defined to be blacktn a given frequency range if all electromagnetic radiationincidentupon it in that range is absorbed. By this definition a ho!e in a cavity isbiack if the hole is small enough that radiation incident through the holewill
t oj Sutj
reflect enough times from the cavity walls lo be absorbed in the cavity with
negligible loss back through the hole.The radiant energy flux density Jv from a black surfaceat temperaturex is
equal to the radiant energy flux density Jv emitted from a small hole in a cavity
al the same temperature. To prove this, let us close the hole wilh the blacksurface,hereaftercalledthe object, in thermal equilibrium the thermal averageenergy fiux from the black object to the interiorof the cavity must be equal,but opposite, to the thermal average energy flux from the cavity to the black
object.We prove the following: If a non-black object at temperature t absorbs a
fraction a of the radiation incidentupon it, the radiation flux emitted by the
object will be a ttnies the radiation flux emitted by a black body at the same
temperature. Let a denote the absorptivity and e the emisstviiy, where the
cmissi\\ity is definedso that the radiation flux emitted by the object is e timesthe fiux emitted by a black body at the sametemperature.Theobjectmust emit
at the same rate as it absorbs if equilibrium is to be mainiamed. H follows that
a~e. This is the Kirchhoir law. For the special case of a perfectreflector,a is
zero, whence e is zero. A perfect reflector docs not radiate.The argumentscanbegeneralizedto apply to the radiation at any frequency,
as betsveen <o and w + ito. We insert a filter between the object and the hole in
the black body. Let the filter reflect perfectly outside this frequency range, and
let it transmit perfectly within this range. The flux equality arguments now
apply to the transmitted spectral band, so that a(w)= eM for any surface
in thermal equilibrium.
Estimation of Surface Temperature
One way to estimate the surface temperature of a hot body such as a star isfrom the frequency at which Ihe maximum emission of radiant energy takes
place (see Figure 4.4). What this frequency is depends on whether we look at the
energy fiux per unit frequency range or per unit wavelength range. For if,,,, the
energy density per unit frequency range, the maximum is given from the Planck
law, Eq. B2), as
\342\226\240\342\200\2420 ,
3 - 3exp(-x) = x.
Chapter 4: Thermal Radiation and Planck DiVnfmrion
This equation may be solved numerically. The rool is
kca^JkgT = xm ^ 2.82 , B?)
as in Figure 4.4.
Example:Cosmicblack body background radiation. A major recentdiscovery is that theuniverse accessible to us is filicd with cudutian approximately like thai of a black bodyat 2.9K. Tlic existence of lliis radiation [Figure 4.5) is important evidence Tor big bangcosmotogiol modelswhich assume ltw! tli^ unhorse is expanding and cooling wiitl liliic.
so ihaf Ihc nimicr ;uid itie bhiirk body radiaiion were in Itiaiipl cqtiitilirium. IJy ihe limeitic universe h;id coo ted lo 300A K,! tie m;iiUv Mas primarily in she farm of atomic liydrogen.This irucructs with bLjck body r^diLitJi^Ji only si the fic^ucOci^b of jlic liydro^jcti &fH:ctriJ
lines. Most of llie biack body r;idi;ition clergy thus was cffctiutty docouptoti from itie
mancr. Tftcrcafier ihe radiaiion evotved with \302\273mcin a very siinpte way: llic plioion gaswas cooled by expansion al constani cniropy io a icmpcraiure of 2.9 K. Tlic pfioion gas will
remain at consent cniropy if Ihe frequency of each mode is towered during ihc expastsionof tnc universe wilfa llic nuoi^er of pljotons it\\ c>icti mooe Kept con^JunlL We show in ^joj
below that ihe entropy is constant if lhe number of pholons in each mode is consiant\342\200\224the
occupancies determine ihe eniropy.After lhe decoupling Urn evoluiion of tnaucr inlo heavier atoms (which are organized
inlo galaxies, stars, and ciuSl clouds) was more complicated lhan before decoupling.Electromagnelicradiaiion, such as slarliuM, radiated by the maiter since lhe decouplingis superimposedon the cosmic black botiy radiation.
ELECTRICAL NOISE
As an important exampleof the Planck law in one dimension, we consider the
sponMucousthermalfluctuations tn voltage across a resistor. These fluctuations,which are called noise, were discovered by J. B. Johnsonand explained by
H. Nyquisi.* The characteristic property of Johnson noise is that ihc mean*
square noisevoltage ts proportional lo llic value of the resistance R, as shown
by Figure 4.6. We shall sec that <K:) is also directly proportional to tlie tem-
\342\226\240H. N!jquisi,p!:js Rlical jijsus, Wiley, !
[ Microwave
| Interstellar CN
Q IR measuremeni
2.9 K Black body-
Frequency (cm\"!)
Figure 4.5 Experimental me
body radiation. Observationssurememsof lhe spcclrum of the cosmic W
nf the flux were made with microwave heicr
oflhe speclrum ofinlcrslettacwiih a bulbon-borne infrared s
Caurlesy of P. L.Richards.
; neiir lhe peak, unJ vvere rneusi
ai frequencies abo\\c lhe peak.
pcrature r and tlie band'-vidili A/of the circuit. (Tliis secifon presumes a knov.1-
edge of cLvironuignetic wave propagation at Hk inicrtncdiiitc level.)Tlie Nyquist tlicorcm gives a quanlhative. expression for ihe llicrma!noise
voltage generated by a resisior in thermai equilibrium. The theorem is lhcr^fore
uccded in any estimate of Hie iinnting signal-io-uoise ratio of an experirr.cnial
Chapter 4: Thermal Radiation and Planck
Zl
\342\200\242Carbon filamen~+ Advance wirexCuSO, m H,OvNaCi in H..O
0 0.1 0.2 0.3 0.4 0.5RcMblancu component, in Mil
kinds of conductors,including electrolytes. Afiei
J.B.Johnson.
apparatus. In ihe original form the Nyqutst theorem states that ihe mean
square voltage across a resistor of resistance ft in thermal equilibrium at
temperature t is given by
B8)
where A/ is the frequency* bandwidth within which the voltage fluctuations
are measured; all frequency components ouisidethe given range are ignored.
We show below that the ihermal noise power per unit frequency range delivered
by a resistor to a matched loadisx;the facior4 enters where it does because in
the circuit of Figure 4.7, the power deliveredto an arbitrary resistive load R' is
'(R + Rf
' B9)
which al maich (R' = R) is <1\">/4J!.
'In this section the word frequency refers to cycles per unit time, and not to radians per unit till
Noise generator
Figure 4.7 Equivalent ciicuii for a resistance ft with
a generatot oftliermal noise that delivers power to aload R'.The cilrrenl
which is a maximum wiih respcel to R' when R' = R.hi this condition the ioad is said to bo matched lo thepower supply. At match, & - (Yi)f4R. The filter
enables us to limit the frequency bandwidth under
consideration; ihal is. [lie bandwidth 10 whicii the mean
square voiiagc fluciuaiion applies.
Consider as in Figure 4,8 a losslesstransmission line of lenglh L and charac-characteristic impedance Zc ~ R terminated at eachend by a resistance R. Thus theline is matched at eachend, in the sense that ail energy traveling down the linewill be absorbed wiihoui reficciion in the appropriate resisiance. The entirecircuit is maintained at temperature t.
A transmission line isessentiallyanelectromagneticsysiemin one dimension.
We follow ihe argument given above Tor the distribution ofphoions in thermal
equilibrium, but now in a space of one dimension instead of three dimensions.Thetransmission line has two photon modes (one propagatingineachdirection}of
frequency 2nfa= 2nn/L from A5), so shat ihere are iwo modes in the fre-
frequency range
Sf~c'lL, C0)
where c' is the propagation velocity on the line. Each mode has energy
exp(ftwAJ
C1)
Figure 4.8 Transmission Hue oficiijjtli L with
derivation of ihc Nyquist liieorcm. The
aclerislic impedance 7,cof ihe [ ra us mission
line has ihe vaiuc R. According lo ihcmdamcniai theorem of Iransrnission iines, [he
erminai resistors are matched to the line ivlicn
heir resistance has ihc same value R.
in equilibrium, according to the Planck distribution. We are usually concerned
with circuits in the classical limit hw \302\253z so that the thermal energy per modeis r. It follows that the energy on the line in the frequency range A/ is
C2)
C3)
The rale at which energy comes off the line in one direction is
The powercomingoff the line al one end is all absorbed in the terminal
impedance R at that end; there areno reflections when the terminal impedance
is matched to the line.In thermal equilibrium the load must emit energy to the
line at the same rate,or elseitstemperature would rise. Thus the power inputto the load is
9 =</2>R
\302\273iA/ , C4)
but V ~ 2R1, so that B8) is obtained. The resulthas been used in low tempera-
temperature(hermomctry, itt temperature regions {Figure 4.9) where it is more con-
convenient to measure (V1} than t. Johnson noise is the noiseacrossa resistor
when no dc current is flowing. Additionalnoise{not discussed here) appears
when a dc current flows.
PHONONS IN SOLIDS; DEBYE THEORY
So I decided to calculate the spectraldistribution of the possible fvee vibrations
for a continuoussolidand to consider this distribution as a good enoughapproximationto the actual distribution. The sonic spectrum of a lattice must,
j in Solids: Dcbye Theory
100T
Figure 4.9 Mean square noise \\ o'uge
flucluations observed cxperimcn::i))y from a3 jiO resistorin ihe mixing chamber of adilution refrigerator as a function of magneticicmpcralurc indicated by a CMN' powdertlidrmometer. After R. R. GiiTarJ, li. A- Webb,
and J. C Wheailey, J. Low Tcnir Physics 6,533 A972).
of course,deviatefront this its soon its t!ie wavelength becomes comparable to
t/ie disittuees of the atoms. . .. The only thing which had to be lione was lo
adjust to she fact that every solid ofjunta dimensions contains ajiuite numfrc'r
of atoms and thereforehas a finite mint her of free vibrations.... At low L'uoiujh
temperatures, and ttt perfect analogy to the radiation htw of Stefan-
Boltzmann ..., the vibrational energy contentof it solid will be proportional
P. Dcbye
The energy of an elasticwave in a solid is quantized just as the energy of an
electro magnetic wave in a cavity is quantized.The quantumof energy of an
clastic wave is calleda phovwn. The thermal average number of pitonons in anelasticwave of frequency oj is given by the Planck distribution function, just
;is for photons:
1C5}
We assume that t!ie frequency ofan elastic wjive is independent of theamiMttmleofthe elasticsixain. We want to find the energy and heat capacityofiheelasticwaves in solids. Several of the resiiks obtained for photons may be carried
over to plionons. The resultsare simple if we assume th:tt the velocities of ailelasticwaves are equal\342\200\224independent of frequency, direction of propagaiion,and directionofpolarization.Thisassumption is not very accurate, but it helps
account for the general trend of the observed results in many solids, with a
minimum of computation.Therearetwo important features of the experimental results: the heat capacity
of a nonmctallic solid varies as tJ at lowtemperatures, and at high temperatures
the heat capacity is independentof the temperature. In metals there is an extracontribution from the conduction electrons, treated in Chapter 7.
Number of Plionon Modes
There is no limit to the number of possible electromagnetic modes in a cavity,
but the number of elastic modesin a finite solid is bounded. If the solid consistsof N atoms,each with three degrees of freedom, lite total number of modesrs
3A?. An elastic wave has three possible po!a matrons, two transverse and one
longitudinal, in contrast to the two possible polarizations of an electromagneticwave.In a transverseclasticwave the displacement of the atoms is perpendicularto the propagationdirectionof the wave; in a longitudinal wave the displace-displacement is paraiicl to the propagation direction. The sum of a quantity over all
modes may be written as, includingthe factor 3,
| JW <*\302\253(\342\200\242*\342\200\242), C6)
by extension of A7). Here n is defined in terms of the triplet of integersnxt i\\yt iu,
exactly as for photons. We want to find ;iralI such that the total number ofelasticmodests equal to 3JV:
C7)
In the photon problemthere was no cor
number of modes. It is customary to writebecomes
espondinglimitationon the total
D, after Debye, for nraaI. Then C7)
in\302\273D3= 3;V; nD = FW/7!I'3. C8)
The thermalenergy of the phonons is, from A6),
,\\umbcr ofPho
or, by C6) and CS),
D0)
By analogy with the evaluation of A9), with the velocity of sound v written in
place of the velocity of light r,
V = {'in2twf2L)(xL/n!n-)-i J*V* \342\200\224~~^. D1)
where X = ji/irii/Lr. For L3 we write the volume V. Here, with C8), the upperlimit of integration is
usually written as
,vfl- 0/7 = fcsO/'t . D3)
where6 is called the Dcbye temperature:
0 = (hv/kB)Fn2N/VI \\ D4)
The result D1) for the energy is of special interest at low temperaturessuchthat T \302\2530. Here the limit .xD on the integral is much larger than umly, and .vo
may be replacedby infinity. We note from Figure 4,4 that there is littlecontri-contribution to the integrand out beyond x = 10.For the definite integral we have
f\"^ __\302\243_-?- , D5)Jo exp.x-1 15
as earlier. Thus the energy in the low temperaturelimitis
proportionalto T4\302\273The heat capacity is, for i \302\253kB0 or T \302\2530,
D7a)
Chapter 4: Thermal Radiation ami Planck Distribution
T 17.78
a
E 13.33i
^ 4.44
0\\A
Y
A
- -\342\226\240\342\226\240\342\200\224
A
A
3.99 5.32
r3, in K3
Figure 4.10 Low tempcralure heat capaciiy of solid argon, pioiicd against
TJ10 show Ihc cxcettent agreement whh !tie Debyc T1 law. The value of 0from these daia is 92 K. Courlesyof L.1 Fincgold and N. E. Phiiiips.
In conventional units,
D7b)
This result h known as the Debye T1!aw.*Ex peri menta! results for argon are
plotted in Figure4.10.Representative experimental values of the Debye tem-
temperature are given in Table 4.1. The calculated variation of Cv versus T/6 is
plotted in Fig-jrc 4.11. The high temperaturelimit T\302\273 0 is the subject of
Problem II. Severalrelatedtiiermodymmiicfunctions for a Debyc solid are
given in TabL* 4.2 and are plotted in Figure 4.12.
13.297 A912): 14,65A9K).
Aiimitr of Photon Modes
Lu 210
Jjg
UJ
X
pTUJ
S
e
Z
(J
7.
1
UJ
U
EQ
<
a.
Chapter 4: Thermal RaJhtion ami Planck Dim
25
- 20 -
Figure-5.11 Heal capacity Cv of a solid,according10ihc Dcbye approximation. The
vertical scale is in J mol~' K\"'. TheIiori^onla!scaieis ihc temperature
normalized to the Debye temperature0.Theregion of ihc TJ law is below 0.10.Theasymptotic value al high values of 7\"/1? is
24.943 Jmor1 K\021.
/
/
L\342\200\224
i
^\342\200\224^\342\200\224
0 0.2 0.4 0.6 0.8 1.0 1.2
'able 4.2
IT
00.10.20.30.40.50.60.7
0.8
0.9
1.01.52345678
9
10
15
Values of C,, S, U, ai
Cv
24.943
24.9324.8924.8324.7524.6324.5024.34
24.16
23.96
23.74
22.3520.5916.5312.559.206.234.76
3.45
2.53
1.8910.576
id F on the Debye ihc
S = k^o
CO
90.70
73.43
63.3456.2150.7046.2242.4639.2236.3833.8724.49
18.30
10.71
6.51
4.082.641.771.220.8740.643
0.192
ory. in unlls J mo
U,0
X2402115.6
74.2
53.5
41.16
32.927.122.819.516.829.15.52.361.130.58
0.323
0.1S7
0.114
0.0730.0480.0096
I\021 K\"
170
- 666.8-251-137-87-60.3-44.1-33.5-26.2-209
-17.0:
-7.2:
-3.6.
-1.2-0.4!-0.2-0.1-0.0-0.0-0.0-0.0-0.0
Summary
20
10
Q
-10
-20
-30
i
0
\024\302\2600 0.5 1.0 1.5
0
FiKiire^.12 Energy t/and free energy f s= I/- roof a
solid, according to tiit DcbyC theory. Tlic Dcbye temperatureof tlic solid is 0.
SUMMARY
1. The Planck distributionfunction is
for the thermal average number of photons in a cavity mode of frequency a
2. The Stefan-Boltzmannlaw is
V 15/iV'
for the radiant energy density in a cavity at temperature t.
Chapter 4: Thermal Radiuiwn and Planck Distribution
3. The Planck radiation law is
h \302\2533
r2c3 exp(/io)/t)\342\200\224i
for the radiation energy per unit volume per unit range of frequency.
4. The flux density of radiant energy is Ju \342\200\224oBTA, where aB is the Stefan-
Boitzmannconstantn1ks*/(i0hici.5. The Debye low temperature limit of the heat capacity of a dielectricsolid
is,in conventional uniis,
where the Debye temperature
0 s (hvjkB)Fn2$fvy'\\
PROBLEMS
1. Number of thermal photons. Show that the number of photonsXXs\")in
equilibrium at temporal ure x in a cavity of volume V is
N = 2.404n~2K(t/Ac)s. D8)
i-rom B3) theentropy is a = Dn2F/45)(r//icK, whence <r/N
=s 3.602. It is
believed that the total number of photonsin the universe is 109 larger than thetotal numberof titiclcons(protons,neutrons). Because both entropies are ofthe order of the respectivenumber of particles (sec Eq. 3.76), the photonsprovide the dominantcontributionto the entropy of the universe, althoughthe particles dominau- ., c total energy. We believe that the entropy of thepllotonsis essentially constant, so that Mie entropy of the universeis approxi-approximately constant with time,
7. Surface temperature ofthi! Sun. The value of the total radiant energy Hilx
density at the Earth from the Sun normal (o the incident rays is called the solarconstant of the Earth. The observed value integrated over all emissionwave-
wavelengths and referred to the mean Eanh-Sim disianceis:
solarconstant = 0.136 J s D9)
(a) Show thai the total rate of energy generation of the Sun is 4 X 10\026 J S\" *.
(b) From this result and .the Stefan-Boltzmann constant <rB = 5.67 x10\"12J s~' cm'2 K\"\"\\ show thai the effective temperature of the surfaceof theSun treatedasa black body is T ~ 6000 K. Take the distanceofthe Earth from
theSunasl.5 x 10!3 cm and the radius of the Sunas 7 x 10l\302\260cm.
3. Average temperature of t/te interior of the Sun, (a) Estimate by a dimen-dimensional argument or otherwise (he order of magnitudeof the gravitationalself-energy of the Sun, with AiQ = 2 x !033g and
RQ=> 7 x 1010 cm. The gravi-
gravitational constant G is 6.6 x iQ\"8dynecm2g~2,Theself-energy will be negativereferred to atoms at rest at infinite separation, (b) Assume ihat ihe total thermalkineticenergy of the atoms in the Sun is equal to \342\200\224
\302\243limes the gravitational
energy. This is the resultof the virial theorem of mechanics. Estimate the averagetemperature ofihe Sun. Take the number of particles as 1 x IG*7.This csUmaiegives somewhat too low a temperature, because the density of tlie Sun is far
from uniform. \"The range in central temperature for different stars, excluding
only those composed of degenerate matier for which the Saw of perfect gasesdoes not hold (white dwarfs) and those which have excessively sniall averagedensities(giants and supergiants), is between 1.5 and 3.0 x iO7 degrees.\"
(O. Siruve, B. Lynds, and H. Pillans,Elementary astronomy, Oxford, 1959.)
4. Age of the Sun. Suppose4 x lQ;6Js~! is the total rate at which the Sunradiatesenergy at the present time, (a) Find i he total energy of the Sun availablefor radiation, on the rough assumptions that the energysourceis ihe conversion
of hydrogen (atomic weight L0Q7S)to helium(atomicweight 4.0026) and that
the reaction stops when 10percentofthe original hydrogen has been convertedto helium. Use the Einstein relation \302\243= (AAf)c\\ (b) Use (a) to estimate thelife expectancy of the Sun. It is believed ihat [lieage of the universe is about
10 x 109 years. (A good discussion is given in ihe books by Peebles and byWcinberg, died in the generalreferences.)
5. Surface temperature of the Earth. Calculate the iemperatureofl hesurfaceofthe Earth, on the assumpiion ihat as a black body in thermal equilibrium it
renijiates us much thermal radiationas it receives from the Sun. Assume alsothat ihe sin face of the Earth is ;it a constant jemporiittini
over ihe day-night
cycle. Use 7\"o- 5S00K; RQ = 7 x 10locm; and ihe I2arih-Sundislanwof
1.5x IOlJcm.
6. Pressure of thermal variation. Show for a photon gas thai:
(a) p = -(cUfcV), - -^s/iiiluij/ilV), E0)
l Radiat and Planck Oiildbnti,
where s; is the number of photons in the moJej;
(b) dojjfilV=
-mjyV;
(c) p = U/iV.
E1)
E2)
Thus the radiation pressure is equal to 3 x (energydensity).(d)Comparethe pressure of thermal radiation with the kinetic pressure of a
gas of H atomsat a concentration of t mote cm'3 characteristic of the Sun.At what temperature (roughly) are the two pressures equal? The average
temperature of the Sun is believed to be near2 x tOT K. The concentration is
highly nonuniform and rises to near rOQmoiecm.\023 at the center, where thekinetic pressure is considerablyhigher than the radiation pressure.
7. Free energyofa p/iot on gas. (a) Show that the parlhion function of a
photon gasis given by
where ihe product is over the modes \302\273.(b) The Helmholtz free energy is found
direclly from E3) as
F-T][tog[t-exp(-AuiA/T)]. E4)
Transform the sum to an integral; integrateby parls to find
F = -n E5)
5. Heatshields.A black (nonreflective) plane at temperature Tu is parallelto a black plane at temperature T(. The net energy flux density in vacuum be-
between the two planes is Jv = aB{Tf - T*),where aB is the Stefan-Boltzmann
constant used in B6). A third black plane is inserted between the other two andis allowedto come to a steady state temperature Tm. Find Tm in terms of 7'Band T,, and show that the net energy flux density is cut in half because of the
presence of this plane. This is the principle of the heat shieldand is widely
used to reduce radiant heat transfer. Comment:The result for N independent
heat shields floating in temperature between the pianes TM and T, is that the
net energy flux density is Jy = ciT^ - T,4)/(N+ !)-9. Photon gas in one dimension. Consider a transmission lineof length L on
which electromagnetic waves satisfy the onc-dimcnsional wave equation
v2d2Ef3x1 =s c2E/ct2l where E is an electricfield component- Find the heat
capacity of the photons ontheline,when in thermal equilibrium at temperature
r. The emiiiicrulioiiof modesproceeds in ihe usual way for one dimension:lake ilicsoiulionsas siundmg waves with zero amplitude ai cacli cud of ilicline.
10.Hcitt capacity of iniergalaciic space. Sntcrgalactic space is believedto beoccupiedby hydrogen atoms in a concentration =laionim~\\ The space isalsooccupiedby thermal radiaiion al 2.9 K, from the Primitive Fireball. Showthai ilic ralio of ilic hcai capacity of mailer lo lhal ofradiaiionis ~- 10'9.
//. Heat capacity of solidsin high temperature limit. Show lhal in ihe iimil
T \302\2730 liie heal capacity of a solid goes towards die limit Cy \342\200\224\342\226\2403A'\302\243B, in
conventional units. To obtain higher accuracy when T is only moderatelylarger than 0, the heat capacity can be expanded as a power seriesin 1/T, of
the form
E6}
Determine the first nonvanishing term in the sum. Check your result by inserting
T - 0 and comparing with Table 4.2.
/2. Heat capacity of photonsandpltonotts. Considera dtelcciricsolidwith a
Dcbye temperature equal to 100K atid with 10\" atoms cm ~3. Estimate the
lemperature at which the photon contribution to the heat capacity would beequal to the phononcontributionevaluated at 1 K.
13. Energy fluctuations in a solid ai low temperatures. Consider a solid of Natoms in the temperature region in which ihe Dcbye Ti law is vaiid. The solid
is in ihermal contact with a heat reservoir. Usethe resultson energy fluctuations
from Chapter 3 to show that the root mean square fractional energy fluctuationIF is given by
(E7)
Suppose that T = I0~2K;0 = 200K;and N * 1O1S for a particle 0.01cm ona side;then $F p= 0.02. At !0\025K the fractional fluctuation in energy is of iheorder of uniiy for a dielectric particle of volume 1 cm3.
14. Heal capacity of liquid *Heat low temperatures. The velocity of longitu-longitudinal sound waves in liquid 4He at temperatures bctaw 0.6 K is 2.383 x 104cms\"'. There are no transverse sound waves in the liquid. The density is
0.145gcm~3. (a) Calculate the Debyc temperature, (b) Calculatethe heat
capacity per gram on the Dcbye theory and compare with the experimentalvalue CK
=\342\226\2400.0204 x T\\tn Jg\021 K*1. The T3 dependenceoftheexperimental
Chapter 4: Thermal Radiation and Pfanek Distribution
value suggests thai phonons are Hit; most important excitations in liquid 4Hebelow 0.6K. Note that the experimental value has been expressed per gramofliquid.The experimems are due to J. Wiebes, C. G. Niek-Hakkenberg,andH.C Kramers, Physica 32, 625 f !957}.
15. Angular distribution of radiant energyflux, (a) Show that the spectra!density of the radiant energy flux that arrives in the solid angle i!Q isfuucos0*</n/47r,where 0 is the angle the normal to the unit area makes with
the incident ray, and i^ is the energy density per unit frequency range, (b)Showthat the sum of this quamity over all incident rays is \\aiu.
16. Image of a radiant object. Let a lens image the hole in a cavity of area
An on a black object of area Ao. Use an equilibrium argument to relate theproduct AuQtl to AaQ0 where Qtl and QQare the solidanglessubtendedby the
leas as viewed from the hole and from the object. This general property offocusingsystemsis easilyderivedfrom geometrical optics. It is also true whendiffraction is important. Make the approximation that all rays are nearly
parallel {al! axial angles small).
17. Entropy and occupancy.We argtted in this chapter that the entropy of thecosmicblack body radiation has not changed with time because the numberofphotonsin each mode has not changed with time, although the frequency ofeach mode has decreasedas the wavelength has increased with the expansionof the universe. Establish the implied connection between entropy and oc-
cupattcyofthemodes,by showing that for one mode of frequencyw the entropy
is a function of the photon occttpancy<<(s) only;
a = <s + l)log<5 + !) - <s)log<s). ES)
It is convenient to start from the partition function.
18. haxiropic expansion of photon gas. Considerthe gas of photons of thethermal equilibrium radiation in a cube of volume V at temperature r. Let the
cavity volume increase;the radiation pressure performs work during the expan-expansion, and the temperature of the radiation wit] drop. From (he result for the
entropy we know titat iV1'3 is constant in such an expatision.(a)Assume that
the temperature of lite cosmic black-body radiation was decoupled from the
temperature of the mutter when both were at 3000K..What was the radius of
the universe at that time, compared to now? If the radtus has increasedlinearly
with time, at wltat fraction of the present age of the universe did the decouplingtake place?(b) Show that the work done by the photons during the expansion
The subscripts i and / refer to the initial and final siatcs.
19, Reflectiveheat shieldandKircbhoff's(aw. Consider a plane sheet of mate-material of absorptivity it, e-mtssjvtty e, and rcllecttvjty r ~ \\ \342\200\224a. Let the sheet be
suspended betweenand parallel with two black sfieets maintained at tempera-temperaturesru und t,. Show that the net flux density of thermal radiation between theblack sheetsis (I -
r) times the flux density when tiie intermediate sheet isalsoblack as in Probfem 8, which means with a = e = I; r - 0. Liquid helium
dewars are often insulated by many, perhaps 100,layers of an alumtntzed
Mylar film called Superinsulation.
SUPPLEMENT: GREENHOUSE EFFECT
The Greenhouse Effect describes the wanning of the surface of the Earth
caused by the interposition of an infraredabsorbentlayer of water, as vaporand in clouds, and of carbondioxidein the atmosphere between the Sun andthe Earth. The water may contribute as much 90 percent of the warmingeffect.
Absent such a layer, the temperature of the surface of the Earth is
determined primarily by the requirement of energybalancebetween the flux
of solar radiation incident on the Earth and the flux of reradiation from theEarth; the reradiationflux is proportional to the fourth power of the tempera-temperatureof the Earth, as in D.26). This energy balanceis the subject of Problem
4.5 and leads to the result Ts \342\200\224{RsI^seV^Ts, where 7'\302\243is the temperature
of the Earth and Tsis that of the Sun;here/fjis the radius of the Sun and DSEis the Sun-Earthdistance.
The result of that problem is TE \342\200\224280 K, assuming T,=\302\273 5800 FC The
Sun is much hotter than the Earth, but the geometry (the smali solid anglesubtended by the Sun) reduces the solar iiux density incidentat ihe E;irth by a
factor of roughly (i/20)*.We assume as an example that the atmosphere is a perfectgreenhouse,
defined as an absorbent layer that transmits al! of the visible radiation that
falls oa it from the Sun, but absorbs and re-emits a!! the radiation (which liesin the infrared), from the surface of the Earth. We may idealize the problem
by neglecting the absorption by the layer of the infrared portion of theincidentsolarradiation,because the solar spectrum lies almost entirely at
higher frequencies,as evident from Figure 4.4. The layer will emit enerry flux
tL up and IL down; the upward Oux will balance the suiur i!ux 1$, Su UiJi
ft **Is- The net downward flux will be the sum of the solarflux Is and the
flux lL down from the layer. The latter increases the net thermal Oux incident
at the surface of the Earth. Thus
lEt^h + h-ns. E9)
wherel\302\243g
is the thermal Oux from the Earth in the presenceof the perfect
greenhouse effect. Because the thennal flux varies as T4, the new temperatureof the surfaceof the Earth is
T\302\243s=
2\302\273/\302\253rfi\302\253=
A.19) 280 K ~ 333 K, F0)
so that the greeahouse warming of the Earth is 333K \342\200\224280 K = 53 K for
this extremeexample.*
\342\200\242For detailed discussions see Climale change and Climate change 1992, Cambridge U.P., 1990
end 1992: i. T. Houghton ct aJ, editors. .
Chapter 5
Chemical Potential andGibbs Distribution
DEFINITION OF CHEMICAL POTENTIAL 119
Example: Chemical Potential of the Idea! Gas 120
Internal and Total Chemical Potential 122Example: Variation of Barometric Pressure with Altitude 125
Example: ChemicalPotentialofMobileMagnetic Panicles
in a Magnetic Field\"'\"
127
Example; Batteries' \342\226\240-
129
Chemical Potential and Entropy 131ThermodymtmicIdentity
133
GIBBS FACTOR AND GIBBS SUM 134
Numberof Particles 139
Energy 140
Example: Occupancy Zero or One 140Example: Impurity Atom !omz;ition in a Semiconductor 143
SUMMARY 144
PROBLEMS 145
1. Centrifuge 1452. Moleculesin the Earth's Atmosphere 145
3. Potential Energy of Gas in a Gravitational Field 145
4. Active Transport 1455. Magnetic Concentration 145
6. Gibbs Sum for a Two LevelSysSem146
7. States of Positive and Negative Ionization 1468. Carbon Monoxide Poisoning 146
9. Adsorption of O2 in a MagneticField 147
10. Concentration Fluctuations 147
11. Equivalent Definitionof ChemicalPotential 148
12. Ascent of Sap in Trees 14813. IsentroptcExpansion
148
14. Multiple Binding of O2 14815. ExternalChemicalPotential .... 149
\342\226\240alPotential and Gibbs Distribution
(H. We found earlier thai Tor a single system S in thermal equilibrium with a
reservoir <2t, ihe Helmholtz free energy of & will assume the minimum value
compatible wiih the commontemperature x and wiiii oiher restraints on the
system, such as the volume and the number of particles. This result appliesy, p pp
equally to ihc combined\302\243,+ S: in equilibrium with (R. in diiTusive equilib-
equilibrium between S, .md ^2, the paiii^lc diaiiibution N,,.V\302\273 between ihi5 sysicms
makes t!io toial Hchulioltz free energy
F =h\\ + F2 - t/( + t/2 -
T(ff, + <T>) A)
a minimum, subject to N = jV, + jV2 = constant. Because N is constant,She
Heimholtz free energy of the combined system is a minimum with respect to
Definition of Chemical Potential
System .Sj
t*\"Ener\302\260y exchange\"\"}
Figure 5.1 Example of Iwo systems,Sx ;mdX., in ihcrmal conlacl with each oihcrand wiih a large reservoir Of, forming a closed total system- I)y opening the take,S{ and ^j can be brouglit in tlillusive contact wliilc remaining at the common
tcmpcraiurc r. Ttie arrows at tlm valve liave been drawn for a net panicle transfer
from \302\243jlo S2.
variations 5Nt = -5N2. At the minimum,
tlF - (cFi/cN^^lNi + lcF2/dN2)sc!N2= 0 ,
with K[, K2, also held constant. With (!Nt= ~-tlN2, WG have
(/F = [(cfj/t'iV,),- (SF2/cNl),yNl - 0 ,
soHutt at cqutlibriuni
((Tj/oV,), - (cF3/cN2)t.
DEFINITION OF CHEMICALPOTENTIAL
We define the chemical potential as
I I
B)
C)
(\342\226\240\302\273)
(/;0
E)
Chapter 5: ChemicalPotential and Gibhs Distribution
where ji is ihe Greek letter niu. Then
/'I'-
t'l
expresses ihe condition for ditTusive equilibrium. If/(, > ;i2, we sec from C)
thai elF will be negative whendiVt is negative: When panicles are transferred
from 5, to 5j, ihe value of dN \\is negative, and tlN2 is positive. Thus the
free energy decreases us particles flow from ^i to &2; ilitit 's. particles (low
from the system of high chemicalpotentialto the system of low chemical
potential. The strict definition of /i is in termsof a difference and not a deriva-
derivative, because particles are not divisible;
)=
F(t,V,N) - F{x.V,N- 1). F)
The chemical potential regulates the particle transfer between systems in
contact, and it isfully as important as the temperature, which rcguhucs the
energy transfer. Two systems ihat can exchange both energyand particlesare
in combined thermal and ditTusive equilibrium when their temperaturesand
chemical potentials are eqtial: i[ = t2;/i; =ji%.
A difference in chemical potential acts as a drivingforcefor the transfer of
particles just as a differencein temperatureacisasa driving force for the transfer
of energy.If severalchemicalspeciesare present, each has its own chemical potential.
For speciesj,
G)
wherein the differentiation!
for the species j.TTumbcrs of all particles are held constant except
Example: Chemical potential of the Ideal gas. In C.70}of Ihe monatomic ideal gas is
showed that ihe free
/logZi- logN!] , (8)
Definition of Chemical Potential
is the partition function for a single panicle. From (8),
A0)
If we use the Stirling approximation for A\"! and assume tlutt v,e can differentiate thefactorial,we fold
= log A\" + [N + -1)-- - 1 = togtf + \342\200\224. (II)
which approaches log iV for large values of \\. llcttcc the chemical potential of the ideal
gas is
,< = -rUogZ, - log.V)
or, by (9),
A2a)
where n \342\200\224NjV is the concentration of particlesami na
= (A/r/2nfi2K2 ts the quantum
concentration defined by C.63).Ifweuseji = F{N)- F{N
- i)froin.{6)as the definition af/j, we do not need to use the
Stirling approximution. From (S) we obtain ji= -i[log2,
- logjV], which agtces with
A2). The result depends on the concentrationof particles, not on their total numheror onthe system volume separately. By use of ths ideal gas law p = nr we can write A2} as
/t= Ttog(p/tiiti). A2b)
The chemicalpotential increases as the concent mi ion of particlesincreases.This is what
we expect intuitively: particles flow from higher to lower chemicalpotential, from higher to
lower concentration. Figure 5.2shows the dependence on concentration of an ideal gascomposedof electrons or of helium atoms, for two temperatures, the boiling temperature
Chapter5: Chemical Potential andGibbs Distribution
Figure 5.2 The concentration dependenceof/*, in units o! r, of an idealgascomposedofdectroio.jrhelium atoms, at 4.2 K and 300 K..Tobe in the classical
regime with n \302\253nQ, a ^as must have a value of -
^i at least t. For electrons this is
satisfied oulv for concentrations apprccwhly le\302\253lluin those in metals, as in ihe
range of lypical semiconductors. For gasesit is always satisfied under normalconditions.
of liquid licimin ill atmospheric pressure, 4.2K, and room icmpcraturc, 300 K. Atomic
and tnolccubr jtascs aiwjjs lave ncgaiivcchemical potentials unJer plijiically tciilii^blccondiiioas: at classical concernrations such ihnt nfnQ \302\253!, we soe from A2) iliat /i is
ncgaiive.
Internal and Total ChemicalPotential
The best way to understand the chemical potential is to discuss diffusive
equilibrium in Ihe presence of a potential step that acts on Ihe particles. This
Internal and Total Chemical Potential
Figure 5.3 A potential step between two
systems of chargedpaniclescan be established
tiie voltage polarity shown, the potential
energy of positive parlic'es with charge q > 0in system i, would be raised by qAY wild
respect !o J2. The potential energy of negative
particles would be loweredin 3, with cespect
problem has wide application and includesthe semiconductor p\342\200\224n junction
discussed in Chapter 13. We again consider hvo systems, Sx und &->t al the
same temperature and capable of exchanging particles,but not yel in diffusive
equilibrium. We assume that initially fix > /(j, and we denote the iniiial non-equilibriumchemicalpotential difTcrence by A/j(iimial) = ;/2 - ^i- Now let a
difference in potential energy be establishedbetweenthe two systems, such
iliat the potential energy of each panicle in sysiem Si is raised by exactlyAf[(im'tial) above its ininal value, if Ihe particles carry a charge qt one simpleway to establish this potentialstepisto
apply between the two systems a voltageA I' such lhat
A3)
with the polarity shown in Figure 5.3. A difference in gravitational potentialalso can serveasa potentialdifference: when we raise a system of particles eachofmassM by the height /i, we establish a potential differenceMgh, where g is
the gravitational acceleration.Once a potentialstepispresent, the potential energy of ihe patrides produced
by this sfcp is included in (he energy U and in the freeenergyF of the system.
If in Mgurc 5.3 we keep the freeenergy
of system S2 fiNed, ihe step raises thefree energy of Sx by /Vs A;i(mitial)
*= AV/ A!7 relative to its initial value. Inihe language of energy states, to the energy of each suite of .Si the potential
energy ,Yt A/i(initial) has been added. The \"insertion of the potential b^rier
Hveificd by (B) mises iliechemicalpotential of $x by A/i(!il!ti;il). to male thefin::!i.-hciiiiL:;il puk-ntuil of ^, c\302\253i\302\273:ilto ih:lt oi\\S,:
/affinal) = ;(,(initial) -f [,..(initial) - /i,(inilial)]A4)
When the barrier was inserted, pj was held fixed.Thus the barrier gA
/^(initial) \342\200\224/(i(initial) brings the two systems into diffusive equilibrium.
The chemical potential is equivalent to a true potential energy; thedifferencein chemical potential between two systems is equal to the
potential barrier that will bring the two systemsinto diffusive equilibrium.
This statement gives us a feeling for the physical effect of the chemical poten-potential, and it forms the basis for the measurementofchemicalpotentialdifferences
between two systems. To measure ft2- /<,, we establish a potential step
betweentwo systems that can transfer particles, and we determine the step
height at which the net particle transfer vanishes.
Only differences of chemical potential have a physical meaning. The absolute
value of the chemical potentialdependson the zero of the potential energyscale.The idealgas result A2) depends on the choice of the zero of energyof afree particle as equal to the zero of the kineticenergy.
When external potential steps arc present, we can expressthe total chemical
potential of a system as the sum of two parts:
/' = iv =;\342\226\240\342\200\236,+ft,,- A5)
Here /jtM is the potential energy per particle in the external potential,and jiial
is the internal chemical potential* definedas the chemical potential that would
be present if the external potential were zero. The term /iCM nmy be mechanical,
electrical, magnetic, gravitational, etc. in origin.The equilibriumconditionPi \342\200\224Hi ca'i be expressed as
Apeil = -A A6)
Unfortunately, the distinctionbetween external and internal chemica! potentialsometimes is not madein the literature.Somewriters, particularly those working
with charged particles in the fieldsofelectrochemistryand of semiconductors,
often mean the internal chemical potentialwhen they use the words chemical
potential without a further qualifier.
The total chemical potential may be called the electrochemical potential if
the potential barriers of interest are electrostatic.Although the term electro-
\342\226\240Gibbs called n the potential and it,*, ihe intrinsic potcnsjal. He recognizedthat a voltmeter mea-measures differences in fi. .
'
Internal and Total Chemical Potential
SysicmB)
Figure 5.4 A model of ihe variationaimcispliericpressure\\>.iih altitude: i
\\olumcs of gas at different heights in
gravitaiiona! field, jn iherma! and di!con wet.
System (!)
chemicalpotential is clearand unambiguous, we shall tisc \"total chemical
potential.\"The use of \"chemicalpotential\"' without an adjective should beavoided tit situations in which any confusion about tis meaningcouldoccur.
Example; Variation of barometric pressure whhahhutte. The simplest example of the
diffusive equilibrium bciwccn sysiems in difTcrcm external potentials is itte equilibrium
bciwccn layers ;lt different llcigliis of ihe Canh'satmosphere, assumed lo be isothermal.Tim rcul uimosplicfc is in imperfect equilibrium: ii is'cunstaiiily upsa by Hitititoroloeicutprocesses, faoih in the form of macroscopic air movemems and of strong temperature
grudiunis from clouii fonn;uion, ;inii becauso of heal input from liiil ground. We maymake ;in ;ippro.\\imaic model of the aimosphcrc by KtMliiia the dilL-iem air layers assyslemsof idea! gases in ilierilia! and diftusive eiiuilihrium ^iih each oilier, in diHerem
exieroul poicnlials (Figure 5.4}.If we place ihe zero of the poieinul energy ai ground \\c\\c\\,
the poiemial energy per moleculeai heighi Ji is Afyfi,whcrc.\\f is ihe particle massand g ihe
gravitational acceleialion. The internal chemical potential of ihe parlidesis given by A2}.
The loial diemical poiemialis
{17}
In equilibrium, this musi be independent of {he hcidil. Thus
Tlog[i.(/t)/H0] + A/y/i-
rlog[ii@)/nQ] ,
and ihc conceniralion \302\253(/t).alhcijjlU h satisfies
(IS)
Chapter 5.-Chemical Potential and Gibls Dhtnliution
1.0
0.5
0.2
figure 5.5 Decrease of atmospheric pressurev-iih ahiiude. The crosses represent Ihc averageatmosphere as sampled on rocket fliyhis.
The concerting siraight Vine lias a slope
corresponding lo a icmpcraiureT = 227 K.
o.i V
2 0.0S
1
\\
\342\200\224
\\
i
h
X
\\
M0-10 20 30 40 50
Heighi,in km
The pressure of an idealgas is proportional lo llieconcenlration; therefore Ihe pressureaullilude li is
p(/i) =. p@)exp(-.V9li/t) = p@)exp(-/i/'iJ- A9)
Thl^ &s itio uuromoLrtc pressure conation Ti i*i\\ os ^Jtc ilc^czitjciicc 01 liic pressiifcon uli^iLide
in an isoihermat anuoipliere of a single chemical species. Al ilie cliaracierisiic Mght hc=
t/A/g ihe .-.unospi^er-c pressure decreasesfay ihe fraciion e~l =3 0.37. To esiimaie Hie
characteristic fietght, consider an isoihermal atTuosphere composed of nitrogen nioiccuies
wiLliiinioiecular weigh! of 28.Tliema^orail .V^ niolecuie is 48 x 10\" \"gill. At a teiliper.l-iureof :90KiIic value of r = kBT is 4.0 x 10\" l4crg. \\W\\nj =
9K0L1HS\021. the d1.1r.1t-
icrisiic iieiijlii Jit is S.5tm, approximately 5 mites. Li^Uiermolecules,H, and He, will
t\\ictul fatthcr dp, iiut these Ivavc hrgeiy cscipctl from ilic iiiiiinspUufc: see Probicm 1.
11UC.U1-.C ll.o UanhS LiiiiiO^pticrc i> not ;n;tiir.ittly iwitlicniiiit, ni/i)ft;is ;i iitoic a'ttipliaital
behavior. Fiyuie 5.5 is :i loyatiifunic jjloi oi\" pressure tfatj bciv^un 10 and 4QKiloiMcttrs,
iaken on rocket flights. The data potnis f;i!i near a siraighi fine, suggesting roughfy iso-
vtl and total Chemical Pot
thermal behavior. Tlie straight line connecting the data points of Figure 5.5 spansa pressurerange jj{/ij):pl/i,J
= 1000:1, over an altitude range from ft,= 2km io A,
= 43km. New,from A9),
so that the slope of the line is Mg/x, which leads to T ~x,'kB
\302\253227 K. The non-intersectionof tltc observed eurve wish !hc point k ~ 0, p(li),'p{0)
\302\253I, is caused by the higher tempera-
temperatureat lower aliiiudes.
Tlic atmosphere consistsof more than one species of gas. In atomic percent, the com-
composition of dry air at ica fevel is TS pet A'j. 21 pet Oz. and 0.9 pet Ar; oilier constiiuemsaccount for iess than 0.1 pci each. The water vapor content of tlie aimosphcre may be
appreciable: at T ~ 300K B7\"C}_ a relative humidity of 100 pet corresponds to 15 pet
H,O. The carbon dioxide concentration varies about a nominal value of 0.03 pet. In an
ideal static Uolhenrtal atmosphere eadi gas would be in equilibrium wftli iisclf. The con-
concentration of each would fall off with a separaie Boltzmann factorof the form cxp( --\\/y/i/t},
with M the appropriate molecular mass. Becauseof tlie JiiTerences in mass, ihe difTcrcnt
p^insiTrii^itic T *}\\ tiff u\\ /iifrprcnl nk*c
Example: Chemicalpotentialof mobile magnetic particles in a magnetic field. Considera system of .V identical particles each with a magnetic moment m. For simplicity suppose
each moment is directed cither parallel f or antiparalld [ io an applied magneticfield B.
Then the poiential energy of a f particle is -iufl, and the potential energy of a 1 particle is
4-ii]\302\243i.We may treat the particles as belonging to the two distinct chemical specieslabelled t
and i, one vhh external chemical potential ;in,(|J = -mBand the other with jjm,U) = \"iB.
The particles f and 1 are as distinguishable as Uvo difTerent isotopes of an clement or as two
different elements; we speak of f and 1 as distinct species in equilibrium with each other.
The internal chemical potentials of the particles viewed as ideal gases with concentrations
n, and iij are
ill potentials are
If iltcrti:iM\302\273\302\253:lK-|k-lJ !t\\:nivs iintwi'iiittKlfovcrtbtf vo!ui\302\273c^iltcsyslciH.llici-.-:-.vt.ij.Hit. mtht vary out \\\\w \\ wlumc in unlcr to ut:tintaii) j cotiMjisl unal clicnnt.it i- \342\226\240\342\226\240Mi ,i
out i|ic volume irigmo 5.6).(Tlie total chemical potential of a species is eoitstam iiiJcdent of posiiion, if there is free diffusion of particles within the volume,) Becausethe
Chapter 5: Chemical Potential and G'thhs Dhtribut
-
\342\200\224\342\200\224\"
---
Iff' 1GIU
ion (i, in particles ciir3 \342\200\242
Figure 5.6 Dependence ofihe chemical potential of a gas of magneticpari ides on ihe concentration, at several values of the magnetic field
intensity. Ifn = 2 x iO7 cm\021 for I! ~ 0, ihcn ai a point \302\253here B \342\226\240=
kilofiauasl2iesb)ihccona:nlraiionwillbe2 x !09cm\023.
20
species in equilibrium have equal diemioil poientiais,
lUl) = consiant- ;i,01(|). B3)
The desired solutions of {22)and B3)are easily seen fay substitution to be:
\302\253,{B)- in{0)exp(mB/t); n^B) =
^\302\253@)exp{-)nB/i) , B4)
where ti@} is tiie total concentration ii[ 4- nt at a point where the field B = 0. The totalconcentrationat a poini at magnetic field S is
n{B) \302\253n,{B) + ji,(B) = Jn@)[exp{jHB/T) +
exp{-\302\273)B/T)];
n(B)= n{0)cosh{mB/T) * n{0)M 4-
^-+ \342\200\242\342\200\242-Y B5)
The result shows [he tendency of magnetic panicles to concentrate in regions of high
magnetic Held intensity- The fujictionai form of ihc result is not limited to atoms wi[h two
magnetic orientauons, but is applicable to fine fecromagncfic particles in suspension in 3
colloidal solution. Such suspensions areusedin the laboratory in the siudy of f he magneiic
flux strucfure of superconductorsand fhe domain strucfure of ferromagnetic materials.In
engineering, the suspensions ace used to tcsf for fine structural cracks in high strength sfcel,
such as furbine blades and aircraft landing gear. When fhese arc coatedwith a ferromagnetic
Internal and Total ChemicalPotential
suspension and placed in a magnetic field, flic particle concentration becomes enhanced atttie intense fields at the edges of the crack.
In the preceding discussion we added to /<\342\200\236,the internal chemical potential of iheparticles.If the particles were ideal gas atoms, /iml would be given by A2). Tlie logarithmicform for /iinl is not restricted to id^algases,but is a consequence of the conditions iliat the
particles do not interact and that their concentration ii sufficiently low. Hence, A2) appliesto macroscopic particlesas well as to atoms that satisfy these assumptions.The only
difference is the s;tlne of the quantum concentration n,,. We can thereforewrite
/'iM= t log it -f constanl , B6}
where the constant (\342\226\240=\342\200\224t log \302\273,,)does not depend on the concentration of the puctielcs.
Example; Batteries. Oneof the most vivid examples of chemicalpotentials and potential
steps w the electrochemical battery, hi the familial lead-acidbattery the negative electrode
consists of metalliclead, Pb, and the positive electrode is a layer of reddish-brownlead
oxide. PbOj, on a Pb substrate. The eltclroiics ari: ininwisiid in ditulod sutfurie acid,
H.SCU, which is partially ionUai into H* ions (protons) and SG4~~ ions (Figure 5.7}.
in the discharge process both tlie metallic Pb of the negative electrode and the PbO.
of the positive electrode arc converted to Ie;td sulfate, PbSO4, via the two reactions;'
Negative electrode:
Pb + SO4\"\"-+ PbSO.,+ 2e~; B7a)
Positive electrode:
PbO2 + 2H+ + HjSO- + 2e\"-+ PbSO4 + 2H2O. B7b)
Because of B7a)the negative electrode acts as a sink for SO*~\"
ions, keeping the internal
chemical potential ji(S04\"\"\") of the sulfafe ions at the surface of the negative electrodelower than inside the electrolyte (sec Figure 5.7b). Similarly, because of B7b) the positive
electrode acts as a sink for H+ ions, keeping the internal chemical potential /((H1\") of 'tie
hydrogen ions lower at the surface of ihe positive electrodethan inside the electrolyte. Thechemical potential gradients drive the tons towards the electrodes, and they drive the
electrical currents during the discharge process.If the battery terminals arc not connected, electrons are depleted from the positive
electrode and accumulate in the negative electrode, thereby charging both. As a result,
electrochennca! potential steps develop at the electrode-electrolyte interfaces, steps ofexactly the correct magnitude to equalize the chemical potential steps and to stop the
diffusion of ions,which stops the chemical reactions from proceeding further. If an external
current is permitted to flow, the reactions resume. Electron fiow directly through the
electrolyte is negligible, becauseof a negligible electron concentration in the electrolyte.
\342\200\242The reactions given are net reactions. The actual reaction steps are more complicated.
(\342\226\240f)clcclrodc
PbO, i l>b
(b).
T
I
T
5-igurc 5.7 (ajThekjJ-aciJ bailcry coniists of a Pb anj a PbO2 cicctrotic ilnmclsed in
parlialiy ioni/cd U^SOj. One SOI\" ion coii\\crfs one Pb atom into PbSOt + 2e~;two H
Tions plus one im-ionizcd HjSO^ molecule convert one PbOj moltrcul^ imo
PbSO4 + 2HjO,Lii isiirning Iwo ciccirons. (b) The eiectrociiemicaipokntials for SOI\"
and H* before the development ofimcfiia! poltnltai barriers thai slop the dilTuston
and ihe chemical reaction, (c) The eicctroiuticpoieuliai y(.\\) after the formation of thebarrier.
Chemical Potential and Entropy
During I he charging process the reactions opposile lo B7a,b) take place, because now ancMornai voiUigc is ;i}>[i!ied Eliai generates electrostalk [wicin b I steps ;H ihc surfiigc t'f Useei^mMc of such mugnituilti us lo reverse ihc s!ytl of Hie (toUi!)ciieiiik;ii poiciiliiit gradients,ami Iwik-u the diction of ion How,
U'c denote by A K_ and A f. the difTcrenccsin dectcostaitc polcmial of the negative andposilncclcclrodcsrelative to ihc common ciectroiyie. Because the suifute ions curry twonegative charges,diffusion will slop when
\342\200\2242gAK_
= A/i(SO4\"\.") BSaJ
DifTusion of the H+ ions wi!i stop when
+ ^AK+ \302\253A/i(//'). B8b)
The two polemics AK_ and Al'+ are called ha!f-cc!i potentials or half-eeil EMF's
(electromotive forces); their magnitudes are known:
AV_ = -0.4voll; Aft \302\273+1.6volt.
Tiic lolai electroslalic polcnliai difference developed across one full cell of ihc bniicry,as requiredto stop the diffusion reaclion. is
AK =AK+
- AK_ \302\2532.0voli. B9)
This is ihc opcn-drcuitioitage of EMFof the battery. Il drives ihc electrons from the ncca-
live lenninal lo the positiveicnr.ma!,when ihc iwo are connected.We have ignored free electrons in tha electroljle. The polenlial steps tend to drive
electrons from the negative elecirodesinto the electrolyte, and from the cleciroiyie into ihe
posiliveelectrode.Such an electron current is present, but the magnitude is so small as to bepractically negligible, because the concentration of elcclrons in the electrolyte is manyorders of magnitude less than lhal of the ions. The only effective electron (low path is
through the external connection betweenihe electrodes.
Chemical Potential and Entropy
In E) we defined the chemicalpotentialas a dertvathe of the Hclmlioltz free
energy. Here we detivc an alternate relation, needed later:
C0)
This expressesihe raliopi/x as a derivative of ihe entropy, similar to tlteway
1/r was defined in Chapier 2.
Clmpler5; Chemical Potential and Cibbs Distribution
To derive C0),considertheentropy as a [unction of the independent variables
U, V, and N. The differential
cU+ ^ <w C1)
gives the differential change ofthe entropy Tor arbitrary, independent differential
changes </t/, t/K, and rfJV. Let dV = 0 for the processes under consideration.Further,select the ratios of rf<r, c!U, and i/N in such a way that the overall
temperature change dx will be zero. IT we denote these interdependent values
otda, dV, and dN by {&a)t, FU)t and (<5N)rl then dx = 0 when
After division by {5N)r,
C2)
The ralio {Sa)J(SN)lis (fo/5N),,and D[/),/(JW), is (dU/dN),, all at conslant
volunie. With the original definition of 1/t, we have
C3)dNj,,
This expressesa derivative at constant U in terms oTderivatives at constantt
By the original definition of the chemical potential,
\023[H,
~
w.,
and on comparison with C3) we obtain
C4)
C5)
The two expressionsE) and C5) represent two different ways to express thesame
quantity ft. The difference between them is the following.In E), F is a
Chemical Potential am! Entropy
Table 5.1Summary of relations expressing ti-.c temperature
partial derivatives of liic cm ropy cr, liic energy U, and liic freeenergy F, witii a, U, and F given as functions of their itatur.il
independent variables
o(U.V,N)
(IU\\ , is independent\\ia)rtl variable
y,,
function of its natural independentvariables x, V, and N, so that n appears asa function of the same variables. In C1) we assumed a =
a(U,V,N), so that
C5) yields n as a function of U, V, N. The quantity fi is the same in both E)and C5),but expressedin terms of different variables. The object of Problem1iis to find a third relation for p:
C6)
and in Chapter 10 we derive a relation for p asa function of t, p, and N. Table 5.1compilesexpressionsTor i, p, and ^ as derivatives of o, U, and F. All forms
have their uses.
Tlicvmodynamic identity. We can generalize the statement of the Ihermo-dynamic identity given in C.34a) lo include systems in which the number of
particles is allowed lo change. As in C1),
C7)
By use of the definitionB.26)of i/r, the relation C.32) for p/r, and the relation
C0)for-
fi/i, we write da as .
da =dU/x + pdV/x
- pdN/x. C8)
Chapter 5: Chemical Potential and Gibbs Distribution
This may be rearranged lo give
dV = xJs -pdV + ;u|jv , I C9)
which is a broader statement of the ihermodynamic identity than we were ableto developin Chapter 3.
GIBBS FACTOR AND GIBBS SUM
The Boltzmannfactor,derived in Chapter 3, allows us lo give the ratio of the
probability that a system will be in a state of energy Et (o the probability the
system will he in a state of energy t2, for a system ii\\ thermal contact with a
reservoir at temperature t:
P(z2)
This is perhajtls the best known resuh of statistical mechanics. The Gibbs factor
is thegenecauzauon of the BoHxmann factor lo a system in thermal and diffusive
contact with a reservoir at temperature j and chemicalpotential p. The argu-
argument retraces much of that presented in Chapter 3.
We consider a very large body with constant energy Uo and constant particlenumberN0.Thebody is composed of two parts, the very large reservoir <R and
the system &, in thermal and diffusive contact (Figure 5-8), They may exchange
particles and energy.Thecontact assures that the temperature and the chemical
potentialof the system are equal to those oftiie reservoir.When the system has
. iV particles, the reservoir has Nu - M particles;when ihe system lias energy e,the reservoir has energy C/o
\342\200\224e. To obtain the statistical properties of thesystem,we make observations as before on identical copies of the system +
reservoir, one copy Tor each accessible quantum slate of the combination. What
is the probability in a given observation that ihc system will be round to con-contain N particles and to be in a stale s of energy \302\243j?
The stale s is a state of a system having some specified number of panicles.The energye](iV)
is the energy of the state s of the A'-particIc system; sometimes
we write only e,, if ihe meaning is clear. When can we wriie ihe energyof a
system having A' particles in an orbital as -V times ihe energy of one pur tidein the orbital'.' Only when interactions between Ihe particlesare neglected,soihat the particles may be treated as independentofeachother.
Figure 5.8 A system iit thermal and diffusive conlacl wiill
a large reservoir of energy anti of panicles. Tile total system<H + S is tnsulaicd from the external world, so thai ihe
total energy and the total number of particlesare consianl.The icmperamre ofihc syslcm is equal to tile lempcrature
oflhc reservoir, and i lie chemical potential of the sysiem is
equal to the chemical potential of Ihe reservoir. The syslcmmay bs as small as one atom or it may be macroscopic, buiihe reservoiris
always to be thought of as much larger lhan
lite system-
Let P[NtEs}denote the probabilitythat the system has jY particles and is in
a particular stale s.This probabilityis proportionalto the number of accessible
stales of the reservoir when the stateof the sysiem is exactly specified. That
is, ir we specify the state of \302\243,the number oraccessible states of (H + ^ is just
the number of accessibleslatesofO?:
\302\273((H+ S) = gift) x 1. {41)
The factor! reminds us that we are looking at ilic sysiem S in a single spiciixd
stale, The y(<Jl)stales of the reservoir have No -
N particlesand Ikivc energy
Uo - e,. Becausethe system probability P{N,i:J is proportional to Ihe number
ofaccessiblestalesof the reservoir.
- NtU0 -zt).
Chapter 5: Chemical Potential and Gibbs Diitrihufit
Particles i\\'o-
,V,
Energy UQ\342\200\224
Ft
(a)
PaniclesA\\
Enemy r,
fit
Panicles ,VU-
Energy Uo-
Particles
Energy ,
(b)
and dirTusive contact with the system. In (a) tile
to it. In {b] the system is in quantum state 2, and the reservoir has g{N0 - A'j, Uo-
\302\243;
stales accessible to it. Because we have specifiedthe exact state of the system, the totalnumber of states accessible 10 01 + i k just the number ofsiulcs accessible tool
Here g refers to the rcsetvoiraloneand depends on the number of particlesin the reservoir and on the energy of the reservoir.
We can express D2) as a ratio of two probabilities, one that the system is in
state 1and the other lhat the system is in state 2:
g(N0 --JV,,[/0- e,)
j
D3)
where g refers to the state of thereservoir.The situation is shown in Figure 5.9.
By definition of the entropy
s(N0,y0)= ejpOfNo.f,,)] ,
so that the probability ratio hi D3) may be written as
D4)
D5)
>ffel)=
eXp[\"(N\302\253\"
N\"y\302\260
D6)
Here, Act is the eniropy difference:
Ac h a(N0 ~NUUO
- e,) - a[N0 -N2,U0
-e2). D7)
The reservoir is very large in comparison with the system, and Ac may be
approximated quite accurately by the firsl order lerms tn a series expansion inthe [wo quantitiesN and \302\243that relate to the system. The entropy of the reservoirbecomes
-N,U0 ~i) D8)
For A<? defined by D7) we have, to the first order in iVE\342\200\224
N2 und in e, \342\200\224c->t
E0a)
by our original defmilion of the temperature. This is written for the reservoir,
bin the system will have the same temperature. Also,
K (?e' E0b)
by C0).The entropy difference D9) is
(N, - Nt)n (t, -E1)
Here Aa refers to the reservoir, but Nit A'^, \302\243ltt] refer to the system. The centralresult of statisticalmechanicsis found on combining D6) and E1):
E2)
Chapter 5; Chemical Potential and Gibbs Distribution
The probability is the ratio of two exponential factors,c;ichof the form
cxp[(A'/j-
e)/t], A term of this form is calleda Gibbsfuctor.TheGibbsfactor
is proportional to the probability that the system is in a state 5 of energy e,and number of particles N. The result was first given by J. YV. Gibbs, who
referred to it as the grand canonical distribution.The sum of Gibbs factors,taken overall states of the system for al! numbers
of particles,is the normalizing facior that converts relative probabilities toabsoluteprobabilities:
. exp[(A'/i-
E],VJ). r]. E3}
This is called the Gibbssum,or the grand sum, or the grand pan it ion function.
The sum is to be carried out overall states of the system for all numbers of
particles:this defines the abbreviation ASN. We have written e, aseJ(:V,
to
emphasize the dependence of the stale on the number of part icles .V. That is,
c,|.v,is ihe energy of the state s(N) of ihe exact A'-particiehamiitonian.Theterm N = 0 must be included; if we assign its energy as zero, then ihe first
term in ^ w'-' be 1.The absoluteprobability that the system will be. found in a state Nlt e, is
given by ihe Gibbs facior divided by lite Gibbs sum:
E4}
This applies lo a system lhat is ai temperaturet and chemical poieinial ji. The
ratio of any two P's is consistentwith our cenlral result E2} for lite Gibbsfactors.Thus E2} gives ihe correct relative probabilities for the stales A',, s,
and jV2,\302\2432.Thesum of ihe probabilities of all stales for al! numbers of par* icles
of tlte system is unity:
f- 1 \342\226\240155)
by ihe definition of 0-. Thus E4) gives the correci iibsuime probability.*
p.iiiictiljrl)- IicIpM. The nuriliod us,cil there to ikme the Potion dimibuiion impends on I
6744s Factor and Gibbs Sum
Average v;i!ucsover ihe sybiems m diffusive and thermal contact with a
reservoir are easily found. If X{N,s) ts ilie value of -V when ihe sysiem lias N
particles and is in ihe quantum slate s, then ihe thermal averageof .V over all
N and all s ts
{56)
We shiill use this result to calculate thermal averages. ,
Number of particles, The number of panicles in the system can vary because
the system is in diffusive contact with a reservoir. The thermal average of thenumber of particlesin ihe system is
according lo E6). To obtain tile numerator, each term in ihe Gibbs sum hasbeenmultipliedby liie appropriaie value of N. More convenient forms of <N>can beobtaftiedfrom the definition of# :
whence
<,V>- l-ll = r^-1 E9)
Tiletiicrma! average number of particles is easily found from the Gibbs sum
J by direct use of E9). When no confusion arises,we shall write jV for the
thermal average <N>.Wlicn we speak later of the occupancy of an orbital,for </>will be written interchangeably for ,V or <.V>.
We often eniploy the handy notaiion
F0)
where k is called the absolute activity. Here /. is IheGreekletter lambda. We
see from A2) that for an idealgas). is directly propcrliona! to ihe concentration.
Chapter5: Chcutkul Potential and Gibbs Distribution
The Gibbs sum is wriiien as
A ' A5S
and ihe ensemble averagenumberofparticlesE7) is
F1}
F2}
This relaiior. is useful, because in many actual problems we determine /. byfinding ihe value ihat will make <N> come out equal to the given numberofpanicles.
Energy, The thermal average energy of the system is
v -<E>
- ^i F3)
wherewe ha\\elemporarilyintroducedthe notation /J = 1/t. We shall usuallywrite V for <\302\243>.Observe that
so that E9} and F3} may be combined lo give
F5}- ;r-\302\253toi
A simpler expression that is more widelyused in calculations was obtained in
Chapter 3 in terms of the partition function Z-
Example:Occupancy zero or one. A red-blooded example of a syslcm ihat may be
occupiedby zero molecules or by one molecule is the heme group, which may be vacantormay be occupied by one Oz molecule\342\200\224and never by more than one Oz molecule(Figure5.10).A single heme group occurs in the prolein myogiobin, which is responsible for thered colorof meal.Ife is ihe energy of an adsorbed molecule of O2 relative to Oa at rest at
D
Figure 5.10 Adsorpiion of an Oj by a hemc
where c is ihe energy of an adsorbed O; rctal10 an Oi al infinite sepatuiion from ihe site.If energy musi be supplied10detacri llie O3from ihe heme, then c will be negative.
infinite distance, ihen the Gibbs sum is
\302\243~I +;.cxp(-\302\243/r). F6)
If energy musl be added io remove the atom from, tile heme, t wilt be negative. The term t
in ihe sum arises from occupancy zero; ihe lerm Aexpf-c/i) arises from single occupancy.Theseare itjc only possibiliiies. We have Mb 4- O2or MbO2 present, where Mb denoies
myoglobin.a protein of molecular weighi 17 000.
Experimental results for Ihe fractional occupancy versus ihe conceniraiion of oxygenarc show a in Figure 5.t t. We compare ihe observed oxygen saturation curves ofmyoglobinand hemoglobin in Figure 5-12. Hemoglobin is ihe oxygen-carrying component of blood,
ti is made up of four molecular strands, each slrand nearly identical wiih ihe single sirand ofmyogiobin,and each capable of binding a single oxygen molecule, Hisiorically, ihe classicwork on ihe adsorpiionof oxygen by hemoglobin was done by Chilian Bohr, the father ofNiels Bohr.The oxygen saturation curve for hemoglobin (Hb) lias a slower rise at low
pressures, because ihe binding energy of a single O, Io a moleculeof Hb is tower Ihan forMb. A! higher pressures of oxygen ihe Hb curve has a region lhat is concave upwards,because trie binding energy per Oj increases after the first O2 is adsorbed.
The Oj molecules on hemes are in equilibrium with the Oj in ihe surrounding liquid, so
thai the chemical potentials of Oj are equal on ihe myoglobin and in solution:
/i(MbO2) \302\273/i(O2); /(MbO2) = ?.{O2) F7)
where A e exp(/i/i}. From Chapter 3 we find the value of / in icrnis of the gas pressure
by the retalion
A = n/nQ -p/t\302\253Q. F3)
We assume ihe ideal gas resuit applies io C), in sotulion. At conslant temperature /(Oj) is
direclly proporiiomUto tjic pressure p.
The fraction/of Mb occupiedby O^ is found from {66}to be
_ Ac\302\273pl-t/i) =1
Fg)I + /.exp(-\302\243/t) r'exp(\302\243/t)+ 1
'
Figure 5.11 The reactiono(u nvyoglobm
(Mb) molecule with oxygen may be viewed as(he adsorptionof a molecule of Gj at a siteon ihe large mjoglobin molecule. Tlieresults follow 3 Langmujr isotherm <{tiitc
accutaiely. Each myoglobin molecule can
adsorb one Oj molecule.These curves show
(he fraction of myogtobin with adsorbed O.
as a function of ihe partial pressureof Oj.The curves are for human myoglobin in
solution. Mvogiobin is found in niuidcs; it is
responsible for the color of slKtfc. After A.
Rosst-Fanellt and E. Antoiiini. Archives of
Btoihciittsiry and Biophysics77, 47\302\253(tySK).
Concentration of O3, rcbt
Fi\302\273ure5-12 Saturation curves of O. boundlo nijoglobin (Nib) and iKmoglobtn (J-ib)molccuks in soluiion in wmer. The partiatpressure of0. is plotted as the Iiorizonl;i!
;t\\is. Tiie vertical a.\\is gives ihe fraction of
the molecules of Mb which lus one bound
0, mokcuk, or the ft action of the suarnis oflib which have one bouiid Qt innlcutle.
Hunioglabm h;is a much Uirgcr change in
oxygen cotitem in ihe pressure range bensecuttw nuerks and the veins. Ihis circumstancefacilitates ihe aciion of ihe heart, viewed as :ipump. The curve for myoglobin has the
pioitual lonn (or the reaction Mb + O2 \342\200\224\342\200\242
MbO.. The curve for lienio^lobin Itits ;t
uiiiCicm forrn because of ifsicracironsbenveen
O, molecules bound to ilic four si rands of ihelib niolccule.The dr^Atntjis after J. S. Frutonand S. Siminondi, Gvm-J bioJtaniMry. Wiley,1961.
G'tbbi Factor and Gtbbs Sut.
which is [he same as the Fcrmi-Dirac distribution function derived in Chapter 7. We
subsiiiute F8) in F9| to obtain
} 4- I\302\273iurexp(E/r}
+ pG0}
or. with p0 \302\256jiqtcxp(\302\243/i).
Po +P'
whac !>0 is constant wiih respect to prcsfiG1}is known as the Langmuir adsorption iseasts on the surfacesof solids.
re, but depends on the temperature. The result
ndtrm when used to dcicribc the adsorptioiiof
Example;Impurity atom ionizatian in a semiconductor. Atoms of numerous chemical
elements when present as impurities in a semiconductor may iose an electron by ionizalion
to the conduction band of the semiconductor crystal- In the conduction band the electron
may oikn be treated as an idea! gas.The impurity atoms are small systems S in thermal
and diffusive equilibrium with the large reservoir fanned by the rest of the semiconductor:the atomsexchange electrons and energy with lite semiconductor.
Lei / be the ionizalion energy of the impurity atom. We suppose that one, but only one.
electron can be bound to an impurity atom; either orientation t or i of the electron spinis accessible.Therefore the system & has three allowed states\342\200\224one without an electron,
one with an electron attached with spin f, and one with an electron attached with spin ].When ^ has zero electrons, the impurity atom is ionized. We choose tiic zero ofenergy of &
as this stale; the other two stales thereforefiave ihc common energy c = \342\200\224/. The accessible
stales of \342\226\240$are summarized below.
Suite number Description
Electron detached
Electron alladicd, s
1-kcifoti ijjnjLlicJ. s
The Gibbs sum is given by
G2)
The probability tliat & is ionized ((V\302\273
0) is
P(ioaized) \302\253P(O.O)
= \342\200\224= -\342\200\224-1- \342\200\224. G3)
The probability tliat $ is neutral (im-ionized) is
P(neutral) = /'(It,-/) + P(U,~/) , G4)
which is just t - PtO.O).
SUMMARY
1. Thechemicalpotential is defined as h{t,V,N) = {tF/tN). v and may also bo
found from /*\342\200\224
{cUJcN)a v \342\200\224-i{cojvN}v_v. Two systems are in difl'usive
equilibrium if jtx = }tz.
2. Tl^cchemicaipotentialis made up of two parts, external ittid internal. The
external part is the potential energy of a particle in an external field of force.The internalpart is ofthermalorigin;for an ideal mooaiomic gas ;t(int)
-
Tlog(u/iiy),whereit is the concentration andnQ ^{Mi'lnh2)*'2 is the quan-
quantum concentration.
3. The Gibbs factor
gives the probability that a system at chemical potential ;i and temperatureiwill have N particles and be in a quantum state s of energy ts.
4, TheGibbssum
is taken over all states for all numbers of particles.
5. The absolute activity X is defined by X ~ expQi/r).
6. The thermal averagenumber of particles is
<A')J kg \302\243
PROBLEMS
1. Centrifuge. A circular cylinder of radius R rotaies about the ions axis with
angular velocity to. The cylinder contains an ideal gas ofatomsof niass SI al
temperature i. Find an expression for lhe dependence of the concentraiionj[(r) on the radialdisiance<\342\226\240from the axis, in terms of *i\302\2430)on the axis. Take
H as for an ideal gas.
2. Moleculeshi die Earth'satmosphere. If a is the concentration of mofcculcsat the surfaceof the Earth, Al lhe mass of a molecule,and g lhe gravitational
acceleration at lhe surface, show that at constant temperature tlic total numberof molecules in the atmosphereis
N =4nn(K)Cxp(-A/yR/t).Jjti </rrJcxp(A/yKV<'0 - f75)
wifh r iiuMstirtiJ from ihc center of flic l-yrih; here R is the railing of ihe liarili.The integral diverges ttf (he tipjwr limit, so that N cannot be bounded and the
aimosphere cannoi be in equilibrium. Molecules, particularly light molecules,are always escapingfrom ihe atmosphere.
3. I'ntuiitiul em-i'ity of gas fa gravitational Jield. Consider a column of momseach of massM at temperature t in a uniform gravitational field y. Find the
thermal average potential energy peratom.Thethermal average kinetic energy
density is independent of height. Find the total heat capacity per atom. Thetotal heat capacityis the sum of contributions from the kinetic energyand from
the potential energy. Take the zero of the gravitational energy at lhe bouomh ~ 0 of the column. Integrate from h = 0 to h ~ oo.
4. Active transport. The concentrationofpotassiumK+ ions in lhe internal
sap of a plant cell(for example, a fresh water alga) may exceed by a factor of !04
the concentration of K+ ions in the pond water in which the cell is growing.
The chemical potential of the K* tons is higher in the sap because their con-concentration it is higher there. Estimate the difference in chemicalpotential
at
300 K. and show that it is equivalent to a voltage of 0.24V across ihe cell wall.
Take ;i as for an ideal gas. Because the values of the chemicalpoiemialsare
different, ihe ions in the cell and in the pondarenot in diffusive equilibrium. The
plant cell membrane is highly impermeable to the passive leakage of ionsihrough tt. Importantquestionsin cell physics include ihese: How is ihe highconcentrationofionsbuilt up within the cell? How is metabolic energy applied
10 energize the aciive ion transpon?
5. Afaguctic concentration. Determine the ratio m/x for which Figure5.6isdrawn. If T = 300K, how many Bohr magnetonsfiB
= eh/Zinc ivould the
particles contain to give a magnetic concentration effect of the magnitudeshown? .\"-\342\226\240..\342\226\240
6, Gibbs sum for a two level system, (a)Considera system that may be un-
unoccupiedwhh energy zero or occupied by one particlein either of two states,
one of energyzeroandoneofenergy s. Show that the Gibbs sum for ihis system is
3-= i + X + ;.exp(-E/r). G6)
Our assumption excludes thepossibility of one particle in each state at the same
time.Notice ttuitn c include in the sum a term forN \302\253=Oasa particular state of a
system of a variable numberofparticles,(b) Show that the thermal average occupancy of the system
is
G7)
(c) Show that the thermal average occupancyof the state at energye is
(d) Find an expression for the thermal average energyof the system.
(e) Allow the possibility that theorbitalat 0 and ate may be occupied each by
one particle at the same time; show that
3.= 1 + I + Aexp(-\302\243/i) + A2exp(-\302\253/t>-\302\253 A + /.)[! + Aexp(-e/i)]. G9)
Because^- canbefactored as shown, we have in effect two independent systems.
7, Statesofposttiw uitd negative iotuzation. Consider a lattice of fixed hy-diogenatoms;supposetliat each atom can exist in four states:
Stale Number of electrons Energy
Ground i -JA
Posiiivc ion 0 -if)
Negative ion 2 J3
E\\cii<.-d 1 U
FioU rlic conUiiion that the Livci;t^^ number of electrons per mom be unity.Theconditionwill involve 5,)., and r.
X. Cmtitm mtmoxitle luihuning. lu catbon monoxide poisoning the CO
replacesthe O2 ad^oibed on hanoglohin (lib) mulcculcs in ilic Wuoil. To hhnw
tl-.e clTect,consider a model for which each adsorption site on a heme may be
vacant or may be occupied either with energy ea by one molecule O2 or with
energy eh by one molecule CO. Let A' fixed home sites be in equilibrium with
Froblcn
O2 and CO in the gas phases at concentrationssuch that the activities are
X(OZ) = 1 x SO\025 and X(CO) - 1 x 10\021, all at body tempcraiure 3TC.Neglect any spin muhiplicity factors, (a) First consider the system t\" the absence
of CO. Evaluate \302\243Asuch that 90 percent of the Hb sitesareoccupiedby O2.
Express the answer in eV per O2. (b) Now admit the CO under the specified
conditions. Find Sgsuch that only 10 percent of the Hbsitesare occupied by O2.
9. Adsorption ofO2 in a magnetic field. Suppose that at most one O2 canbebound to a heme group (see Problem 8), and that when /.{O2)~ iO\025 we have
90 percent of the hemes occupiedby Oj. Consider O2 as having a spin of iand a magneticmomentof i (ts.How strong a magnetic field is needed to chancethe adsorption by 1 percental T = 300 K.?(TheGibbssumin the limit of zero
magnetic field will differ from that of Problem 8 because there the spin muhi-
muhiplicity of the bound slate was neglected.)
10. Concentrationfluctuations. The number of particles is not constant in a
system in diffusive contact with a reservoir. We have seen that
<,v>\342\200\242
re:
from E9). (a) Show lhat
<lY!>=LLi. (81)3-^
The mean-square deviation ((AWI) of N from <iV> is defined by
<(ANJ> = <(N -<.V\302\2732>
= (N>)- 2<N><N> + (N}' =
</V!>- <,V>:;
(b) Show that this may be written as
<(A.V)'> = tc<,V>/c> (83)
In t'liaptcrfi xvi: ;u>rly iliis n-iilt in llio i.K'iilu:isto liiul liial
is the mean square fractional fluctuation in the population of an idea! gas in
diffusive contact with a reservoir. If <A'> is of the order of \\02Q atoms, then ihe
Chapter 5: ChemicalPotential and Gibbs Distribution
fractional fluctuation is exceedinglysmalt. In such a system the number ofpanicles is well defined even though it cannot be rigorously conslatil becausediffusive contacr is allowed wirh ihe reservoir. When <iV) is low, this relationcan be usedin theexperimentaldetermination of the molecular weight of largemoleculessuch as DNA of MW 103 - [O10;see M. Weissman, H. Schindler,
andG. Feher, Proc. Nat. Acad.Sci.73,2776A976).
11. Equivalent definition of chemical potential* The chemical potential was
defined by E) as (cF/cN)tiy. An equivalent expression listed in Table 5.1 is
fi*=
(cU/cN)ay. (S5)
Prove ihat this relation, which was usedby Gtbbs to define ji, is equivalent to
the definition{5}that we have adopted. It will be convenient to make useof theresultsC1)and C5). Our reasons for treating {5}as the definition of/i, and (85) as
a mathematical consequence,are two-fold. In practice, we need the chemical
potential moreoften as a function of the temperature tihanasa funciion of the
entropy a. Operationally, a process in which a particle is added to a systemwhile the temperature of the system is kept constant is a morenatural process
shan one in which the entropy iskept constant;Adding a particle to a system ata finite temperature tends to increase its entropy unless we can keepeachsystem
of the ensemble in a definite, although new,quantumstate.There is no natural
laboratory process by which this can be done.Hcnee the defmtiton E) or F),in which the chemical potential is expressed as the changein free energy per
added particle under conditions of constanttemperature,is operationally the
simpler. We point out that (85) will not give U ~ /tN on integration,becauseH{N,c,V) is a function of N; compare with {9.13}.
12. Ascent of sap in trees. Fitsd the maximum height to which water may rise
in a iree under the assumption lhal the rools stand in a pool of water and Iheuppermostleaves are in air containing waier vapor at a relative humidity
r =
0.9. The temperature is 25\"C.If the relative humidity is r, the actual concentra-concentrationof water vapor in the air at the uppermostleavesis rii0,where n0 is the
concentration in ihe saturated air lhal slands immediatelyaboveIhepoo!ofwater.
13. hentropic expansion, (a) Show thai Ihe entropy of an idea!gas can be
expressed as a function only of the orbital occupancies, (b) From this resultshow ihat xVin is constant in an isentropic expansion of an idea! monatomicgas.
14. Multiple binding of Oi. A hemoglobin molecule can bind four O2
molecules. Assume lhal e is the energy ofeach bound O2, relalive to Oaat restatinfinite distance. Lei / denote the absolute activity exp(ji/T} of the free Oi (in
. solution), (a) What is the probability that one and only oneO2isadsorbedon a
hemoglobin moiecule? Sketch the result qualitatively as a functionof/.,(b) What
ts the probabiiity Uiui four and only four O2 arc adsorbed? Sketch this result
also.
15. External chemical potential. Consider a system at temperature t. withN atoms of mass M in volume V. Let }i[Q)denote the value of the chemicalpotentialat the surface of the earth, (a) Prove carefully atid honestly that the
value of the tola! chemical potential for the identical system when translatedto altitude h is
p(h) =y Mgh ,
where g is the acceleration of gravity, (b) Why is this result different from that
applicable to the barometric equation of an isothermalatmosphere?
Chapter 6
Ideal Gas
FERMI-DIRAC DISTRIBUTION FUNCTION 153
BOSE-EINSTEIN DISTRIBUTION FUNCTION 157
CLASSICALLIMIT 160
Chemicai Potential 161
Free Energy 163Pressure 16-\",
Energy 164
Entropy 165Heat Capacity 165Example:ExperimentalTestsof the Sackur-Tetrode Equation 167
Chemtcai Potential of Ideal Gaswith internal Degrees of Freedom 169
Example: Spin Entropyin Zero Magnetic Field \342\226\240 J70
Reversible Isothermal Expansion 17'iReversible Expansionat ConstantEntropy
IT.t
Sudden Expansion into a Vacuum i \025
SUMMARY 176
PROBLEMS 177
1. Derivative of Fenni-DiracFunction 177
2. Symmetry of Filled and Vacant Orbilals 1773. Distribution Function for Double Occupancy Statistics 1774. Energy of Gasof Extreme Relativistic Particles 177
5. Integration of the Thcrmodynamic Identity for an Ideal Gas 1776. Entropy of Mining
17S
7. Relation of Pressure and EnergyDensily i 7S
8. Time for a Large Fluctuation 17S<J, Gas of Atoms with Internal Degree of Freedom 17')
10. IsentropicRelations of Ideal Gas 179
11. Convcelive isentropic Equilibriumof the Atmosphere J~0
\\2. Ideal Gas iit Two Piitu'itsiuns 1^'13. GibbsSum for Ideal Gas 1SU
14. Idcai Gas Calculations ISO
15. Diesel Engine Compression ISO
Chapter 6: IdealGai
The ideal gas is a gas of noninteractingatomsin the limit of low concentration.The limit is defined below in terms of the thermal average value of the numberof particleslh;it occupy an orbital. The thermal average occupancyis calledthedistribution function, usually designated as /(ej./i), where t is ihc energy ofthe orbital.
An orbital is a stale of [he Schrodingcrequation for only one particle. This
terns is widelyusedparticularly by chemists. !fthc interactions between particlesare weak,Ihcorbitalmodel allows us to approximate sin exact quantum state
of the Schi'6'tlingcr equation of a system of A' particles in terms of an approxi-approximate quantum state thai we construct by assigning the N panicles lo orbituls,with each orbital a solution of a one-particleSchroOingerequation.There are
usually an infinite number of orbitals available for occupancy. The term
\"orbital\" is used even when there is no analogy lo a classical orbit or to aBohrorbit.Theorbitalmodel gives an exact solution of ihe N-particlc problemonly if there are no interactions between the particles.
It is a fundamental result of quantum mechanics (ihe derivation of which
would lead us astray here) thai ail species of particles fail into two distinct
classes, fermions and bosons. Any particle with half-integrai spin is a Fermion,and any particle with zero or integral spin is it boson. There are no interme-intermediate classes. Composite particles follow the same rule: an atomof Hie iscomposedof an odd number of particles\342\200\2242 electrons, 2 protons, I neutron-
each of spin j, so that Hie must have half-integral spin and must be a fermion.An aloin of 4He has one more neutron, so ihcreare an even number of paniclesof spin |-,and 4Hemust be a boson.
The fermion or boson nature of the particlespeciesthat make up a many-
body system has a profound and important effect on the states of the system.The resultsofquantum theory as applied to the orbital modelofnoninleractingparticles appear as occupancy rules:
1. An orbital can be occupied by any integral number of bosons of the samespecies,including zero.
2. An orbital can be occupied by 0 or! fermion of the same species.
Thesecondrule is a statement of the Pault exclusionprinciple.Thermalaverages
of occupancies need not be integral or half-integral,but the orbital occupancies
of any individual system must conformto oneor the other rule.
Fa-mt-Dirac Distribution Function
The two different occupancy rules give rise to two different Gibbs sums for
each orbital: there is a boson stun over all integral valuesof the orbital occu-
occupancy A', and there is a fermion sum in which N = 0 or N = !only. Different
Gibbs sums lead to different quantum distributionfunctions /{e.t./i) for the
thermal average occupancy. Ifconditions are such that/ \302\2531, it will notmaitcr
whether the occupanciesN = 2,3,... are excluded or are allowed. Thus when/ <:< I the fermion and boson distribution functions must be similar.Thislimit
in which the orbital occupancy is small in comparison with unity is c;ii!cd theclassical regime.
We now treat the Fcrmi-Dimc distribution function for the thermal average
occupancy of an orbital by fcrmtonsand the Bose-Einstein distribution functionfor the thermal averageoccupancyof an orbital by bosotis. We show the
equivalenceof the two functions in the limit of low occupancy, and we go on
to treat the properties of a gas in this limil.-ln Chapter 7 we treat the propertiesof famioit and bosoti g;ises in the opposite limit, where the nature of the
particlesis absolutely cruchi for the properties of the gas.
FERMI-DIRAC DISTRIBUTIONFUNCTION
We consider a system composed of a single orbital that may be occupied by a
fermion. The system is placed in thermal and diffusive contact with a reservoir,
as in Figures 6.i and 6.2.A real system may consist of a large numberNo of
fcrmions, but it is very helpful to focuson oneorbitaland call it the system.
All other orbitalsofthe real system are thought of as the reservoir. Our problem
is to find the thermal averageoccupancyof the orbital thus singled out. An
orbital can be occupiedby zero or by one fermion. No other occupancyisallowed by the Pault exclusion principle. The energyofthe
systemwill be taken
to be zero if the orbital is unoccupied. The energy is c if the orbiial is occupied
by one fermion.The Gibbs sum now is simple: from (he definition in Chapter 5 we have
\302\243=1 + ;.exp(~f./r). (I)
The term 1 comes from the configuration with occupancy A' = 0 and energy\302\243= 0. The term -Uxp(-~-\302\243/r) comes when the orbital is occupied by one fermion,so that N *= 1 and the energy is e.The thermalaveragevalue of the occupancy
of the orbital is the ratio of the term in (he Gibbs sum with N = 1 to the entire
Gibbs sum:
B)
Chapter 6: Ideal Gm
Occupied
,VQl Wo)= log \302\243<*,\342\200\236G,,) o(/V0
_ I, Uu- *) = a{A'u. UJ _(|\302\243^-t(J^
Figure 6.1 We consider as the system a singleotbhai that may be occupied a! most by one
fermion. The system is in thermal and diflusixc contact v-uh ihe reservoir at temperature t. Theenergy e of the occupied orbital might be the kinetic encigy of a freeelectron of a definite
spin orientation and confined to a fixed volume- Other allowed quantum states may be
considered as forming the reservoir. The reservoir will contain JVO fcrmions if the sysiem is
unoccupiedand JVD- i fermions if the system is occupied by one fermion.
We introduce for the average oecupancythe conventional symbol /(e) that
denotes the thermal averagenumberofparticlesin an orbital of energy e:
fie) s<.V(\302\253)>. C)
Recaii from Chapter 5 that /. = cxp(/i/i),where p is the chemical potential.We may write B) in the standard form
\"exp[(\302\243 -,0/r] + 1\"D)
Tills result is known as the Fenm-Dirac distributionfunction.* Equation D)
gives Ihe average number of fermions in a single orbital of energy e. The value
O2 119261, and P. A. M. Dirac,Proceedings of i!:c Royal Sociciy of London AI12. 661[
iicovcrcd. The paper by Dirac is concerned with ihe new quantum meclianits a'ld cont
general itaicmenl of lln: form assumed b>- iht P^ulj principle on ihis iheory.
Fertni-Dhac Distribution Functlor
System
Figure 6.2 (a) Tlic obvious method of viewing a system of noninicracling paniclesshown here. The energy levels each refer to an orbital thai is a solution of a single-
parliclc Sclirodinger equation. Ttic loial energy of the sysiem is
where JVB is ihc number of panicles in the orbital n of energy \302\253\342\200\236.For fcrmtons .V. = 0or 1.(b) Ii is much simpler ihan (a), and equally valid, to treai n single orbital as the
system. The system in this scheme may be itic orbital a of energytn. All other orbitals
are viewed as ihe reservoir. The lotal energy of this o\302\273e-orbital system is jVoca. whert
Nn is the number of panicles in the orbital, This device of using one orbital as ihe
system works becauseihe particlesare supposedto interact only weakly with each
oiher. If we think of the fennion sysiem associated with the orbital n, these arc two
possibilities: dsher the system h;is 0 panicles and energy 0, or the system lias t pariki--
and enesgy \302\243\342\200\236,Thus, the Gibbs sum consists of only {wo terms:
3- ~ I + Acxp{~ejti
Tltc firsi scrm arises from the orbital occupancy JVH= 0, aad the second tcnti arises
from ;Vq = 1.
of/always lies between zero and one.The Fermi-Dirac distribution function
is plotted in Figure 6.3.
In the field of solid siaie physics the chemical potential <* is often called the
Fermi level. Tlie chemicalpotentialusually depends on the temperature. Th\302\243
value of ft at zero temperature is often written as ef; ihai is,
Mi - 0) & fi(Q)= Ef. E)
We call cF the Fermi energy, not to be confused*uiih the Fermi level which
*En ilie semiconduUor liseraiure ihe symbol cF is often used for ;( a! any lempfniurt.-. a:-d ef i
ihen called she Fennt level.
Chapter 6; Ideal Gus
........
1
A'
1
N
I 1
\\
\\
\342\226\240~m.
-7 _6 -5 -4 -3 -2 -1 0 I 2 3 4 5 6
e \342\200\224ji, in units of t
Figure 6.3 Plot of the Fermi-Dkac distribution function /(e) versus e - p in unils of
the temperature r. Tlie value of/(c) gives the fraction of orbitals at a given energywhich are occupiedwhen the system is in thermal equilibrium. When tlfe system is
heated from absolute zero, fcrmions are transferred from the shaded region atE/p < 1
to the shaded region at c/ji > 1.Forconduction electrons in a metal, ji might
correspond lo 50 000 K.
is !he temperaturedependent /t(t). Consider a system of many independentorbiials, as in Figure6.4.At the temperature t = 0, ali orbitals of energybelow!he Fermi energy are occupied by exactly one fermion each, and ail orbitalsof higher energy are unoccupied. At nonzero temperatures the value of thechemicalpotential/i departs from Ihe Fermi energy, as we will see in Chapter 7.
If there is an orbilalof energy equal to the chemical potential (e *=//), the
orbital is exactly half-filled, in ihe sense of a thermal average:
F)
Orbitalsoflower energy are more ihan half-filled, and orbilals of higher energy
are less than half-filled.
We shall discussthe physical consequences of the Fermi-Dirac distributionin Chapter 7. Righi now we go on to discussthe distribution function of non-
-I Distribution function
Figure 6.-1 A convenient pictorial way tothink of a system composed of independentorbitalslliat do not interact with eutli oilierbut iutecacl with 3 common tcservoir.
interacting bosons, and then we establish ihe ideal gas law for boih fcrmionsand bosons in the appropriate limit.
BOSE-EINSTEIN DISTRIBUTION FUNCTION
A boson is a particle wtjh an integral value of the spin. The occupancyrulefor bosons is thai an orbital can be occupiedby any number of bosons, so lhatbosons have an essentially different quality than fermions. Systems of bosonscan have rather different physical properties than systems of fermions. Atomsof 4Hc are bosons;atoms of 3He are fermions. The remarkable superfluidpropertiesof the low temperature (T < 2.17 K) phase ofliquid heliumcan beattributed to the properties of a boson gas. There is a sudden increasein the
fluidity and in ihe heat conductivity of liquidJHe below this temperature- In
experiments by Kapitza ihe flow viscosity of 4He below 2.17 K was found to
be iess than IO~7 of the viscosity of the liquid above2.17K.Photons(the quanta of the electromagnetic field) and phonons (the quanta
of elastic waves in solids) can be considered10be bosonswhose number is
not conserved, but it is simpler to think of photons and phononsas excitations
of an osciiiator, as we did in Chapter 4.We consider the distribution function for a system ofnoninteractingbosons
in thermal and diffusive contact with a reservoir. We assume the bosonsare
all of the same species. Let \302\243denote the energy of a single orbital whenoccupiedby one particle; when there arc N particles in ihe orbital,the energy is N\302\253,
as m Figure 6.5. We ireat one orbitalas the system and view all other orbitals
Chapter6: IdealGo:
aspan ofIhereservoir. Any arbitrary number of particles may be in iTheGibbssum taken for the orbital is
: orbital
G)
The upper limit on N should be the tola! numberofpaniclesin the combined
system and reservoir. However, the reservoirmaybe arbitrarily large, so that
N may run from zeroto infinity. The series G) may be summed in dosed form.
Let x s Aexp(~E/t);then
i
\342\200\224A\"
i
i -;.cxp(-\302\243/t)
(8)
provided that kcx.p{~t/z) < 1. In all applications,/,exp(\342\200\224e/i)will satisfy this
inequality; otherwise the number ofbosonsin the system would not be bounded.
The thermalaverageof!henumber of panicles in the orbital is found from
the Gibbs sum by use ofE.62):
-(\342\226\240)/<]-
A0)
L\342\204\242-___ __
Bosc-\302\243insicin Distribution Funclio
Figure 6.6 Comparison of Bose-Einsicinand Fenni-Dirac
dislribuiion funclions. Tlie classical regimeis aiiafoeii for
(e \342\200\224/i) \302\273r, where ihe two distributions become nearly idciuit
We shall see in Ciiapier 7 that in the degenerate regime ai lowtemperature ihe chemical potenijai ft for a FD distribution is
postnve, and changes10negative at high lemperaiure.
This defines the Bosc-Efnstcindistribution function, Et differs malhcmaticaHy
from the Fermi-Dirac distribution function only by having -1 tnsicad of + iin the denominator.Thechange can have very significant physical consequences,as we shall sec in Chapter 7. The two distribution funclions are comparedin
Figure 6.6. The ideal gas represents the limit e \342\200\224/i \302\273r in which the two distri-
distribution functions are approximately equal, as discussed below.Thechoiceof ihe
zero of ihe energy e is always arbitrary. The particularchoicemade in any
problem wiU affect ihe \\a\\uc of the dieniic.il potential /i, buE the vaU:e of ihe
difference e \342\200\224/; has to be independent of the choiceof the zeroofe.This point
is discussed iuriher in B0) below.
A gas is in the classical regime when the average number of atomsin each
orbital is much less than one. The averageorbitaloccupancyfor a gas at room
temperature and atmosphericpressureisof the order of only 10\026,safely in the
classical regime. Differences between fcrmions (half-integral spin) and bosons
Chapter 6: Ideal Gas
and liicq
Regime
Uau
___
iuiu reg
Class
parlic
imcs
ofIc
T!
\342\200\224
Krma! it
\342\200\224\"~
ny
i*: o
III
CCup.iticy
Boson Aluays much less lhan one
Quantum Fcfmioii Close 10 bul less ihan one.Boson Orbital of lowesi energy lias
an occupancy much greater than one.
arise oniy for occupancies of the order of oneor more, so that in the dassieai
regiir.e their equilibrium propertiesare identical. The quantum regime is lhe
opposite of the classical regime. These characteristic features are summarizedin Table 6.L
CLASSICAL LIMIT
An ideal gas is defined as a systemof freenomuteractingparticlesin ttie classical
regime. \"Free\" means confined in a box with no restrictions or e>uernaiforcesactingwithin the box. We develop the properties of an idea! gas with the use
of the powerful method of the Gibbssum.inChapter3we treated the ideal gas
by use of the partition function, buE the identical panicle problem encounteredthere was resolved by a method whose validity was not perfectly
clear.
The Fermi-Dirac and Bosc-Einstein distribution functions in the classicallimit iead to the identical result for the average numberof atoms in an orbital
Write /(e) for the average occupancyof an orbital at energy e. Here e ts the
energy of an orbital occupiedby one particle; it is not the energy of a system of
N particles. The Fermi-Dirac (FD) and Bosc-Einstcin(BE) distribution func-
functions are
where lhe plus sign is for the FD distribution and the minus sign for the BEdistribution.In orderthal/(c) be much smaller th;m unity for ail orbitals, wemust have in this classical regime
; exp[(e-//)/r]\302\273l.,- . . . A2)
for all e. When this inequality is satisfied wemay neglect the term +! in the
denominator of (tt). Then for either fermionsor bosons,the average occupancy
of ail orhiiat of energy r. is
/(i.) * cxp[{/i - f
wiih /. hs exp(/j/r). The limiting result A3) is called the classical dfstnbulionfunction. !t is the limit of the Fcrmi-Dirac and Bosc-Einstein distributionfunctions when the average occupancy /(s) is very small in comparison with
unity. Equation A3), although called classical,is still a result for particles
described by quantum mechanics: we shall find that the expression for A or /.i
always involves the quantum constant /i.'Any theory which, contains h cannot
be a classical theory.We use the classical distribution function /(r) = Aexp(-r/i) to study the
thermal properties of the ideal gas. Therearemany topics of importance: the
entropy, chemical potential, heat capacity, the prcssure-volumc-temperaiure
relation, and the distribution of atomic velocities.To obtain results from the
classical distribution function, we need first to find the chemical potential in
terms of ihc concentration of atoms.
ChemicalPotential
The chemicalpotential is found from the condition that the thermal averageof the total number of atoms equals the numberofatomsknown to be present.
This number must be the sum over allorbitalsof the distribution function/(eJ:
N = A4)
where s is the indexof an orbitalof energy es. We start with a monatornic gasof N identical atoms of zero spin, and later we include spin and molecularmocksofmotion.The total number of atoms is the sum of the averagenumberofatomsin each orbital. We useA3) rn A4) to obtain
JV-/yexp(-\302\243,/T). AS)
To evaluate this sum, observe that the summation over free particle orbitalsis just the partition function Zx for a single free atom in volume V, whence
Chapter6: Idea!Gils
In Chapter 3 it was shown that Zt =HqV,
where nQ s (Mrjlnh2K11 rs thequantum concentration.Thus
A' = ;.Z, = /.,,QK; ;. =W/i?QK
= i?/hq , A6)
in terms of the number density n = NJV. Finally,
/ \302\253=exp(/i/T)
=nftiQ ,
jA7)
which is equal to the number of atoms in the quantum volume l/nQ. in theclassicalregimen;nQ
is \302\253I. The chemical potential of the ideal monatomicgas is
P =\302\273T tog(n/iiQ) , (IS)
in agreement wiih E.12a) obtained in another way. The result may he written
out to give
\302\253i[)ogN
- log V -Jiogt + *SogBn/i2/A-/)]- A9)
We see that the chemical potential increases as the concentrationincreasesand
decreases as the temperature increases.
Comment: The simple expression (IS) for (lie chemica!polentiaf can be subject lo sevcra!modifications.We mention four examples.
(a) If the zero of ihe energy scale is shifted by an energy A so that (he zero of the kinetic
energy of an orbila! falls ill e0 = A instead of at e0 = 0, then
i =. A + Tlog(n/nQ). B0)
(b) If lite atoms \\\\;i\\e spin S, the number of orbiiafs in ihe sum in(iS) i\302\273mufliplicd by ihe
spin ;:iuliipiidiy^S 4- I. For s.pin \\ il isdiJublcdLlhevalucof iheparlition fund ion Z^is JoublcJ; nQ
will be replaced t\\eryw[icre by 2\302\273G,and Ihe nght-hand side of{18)
wifl h3\\c an added ferm -rlog2. Tile tfTctt of Shi: spin on She enSropy is Srcalcdbelow.
fc) If the &3s is not fTiOnatonuc, t^e iiitern^J energy sS^ies associated with ro^itionaj -ind
vibraiional moiion wifl enter ihe partition function, and tfic chemical potential wilt
free Energy
have an added term - t log Zjni, per D8) below, where 2ial is the partition function of
the internal degrees of freedomof one molecule.
K the gas is nonideal, the resuK for /t may be considerably more complicated;secChapter 10 for tk relatively simple van der Waals approximation to j g;is of inter-
Free Energy
The chemical potential is related to the free energy by
(CF/dhr)t_y = n , B1)
according to Chapter5. From this,
F{N,t,V) =j*JN}j{N,t,V}
=tJ^N [log <V +
\342\200\242\342\226\240\342\226\240],B2)
where the integrand is found in brackets in A9). Mow Jt/.vlog.v = xlog x \342\200\224x,
so [hat
F - iVi[logW - I - fog V - Jlogr + JlogBn/iVA')] . C3)
F =Wr[iog(n/nQ)
- 1]. B-t)
Tile free energy increaseswith concentration and decreases with temperature.
Comment: The integral in B2) should strictly be a sum, because A' is a discrete variable.
Thus, from E.6),
f(N,r,l') -f Ji(-V) . B5)
which differs from Ihe integral only in ihc lenn in log ,V in A9), for
f. log.V = Iog(l x 2 x 3 x \342\226\240\342\226\240\342\226\240x N) = iog/V! , B6)
ChapM6:UtaICai
wHcfg the inic^EiiI \302\2433\\eA1 log A' \342\200\224_V in B3). Bui for targe N she Stifling approximation
log,V! ~ MogN - N , B7)
may be used, and now B5) is ihc same as B3).
Pressure
The pressure is relatedto the free energy by C.49):
With B3) for F we ha
Nt/V; pV= Nz ,
which is the idea! gas law, as derived in Chapter 3.
Energy
The thermal energy U is found from F s U ~iff, or
U-F + ta-F~t(cFi<\"t)|.,\302\273
= -t:
With B3) for F we have
B9)
2t'
so that for an idealgas
C0)
CD
C2)
The factor | arises from the exponent of x in\302\253Q
because the gas is in three
dimensions; if iiq were in one or two dimensions,the factor would be | of 1,
respectively. The average kinetic energy of translational motionin the classical
limit Is equal to ^c or %kBT per translaiional degree of freedom of an atom.Theprincipleofequipartitionofenergy among degrees of freedom was discussedin Chapter 3.
Heat Capacity
A polyatomic molecule has rotational degrees of freedom,and the average
energy of each relational degree of freedomis%r when ihe lemperature is highin comparison with the energy differencesbelwcenlheroialiorta!energy |e\\els
of ihe molecule. The rolaliona! energy is kinetic.A linear molecule has two
degrees of rolationa! freedom which can beexciled;a nonlinear molecule has
tliree degrees of rotalional freedom.
Enlropy
The enlropy is related to the free energy by
a = -(oFJdT)v,K. [33)
From B3) for F we have the entropy of an idealgas:
This is idenlica!with our earlier result C.76). In the classical regime h/uqh
\302\2531, so lhiil \\o^n0/n) is positive. The result C4)is known as lhe Siickur-Tflrode
equation for the absolute entropy of a monatomic idea! gas. It is imporianihistorically and is essential in the thermodynamics of chemical reactions. Even
though the equation containsh, the result was inferred from experiments onv;ipor pressureand on equilibrium in chemical rcaclions iong before thequantum-mechanicalbasiswas fuliy understood, it was a greal challengetotheoretical physicists to explain the Sackur-Tetrode equation, and many
un-
unsuccessful attempts to do so were made in the early years of this century. We
shall encounter applicationsof the result in later chapters.The entropy of the idealgasisdirectlyproportionalto the number of particles
N if their concentration n is constant,as we see from C4). When two identical
gasesat identicalconditionsare placed side by side, each system having entropy
ffj, the total entropy is 2al becauseJV is doubled. Jf a valve that connects the
systems is opened,the entropy is unchanged. We see that the entropy scalesasthe size of the system: the entropy islinear in the numberofparticles,at constant
concentration- If the gases are not identical,the entropy increases when the
valve is opened (Problem6).
HeatCapacity
The heat capacity at constant volume is defined in Chapter 3 as
C5)
Chapter6: IdealGos
We can calculate the derivative directly from the entropy C4) of an idea! gas
wlien the expression fornQ
is wn'tien out:
IE
From this, for an idea! gas
C6)
or Cy =%NkB in conventional units.
The heat capacity at constant pressureis largerihan C\302\245because additional
heat must be added to performthe work needed to expand the volume of the
gas againstthe constant pressure p, as discussed in detail in Chapter 8. We use
lhe thermodynamicidentity xda = dU -f pdV to obtain
/eup C7)
Theenergy of an ideal gas depends only on Hie temperature,so tliat
ldU/i)z)p
will have the same value \342\226\240dSidU/di}^, whieh isjust Cy by the argument ofC.17b).By lhe ideal gas law V - Nt/p, so that the terra
p(dVldt\\= N. Thus C7)
becomes
in fundamental uniis, or
C, = Q. + N
C, = C, + Nk,
C8a)
C8b)
in conventional units. We notice again the different dimensions that heat
capacitieshave in the two systems of units. For onemole,NkB is usually written
as R, called the gas constant.TheresultsC8a,b)are written for an ideal gas without spin or otherinlernal
decrees of freedom of a molecule. For an alom Cy\342\200\224
:N, so that
c,= In + n = |,v C8c)
Heat Capacity
in fundamental units, or
Cp \302\273$NkB C8d)
in conventional units. The ratio CpjCv is written as 7, the Greek letter gamma.
Example; Experimental tests of the Sackur-TeiroJc equation. Experimental values of ihe
entropy are oflen found from experiment! values of Ct by numerical integration of C7)
C9)
Here <j@) denotes ihe entropy at the lowest temperature attained in rhe measurements
of C,,. The fhird law of thermodynamics suggests rhat o@) may be set equal ro icta unlessfhcre are mullipliritics not removed at the lowest temperature attained.
We can calculate the entropy of a monatomic: ideal gas fayuse of rhe Sackur-Teirodc
equation C4). The value thus calculated at a selected temperature and pressure may be
compared with the experimental value of the entropy of ihc gas. The experimental value
is found by summing ihe follow ing confributions:
1. Enlropy increase on healing solid from absolufe zero to the melling point.2. Entropy increase in ihe solid-to-liquid transformation (discussedin Chapter 10).
.1. Entropy increase on hearing liquid from niching [mini to ihe boiling point.4. Lntropy increase in ihe liquid-to-gas transformation.
5. Enlropy change on heating gas from the boiling point to ihe selected temperatureand
pressure.
There may further be a slight corrctiion lo C4) for ihe nonidcalilyoflhe gas.Comparisons
of experimental and theoretical values have now been carried out for many gases,and very
We give delails of the comparison for neon, afler ihe measurements of Clusius.Theenlropy is given in terms oflhe conventionalentropy S =s kua.
I. The heat capacity of ihe solid was measured from 12.3K to the melting point M.55 K
under one atmosphereof pressure. The heat capacily of ihe solid below I2.3K was
eslimaicd by a Debyc law (Chapier 4) extrapolation to absolute zero of the me^iuc-menlsaboie IJJK.Theeniiopyof ihe solid at the melting point is found bj nuir.e.'tcal
inlcgralion of UlTiCp/T) 10be
5lolid=, i4.29Jmor'K~t.
\342\200\242A classic vtudy is \"The heal rapacity of oxygen from 12 K to its boding point ,ind its hiat of
ihe American Chemical Society 51.2300 A939).
Chapter 6: idcutGa:
1 able
Gas
Nl-
ArKr
wide
r.
xaluaot
lp,lnK
21287.29
119.93
iiosph
E
uropy ullli
Enlropy in
Nperimenia
96.40
129.75144.56
c boilin
Jmol\"
I Ca
'K\"
Iculalo
96.451292414506
sul'rce:From Lumhh BonMvin labics, 6lh cd., Voi
Pan 4, pp. 394-399.
. 2. The hc.it inpul required to mc!l the solid ai 24.55 K is observed lo be 335 J mol\"'. The
associated enlropy of melling is
3. The heat capacilv of the liquid was measured from the inching poim [o ihe boiling
point of 27.2 K under one atmosphereof pressure. The entropy increase was found 10 be
ASIJ4Uid - a.SSJmol-'K-1.
4. The heat input required to vaporize ihe liquid at 27.2K was observed lo be 1761Jmol\021. The associaied enlropy of vaporization is
The cxpertmenlal value oflheeniropy of neon gas a! 27.2K af a pressure ofoneafmo-
sphere adds up to
sK3i= ^.oiid + ASmelling
+ &SUvAi + ASViPQliMlion\302\25396.40Jmor'K\021.
The calculafed value of the entropy of neon under ihe same condifions is
5g\302\273,= 96.45 Jmor'K\021 ,
from ihe Sackur-Tefrodeequation. The excellent agreemenl with fhe experimental value
gives us confidencein the basis of the entire theoreticalappamlus that led to the Sackur-Tctrodc equation.The rcsuh <31) coutd Unrstly l[:ivc been guessed; to find it verified byobservation is a real experience.Results for argon and krypton are given in Table 6.2.
Chemical Potential of IdealGas with Infernal Degrees of FreeJoin
Chemical Potential of IdealGaswith
Internal Degrees of Freedom
We consider now an idealgasof identical polyaiomic molecules. Each molecule
has rotational and vibrationaldegreesof freedom in addition to the transla-tional degrees of freedom.Thetotalenergy e of the molecule is the sum of two
independent parts,
\302\243\302\253en + eifli , D0)
where einl refers to the rotational and vibrational degrees of freedomand r.n
to the translations! motion of the center ofmassofthe molecule. The vibrational
energy problem ts the harmonic oscillator problem treated earlier. The rota-rotational energy was the subject of Problem '3.6.
In the classical regime the Gibbs sum for the orbital n is
\302\243\302\2531 + ;.exp(-\302\243fl/t) . D1)
where terms in higher powers of ), are omitted because the averageoccupancyofthe orbitaln is assumed to be \302\253I. That is, we neglect the terms in 3- which
correspondtooccupanciesgreater than unity. In the presence of internal energystates the Gibbs sum associated with the orbital n becomes
D2)
D3)
Thesummation is just the partition function of the internal states:
(\024)
which is related to the internal free energy of the one molecule by /\"in, =
~tlogZ;ol. From D3) the Gibbssum is
D5)
Chapter 6: Idea! Gas
The probability that the transtationat orbita! n is occupied, irrespective ofthe stateof internal motion of the molecule, is given by the ratio of the term
in X to the Gibbs sum 3-:
The classical regimewas defined earlier as /(\302\243\342\200\236)\302\253i. The result D6) is entirelyanalogous to A3) for the monatomic ease, but XZinl now plays the role of A.
Several of the results derived for the monatomie idealgas are different for
tlie polyatomic ideal gas:
(a) Equation A7) for,* is replaced by
D7}
with<iq
defined exactly as before (We shall alwaysuseneas defined for the
monatomic ideal gas of atoms with zero spin.) Because X = e\\p(/i/i) we
have
p. =r[log(ri/nQ)
~IogZiol].
(b) The free energy is increasedby, for JV molecules.
(c) Thecniropy is increasedby
D8)
D9)
E0)
The former resuU U \342\200\224jh'z applies 10 die iranslalional energy alone.
Example:Spin enmtpy in zero ntasnrtic field. Consideran atom of Spin /, where / may
represcnl boih electronic and nuclear spins. TIic iniernal partition funciion associaied
Reversible Isothermal Expansion
with the spin alone is
ZiM = {2/ + t) , E1)
this being the number of independent spinsiaics.The spin contribution to the free energy is
Fial = -tlogB/+ 1) , E2)
and the spin entropy is
\302\273,.,= logB/ + 1) . E3)
by E0). TheefTec! of the spin entropy on the chemicalpotential is found with the help ofD8):
}i = T[Iog(n/ii0)- logBi+ I)]. E4)
Reversible Isothermal Expansion
Consider as a modelexample1 x 1022 atoms of 4He at an initial volume of
!03cm3 at 300K. Let the gas expand slowly at constant temperature until the
volume is 2 x !0Jcm\\ The temperature is maintained constant by thermal
contact with a large reservoir.In a reversibleexpansion the system ai any
instant is in its mostprobableconfiguration.
What is the pressure after expansion?The final volume is twice the initial volume; the final temperature
is equal
to the initial temperature. From pV\342\200\224S'x we see that the final pressure is
one-halfthe initial pressure.
What is the increase of entropy an expansion?
The entropy of an ideal gas at constanttemperature depends on volume as
a(V) = A'log V 4- constant , E5)
= jVIog2 = (ix l0\{0.69?)")*= 0.069 x 1033. E6)
Notice that the entropy is largerat the largervolume, because the system h.TS
more accessible states in the largervolume than in the smaller volume at the
same temperature.
Figure 6.7 Work is done by (he gas in an isothermalexpansion.Herethe gas docs work by raising llic weights.Under isothermal conditions pi' is constant for an ulcal gas,so thai the pressure mus! be reduced to affow !hc volume to
expand. The pressureis reducedby removing llio load of
weights a little at a time.
How much work is done by the gas ht the expansion ?When the gas expands isothermally, it does work against a piston, as in
Figure 6.7. The work done on the piston when the volumeisdoubledis
=H\\NxjV)dV Ntlog2. E7)
We evaluate/Vt directly as4.!4 x 30sergpistonis,from E7),
= 43.4J. Thus the work done on the
Ni!og2 \302\253DJ.4J){0.693)
= 2S.7J. E8)
The assumption that the process is reversibleentersin E7) when we assume
that a knowledge of Vat every stage determines p at every stage oftheexpansion.We define W as the work done on the gas by external agencies. This is the
negative of the work done by the gas on the piston. From E8),
w = -jpdV= -28.7J. E9)
ruble Expai mt Entropy
What is the changeof energy in ihe expansion?The energy of an idea!monatomicgas is U *= jNx and does not change in
an expansion at constant temperature. However, the Helmhoitz free energy
decreases by Afilog2, which is the work done.TheconnectionisdiscussedinChapter S.
How much heal flowed inio the gasfrom the reservoir?We have seen that ihe energy ofihc idea!gas remainedconstant when the
gas did work on the pision. By conservation of energy it is necessary thai a(low of energy in ihe form of heat into the gas occur from ihe reservoir throughthe walls of the container.The
quantity Q of heat added to the gas must beequal,but be opposite in sign, to the work done by the piston, because
Q + W = 0. Thus
Q = 28.7 J , F0)
from the result E9).
Reversible Expansion at Constant Entropy
We considered above an expansion at constant temperature. Supposeinsteadthat the gas expands reversibly from I x 103cmJ to 2 x 10Jcm3in an in-
insulated container. No heal flow to or from the gas is permitted, so that Q= 0.
The entropy is constant in a system isolated from the reservoir if the expansionprocessis carried out reversibly (slowly). A process without a cluvnge of entropy
is called an isentroplc processor an adiabaticprocess.The term \"adiabalic\"
has the specific meaning that there is no heat transfer in the process. For
simplicity, we shall stick wilh\"isentropic.\"
What is the temperatureof the gas after expansion ?
The entropy of an idealmonatomicgas depends on the volume and ihe
temperature as
a{z,V)- N{iogT3;I + logK + constant) , F1)
so that the entropy remains constant if
Iogt3/2K= constant; t3/2K = constant. F2)
In an expansionat constant eniropy from V1 to V% we have
'
Tl3'2f, = rivlV1 F3)
for an ideal monatomic gas. . .
Chapter6: IdealGas
We use the idea! gas law pV = Nz to obtain two alternate forms. We insertV - Nxjp into F3) and cancelN on both sidestoobtain
Similarly, we insert r = pV/N in F3) to obtain
F4)
F5)
BothF4) and F5) hold only for a monatomic gas.It is ihesubjectof Problem 10 to generalize these results for an idealgas of
molecules with internal degrees of motion {rotations, vibrations). We obtain
for an iscnlropic process
t.'\"'\"\"'V. =r2\"\"- F7)
Here'/ sCp/Cy
is the ratio of the heat capacities at constant pressureand
constant volume.
With Tx - 300K andVs/V2
= |- we find from F3):
\302\253189 K. F9)
Tt\\k is ihe final temperature after the expansion at constant entropy. The gasiscooledin the expansion process by
Tt - Tt = 300K- 189 K - !!IK. G0)
Expansionat constant entropyis an important method of refrigeration.
What is ihe change hi energy in ihe expansion?The energy change is calculatedfrom the temperature change G0). For an
ideal monatomicgas
U2 \302\243/,= Cy{x2 - r2 - r,) , G1)
fi Expansion into
or, in conventional units,
a2- ut =ivA-B{ra- r,)\302\273i(l x 10IZ)(j.38 x JO\0216ergK\"!){-IHK)
=\302\273-2.3 x 108erg \302\273-231 G2)
The energy decreases in an expansion at constant entropy. The work done by
the gas is equal to the decrease in energyof the gas, which is Ul -U2
= 23 J.
Sudden Expansion inlo a Vacuum
Let the gas expand suddenly into a vacuum from an initial volume of I liter toa Hn:t!vohmicof 2 liters.Thisis an excellent cxiimpfe of an irreversible process.When a hole is opened in the partition to permit ihc expansion, the first moms
rush through the hole and strike ihcoppositewall. If no heal How through thewalls is permitted, there is no way for the atoms to lose their kinetic energy.
The subsequent flow may be turbulent (irreversible),with different parts of tht!
gas at different values of the energy density. Irreversible energy flow between
regions will eventually equalize conditions throughout the gas. We assume the
whole process occurs rapidly enough so that no heat flows in through the walls.
Hois1much work is done in the expansion ?No meansofdoing external work is provided, so that the work done iszero-
Zero work is not necessarily a characteristic of all irreversibleprocesses,but
the work is zero for expansion into a vacuum.
What is the temperature after expansion?No work is doneand no heat is added in the expansion: W = 0, Q ~ 0, and
U2~ t/j = 0. Because the energy is unchanged,the temperature of the ideal
gas is unchanged.The energy of a real gas may change in the process because
the atoms are moved farther apart, which affects their interaction energy.
What is the change of entropy in the expansion ?
The increase of entropy when the volumeis doubled at constant temperature
is given by E6):
Au w a2~
a, - Nlog2 = 0.069x 1023. G3)
For the expansion into a vacuum 2 = 0.
Expansion into a vacuum is not a reversible process: the system is not in the
most probable (equilibrium) configuration at every stage of theexpansion.Only
ChapterS: Idea! Gas'
Reversible ] y t- I-
isothermal V 0Nlog^ -.Vtlog-^ A'rtog-
expansion jReversible
]
isentropic ]
expansion J
Irreversible \021 y
expansion into V 0 Wlog \342\200\224
vacuum J'
-J.Vt,I-(\302\243)L. \\vi/ j
the initial configuraiion before removal of the partition and the final con-
configuration after equilibration are most probable configuraiions. Al intermediate
stages the distribution in concentration and kinetic energy of atoms between
the two regions into which the system is divided does not correspond to an
equilibrium distribution. The central results of these calculationsare sum-
summarized in Table 6.3.
SUMMARY: STEPS LEADING TO THE IDEAL
GAS LAW FOR SPINLESS MONATOMIC GAS
(a) /(e) =Aexp(-\302\243/r) Occupancy of an orbital in the
classical limit of/(\302\243)\302\2531.
N
(b) X = =\342\200\224-\342\200\224-\342\200\224~~~ Given N, this equation determinesi e*P(-\302\243nA)
x -[n the dassical limit.
(c) En=
-\342\200\224-177x73 j Energy of a free particle orbital of^ '
quantum number n in a cube ofvolume V.
(d) X!CXP(\342\200\224\302\243JT)= Ik
j'/ii n2 exp(\342\200\224e/t) Transformation of the sum to an
\"integral.
(e) X = N/nQV Result of the integration (d)after
subsitution in (b).
(f) nQ = {Mi/27rfi2K/2 Definitionof the qaantam. concentration.
(g) H=
T!og(ll,'!!Q)
(h) F =J(/,V/i(.V,t,K)
=NiDog(tr/HU)
-1]
(i) p - -icFfcV)t,N-Nt/K
PROBLEMS
7. Derivative of Fermi-Dirac function. Show that\342\200\224cf/vc
evaluated at ihc
Fermi level e = ji has the value Dt)~'. Thus the lower the temperature, thesteeperthe slopeof the Fermi-Dirac function.
2. Symmetry of 'filled and vacant orbitah. Let e = /i + 5, so that /(c) ap-
appears as f{fi + i5). Show that
fin + 5) = 1 -'/(/< - 3). G4)
Thus the probability that an orbital 5 above the Fermslevel is occupied is equalto the probability anorbital6below the Fermi level is vacant. A vacant orbital is
sometimes known as a hole.
3. Distribution function for double occupancy siaiist'tcs. Let us imaginea new
mechanics in which the allowed occupancies of an orbital are0, I, and 2. The
values of the energy associated with these occupancies are assumed lo be 0, e,and It, respectively.
(a) Derive an expression for the ensemble average occupancy (N>, when the
system composedof this orbital is in thermal and diffusive contact with a
reservoir at temperature t and chemical potential/t.(b) Return now to the usual quantum mechanics,and derivean expression
for the ensemble average occupancy of an energy level which is doubly de-
degenerate; that is, two orbhals have the identical energy e. If both orbitals areoccupiedthe total energy is 2e.
4. Energy of gas of extreme relativistk particles. Extreme relativistic parti-particles have momenta p such that pc \302\273Me2, where M is the rest mass of theparticle.ThedeBroglierelation /. = h/p Tor the quantum wavelengthcontinuesto apply.Show that the mean energy per particle ofan extremerelativistic ideal
gas is 3t ift S pc,in contrastto ir Tor the nonrclativistic problem. (An interesting
variety of relativisticproblemsare discussed by E. Fermi in Notes on Thermo-ttynumU-s wul Suiiisiks, University of Chicago Press, 1966,paperback.)5, Integration of the tlwrntodynanric identity for an ideal gas. Fromthe thcr-
modynamic identity at constant number of particleswe have
+ ^. G5,X
Chapter 6; Ideal Gas
Show by integration thai for an ideal gas the entropy is
a =Cv logT + N log V + at , G6)
where crj is a constant, independentoft and V.
6. Entropy of mixing. Suppose that a system of jV atoms of type A is placedin diffusive contact with a system of N atomsof
typeB at the same temperature
and volume. Showthat after diffusive equilibrium is reached the total entropyis increased by 2N log 2. The eniropy increase2Nlog2isknown as [he entropy
ofmixing. if ihe atoms are identical{A s B), show that there is no increasein
entropy when diffusive contact is established. The difference in the results has
been called ihc Gibbs paradox.7. Relation of pressure and energy density, (a) Show ihat ihe average pres-
pressure in a system in thermal contact with a heat reservoir is given by
G7)
where thesumisoverallstates of the system, (b) Show for a gas offreeparticlesthat
;-r77 , G8)
as a result of the boundary conditionsof the problem. The result hoids equallywhether esrefersto a stateofJV noninteracting particles or to an orbital (c) Showthat for a gas of free npnrelativistic particles
p= WjlV , G9)
where U is the thermal average energy of the system.This result is not limited
to the classical regime; it holds equally Tor fermion and boson particles, aslongas they are nonrelativistic.
8. Time for a largefluctuation. We quoted Boitzmann to ihe effect that twocasesin a 0.1 liter container wiH unmix only in a lime enormously long compan^to 10\"\"'\"'years.
We shall investigate a related problem: we let a gas ofatomsofJHeoccupy a container of volume ofOJ liter at 300 K and a pressure of 1 aim,
and we ask how long it will be before the atoms assume a configuration in
which all are in one-half of the container..(a) Estimatethe number of states accessible to the system in this initial
condition.
(b) The gas is compressed isothermallyto a volume of 0.05 liter. How manystales are accessiblenow?
(c) For the system in the 0.1 liter container, estimate the value of the ratio
number of states for which all atoms are in one-half of the volume
number of states for which the atoms are anywhere in the volume'
(d) If the collision rate of an atom is % 10lC)s\"', what is Ihe total number ofcollisionsof all atoms in the system in a year? We use this as a crude estimateofthe frequency with which the slate of the systemchanges.
(e) Estimale the number of years you would expectto wait before all atoms
are in one-half of the volume, starting from ihc equilibriumconfiguration.9. Gasofatoms m'th iutermil degree of freedom. Consider an ideal man-
atomicgas,but one for which the atom has two internal energystales,one an
energy A above [he oiher. There ;ire H aiomsin volume V at temperature r.l-'nut ttic (a) chemical puk'iiljai; (h) live energy; (c) entropy; (t|> pic.^uic;(o)heatcapacity;it constantpressure.10-hentropic relations of ideal gas. (a) Show that the differential changes for
an tdcal gas in an isentropic process satisfy
(SO)
where y=
CpjCv\\ these relations apply even if the molecules have internai
degrees of freedom,(b)Theisentropicand isothermal bulk moduli are defined as
BB= ~V{cp!cV)a; Bs =* -V(dp/dV)t. (81)
Show that for an idea! gas Ba = yp; B, = p.The velocity of sound in a gas is
given byc - (Ba/p)\022; ihcre is very iittie heat transfer in a sound wave. For an
idealgas of molecules of mass M we have p = pi/M, so that c = (yr/'A/I'' .
Here p is the massdensity.
//. Convcaire iscntropk equilibrium of the atmosphere. The iower iO-15kmof the atmosphere\342\200\224the troposphere\342\200\224is often in a convccijve sieady stateat constant entropy, not constant temperature. \\\\\\ such equilibrium p\\\"' is
independent of altitude, where -,\342\226\240= CplCr. Use the condition of mechanical
equilibriumin a uniform gravitational field to: (a)Show that dTjdz = constant,where r is the altilude. This quantity, important in meteorology,is calledthe
dry adiabatic lapse rate. (Do not use ihe barometricpressurerelation that was
derived in Chapter 5 for an isothermal atmosphere.) (b) Estimate JT/i/r, inCC per km, Take y
= 7/5. (c) Show that p x p1,where p is the mass density.
Chapter6: IdealGas
If the actual temperature gradient is greater than the isentrapic gradient,the
atmosphere may be unstabic with respect to convection.
12. Idealgas in ttro dimensions, (a) Find ihe chemical potential of an idealmonatomicgas in two dimensions, with N atonis confined to a square of areaA = L2. The spin is zero, (b) Find an expression for the energy V of the gas.(c)Find an expression for the entropy a. The temperature is r.
13. Gibbi sum for ideal sas. {a) With the help of Zs ^(i>QV)sf,\\\\
from
Chapter 3, show that the Gibbs sum for an ideal gas of identical atoms
is 3-= exp(;.iiQK).(b) Show that Hie probability there arc ;V atoms in Hie gas
in volume V in diffusive coniact wiih a reservoir is
P(N}=<N/exp(-<N\302\273/N! , (82)
which is just ihe Poisson distribution function {Appendix C). Here <jY> is thethermal average number of atoms in ihe volume, which we have evaluated
previouslyas <N) =XVnQ, (c) Confirm that P[N) above satisfies
X P(N) = 1 and\302\243NP{N)
= <N>.
14, Ideal gas calculations. Considerone mole of an ideal monatomic gas ai300K and 1atm. First,let the gas expand isothermally and revcrstbly to twicethe initial volume; second, let Hits be followed by an iseniropicexpansionfrom
iwice io four limes ihe initial volume, (a) Howmuch heat (in joules) is added iothe gas in each of these two processes? (b) Whai is ihe temperature at ihe end ofthe second process?Supposethe first process is replaced by an irreversibleexpansioninto a vacuum, to a total volume twice the initial volume, (c) What
is ihe increase of emropy in ihe irreversible expansion, in joules per kelvin?
15, Dieselengine compression. A diesel engine is an internal combustionengine in which, fuel is sprayed into the cylinders after ihe air chargehas beenso highly compressed that it has attained a temperature sufficient !o tgntte the
fuel. Assume that the air in the cylindersiscompressedi sen tropically from an
initial temperature of 27\302\260C C00 K). If the compression ratio is 15,what is the
maximum temperature in \302\260Cto which the air is healed by the compression?Take y
= 1.4.
Chapter 7
Fermi and Bose Gases
FERMI GAS 183
Ground Slale of Fermi Gas in ThreeDimensions 185Density of Simcs 1S6
Heal Capacity of Electron Gas IS9FermiGasin Metals 194
White Dwarf Stars 196Nuclear Mailer
' 19S
BOSONGAS AND EINSTEIN CONDENSATION 199
Chemical Potential Near Absolute Zero 199Example: Spacing of Lowest and Second Lowest Orbilals
of FreeAtoms 201
Orbital Occupancy Versus Temperature 202EinsteinCondensationTemperature 205Liquid \"He 207
Phase Relations of Helium 210Quasiparticlesand Superfluidity, *He 212
Superfluid Phases of 3He 217
SUMMARY 217
PROBLEMS 218
1. Density of Orbitals in One and Two Dimensions 218
2. Energy of Relativists FermiGas 2183. Pressure and Entropy of Degenerate Fermi Gas 2194. ChemicalPotential Versus Temperature 219
5. Liquid'He as a Fermi Gas 2196. Mass-Radius Relationship for White Dwarfs 2197. Photon Condensation 2218. Energy, Heat Capacity, and Entropy of Degenerate Boson Gas 2219. Boson Gas in One Dimension 222
10. Relativistic White Dwarf Stars 222
!!. Fluctuationsin a FermiGas 22212.Fluctuations in a Bose.Gas 222
13. Chemical Potentia!VersusConcentration 22214.Two Orbital Boson System . 223
It is a fundamental resultofquantum theory that all particles, including atomsand molecules,areeither fermions or bosons. They behave alike in the classicalregimein which the concentration is small in comparison with the quantum
concentration,
(i \302\253ifQ
s (Mt/2tt/ijK'2. A)
Whenever n >iiq
the gas is said to be in the quantum regime and is called a
quaniutn gas. The difference in physical properties between a quantum gas oflemi ions and one of bosons is dramatic, and boihareunlike a gas in the classical
regime. A Fermi gas or liquid has a high kineticenergy, low heal capacity, low
magnetic susceptibility, low interparticlecollisionrate, and exerts a high
pressure on the container, even at absolute zero, A Bose gas or liquid has ahigh concentration of panicles in the ground orbital, and these particles\342\200\224
called the Bose condensate-\342\200\224may act as a superfluid, with practically zero
viscosity.
For many systems the concentration n is fixed, and the temperature is the
important variable.The quantum regime obtains when the temperature isbelow
t0 s {2nh2/M)n213, B)
defined by the condiiion n ~ iiq. A gas in the quantum regime with r \302\253ro
is often said to be a degenerate gas*.h was realized by Nernst that theentropy of a classical gas divergesas logr
as t -* 0. Quantum theory removes the difficulty:both fermion and boson
gases approach a unique ground stateas i -+ 0, so that the entropy goes tozero.We say that ilie entropy is squeezed out on cooliuga quantum gas (see
Problems 3 and S).in the classical regime{Chapter6) the thermal average number of particles
in an orbital of energy \302\243h given by
Wilh Ihe result for^ appropriateto this regime,
J{z)-
(n/HG)exp(-\302\243/i) , D)
with the usual choice of the originofeat zero for the energy of the lowest orbital.The form D) assures us that the average occupancy of any orbital is always
<it/nQt which is \302\253!,consistent with our original picture of the classicalregime.A fcrniion is any panicle\342\200\224elementary or composite\342\200\224with a half-iniegra!
spin. A fermion is limited by ihe Pauli exclusion principleto an orbital occu-
occupancy of 0 or I, with an average occupancy anywhere between these limits. Ai
low temperatures it is dear that many low-iyingorbiials will have one fermitm
in each orbital. At absolute zero all orbitals with 0 < e < eF will be occupiedwith / = I. Here ef is the energy belowwhich there are just enough orbitalsto hold the number ofparticles assigned to the system. This energyiscalledihe
Fermi t-iierfiy. Abo\\cr-fall orbiusls will have/ = Oiit r = 0. As t increases thedistribution function will develop a high energy mil, as in Figure 7.3.
Bosons have integral or zero spin. They may be elementary or composite;if composite, they must be made up of an even numberof elementary particles
if these have spin \\, for there is no way to arrive at an integer from un odd
immber of half-integers. The Pauli principle doesnot appiy to bosons, so there
is no limit on ihe occupancy of any orbital. At absolute zero the ground
orbital\342\200\224the orbital of lowest energy\342\200\224is occupied by all the particles in the
system.As the temperature is increased the lowest singleorbital ioscsitspopula-populationonly slowly, and each excited orbital\342\200\224any orbital of higher energy\342\204\242will
contain a relaiively small number of particles. We shall discuss this point
carefully. Above r = r0 the ground orbital losesitsspecialfeature, ;md its
occupancy becomes much like that of any low-lying excited orbital.
FERMI GAS
A Fermi gas is called degeuerate when the temperature is low in comparisonwith ihe Fermi energy. When the inequality i \302\253e^ is satisfied the orbitals of
energy lower than ihe Fermi energy ef will be almost entirely occupied, and the
orbitals of higher energy will be almost entirely vacant. An orbital is occupied
fully when it contains one fermton. A Fermi gas is said to be noudegeneratewlicrt the temperature ishigli compared with the Fermi energy, as in the classical
regime treated in Chapter 6.
The unportam applications of tlie theory of degenerateFermi gases include
conduction electrons in metals, the wliiie dwarf stars; liquid 3He; and nuclearmatter. The most striking property of 3 fermion gas is the high kinetic energy
\342\226\240|60
\"a
Fermi level f, for16eleclrons;in Hie
gTOUnd slalc thelouesi eight levels(!6orbilals) areoccupied
(a)
Figure 7.1 (a) The energiesof the orbhals n = i,2 10for an etecfron
confined to a fine of lenglh L. Each levelcorrespondsto two orbitafs, one for
spin up and one for spin down, (b) The ground siaicof a system of t6 electrons.
Orbitals above the shaded region are vacant in the ground Slate.
of the ground state of the system at absolutezero.Suppose that it Is necessaryto accommodate N nomnteracting electronstn a length L in one dimension.What orbitals will be occupied in the ground state of the N electron system?In a one-dtmenstonalcrystal the quantum number of a free electron orbitalofform stn{fJ7ix/L) is a positive integer it, supplemented by the spin quantumnumber ms \342\200\224
\302\261j- for spin up or spin down.If the system
has 8 electrons, then in the ground state the orbttais with
(i = 1, 2, 3, 4 and with in,=
\302\261\302\243are fiHed, and the orbitais of higher n are
empty.Anyother arrangement gives a higher energy. To constructthe ground
state we fill tire orbitals starting from it = 1 at the botiom, and we continue
filling higher orbitals with electrons until all N electrons are accommodated.
The orbitais that are filled in the ground state of a systemof 16electrons are
shown in Figure 7.1.
Ground Sttite of Fermi G&s in Thtee Dimensions
Ground Slate of Fermi Gas in Three Dimensions
Let the system be a eube of sideL and volume V = L3. The orbilais have the
form of C-58) and their energy is given by C.59). The Fermi energy Efis the
energy of the highest filled orbiiai at absolutezero;it is determined by the
requirement that the system in the ground stale Iioid N electrons, with each
orbital filled wiih one electron up to the energy
E)<5)
Here tiF is the radius of a sphere (tn the space of the integers nx, tty, \302\273r)ihai
separates filled and empty orbitals. For the systemto hold N electrons the
orbitais must be filled up to nF determined by
yn, = {lNfn)ll\\ F)
The factor 2 arises because an electron has two possiblespin orientations.Thefactor| arises because only triplets nx, ny, nz in the positive octant of the spherein n space are to be counted. The volumeof the sphere is 4nn//3. We maythen write E) as
This relates the Fermi energy to the electron concentration N/V= n. The
so-called \"Fermi temperature\"tf is definedast> s ef.The total energy of the system in the ground state is
[/\342\200\236= 2
\302\243e,= 2 x | x 471
\\^reln n2 eB *=--1 j j P' (in \302\273a, (S)
with En= (h2/2m)(n}t/LJ. In (8) and (9),n is an integerandisnot N/V.Consistent
with F), we have let
21 (\342\226\240\342\226\240\342\226\240)->2(fcX4n)JdrtirV-)
(9)
in the conversion of (he sum into an integral.Integration of (8) gives the total
ground state kineticenergy:
l0m\\L
0 10 20 30 40 50 60 70 80 90 100Volume, in cm3
Figure 7,2 Total ground stale energy Uo of one mole ofelecirons, versus volume.
using E) and F). The average kinetjcenergy per particle is UJN and is f ofthe Fermi
energy cF. At constant N the energy increasesas the volume decreases
(Figure 7.2), so that the Fermi energy gives a repulsive contribution to the
binding of any material; in most metals and in white dwarf and neutronstarsit is the most important repulsive interaction. That is, the Fermienergy tends
to increase the volume. It is balanced in metals by the Coulomb iiltractionbetween decuoos and ionsand in she stars by gniviimicurjS attraction.
Density of States
Thermalaverages for independent particle problems have the form
where rr denotes the quantum orbital; XR is the value of the quantity A' in
the orbita! n; and f(t:a>T,!i.)is the thermal averageoccupancy,calledthe dis-
distribution function, of the orbital n. We often express <.Y) as an integral over theorbital energy
\302\243.Then A1) becomes
A2)
Density ofStates
wherethe sum over orbitais has been transformed to an integralby the sub-
substitution
X (\342\226\240\342\226\240\342\226\240)->Jrfs
*>(eH' \342\226\240\342\226\240)\342\226\240 A3)
Here <D(\302\243}^\302\243is the number of orbitais of energy betweent and t 4- dt. The
quantity 'D(e)is nearly always cailed the density of slates, although it is more
accurate to call it the density of orbitais because it refers Jo the solutions of a
one particleproblemand not to the states of the N particlesystem.Consideran example of Ihe calculalion of \302\251(e). We see from G) thai the
number N of freeelectronorbitaisofenergy iess than or equal lo some e is
N{e)=>(V/in2)BM/h2)il2til2 , A4)
for volume V. Take the iogarithm of both sides:
logN s= flogs + constant, A5)
and take differentials of log N and loge:
The quantity dN = (iN/lsjdt is the number of orbifals of energy between c
and e + (/e,so that
C(e) ~dNltlt = ZN{t)llt A7)
is the density of orbitais. The two spin orientations of an electronhave been
counted throughout this derivation because they were counted in F). We can
write \302\251(e)as a function of e alone because
JV(\302\243)A= (V/3n2){2m/h2)y'2F.in , (IS)
from A4).ThenC7) becomes
A9i
Chapter 7: Fermi and BaseGm
Figure 7.3 Density of orbitals as a function
of energy, for a free electrongas in three
dimensions. The dashed curve represents the
densiiy /(eVD(e) of occupied orbilals ai a finile
temperature, bm such thai r is small in
comparison wiih cF. The shaded area rcprcsetusthe occupied orbiiah ai absolute zero.
Energy, e *\"\342\226\240
When multiplied by the disiribulion function (Figure 6.3),the densiiy oforbilals*D(e) becomes <D(\302\243)/{e), ihe dens'ny of occupied orbilals (Figure 7.3).Thetotal number of electrons in a sysiem may now be written as
B0}
where f{t) is Ihe Fcrmi-Dirac distributionfunction described in Chapier 6. In
probkms where we know the total number of patiides, we determine }t by
requiring ihal the total number of particlescalculatedfrom B0) be equal to Ehe
correct value. The total kinetic energy of iheelectronsis
V = B1)
If Jhe sysiem is in the ground stale, all orbitais are filled up to the energy \302\243F,
above which they are vacani. The number ofekcirons isequalto
B2)
and ihe energy is
B3)
Heat Capacityof Electron Gas
Heat Capacity of Eleciron Gas
We derive a quanliiaiive expression for the heal capacity ofa degeneraie Fermi
gas of electrons in three dimensions. The calcufaiionis perhapsthe most im-
impressive accomplishment of the theory of (he degenerateFermi gas. For an
ideal monalomic gas the heat capacityis \302\247W,bui for elecirons in a meial very
much lower values are found. The calculation that follows gives excellent agree-
agreement wiih theexperimemal results. The increase in (he total energy of a systemof N electronswhen healed from 0 10 i is denoted by AU ~ U{x) \342\200\224
L'(Q),
whence
AU =JjVcrfWO:)
-J0\"<fcrf>(E).
B4)
Here f{c) is (he Fermi-Dirac fimetion, and O(c) is the number of orbilals perunit energy range. We multiply the identity
N \"Jo\342\204\242
</e/WW ~J^W
B5*
by tf to obtain
{jo+
ifjAttrfteW- f/dttfiiz). B6)
We use B6) fo rewrite B4) as
AU =JJ Je(b - tr\\n^y0{c)+ fc'M*r - $0 ~
fW&iz)- W)
The first integral on the right-hand side of B7) gives the energy neededlo take
electrons from ef to the orbitals of energy \302\243> \302\243f,and the second integral givesihe energy neededto bring ihe eiecirons to ef from orbifals below ef. Bothcontributionsto !he energy are positive. The product f(c)'D{c)de in the firsi
integral is the number of electrons elevated to orbitals in ihe energy range
dc at an energye.Thefactor[1 \342\200\224/(\302\243}]
in the second integral is ihe probabilitythai an elecironhas been removed from an orbilal c. The function A (/is plotted
in Figure 7.4. In Figure 7.5we plot ihe Fermi-Dirac distribution function versus
\302\243,for six values of the lemperalure. The electronconcenlrationof the Fernii
gas was laken such that tfjkB ~ 50000K, characteristic of ihe conduction
electrons in a meial.
The heat capacity of the electron gas is found on differentiating AU with
respect to r. The only temperature-dependentterns in B7) is /(e), whence we
Figure 7.4 Temperature dependence of the energy of \302\243\342\226\240
a noninteracting fermian gas in three dimensions. The S; 05energy is plotted in normalized form as AU/NeT, ^uhere N is the number ofdecirons.The temperature
is ploued as xjtF.
0.4 0.6
3 4 5 6 7
e/A'b, in uniis of 104K
Figure 7.5 Fermi-Dirac disiribuiion function ai various lempcraiures, for
TF =\302\243F/kB
= 50000 K. The resuhs apply 10 a gas in ihreedimensions.Thnumber of particlesis constani, independent of temperature. The chemicaleach temperature was calculated with the help of Eq.B0)and may be read
graph as the energy at which/ = J. Courtesyof B. Feldman.
Heat Capacity of ElectronGas
-Region of UcMsieraicquantum gas
/
/ \342\200\224-^ Rc\302\253ion
\\
of classic-,
Figure 7.6 Piol of ihe chemicalpoiemiai /i versus temperature T for a
gas of noninteraaing fcrmions in ihrcc c!iincisions. For convenience in
plotting, ihc units of ji and i arc 0.763cf.
can group terms to obtain
B8)
At (he tempera lures of interest in nseiats x/eF < 0.01,and we see from Figure 7.5
that the derivative df/dx is large only at energies near er. It is a goodapproxi-
approximation to evaluate the density of orbitais <O(e) at sf and take it outside of the
integral:
Ccl S B9)
Examinationof the graphs hi Figures 7.6 and 7.7 of the variation of/i with x
suggests that when t \302\253\302\243fwe ignore the temperature dependence of thechemica!potential ji in the Fermi-Dtrac distribution function and replace ;i by the
constant cF. We have then:
C0)
Chapter 7; Ftrmi and Bost Gas
Figure 1.1 V&riaiiors with temperature of shechemical potential ji, for free electron Fermigasesin one and three dimensions. In commonmeials t/nF a= 0.01 at room (empcraiure, sothai ;i is closely equal lo cF.Thesecurves were
calculated from series expansions of the integral
for the number of particles in ihe system.
We set
x a (e.- e,)/t ,
and it follows from B9) and C0) that\"
We may safely replace the lower limit by -co because the factor
integrand is already negligibleat x = -ef/r if we are concerned
temperatures such that cF/x -~ f 00 or more. The integral* becomes
+ iI~
3
\342\200\242The inicgta! is not demciuary, but may be cvafua^d from ihe n
p,-V it2
ondilTeremhuion of bsjih sidesuiih rcspcci lo ihc parameter a.
Heat Capacity of Electron Gas
whence we have for she heat capacityof an electron gas, when r \302\253xf,
Cel - WUEfU. I
'
C4)
In conventional units,
Ctl
We found that the density of orbitajs at the Fermi energy i,
\302\253(\302\243,)- 3N/lcf = 3iV/2tf
C5)
06)
for a free electron gas, with xF s eF. Do not be deceivedby the notation rF:
if is not the temperature ofthe Fermi gas, hut only a convenient reference point.For r \302\253xF the gas is degenerate; for r \302\273if (he gas is in the classical regime.ThusC4)becomes
C7)
C8)
in conventional units there is an extra factor kBl so that
C\\,
where A'flTFs sF. Again, TF is not an actual temperature,but only a reference
point.We can give a physical explanationof the form of the result {37). When the
specimen is heatedfromabsolutezero,chiefly those electrons in states within
an energy range r ofthe Fermi level are excited thermally, because the FD
distribution function is affected over a region of the order of r in width, illus-illustrated by Figures 7.3 and 7.5. Thus the number ofexcitedelectronsisofthe order
ofA'i/\302\243F, and each of these has its energy increasedapproximatelyby
r. The
total electronic thermal energy is thereforeofthe order of Uci as Ni2/eF. Thus
the electronic contributionto the heat capacityis given by
A'r/rF C9)
which is directly proportional to t, in agreement with the exact result C4) andwith the experimental results.
Chapter 7: Fctmi and Bos
Table l.\\ Cuiculaied Fermi energy parameters for free eleciroi
Li
NaKRbCsCuAg
Au
Comiuciion
eieclron
concemralion
NiV. in cm'3
4.6 x 10\"2.51.341.0S0.568.505.76
5.90
Velocity
SV, in cm s~'
1.3 x 10\"
1.1
0.85
0.79
0.73
1.561.381.39
Fermi
energy
\302\243F,in eV
4.7
3.S2t1.81.57.05.55.5
Fermi
iempcratur
T., = [. k,inK
5.5 x 10J3.72.42.11.88.2646.4
Fermi Gas In Metals
The alkali metals and copper,silver,and gold have one valence electron peratom, and the valence electron becomes the conduction electron in the metal.
Thus the concentration of conduction electrons is equa! to the concentration
of atoms, which may be evaluated eilher from the densiiy and ihc atomic
weight or from the crystal lattice dimensions.
If the conduction electronsact as a free fermion gas, the value of she Fermienergy eF may be calculated from G):
ef = {hl12m)Cn2uI!\\ D0)
Valuesof n and of ef are given in Table 7.1 and in Figure7.8.The electron
velocity vF at the Fermi surface is also given tn (he \\ab!e; it is defined so that
the kinetic energy is equal to ef:
^ttuy3-
\302\243f. HO
where nt is the mass of the electron. The values of (he Fermi \\emperatureTF ~
eFjkBfor ordinary metals are of the order of 5 x l04K,so that the assumption
T \302\253TF used in (he derivation of C5) is an excellentapproxima(ionat room
temperature and below.
The heat capacity of manymetals at constant volume may be written as
the sum of an electronic contribution and a lattice vibration contribution. At
Sow temperatures the sum has the form
Cv ~ yi + At3 D2)
JRb
Na
a functio
monovaienl mclals. The siraighl line is dra
!brEf ^ 5.835 x 10\023'n1/J ergs, whh iiin
5 10\" 2 5 10\302\260
Eleciron concentration, in cm~3
Figure 7.9 Expcrimcnlalheal capacity values for polnssium, plolled as C/TvTK After W. H. Lien and N. E. PhiJHps, Phys. Rev. 133, AI37O A964}.
where y and A are constants characteristic of the material. Here ys jn2N/iF
from C7), and the lattice vibrationterm-4i3was discussed in Chapter 4, The
electronic term is linear in t and is dominant at sufficiently iow temperatures.Ii is helpful to display the experimental values of the heat capacityfor a given
material as a plot of Cvjx versus t2:
Cy/t~ y + Ax2 D3)
for then the pointsshould lieon a straight line. The intercept at t \342\200\2240 gives
the value of y. Sucha plot isshownforpotassiumin Figure 7.9. Observed values
of y are given in Tables 7.2 and 7.3.
Table 7.2 Experiment! and free eteciron eSecironicheat capacities of monovalent metals
y Cexp),
mJmol\"'K\"
e electron},1
y/y0
Li
Na
KRbCs
CuAg
Au
oni:
iUlU
1.631.382.082.413.200.6950.646
0.729
The values of \342\226\240/nud yo arc in i
u: Oluilcsy of N. li. 1'hillim.
0.751.141.691972.360.500.65
0.65
2.17
1.211.231.221.351.391.001.13
Table 7.3 ExpenmenUt values of declronic heat capaciiyconsiain y of mcials
Li1.63
Na1.38
K2.08
Rb
2.41
Cs
[3.20
Be0.17
mb1.3
Ca2.9
Sr3.6
Ba
2.7
Sc
107
Y10.2
La10.
Ti \021
3.35
2r
2.80
Ht2.16
V
9.26
Nb
7.79
T*5.9
Crt.40
Mo
2.0
W
1.3
Mr9.2
Tc_
Re
2.3
sOTE;Thc value of y is in
4.73
Rh4.9
lr3.1
Ni
7.02
P<!
9.42
Pt6.8
Cu0.695
Ae
0.646
Au
0.729
I
Zn
0.64
a0.638
He1.79
Al
1.35
Cra
0.596
In1.69
Tl1.47
C
Si
Gc
Sn1.78
jpb2.98
N
P
As
0.19
Sb0.11
Bio.oos
,s fuiniihed by R E Phillips and N. Pear
White Dwarf Stars
White dwarf slars have masses comparable to that of the Sun. The mass andradius of Itic Sun arc
Q* 2.0 x 10!3 g; Ko = 7.0 x 10'\302\260cm. D4)
The radii of white dwarfs are very small, perhaps O.Oi that of the Sun. The
densityof the Sun, which is a normal star, is of the order of 1 gem\023, like that
of water on the Earth. Thedensitiesofwhite dwarfs are exceedingly high, of the
White Dwarf Stars
order oflO'1 to 107gem\023. Atoms under the densities prevalent in white dwarfsare entirely ionized into nuclei and free electrons, and the electrongas is adegenerategas,as will be shown below.
The companion of Sinus was the first white dwarf to be discovered. In 1844Besselobservedthat the path of the star Siritis oscillatedslightly about a straight
lincasifithadan invisible companion.The companion,SiriusB,was discovered
near its predicted position by Clark in 1862.ThemassofSiri us B was determined
to be 2.0 x 3O33 g by measurements on the orbits. The radius of Sirius B ts
estimated as 2 x 109cm by a comparison of the surface temperature and the
radiant energy flux, using the properties of thermal radiant energy developedin Chapter 4.
The mass and radius of Sirius B lc;id to the mean density
D5)
This extraordinarilyhigh density was appraised by Eddington in 1926 in the
following words; \"Apart from the incredibility of the result, there was no
particular reason to view the calculation with suspicion.\" Other white dwarfs
have higher densities; that named Van Maanen No. 2 has a mean density
100 times higher.
Hydrogen atoms at a density of 106gcni~3have a volume per atom equal \\o
=s 2 x iO~30cni3pcratornA~
(I06molcm\023){6 a 10\" atoms mol\021)
or 2 x 10~6A3peratom. The average nearest-neighborseparationis ihen of
the order of 0.01 A, as compared with the internuclcar separation of 0.74 A in a
molecule of hydrogen. Under conditions of such high density the atomic
electrons are no longer attachedto individual nuclei. The electrons are ionized
and form an electron gas. The matter in the white dwarfs is held togetherby
graviiational attraction, which is the binding forcein all stars.In the interior of white dwarf stars* the electron gas is degenerate;the
temperature is much less than the Fermienergy ef. The Fermi energy of au
electron gas at a concentrationof 1 x 103Oelcetronscm~3 is given by
cr * (h2/2i}i){3n2nI13as 0.5 x 10~6erg as i x 10s eV , D6)
'A \302\243ooddiscussion of while dwaif siais is given by W. K Rose, Aitropkvsics, Hotf. R
WinHon, 1973.
\"able 7.4 Fermi energy of degenerate fsmiion g
characierisiic values)
of mailer Particles Tf, in K
Liquid 3Hc atoms 0.3Metal demons 5 x tQ1
While dwarf stars electrons 3 \302\253!09
Nuclear matter nuctcons 3 x \302\2730u
Ncujron stars neutrons 3 x 10!I
about 10' higherthan in a typieal metal. The Fermi temperature zFikB of the
electrons is =s3 x 109K,as in Table 7,4. The actual temperature m the interiorof a while dwarf is believed to be of the order of I0\021 K. The electron gas in the
interior of a white dwarf is highly degenerate because the thermal energy is muchlowerthan the Fermi energy.
Are the electron energies in the relativistie regime?This questionarisesbecause our theory of the Fermi gas has used the nonrelativistic expression
p2/2m for the kinetic energy of an electronof momentump.The energy equi-
equivalence of the rest mass of an electronis
\302\243Q= me2 \302\273A x 10\"\"g)C x I010cins-!): * 1 . D7)
This energy is of the same order as the Fermi energy D6).Thus relativistic
effects will be significant, but not dominant. At higher densities the Fermi gasis reiativistic.
NuelearMatter
We consider the state of matter within nuclei. The neutronsand protons of
which nuclear matter is composed form a degenerate fermton gas, at least
qualiiaiively.We estimate here the Fermi energy of tbenucleongas;The radiusof a nucleus that contains A nucleons is given by the empiricalrelation
R a A.3 x ltT13cm) x Al!\\ D8)
Accordingto this relation the average volume per particle is constant,for the
volume goes as R3, which is proportional to A. The concentration of nucleons
b nuclear matter is
?S 0.11 x 1039cm-3 , D9)
Chemical Potential Near Absolute Zero
about 103 times higher than the concentration of nudeons in a white dwarfStar.Neutrons and protons are not identical particles. The Fermi energy ofthe neutronsneednot equal the Fermi energy of the protons. The concentra-concentrationof one Or the oilier, but not both, enters the familiar relation
\302\243,=\342\204\242C>r'\302\273)M E0)
For simplicity let us suppose that the numberof protonsis equal to the
number of neutrons. Theit
'W>n>* \302\253\342\200\236\302\253\342\200\236\342\200\236(,\302\253a 0.05 x 1039 cm\023 , {51}
as obtained from D9) ondividing by 2. The Fermi energy is
\302\243f\302\243C.17 x 10\023>1/3a;0.43 x 10\024erg \302\25327 Mev. E2)
The average kinetic energyof a particlein a degenerate Fermi gas is -J of ihe
Fermi energy, so that in nuclear matter the averagekineticenergyis 16 Mev
per nucleon.
BOSON GAS AND EINSTEIN CONDENSATION
A very remarkable effect occurs in a gas of nonintcmcting bosons at a certaintransitiontemperature,below which a substantial fraction of the total numberofparticlesin the system will occupy the single orbital of lowestenergy, called
the ground orbital. Any other orbital, including the orbital of second lowestenergy, at the same temperature will be occupied by a relatively negligible
number of particles. The total occupancyof all orbitalswill always be equal to
the specified number of particlesin the system. The ground-orbital effect ts
called the Einstein condensation.
There would be nothing surprising to us in this result for the ground stale
ccupaney if it were valid only below I(T14K.. This temperature is comparable
whh the energy spacing between the lowestand next lowest orbitals in a systemof volume 1cm3,as we show below. But the Eitistein condensation temperaturefor a gas of fictitious noninteracting helium atoms at the observeddensity
of
liquid helium is very much higher, about 3K. Helium is the most familiar
example of Einsteincondensationin action.
Chemical Potential Near Absolute Zero
The key to the Einstein condensation is the behavior of the chemicalpotentialof a bosonsystem at low temperatures. The chemical potential is responsible
occ
for the apparent stabilization of a large population of particlesin the ground
orbital. We consider a system composedof a hrgc number N of nonintcractmgbosons.When the system is at absolute zero all particlesoccupythe lowest-
energy orbital and the system is in the state of minimum energy. It is ceriainiynoi surprising dim at i ~ 0 ail particles should be in the orbital of lowestenergy. We can show that a substantial fraction remains in the ground orbila!
at low, although experimentally obtainable,leniperaturcs.If we put the energy of the ground orbitalat zero on our energy scale, then
from the Bose-Einstein distribution function
<53)
e obtain the occupancy of the ground orbitalat e = 0as
When i\342\226\240-+0 the occupancy of the ground orbital becomesequalto the total
number of particles in the system, so ihal
Herewe have made use of the series expansion cxp{\342\200\224x)
*s I ~ x + \342\200\242\342\226\240\342\200\242.We
know dial .v, which hji/x, must besmallin comparison wiih unity, for otherwise
the total number of particlesN couldnot be large. From this result we find
E5)
asT -.0. ForN = 1022at T = IK, we have /i s -1.4 x lO'38erg.We noic
fromE5) that
E6)
as i ~+ 0. The chemical potential in a boson system must always be lower in
energy than the ground state orbiral, in order tirar rhc occupancy of every orbital
b\302\243non-nesative.
Chemical Potential Near Absolute Zcr
Cxampte;Spacingoflonat and second lowest ttrbiiuh of free atoms. The energy of an
orbital of an atom free to move in a cube of volume (' =\302\273/-1 is
where n,, n>4 \302\273,are positive integers. The energy e(I 11)of ihe lowest orbital is
+ 1 + 1) , E8)
ind the energy eB1 i) of one of the set of next lowest orbitais is
+ 1+1). E9)W
The lowest excitation energy of the atom is
As =*\302\243BH)-e(III) =* 3 x \342\200\224
(-). F0)
lfA/{4tie} - 6.6 x lO'^gandi, = I cm,
Ae = C){8.4 x IO-3!)(9,S6) = 2.4Sx NT30erg. F1)
In temperature units, Ae/*8 - 1.S0x KT1*K.This spiining is extremely small, and it is tiiRtcuU to conceive that it can play unimportant
part in a physical problem even at the lovvest reasonably accessible temperatures such as1 ijtK, whicli is I0\021 K. However, at the I mK temperature {55} gives )i ^ -1.4 xiO\0211 erg for N = \\0:i aiouts, referred to the orbital ES) as the zero of energy.Titus /i is
much closer to the ground orbjia! than is the nexl lowest orbital E9), and cxp{[t(J 11)-/i]/r} is ntucit closer io l than is ap{[t[2li}~n\\h}t so that t(Ill) dominates the dis-distribution function.
The Boltnnann factor exp{~A\302\243/i) at 1 mK. is
exp(-1.8 x 10-\") s I - 1.Sx I0\0211 , F2)
which is essentially unity. By D) we would expeel that even if/t *\302\253cilte occupancy of tlie
first excited orbit;il would only be of the order of 1. However, ihe nosc-Eiiliit-'itidis.tr ibutioii
gi\\cs an entirely dilfctcni vuluti for the occupancy of ihe first exched ofhii;il:
because Ae \302\273p.. Thus the occupation of the fust exciied orbiial ai 1 mK i:
F4)
so thai the fraction of the N particles thai are in this orbital is/.iV * 5 x 10i0v 10ir =5 x 10\" 13t which is very small. We seethai the occupancy of the first exciied oibiialai low
temperatures is [datively very much lower than would be expeciedat first sight from the
simple Boltzmann factor F2). The Bose-Einstein distribution is quiie strange; ii favors a
situation in which ihe greatest part of tlie population is left in the ground orbiiai at sufi'l-
cienily low temperatures. The particles in ihe ground orbiial, as long as iheir number is \302\2731,
are called the Bose-Einsicin condensate. The atoms in ihe condensate act quiie differently
from the atoms in excited stales.
How do we understand the existenceof the condensate? Suppose ihe aioms weregoverned by ihe Planck distribution (Chapter 4), which makes no provision for holdingconsiani the loial number of particles; instead, the thermal average number of photonsincreases wiih temperature ai i\\as found in Problem 4.1. If the lav. s of nature restricted iheloiiil *iumbcr of photons to *i vliIuc $i we wotild suy thitt the i^rousid orbital of ii\\c plioion
gas contained the difference No = S* ~ N(r) between the number aiioued and ihe number
thermally excited. The ,V0 noncxciK'd photons would be described as condensco1 into tjic
ground orbital, but A'o becomes essentially zero at a temperature i, such that ;ill ,V photonsarc excited. There is no actual constraint on the totiil number of photons; however, thereis a constrain!on the total number A' of material bosons,sucli as MIe atoms, in n sysiem.
This consiniint is tht' origin of the condensation into the ground oibiul. The diiTcrctKcbetween the Planck distribution and the Bosc-Einstein distribution h lliat the laner will
conserve the tot;tl number of particles, independent of icmperaiurc,so (hat none\\ciicd
atoms are really in the ground ilalc condensate.
Orbital Occupancy VersusTemperatureWe saw in A9) that the number of free particle orbitalsperunit energy range is
fora particle of spin zero.Thetotal number of atoms ofheliutn-4 in the ground
and exciied orbitals is given by the sum of the occupancies of all orbitals:
N . F6)
We have separated the sum over n into two parts. Here N0(t) has been written
for /@,t), the number of atoms in the ground orbital at temperature t. The
integral in F6) gives the number of atoms NJ,i) in all excitedorbitals,with
/(\342\200\242.t
\\
o!
\\r = 0.5
\\
-\342\200\224\342\200\224
\342\200\224\342\226\240\342\200\224__
Figure 7-10 Plot of Ihe boson distribution funciion for two temperatures, wilh sufficient
particles presentto ensure). a I.The integral ofthe distribution times the density of
slates gives the number N. of particles in exciied orbitals; the rest ofthe particlespresent
arc condensed into the ground slate orbital. The value of No is loo large to be shown on
Hie plot.
/(e,i) as the Bose-Einstetndistributionfunction. The integral gives only the
number of atoms in excited orbttals and excludes the atoms in the groundorbital,becausethe function D(t) is zero at e = 0.To count the atoms correctly
we must count separately the occupancyNaof the orbital with e = 0. Although
only a singleorbitalis involved, the value of No may be very large in a gas of
bosons. We shall call NQ the number of atoms in the condensed phase and Nt
the number of atoms in the normal phase. The whole secret of the result wluch
follows is that at low temperatures the chemicalpotentialp isvery much closer
in energy to the ground state orbital than the first excited orbital is to the
ground state orbital.ThisCloseness of p to the ground orbital loads mostofthe population of the system into the ground orbital(Figure7.10).
Chapter 7: Fermi and Base Gases
The Bose-Einstein distribution function when written for the orbilal ate - 0 is
NM=V^~, F7)
as in E4), where X will depend on the temperature x. The numberof particlesin all excited orbitals increases as tm:
or, with x s e/z.
Nolice the facfor zil2 which gives Ihc temperature dependence of Nc.At sufficiently low temperatures the number of particles in Ihe ground si ale
wiUbe a very large number. EqualionF7)tellsus that / must be very close tounity whenever No is \302\2731. Then / is very accurately constant, becausea mac-macroscopic value of ;V0 forces /. to be closeto unity. The condition for the validityof the calculation is that No
\302\273I, and it is not required that Ne\302\253 N, When
g s\302\273i in the integrand, the value of the integrandis insensitive to small devia-
deviations of a from 1, so that we can set /. -' 1 tn F8), although not in F7).
The value of the integral*in F8) is, when ). = 1,
The infiniic
>\342\226\240= U1 10 gi
it ton Temperature
Thus ihe numberofatomsin excited states ts
.. \\.IO6VB\\H\\3;:iG0)
where /iQ = (Af r/2rr/i2)a/2 is again thequantum concentration. We divide Nf by
N to obtain the fractionofatomsin excited orbitals:
N./N = 2.6l2nqV/N = 2.6l2na,'n. G1)
The value X = 1 or 1 \342\200\224\\/N which fed to G!) is valid as long as a large
number of atoms are in the ground state. A!I particles have to be in some
orbital, either in an excited orbital or in the ground orbital- The number in
excited Orbitais is relatively insensitive to small changes in X. but the rest of
tlie particles have to be in the ground orbital.Toassure this we must take /.
very close to 1 as longas NQ is a large number. Even 103 is a large numberforihc occupancyof an orbital. Yet witliin Ar/rE = 10\"fi of the transition,where
r\302\243is defined by {72) below, [he occupancy of the ground orbilal is > !015 a loinscm\023 at the concentration of liquid 4He. Thus our argument is highly accurate
at Ar/r\302\243= 10~5.
Einstein Condensation Tcmperaiure
We define Ihe Einstein condensation temperature* i\302\243as (he (cmperaiure for
which the number of aionis in excited states is equal to (he total numberofatoms.That is, A^frJ
\302\253N. Above r\302\243ilie occupancy of the ground orbital isnot a macroscopicnumber;below n the occupancy is macroscopic. From G0)witli A7 for Ne we find for the condensation temperature
M \\2.6I2k
Now Gt) rimy be written as
G2)
G3)
wlicrcjV is tlic tola! number of atoms. The numberofatoms in excited orNuils
v;irtes ;is ii:2 a I Ictuperiitures below if;, as sliown in Figure 7.11.Tlicuik'iilaledvalue of T^foralomsof 4Hc is a=3K.
in, Atademic dcr W'issenschaficn, Berlin, Siuunesbcricliie 152-1,261;1925.3.
Chapter 7: Fermi aid Hose G:
1.0
0.8t
\\
Superfluid
component'
ormal flui
omponen
y
\\
/
/
\"n
\\
/
\\|0 0.2 0.8 i.o0.4 0,6
Figure7\302\273tl Condensed boson gas: tempcraiuce dependenceof the proportion No/N ofaioms in ihe ground orbiial andof ihc pfoponion NJN of aloms in all exciled orbilals. We
have labeledihe two components as normal and superftuid
!o agree wilh the cusiomary descriptionof liquid helium.
The slopes of all Hucccurves arc intended !o be zero at x = 0.
The number of particles in the ground orbital is found from {73):
No= N - Ne = N[l -
{x/x\302\243?ir\\. G4)
We note that N may be of the order of 102\\Forx even slightly less than te a
large numberofparticleswill be in the ground orbital, as we see in Figure7.11.We have said that the particles in the ground orbitalbelowt form the condensed
phase or the superfluid phase.The condensationtemperature in k'elvin is given by the numerical relation
M) . G5)
where VM is the molar volume in cm3 mol\021 and M is the molecular weight.For liquidhelium Vu~ 27.6cm3mol\021 and M = 4; thus TE = 3.1K.
Liquid4 He
Liquid 4He
The calculated tempera! ure of 3 K issuggesiivelycloselo j he actual tempera Jure
of 2.17K a! whicha transitionlo a new stale of matter is observed to Jakeplacein liquid helium (Figure 7.12). We believe that in liquid 4Hebelow2.17K there
is a condensation of a substantial fractionofiheatomsof4He inlo the ground
orbiial of jhe system. This is different from the condensation in coordinate
space that occurs iit the condensation of a gas to a liquid. Evidently ihe j'nicr-
alomic forces that lead to ihe liquefactionof4Heat 4.2 K under a pressure ofone atmosphereare tooweak to destroy the major effects of the bosoncon-condensation at 2.17 K. In this respect tlte liquid behavesas a gas. The condensa-
condensationinto the ground orbital is certainly connected with the properties of bosons.
2.5
\"\302\253,2.0
s t.o
X0.5
0
y
1.6 1.8 2.0Tempera!
u;
2.2 2.4 2.6
Figure 7.12 Heat capacity of liquid *He. The sharppeak near 2.17 K is evidence of an important transition
in the nature of the liquid. The viscosity of the liquidabove the transition temperature is typical of normal
liquids, whereas the viscosity belowtlic transition as
determined by rate of flow through narrow slits is
vanishingly small, at least I06 times smaller than the
viscosity above the transition. The transition is often
called a lambda transition merely becauseof tlte shape
of the graph. After Kccsom et al.
Chapter It Fermi twd Base Gases
The condensation is normally not permitted for fennions, but pairs of fermions
may act as bosons,as in the superconductivity of electron pairs (Cooper pairs)in metals.A different type of transition to complex phases with superflutd
properties has been observed in liquid 3He below 3mK. Atoms of 3He havespin \\ and are fermions, but pairs of 5Hcatomsactasbosons.
We can give several arguments in support of our view of liquid helium as a
gas of noninteracling particles.At first sight this is a drastic oversimplificationof fite
problem, bur there are some important features of liquid helium for
which the view is correct.
{a) The molar volume of liquid 4He at absolute zero is 3.1timesthe volume[hat we calculate from the known interactions ofhelium atoms.Titeinteraction
forces between pairs of helium atoms are we!!known experimentally aud
theoreticaily, and from these forces by standard elementary methods of solid
state physics we can calculate the equilibrium volume of a static lattice ofhelium atoms. In a typical calculation we find the molar volume to be 9 cm3mo!~', ascompared with the observed 27.6 cm3 mo!\"l. Thus thekineticmotionof the helium atoms has a large elTccton ihe liquid siale and leads to an ex-
expanded structure tn which the aloms to a certain extentcan move freely over
appreciable distances. We can say ihat ihe quantum zero-point motionisresponsiblefor the expansion of ifie inoiar volume.
(b) The transportpropertiesofliquid helium in ihe normal state are notvery
different from ihosc of a normal classicalgas.In particular,the ratio of the
thermal conductivity K to the product ofthe \\iscosity tj times the heat capacityper unit mass has the values
JC__ _ |16,at 2.SK
JJCV
~
[3.2, at 4.0 K
These vaiues are quite closeto ihoseobserved for normal gases at room
temperature\342\200\224see Table 14.3- The values of the transport coefficientslliem-selvesin the liqirid are with in an order of magnitude of those calculated for the
gas at the same denstly. Normal liquids act quite differently.
(c) The forces in [he liquid are relalively weak, and I be liquid does not existabove the critical temperaiureof 5.2K, which is itie maximum boiling pouitobserved. The binding energy vvotiki be perhaps ten limes stronger in Iheequilibriumconfijjuralion of :i si:uic lull ice, hul the expansion of ilte molarvolumeby the quanlum zero-poml motion of the atoms is responsiblefor tile
reduction in the binding energy to the observed value. The value of ihe criticaliemperaliireisdirccliyproportionalio the binding energy.
(d) The ikjuid is slablc at absolutezeroai pressures muter 25atnt; nbuvc
25 aim the solid is morestable.
Liquid 'He
as g
r
1
I
\\r'Hz 'lie
T, in K-
Fi\302\253iire7.13 Comparison of rales of flow of liquid 3He and
iiqujd 4He uuder gravify through a fine hole. Noiice fhc suddcronset of high fluidify or superfluidity in \"He. After D. W.
Osborne, B. Weinsiock, and B.M. Abraham, Pliys. Rev.75,9S8 A949).
The new stale of ntallcr inio which liquid 4He enters when cooled beiow2.17K. has quhe asionishing properlies. The viscosity as measuredin a flow
experiment* is essenlially zero (Figure 7J3), and the [hernia! conduciiviiy is
very high. We say thai liquid 4He below Ihe [ransilion icmperalureisa super-fluid. More precisely, we denole liquid *He below ihe transiUon lempcralurc
as liquid He U, and we say llial liquid He II is a mixlurc of normalHuid and
Supcriiuid coniponcnls. Tlte normal fluid component consisfs of Hie helium
aloms in thermally excited orbiials, and the superfluid component consisis of
life helium atoms condensed into the ground orbital. It is known lhat liieradioacljveboson6He in solution in liquid 4He does not tat:e part in Ihe
supijrflow of the latter; neither,ofcourse,doesthe fermion 3Hc in solution in
4llc kike part in ihe super/low.
We speak of liquid \"Mle ;ihovc liie iransilion leniper^lnrc as liquid ik i.Thereis no supcrdnid coiliponcnt in Ikjuid lie 1, for here the grama! oihii;ilot-vupiuicy is ucgti^ihlc, being uf l\\ic suim: order of uKiyuiiude as ihe oct.'up:mt;y
fliii.ls ofiliirftciil visttiMlics, sums c\\fKfinn:liii iiicisuic ifJiraif(Jj;t; i imnily. .t\302\273Juliet l'vjil'Ijjjk'j
Bic.isufi; the .ivct.igc of K'ii, i)f 'be aierjgt* lluidiiy.
Kjurc 7.14 The mcUing curve of liquid and
solid helium (*Hc). and the transition curve
between itie two forms oftiquid helium, He I
and He II. The liquid He ll form exhibits
sispcrliow ptopciiics as a consequence of Uiccondensarionof aroms into the ground orblialof the s; stem. Note ihat licl'mm h a liquid at
absolute zero ai pressures below 25aim. The
tiquid-\\apor boiling curve is not included in
this gcaph as ii would ractgc wjih [he zcco
pressuic line.After C. A. Swenson, Phys. lev.19,626E950).
of any other low-iying orbital, as we have seen. The regions of pressureand
temperature in which liquid He { and ({exislare shown in Figure 7,14.
The development of superftuid propertiesis no[ an automatic consequence
of [he Einstein condensation of aloms into the groundorbira!.Advanced cal-
calculations show that il is [he existence of some form (almost any form) of inter-
interaction among aloms [hat leads [o the development of superfluid propertiesin
[he aloms condensed in the ground orbital.
Phase Relationsof Helium
Thephase diagram of 4He was shown in Figure 7.14. The iiquid-vapor curve
can be followed from Ihe crilical poinl of 5.2 K down to absoiule zero without
any appearance of the solid.Al ihe transilion temperature Ihe normal liquid,called He I, makes a transit ion to the form wilh superfluid properties,calledHe II. A temperature called the/ poinl is the triple point al which liquid He (,
liquid He ii, and vaporcoexist.Keesom, who first solidified hciium, found thatthe solid*did not existbeiowa pressureof25alm.Another triple point exists
\342\200\242An tnlercsring discussion of solid heli
can, August 1967, pp. 85-95. Solid 'Hem is given by B. Bertram and R. A. Guyed Scientific Am
:xisis in three crysral structures according ro rhc condili
; of Helium
\342\226\240i
5 40
'He
Solid
/
/Liquid
/
Gas
Figure 7.15 Pliase diagrams for liquid 3He, (a) in
kelvin and (b) in miKikeivin. In rhc region of negaiiveslopeshown in (a) on ihc phase boundary [he soiid has
a higher eniropy ihan liie liquid, and we have io addheal lo 'he liquid io solidify it. Superfluid properiics
appear in(b) in ihc A and B phases of liquid JHc. The
A phase is double\342\200\224in a magnetic field [lie phase dividesimo hvo componenls wiih opposiie nuclear magnciicmomenis.
at 1.743 K: here the soiid is in equilibriumwiih ihe iwo liquid modifications.He I and He II.The two triple poinis are connecicd by a line !ha!separatesihe
regions of existence of He II and He I.Thephasediagram of 3He differs in a remarkable way from the phasediagram
of4He.Figure 7.15 exhibiis the importance of ihe fermion nature of3He.Note
ihe negaiive slope of ihe coexistence curve at low temperatures.As explained
in Chapter 10, the negative slope means thai !hc entropy of! he liquid phaseISlower lhan !heentropy of the solid phase.
Qunsipariieies and Superfluidity, 4He
For many purposes the superfluid component of liquid helium il behavesasif it were a vacuum, as ifi! were not thereat all.The No atoms of the supcrfluidare condensed into the groundorbila!and have no exeitation energy, for the
ground orbila! by definition has no excitation energy. The superfluidhas energy
only when I he center of mass of the superfluid is given a velocity relative to tlic
laboratory reference frame\342\200\224as when Ilie superfltiid is set inlo flow relative to
the laboratory.
The condensed component of Na atoms will flow with zero viseosity so longas the flow does not ereate cxcitatiosis in tile superfluid\342\200\224that is, so long as noatoms make transitionsbctwecilthe ground orbital and !ho excited orbitals.Such iransitions might be caused by collisions of helium atoms with irregu-
irregularities in the wall of the tube through which the helium atomsare Rowing.
The transitions, if they occur, are a cause of energy lossand of momentum
loss from the moving fluid, and the flow is not resistanceless if such collisions
can occur.
The criterion for superfluidity involves the energy and Momentum relation-
relationshipof the excitations in liquid He !!. If the excited orbitals were really like theorbilais of freeatoms,with a free parliele relation
E = l\\.ivi = -_(/,/,-)* G6)
between the energye and the momentum Mo or /ik of an atom, then we can
show that superfluidity would not be expected.Here k = ^/wavelength. But
bceause of the existence of interactions between the atoms the low energyexcitations do not resemblefree particle excitations, but are longitudinal sound
waves, longitudinalphonons(Chapter4).After all, it is not unreasonable that a
longitudinal sound wave should propagate in any liquid, even though we have
no previous experience of superliquids,A language has grown up to describe the low-lyingexciiedstatesofa system
of many atoms. These slates are called elementaryexcitations and in their
particie aspect the states are called(jiiasip articles. Longitudinal phonons are
the elementary excitations of liquid He II.We shall give the clear-cut experi-menial evidence for this, but first we derive a necessary conditionfor super-
superfluidity. This condition will show us wiiythe phonon-tfke nature of the
elementary excitations leads to the superfluid behavior of liquid He II.
Quaupanhlts andSuptrJluMty, 'lie
\342\226\240\342\200\242a
Figure 7. [$ Body of massMovelocity
V down a cylinder th;it
He It at absolute zero.
We consider in Figure 7.16 a body, perhaps a steel ball or a neutron, ofmass Mo falling with velocity V down a column of liquid helium aJ rest atabsolute zero,so that initially no elementary excitations are excited, if thenioliou of the body generates elementary excitations, Jhere will be a dampingforceon Jhe body, in order to generate an elementary excitation of energy ck
alid momentum hk, we must satisfy the law of conservationof energy:
\\M0V2*= |A/0F'2 + \302\243k, G7)
where V is the velocity of the body aftercreationof the elementary excitation.
Furthermore, we must sarisfy the law of conservation of momentum
=Ma\\\" + hk. (IS)
The two conservation taws cannot always be satisfied at the same time evenif the direction of Jhe excitation created in the process is unrestricted. To show
Af0V
Chapter 7; Fermi and Base Gases
this we rewriteG8)as
A/0V- l 0
and lake the square ofboihsides:
M02V2- 2A/0/iV-k + h2k2 = MQ2V'2.
On multiplicationby l/2Af0 we have
|A-/K2 /V k + \\M0V'\\
We subtract G9) from G7) to obtain
1\342\226\240
'
2mTq
G9)
(SO)
There is a lowest value of the magnitudeof the velocityV for which this
equation can be satisfied.Thelowest value will occur when the direction of kis parallel to that of V. This critical velocity is given by
-h2k2
Vs- minimum of- (81)
The conditionis a little simpler to express if we let the mass Mo of the body
become very large, for then
(82)
A body moving with a lower velocity than Vc will not be able to create excitationsin the liquid,so that the motion will be resistance less.The viscosity will appear
to be zero. A body moving with higher velocity will encounter resistance
Quusipartfctes and Superfluidity, *lle
the curve from below.The slopeof this line is equal to the critical velocity, ifck
s= h2k2/2M, as for the esciiation of a free atom, [he straight line has zeroslopeand the crilical velocity is zero:
Free atoms: Vs= minimum oUik/2M = 0. (83}
The energy of a low energy phonon in liquid He II is \302\243k= tui>k = tusk in [lie
frequencyregionofsoundwaves where the product of wavelcnglh and frequencyis equalto the velocity of sound t'a, or where the circular frequency mk is equal
to [he product of vs times the wavevector k. Now the critical velocity is
Phonons: Vc = minimum offtr^/hk = v,. (84)
The critical velocity Vc is equal to the velocity of sound if (84} is valid for ;ill
wavcveciors, which it is not in liquid helium ii. The observedcriticalflow
velocities arc indeed nonzero, but considerably lower than the velocity of soundand usually lower than the solid straight line in Figure 7.i 7, presumably beeausethe plot of Ek versus lik may turn downward at very high hk.
The actual spectrum of elementary excitationsin liquid helium II has been
determined by the observations on the inelastic scattering of slow neutrons.
The experimentalresultsareshown in Figure 7.17. The solid straight line is theLandau critical velocity for the range of wavevectots coveredby the neutron
experiments, and for this line the critieal velocity is
Vc\302\253
&/hk0~ 5 x I03 cm s\021 , (85)
where A and k0 arc identifiedon the figure.
Charged ions of helium in solution in liquid helium 11 under eertain experi-experimental conditions of pressure and temperature have been observed*to move
almost like free partieles and to have a limiting drift veloeity near 5 x 103 cm s\"*
closely equal to the calculated value of (85).Underother experimental condi-
conditions the motion of the ions is limi(ed at a lower velocity by the ercation of
vortex rings. Sueh vortex rings are transverse modes of motion and do not
appear in the longitudinal modes covered by Figure 7.17.
Our result (84) for a neeessary conditionfor the critieal velocity is more
general than the calculation we have given.Our calculationdemonstrates that
a body will move whhout resistance through liquidHe II at absolute zero if
the velocity V of the body is less than the critieal velocity Vc. However, at
* L. Meyer and F. Reif, Phys. Rev.'123,727 U96t|; G. W. Rajfiekt, Ph)s. Rev. Lcllcrs 16,934 A966).
1.0 2-0 3.0
WavevccioT, in units of 50s cm\021
Figure 7.17 Energy ck versus wavevecior i of elementary
exciiaiioHS in liquid helium ai t.JJK. Theparaboliccurve rising
from lhe otigin represents the iheoreiicaiiycalculated curve for
free helium aioms ai absolutezero.Tlic open circles correspond lothe energy and momentum of the measured exciutions. A smooshcurve has beendrawn through ihe poinls. Hie broken curverising iineariy from lhe origin is she tiieorcsiealphonon brynch
wfili a vclodfy of sound of 237 m s\"'. The solid straight line gives
the crmcai vdocily, in appropri^se uniii: Ttie line gi\\e$ jh\302\253imxn-
mum ofij/A\" over ihe region of k covered in ihcse expcrimenls. After
D.G. Hcnshaw and A. D. B.Woods. Phis. Rev. t2l. 1266A960-
femperutures above absolute zero, but below the Einstein temperature, there
wili be a normal fluid component of demcnlary excitations that are thermallyexciied.Thenormal fluid component is the source of resistance to the motionof the body. The supcrflow aspect appears first m experiments in which the
liquid flows out ihrough a fine lube in the side of a container. The normaifluid component may remain behind in lhe container while the superfluid
component leaks out without resistance.The derivation we have given of the
critical velocity also holds for this situation, with V as the velocity of the super-fluid relative to the walls of the tube; A/o is the niass of the fluid. Excitationswould be createdabove V( by the interaction between the flow of the liquid
and any mechanical irregularity in the walls.
Supcrfluid Phases of 3He
Three superfluid phases of liquid 3He are known* (Figure 7.15b), but\342\200\224in
contrast to liquid 4He\342\200\224with transition temperatures of only a few milhkelvin.The superfluid phases are beiieved to be qualitatively similar to the super-superconducting slate of electrons in metals, where pairs of particlesin orbiials near
[he Fermi surface form a type of boundsiaieknown as a Cooper pair. Such a
pair is qualitatively like a diatomic molecule, but the radius of the moleculeis much larger than the average iniereleciron spacing in a metal or the average
mlerparticle spacing in liquid 3He.In metallic superconductivity [he two electrons [hat form a Cooper pair are
in a nonmagnetic (singlet) spin staie. In the superfiuid siatesof liquid 3He the
Uvo aioms [hat form a pair are in the triplet spin states of the two JHe nuclei,so thai ihree magnetic supcrfiuids are possible, corresponding to spin orienta-orientations M,
~ 1, 0, and -1, or mixturesof these three states. The magnetic
superfluids have been exploredexperimentally,and both the magnetic \302\243nd
superfiuid properties have been confirmed.
SUMMARY
1. Comparedto a classicalgas,a Fermi gas at low temperature lias high kinetic
energy,high pressure, and low heat capacity. The entropy of the Fermigasis zero in the ground slate. The energy of the highest filled orbiial in the
ground state of a free particle gas of ferniions of spin j is
' ~ut\\ v )
\342\226\240
2, The lota! kinezic energy in the ground stale is
\342\200\242For elementary roif^s, see j. C. U heat icy, Physics Today, February ly/b. p. i
PU)sicsfluiic!tn25. 311ii97j|;aritt J.R. Hook, i%sici Bulletin 29. 5i3A97SJ. For
C Wll f Ph R \\d Ph
Chapter 7: Fermi and Base Cases
.3. The density of orbitats at r,f is
\"D{Cf) ~ 3,V/2c^.
A. The heat capacity of an electron gas at r \302\253i> is
in fundamental units.
5. For a Boscgasat r < rE the fraction of atoms in excited orbitiiis is
fi. The Einstein condensation temperature of a fj;iS of nuniiitenicthig bosons
__2nftVN Y>
PROBLEMS
i. Density of orh'ttah hi one and two dimensions, (a) Show that the densityof orbiiafs of a free electron in une dimension is
O,(e) =\302\273(t,/rc)t2*rt/Aa\302\243I
2, (86)
where i. is the length of the line,(b) Show thai in two dimensions, for a squareofareaA,
\302\251i(e)= ^tiii/Tcft2 , (87)
independent o(e.
2.Energy of refarivistic Fentu gas. For electrons with an energy e \302\273me1,
where in is the rest mass of the electron,the energy is given by e = pc, wherep is [hemomentum.Forelectrons in a cube of volume V = i.3 the momentumis of [he form (nh/L), multiplied by (n/ + it/ + n:2)ll2> exactly as for the
nonrelativistic limit, (a) Show [hat in this extreme relativistic limit the Fermi
energyof a gasofN electronsisgiven by
zr \302\273AncCii/n)\025 , (88)
Problems
where u = N/V. (b) Show that the total ener\302\273y of the ground stale uf ihc gas is
(/\342\200\236= J,Vef. (S9)
The general problem is Ireaied by F.Jutlner,Zeilsduifl furPhysik47,542({928).
3. Pressure and entropy of degenerate Fermi gas. (a) Show that a Fermi
electron gas in the ground stale exerts a pressure
In a uniform decrease of the volume of a cubeevery orbital has i!s energy
raised; The energy of an orbital is proportionalto XjL1 or to l/f/2|J. (b) Kindtin expression for Use entropy of -a Fermi electron gas in the region r \302\253t>.
Notice that a -* Oast-0.
\"/. Chemical potential verms temperature. Explain graphically why the initial
curvature of p. versus t is upward for a fermion gas in one dimension anddownward in threedimensions(Figure 7.7). Him: The C,(e) and *D3(e) curves
are different, where *D, is given in Problem t. It wiii be found useful So set upthe integral for N, the number of particles, and to consider from the graphs
the behavior of the integrand betweenzerotemperature and a finite temperature.
5. Liquid 3He as a Fermigas. The atom 3He has spin 1 =\\ and is a fermion.
(a) Calculate as in Table 7.1 the Fermi sphere parameters vF, ef, and TF for
3He at absolute zero, viewed as a gas of nan interacting fermions.The density
of the liquid is 0.081 g cm\"\\ (b)Calculatethe heat capacity at low temperaturesT \302\253TF and compare with the experimental value Cr
= 2.89NfcBT as observedfor T < 0.1K by A. C. Anderson, W. Reese, and J. C. Wheatley, Fhys. Rev.
130, 495 A963); see also Figure7.18.Excellent surveys of the properties of
liquid 3He are given by J. Wilks, Properties of liquid and solid helium,Oxford,1967,and by J. C. Whealley, \"Dilute solutions of JHe in \"Heat low tem-
temperatures,\" American Journal of Physics 36, 181-210A968).The principles
of refrigerators based on 3He-*He mixturesare reviewed in Chapter 12 on
cryogenics; such refrigerators produce steady temperatures down to 0.01 K.
in continuously acting operation.
6. Mass-radius relationship for white dwarfs. Consider a white dwarf of massM and radiusR.Let the electrons be degenerate but nonrelativistic; the protonsare nondegenerate.(a) Show that the order of magnitude of the gravitationalself-energyis -GM2jR, where G is the gravitational constant. (If the massdensity is constant within'the sphere of radius R, the exact potential energy is
Chapter 7: Fermi and Sose Gases
5.0
20 SO $00 200
Temperature, in K
Figure 7.18 Heat capacity of liquid 3He and of a 5 percent solution of3Hein liquid *Hc The quansisy plotSedon she vcrSica! axis is C/T, andshe horizontal axis is T. Thus for a Fermi gas in the degenerate temperaturefegion the theoretical cufves of C/T at constant volume are horizontal.
The curve for pure 3Heis taken at constant pressure, which accounts fof
the slight slope. The curve for the solution of Hie in liquid 4He indicates
thai she 3He sis solution acts as a Fermi gas; she degenerafe region at low
temperature goes over to the nondegene/ate region at higher temperature.The solid Sine through she experimental possHsfor the solution is drawn
hrTf - 0.331K, which agrees with the calculation for free atoms if the
effective mass is taken as 2.38times the mass of an atom of 3He.Curves
after J. C. Wlicatley, Amer. J. Physics36 A968).
-3GM2f5R). (b)Showthat the order of magnitude of the kiiicsicesiergyof the
electrons in the ground state is
w here m is she mass of an clcclrosiand Mit is the mass of a proton, (c) Showlhai if she gravitational and kitsctic energies are of the sameorderofmagnitude
(;ss required by ihe virsal theo/esiiofsnechanics), Mll3R ~ 10;ogW2cm.{d)Iflhe
Figure 7.19 Heal capacity of ;tn
Eimtcin gas ut cortilam volume.
mass is equal to that of she Sun B x lQ33g), what is the density of the while
dwarf? (e) H ss believed thai pulsars ate starscomposedofa colddegenerate
gas of neuUons. Show that tot a neutron star Mll3R =s 10!7 g\023ciyi. What is
she value of the radius for a neutron star with a mass equal so that of the Sim?Express she resuls tn km.
7. Photon condensation. Considera science ftclton universe in which shenumber ofphotonsN isconstant,at a concentration of I0Iocm\"s. The numberof thermally excited photons we assume is given by the result of Problem4.1,which is Ne = 2.404 Kt3/*2/] V. Find the criticaltemperaturein K below which
jVe < JV. The excess N \342\200\224Nr will be in the photon mode of lowest frequency;
the excess might be described as a photon condensatein which there is a largeconcentration ofphotonsin the lowest mode. In reality there is nosuchprinciplethat the loui! number of photons be constant, hence there is no photon
A1. Energy, heat capacity, ami entropy of degeneratebosongas. Find ex prcs-
sions ;is ;i function of tcmpcf.ifun; in the region x < th for the cne-igy. hc;ilop.idiy, ;mii cn[[opy of ;i y;is o\\ ,V noniiilcMlcling bosons of hpiil zero confiiialto ;i volume V. Put the del'milc imegnil in dim ens ion less form; it need not be
evaluated. The calculated heat capacity above and below r\302\243is shown in
Figure 7.19. The experimental curve was shown in Figure 7.12. The dilTcrcnce
Chapter 7: Fermi and Base Gases
between the two curves is marked;It is ascribedlo the effect of interactions
between the atoms,
9. Bosongasin one dimension. Calculate the integral for Nt{x) for a onc-
dimensional gas of noninteracting bosons, and show that the integral does
not converge. This result suggeststliala bosonground stale condensaie does
not form in one dimension.Take/. ~ I for the calculation. (The problem should
really be treated by means of a sum over orbitats on a finite line.)
10. Relativhtk white dwarf stars. Consider a Fcrniigas of A' electrons eachof rest mass m in a sphere of radius R. Conditions in certain white dwarfs aresuch that the great majority of electrons have extreme relativislic kineticenergies e = pc, where p is the momentum. The de Broglie relationremains). ~
Inhjp. Problem 2 gives the ground stale kineticenergy of the A1 electrons
on the assumption that e = pc for ali elecrrons. Treat the sphereas a cubeof equal volume, (a) Use the standard viria! theorem argumentto predictthe
value of N. Assume that the whole star is ionizedhydrogen, but neglect the
kinetic energy of the protons comparedto that of the electrons, (b) Estimatethe value of N. A careful treatment by Chandrasekhar leads not to a singlevalue of N, but lo a limil above which a stable while dwarf cannot exist: see
D. D. Claylon, Principlesof stellarevolution and nucleosynthesis, McGraw-Hill,
1968, p. 161; M- Harwii,Astrophysicat concepts, Wi!ey, 1973.
11. Fluctuations in a Fcwri gas. Show for a single orbisa! of a fermion system
that
<(ANJ> - <N>A -<N\302\273 , (91)
if \342\226\240GAOis the average number of fermions in thai orbiial. Notice that the
fluciuation vanishes for orbitals with energies deep enough below the Fermi
energy so thai <N>= I. By definition, AN s N ~ <N).
12. Fluctuationsin a Basegas. If </V) as in (I!) is the average occupancyofa single orbiial of a boson system, fhen from E.83) show iiiat
<(ANJ> =<JV>(I + <JV\302\273. (92)
Thus if fhe occupancy is large, with <N> \302\273I, ihe fractional fluctuafions areof the order ofuntty; <(ANJ>/<N>2\302\2531, so mat the actual fluctuations canbe enormous.It has been said that \"bosons travel in flocks.\" The first edition
of thrs text has an elementarydiscussionofthe fluctuations of photons.
13, Chemical potential versusconcentration* (a) Sketch carefully the chemical
potential versus the number of particlesfor a boson gas in volume V at
icmpcraiure r. Include both classical and quantum fegimes. (b) Du the same
for a syssem of fermions.
14. Two orbilai boson systent. Consider a system of A' bosons of spin zero,with orbitals at (he single particle energies 0 and e. The chemical potential is
/x, and the temperature is r. Find r such, that the fhefma! average populationof the lowest orbital is twice the population of the orbital at e. AssumeJV 3> | and make what approximations are reasonable.
If the atoms in a gas have integral spin (counting the sum of electronic
and nuclear spins), they can form a boson condensate when the gas is cooled
below the Einstein condensation temperature te given by G2):
rE~ BTTh2iM){Ni2.6\\2VJ/\\
For atoms in the vapor phase she Einstein'condensation temperature is verySow because the number densities are very low\"; In A995) early successful
experiments were carried out at Boulder, MIT, and elsewhere. Such experi-experiments, which are extraordinarily complex, mark the exciting forefront of the
quantum gas field. A large literature on BEC experiments and theory is on
the Web.
One set of experiments (MIT) started with a beam of sodium atomsexiting an oven at 600K at a concentration N/V of SO14cm\023. Whas happens
next is the result of a numberof clever tricks with laser beams directed onone pint or another of the beam of atoms. First file atoms are slowed by onelaser bcum from an exit velocity of 800 in jt1 to about 30 in s~'. This isslow eiioug.lifor !0!\" ntonis to be trapped within a magneto-optical trap.Fusthcr tricks, including evaporation, reducedthe temperature of the gas to2 /nK, the uhraiow temperature rE at whieli the condensate was formed. The
concentration at rE was again !014atoms/cmJ.The atoms in the condensed phase are in tlie ground orbital and expand
Giiiy slowly once released from the trap. The atoms in excited states move
relatively rapidly out of their steady-state positions.The positions of the
asoms can be recorded as a funcsion of sime after release, using a laserbeam.The number of asoms in excited orbitals is in good agreement with she t3
law, G3). With this sechnique the signasure of Bose'Einssein condensation isshe sudden appearance of a sharp peak of atoms as the semperature is
decreased through rE. The peak comes from lighs scattered by atoms in the
condcusase; she wings of she line from light scattered by atoms in excited
orbitals.
Chapter 8
Heat and Work
ENERGY AND ENTROPY TRANSFER:
DEFINITION OF HEAT AND WORK 2-7
HEAT ENGINES: CONVERSIONOF HEAT INTO WORK 2IS
OirnOl Inequality 2-S
Sources of Irrevcrsibiliiy 252Refrigeraiors 233Air Condiu'oners and Heat Pumps 255Carnot Cycle 2?6Example:CarnolCycle for an Idea! Gas 237
Energy Conversion and the Second Law of Thermodynamics 240Palh Dependenceof Heat and Work 240
Irreversible Work 242
Example: Sudden Expansion of an Ideal Gas 245
HEAT AND WORK AT CONSTANTTEMPERATURE OR CONSTANT PRESSURE 245
Isothermal Work 245IsobaricHealand Work 245
Example: Elecirolysis and Fuel Cells 247ChemicalWork 250
Example: Chemical Work for an Ideal Gas 25!Magnetic Work and Superconductors 2--
SUMMARY 257
I'UOBLEMS 2?7
1. Heat Pump :::2.
Absorption ReiYigerator2!:<
). I'lwicm Carnol Hnginc 25S4. IlealEngine\342\200\224Kel'rigcralor Cascade 25S
5. Thermal Pollution 2586. RoomAir Condilioner 258
7. Light Bulb in a Refrigerator8. GcotlierrnalEnergy
9. Cooling of Nonmetallic Solid to T= 010. IrreversibleExpansion of a Fermi Gas
259259259259
Energy and Entropy Transfer: Definition of Heat and Work
ENERGY AND ENTROPY TRANSFER:
DEFINITION OF HEAT AND WORK
Heatand work are two different forms of energy transfer.Heatis the transfer
of energy to a system by thermalcontactwith a reservoir. Work is the transferof energy to a system by a change in the external para meters that describe thesystem. The parameters may include volume, magnetic field, electricfield,orgravitational potential. The reason we distinguish heal from work will be clear
when we discuss energy conversion processes.The most important physical process in. a modern energy-intensive civiliza-
civilizationis the conversion of heat into work. The IndustrialRevolution was made
possible by the steam engine, which convertshealto work. The internal com-
combustion engine, which seems to dominate man as much as It serves him, is a
device to convert heat to work.The problem of understanding the limitations
of the steam enginegave rise to much of the development of thermodynamics.Energy conversion remains one of the central applications of thermalphysics
because most electrical energy is generated from heat.The fundamental difference between heal and work is the difference in the
entropy transfer. Consider the energy transfer dtl from a reservoir to a systemwith which the reservoir is/in thermal contacl at tcmperaiure t; an entropy
transfer da ~ dU/x accompanies ihe energy transfer,accordingto the argument
of Chapter 2. This energy transfer is what we defined above as heal, and we seeit is accompanied by entropy transfer- Work, being energytransferby
a change
in external parameters\342\200\224such as the position of a piston\342\200\224does not transfer any
entropy lo the system.Thereisnoplacefor entropy to come from when onlyworkis performedor transferred.
However, we must be careful; the toial energy of two systems brought intocontactis conserved,but their total entropy is not necessarily conserved and
may increase. The entropy iransfer between two systems in ihermalcontactiswcl! defined only if theenlropy of one system increasesby as much as the entropyof the other decreases. Let us restrict ourselves for the present to reversible
processes such that the combined entropy of the interacting systems remainsconstant.Laterwe will generalize the discussion to irreversible processes whichareprocesses in which the total entropy of the two systems increases,as in the
heat flow example in Chapter 2.
CkapterS:Heat and Work
We can give a quantitative expressionto the distinction between heat andwork. Let dU be the energy change of a system during a reversibleprocess;daisthe entropy change, and t is ihe icmperitture.We define
i!Q s xda (I)
as the heatreceivedby the system in the process. By the principleofconservation
ofenergy,
dV = aw + dQ , B)
which saysthai the energy change is caused partly by workdoneon the sysictn
and partly by heal added to ihe System from the reservoir. Then
t1W = dU ~(tQ
= dV ~ zda 0)
is (be work performed on the system in the reversible process. Our reasons for
designatingheat and work by iftjand dW rather ihan dQ and d\\V arc explainedbelow. For da \342\200\224
0, we have pure work; for dV = xda,pureheat.
HEAT ENGINES: CONVERSION
OF HEAT INTO WORK
Catnotinequality. Heat and work Have different roles in energy conversion
processes because of the difference in entropy transfer. Consider two conse-
consequences of the difference:
(a) All types of work are freely convertible into mechanical work and into
each other,becausethe entropy transfer is zero. An ideal electrical motor,
without mechanical friction or electricalresistance,is a device to convert
electrical work into mechanical work. An ideal electrical generator converts
mechanical work into electricalwork. Becauseall forms of work are freely
convertible, they are thermodynamically equivalent to each other and, in
particular, equivalent (o mechanical work. The termwork denotes all types of
work.
(b) Work can be completely converted into heat, but the inverse is not irue:heat cannot be completely converted into work. Entropyentersthe system with
the beat, but docs not leave the system with the work. A device that generateswork from heal must necessarily strip the entropy from the heat that has been
converted to work. The entropy removed from the converted input heat cannot
be pennittedto pile up inside the device indefinitely; this eniropy must ulti-
ultimately be removed from the device. The only way to do this is to provide more
Heal\302\243itgines: Conversion of Heat into Work
Entropy
Figure 8.1 Eniropy and energy flow in any continuously operatingreversible devicegencralmg work from \\\\ca.t. The entropy outflow
must equal ihc entropy inflow.
input heat than the amount converted to work, and 10ejectthe excess input
heat as waste heat, at a tempcrature lowerthan thai of the input heal (Figure 8.1).BecauseiiQ/ifc= r, the reversible heat iransfer accompanying one unit of
entropy is given by the temperature .it which the heat is transferred. It follows
that only part of the input heat need be ejected at the lower temperatureiocany ;t\\\\ay all the entropy of the input heat. Only the difference bctv^^.i inputand outpul heal can be convened to work. To prevent ihc accumulation of
eittropy there must be some output heat; therefore h is impossibleto convert
ail the input heal to work!A prohibition against unlimited entropy accuniLiiaiion tn a device does not
mean entropy cannot accumulatetemporarily,providedthat it is ultimately
removed. Many practical energy-conversiondevicesoperatein cycles, and the
entropy contained in the device varies periodically with time. Such a cyclicdeviceis calleda heat engine. The internal co'rnDiisTion engine is an example:The entropy contained in each cylinder is at a minimum near the beginning of
the intake stroke and a maximum near the beginningof the exhauststroke.There is a value of the entropy eontent to which the devicereturnscyclically;the entropy does not pile up indefinitely.
What fraction of the input heat Qh taken in during one cyele at ihe fixed highertemperatureth can be convened into work.? The input entropy associatedwith
the inpui heal is ak \342\200\224QjTh. To avoid confusing signs, we define in this discus-
discussion all energy, heat, and entropy flows as positive whether the flow is into or
out of the system, rather than following ihe usual convention according towhich a flow is positive into the system and negative out of the system. If Q, is
the waste heat leaving the system per cycle ;j{ the fixed lower temperature t,.the output entropy per eyclc is as = Qjxy In a reversibleprocessthis output
entropy is equal to the input entropy:
Q,fa = QJik , 14)
Qi= ir,/rh)Qh. E)
The work generatedduring one eyeleof a reversibleprocessis the difference
between the heat added and the wasteheatextracted:
W = Qh- Q, = [1 -
( QA= -\302\261\342\200\224>-
Qh. F)
The ratio of the work generatedto the heat added in the reversible process isealledthe Carnot efficiency:
G)
This quantity is named in honor of SadiCarnot,who derived it in 1824. It wasa remarkable feat: the concept of entropy had not yet been invented,and
Carnot*s derivation preceded by some 15 years the recognitionthat heat is a
form of energy.The Carnotefficiency is the highest possible value of the energyconversion
efficiency ij= W/Gh, the output work per unit of input heat, in any cyclic heat
Heat Ens'\"\": Conversion of Heat into Work
input
Output
Figure 8.2 Entropy and energy (low in a teal heat engine containingirreversibilities ihal generate new entropy inside the device. Theentropy outflow at the lower temperature is larger than the entropyinflow at ihe higher temperature.
engine that operates betweenthe temperatures t,, and t,. Actual heat engineshave lowerefficienciesbecausethe processes taking place within the device arenot perfectly reversible. Entropy will be generated inside the deviceby irrevers-
irreversibleprocesses. The energy-entropy flow diagram is modifiedas in Figure 8.2.
We now have three inequalities
ft S:&(r,/Tfc); (9)
A0)
The actual energy conversion efficiency j; obeys the Carnal inequality
W/Qk
We can have i; = ijc only in the limit of reversible operation of a device thattakes in heat at ih and ejectsheal at rt.
The Carnot inequality is the basic limitationon any heat engine that operatesin a cyclic process. The result te!!s us that it is impossible to convert all inputheat tnto work. For a
given temperature ratio tJt, the highest conversionefficiency is obtained under reversible operation. The limiting efficiency in-
increases with increasing xhfxit but we attain !00 percent efficiency only whentJt, -* go.
The low-temperature waste heat of any heist engine mustultimately be
ejected into the environment, so thai r, c;innoi be below the environmental
temperature, usually about 300 K.Higheilicieneyrequires an input temperature
Th high compared to 300 K.The usable temperatures in practice are unfortu-
unfortunately limited by various materials constraints. In power plant steam turbines,which are expected to operate continuously for years, the upper temperatureis
currently limited to about 600 K\"by problems willi the strength and corrosionof steel.With r, = 300 K and Th
= 600 K, the Carnot efficiency is i;c = $, or50 percent. Lossescausedby unavoidable irreversibiliiies reduce this ellkicticytypically to about 40 percent.ToobtainHigher eilicicncies is a problem in high
temperature metallurgy.
Sources of UreversibHity. Figure S.3 illustratesseveral common sources of
irrevcrsibility:
(a) Part of the input heat Qh may flow directly to the low temperature,by-
bypassing the actual energy conversion process, as in the heat flow into the
cylinder walls duriiig the combustion cycleof the internal combustion
engine.
(b) Part of the temperature differencex,,- r, way not be availableas tem-
temperature diflcrencc in the actual energy conversion process,becauseofthe tesnperature drop across thermal resistances in the path of the heatflow.
(c) part of ilic work generated may be convertedbackto heat by mechanical
friction.
(d) Gas may expand irreversibly without doing work, as in Ihe irreversible
expansion of an idealgas into n vacuum.
Irreversible expansionwfthout work or heal
bypass
Figure 8.3 Four sources of trreversihilily in heal engines: heal How
b> passing the energy conversion process,ihermai resislance in the palhof tile heal flow, frtclional losses, and entropy generation during
irreversible expansions.
Refrigerators
Refrigerators are heat engines in reverse.Refrigeratorsconsume work to move
i'^ai from a low temperature r, to a highertemperaturerA. Consider the energy-
entropy flow diagram of a reversibleheat engine in Figure 8.1. Because no
enlropy is generatedinsidethe device, its operaiion can be reversed, with ane*act reversal of the energy and entropy Ot>-\\i. Equations D) ihrough F)remam valid for the reversed flows.
Chapter S: Ihat and Work
The energy ratio of interest in u refrigerator is itot tile energy conversionefficiency G), but tlic ratio y
s Q,/W of iho heat extracted at tlic low tem-
temperature to the work consumed. This ratio iscalledthe coefficient of refrigerator
performance; its limiting value in reversible operation is called the Carnotcoefficient of refrigeratorperformance,denoted by yc. Do not confuse*/ = 0,/Wwith \302\273;
52 W;Qh f\302\260r'he energy conversion efficiency of a heat engine;although
i; < i always, y can be > 1 or < 1.From Eq. E) and W = Qh -Qh the work
consumed is
A2)
The Carnot coefficientof refrigeratorperformance is
Th- T,'
A3]
This ratio can be larger or smaller than unity.
Actual refrigerators, like actual heat engines, always contain irreversibilities
that generate entropy inside the device.In a refrigerator this excess eniropy is
ejected at the higher temperature,as in the ettcrgy-entropy flow diagram of
Figure 8.4.With the convention that all energy and entropy flows are positive,
we now have
ah > e, , A4)
in place of (8). Further,
& 2 (V'Jfl , A5)
W = Qt - Q, > \342\200\2241]Q,
=\342\200\242 - Q, \302\273Q,/yc . A6)
Q,iW < yc.
Air Conditioners ami Heal Pum
Ouipui
Input
Figure 8.4Entropy
and energy flow in a refrigerator.
The Carnot coefficientyc is an upper limit to the actual coefficientofrefrigerator
performance y, just as the Carnot efficiency t\\c is an upper limit to the actual
energyconversionefficiency ;; of a heat engine.Both heat enginesand refrigerators are subject to restrictions imposed by the
law of increase of entropy, but the device designproblemsare totally different.
In particular, the design of refrigerators to operateat the temperature ofliquid
helium or below is a challengingproblemin thermal physics (Chapter 12).
Air Conditioners and Heat Pumps
Air conditioners are refrigerators that cool the inside of a building or an auto-
automobile; the heat is ejected lo the outside environment.If we interchange the
inside and outside connections, an air conditionercanbeusedto heat a building
during the winter, Such a device Is calleda heat pump. If r\302\273\342\200\224
t| \302\253xk a beat
Chapter Si Hear and Work
pump can heat the building with a lower consurnption of energy than by directhealing (Problem I).
The limitations on the use of heat pumps are largely economical, They aremuch more costly to install and to main tain than are simpleheatersor furnaces.
Heat pumps make economic sense primarily in eiimatic conditions in which
air conditioning is required anyway.
Carnot Cycle
The derivation of the Carnot energy conversion efficiency and of the Carnotcoefficient of refrigerator performance made no statement about how torealizea process by which work is generated from heat, or about how refrigera-refrigerationis achieved. The simplest and best known suchprocessis the Carnot cycle.
In the Carnot cycle a gas\342\200\224or another working substance\342\200\224is expanded and
compressed in four stages, two isothermal and two isentropic, as in Figure 8.5.
At point 1 the gas has the temperature xk and the entropy aL. The gas isexpandedat constant r until the entropy has increased to the value a,,, at point 2. In the
second stagethe gasis further expanded, now at constant a, until the temperature
has dropped to the value r,, at point 3. Thegas is compressedisothermally to
point 4 and then compressed isentropicallyto !heoriginals!a!eI. We write aL
and c,, for the low and high values of the entropy contained in the working
substance, to distinguish these values from a, and ch, which are the entropyflows per cycle at !hc low and high temperatures r, and rv For tlic Carnot cycle,
G\\~
\302\260h= &I1
\"\302\260L-
The work done by the system in one cycle is [he area of the rectangle in
Figure 8.5:
W~{Th~ T,)(tfn - ffj. A3)
whicii follows from
SdU \342\226\240\342\226\240=0 =<\302\243uta -SpJV ,
where jjklV is the work done by the system in one cycle. The boat taken up at
T = ts during the first phase is
We combine (IS) and A9) to obtain the Carnot efficiency ;/c. Any process
described by Figure 8.5 is calleda Carnot cycle,regardlessof the working
substance.
,lCyd.
Figure 8-5 A Carnot cycle, for the conversion of heat into work,
illustrated as a plot of entropy versus temperature, for an arbiiraryworking subsiance. Thecycleconsistsof two expansion phases(t -* 2 and 2-<3) and wo compression phases <J \342\200\224\302\2734 and 4 -+ 1).Cjncof the expansion ano one of ihc compressionolsascsiireisothermal (I - 2 and 3 -
4), and one phase of each kind is
iscntropic B -> 3 and 4 -* 1}.The nei work done is ihc area of ihcloop.The heal consulted at r^ is the area surrounded by ilic
broken line.
The Carnol cycle is a point of reference!o indicate what could in principlebe done, rather than what in fact is done. All energy conversion cycles need a
high temperature input and a low temperature output of heal, bu! often (lie
heat inputs and outputs are not well-definedreservoirsat constanttemperatures.Even where such reservoirs exist, as in steam turbines, (here is invariably ;s
temperature difference presenl between the working substance and the reser-
reservoirs. The healing and cooling processes are never trulyreversible.
Example: Caettot cycle for an idea!gas. We carry an ideal monaionjic gas through a
Carnot cycle- initially ihc gas occupies a volume t', and is in thermal equilibrium with j
reservoir <i\\h at the high temperature tv The gas isexpanded isolherlttally to the volume
K, as in Figure \302\243.6a.In Hie process the gas absorbs Die heal Q, from (Rt and delivers ii as
\\vo[k *jj 10 ^n c\\lern>ii n^cdmmcal sy ^Icm conncclco to Die pislon. i~oran sdcaiQas the
heai absorbed from (he reservoiris
Qh= II'l3 =.
JpJK=
Nr^dV/V= A'r B0}
This work is indicated by the area labeled \0212\"
Next, the gas is disconnect! from (SI* and
furlhcr expanded, now iscntropically, until ihe temperalure has dropped to the low tem-
peralure :,. In Ihe process the addilional work
is delivered by the gas. The volume V3 at the end of ihe iseniropic expansionis related lo
TlV32:3 = TftV22'3 , or VJVj.= (tJt,}*2 . B2}
from F.63). Afler point 3 the gas is brought into contact with a temperature reservoir <Slf
ciioscnlo satisfy
KfVx = (t*A,K'2 -VJV2 , B3)
so thai Kj/V'4\342\200\224
i'j,- f,. To accomplish this compression, the work
ust be done on ihe gas.This work, is ejected to <R, as heai:
ft = ^34- B5)
Finally, the gas is disconnected from CH, and recompressed iseniropicatly until its tempera-temperaturehas liseii to ihe initial temperature i*. Because of the choice B3}of K4l the gas volume
at this point has returned to its initial value Vlt and the cyde is completed,in this last slage
the work
jy4! \302\253|N(rh
_T() B6)
is performed on ihe gas;this cancels the work W2i done by the gas during the iseniropic
expansion2 -* 3, by B1).
The ncl work delivered by the gas during the cycle is given by ihe difference in shaded
areas in Figures 8.6a and 8.6b, which is the enclosed area in Figure 8.6c. The isenfropiccurves in the p-V diagram are steeper lhan the isothermalcurves, so thai Ihe area of the
y3v
Figure 8.6 The Camot cycle for an ideal gas, as a p-V plot. An ideal gas is expandedand riicompressed in four stages. Two of them are isothermal, at the temperature I* and
\342\226\240t,(xk> r,}. Two ofthem arc isentropic,ffomtt to ti.and back. The shaded areas show(a) the work done during the two expansion stages, (b) the work done during the two
compression stages,and (c) the net work done during the cycle.
Chapter 8: Heat and Work
loop is nnitc 2nd i^ ctintii to trie 3r\302\243*iof jtic rccmn^lc m F^Rurc 8,3. iVc Imvc
-W(rfc
-TjloaflV^). . B7)
The heat absorbed from Cftk was given in B0), so that W/Qj, - (ij ~-r,)/Tj,, which is jusi
the Camot relation G).
Energy Conversion and the Second Law of Thermodynamics
The Carnoi limiison the conversion of heat into work and on the performanceofrefrigerators are direct consequences of ihe law of increaseof entropy. The
second law of ihermodynamics usually is formuiaiedwiihom memionofeniropy, We stated ihe classical Kelvin-Planck formulation in Chapter 2:\"Ii is impossible for any cyclic process lo occurwhose sole effcci is ihe exiractionof heai from a reservoir and the performance ofan equivalent amountofwork.\"
All reversible energy conversion devices thai operaie beiweenihe same ieni-
peratures have the same energy conversionefficiency ;j= W/Qh- Were ihis not
so, We could combine two reversible devices with different efficiencies, qt < ij2,in such a way (Figure 8.7)ihat d\302\243\\ice I with ihe lower efficiency is operaied in
reverse as a refrigeralor (hat mows not only ihe entire wasteheat Ql2 from ihe
more efficient device 2 back io thehigher iemperaiure th, but an additional
amouni Q{\\n) of heai as well. The overall result would be ihe conversion of
ihe heal Q{in) to work li^otit), uiihout any net waste heai. This would requireihe annihilationof
entropy and would violate ihe law of increase of entropy.Now thai we have eslablished jhai all reversibledevicesIhatoperatebeiween
the same temperatures have the same energyconversionefficiency, ii is sufficient
to calculate this cfiiciency for any particular device to find the common value.TheCarnotcycledeviceleadsloi;c=
(xh~
ri)/rft for the common value.
Path Dependenceof Meat and Work
We have carefully used the wordsheat and work to characterizeenergy transfer
processes, and not to characterize properties of the system iiself. It is not
meaningful to spe;ik of the heat contentor of the work content of a system.We look at the Cinnot cycle once more: Around a closed loop in the p-V
plane, a net amount of work is generatedbythe system, and a net amount of
heat is consumed. But the system\342\200\224on being taken once around Ihe loop\342\200\224is
returned to precisely the initial condition; no property of the system haschanged.Thismeans that there cannot exist two functions Q{a,V)and W{a,V)
Path Dependence of Ihat and Work
Figure 8.7 If uvo different reversible energy conversiondevicesoperating between the
same temperatures tj, and r, coutd have diiTercni energy conversion efiiciencies f'jj > >ji
h would he possible to combine them into a single device wilti 100 pd cfikictio '\342\226\240>\342\226\240
using the lessdiicienl deuce ! us a refrigerator that moves not only ihc eniire w;i>:cficas Qti ofstie more efficient device 2 backto the higher semporaiure, bui an addiriotiii
amount Q[in) of licaE us wt;il. Tlifs additionat huat would shcii be completely convertedto work.
sudi th;n the heal Q^ and the work Wub required to curry the system from a
siSilc(aa,VB)toa slate {ab,Yb) are given by the differences in Q and W:
If such fimcitons extsicd, the nut transfers of heat and of work aroumla closedloopnecessarily would be zero, ;md we h:ivc shown that the transfers ;tre not
zero.
The transfers of hc:itand work between state (a) ;ittd slate (b) depend on the
path taken between the two states. This path-dependenceis expressedwhen
we say (hat hwt and work' are noi siaic functions. Unlike temperature, entropy,
and free energy, heat and work are not inlrtnsic atlributes of (he system. The
increments dO and (fIS'that we introduced in (l)and B) cannot bedillerentials
Chapter 8: that and Work
Figure 8.8 Twotrr sibieproceisesin which mechanical or electrical potemial
of mathematical functions Q(a,V)and \\V(o,Vy For this reason we designatedthe increments by dQ and d\\V, rather than by itQ and dW. Without the path
dependence of heat and work there would not exist cyclical processes that
permit the generationof work from heal.
Irreversible Work
We consider the energy transfer processes of Figure 8.8. In each process &i
is a purely mechanical or electrical system that delivers pure work with zero
entropy change. The eneray transferred to &2 is converted to heat, either by
mechanical friction or by electrical resistance. The finai state of S2 ^ J^esame as if the energy had been added as heat in the first place. The entropy
of &2 is increased by do2\302\253
ilU2fc- This entropy is newly created entropy.Processesin which new entropy is created are irreversible becausethereisno way to reverse the process in order to deslroy the newly created entropy.
If newly created entropy arises by the conversion of work to heat, we say that
irreversible work has been performed.If we look only at the net change in a system, there is no way to tell whether
the processthat led to this change was reversible or irreversible.For a change
dU in energy and i\\a in entropy, we can define a reversibleheatdQICV
and a
reversible work tfWtev as the amount of heat and work that would accomplishthis change in a reversible process. If part of the work done on the system is
irreversible, the actual work required to accomplish a given change is larger
Ineveniblt Work
than the reversible work,
tn\\'il
By conservation of energy
dU =if\\Viim +
so that
The actual heal transferred hi the irreversibleprocessmust be less ihan the
reversible heat.
Example:Sudden expansion of an ideal gas. As an example of an irreversible processwe
consider once more llic sudden expansion of an ideal gas info a vacuum. Neither heal norwork is transferred, so thai ttU *= 0 and dx = 0-The final stale is identical with the stale
that refills from a reversibleisothermai expansion with the gas in Ihermai equilibriumwith a reservoir. The work IV,,, done on the gas in the reversible expansion from volumei\\ :o Vj is, from F.57).
\\V,n= -Nilog(^/K,). C0)
The work done on the gas is negative;the gas does posittve work on the piston in an amount
equai to the heal transfer into the system:
Q[cy= -W[cy > 0; Wtet < 0. C1)
The entropy change is equal lo Q,cJt, or
a1-al'= ~Wlcy/x= NlogfiyV,). C2)
In the ineversible processof expansion into the vacuum this enlropy is newly created
entropy because neiiher heat nor work Rows into the system from the outside: lVitltt*=
Qi\302\253c.= 0- From C1) we obtain
in agreement with B8) and B9)-
Chapter8: Heat and Work
Figure 8.9 Systems between which on!)' work but noheat is transferred need not be at the same temperaturefor the process to be reversible.
In our discussionof irreversible work we assumed that the new entropyis createdinsidethe system during the delivery of work Jo the system by other
systems. This is nol she only sourceof irreverstbility in energy transfer. Pure
heat transfer, nol involvingany work, is irreversible if it takes place betweeniwo systems having different temperatures. We worked out an example in
Chapter 2. In this process heat fs transferred from a system a! rt to a systemai the lower temperature t2. We have.'.
i!U2 =\342\226\240<fQt 0. C4)
The newly created entropy is
= (I/I, -1/I2)rfe, -ii.re.. C5)
The heal flow is from high to low temperature; i!Qt is negative; r3 - ri is
negative, so that chitT > 0. .The energy transfer between two systems with different temperatures need
not be irreversibleif only work but no heat is transferred(Figure8.9).AH actual energy transfer processes are invariably somewhat irreversible,
but reversible processes remain the backbone of the theory of thermalphysics.They
constitute a natural limit, which is the equilibrium limit of vanishing
entropy generation. We shall assume hereafterthat the words heat and work,
without a further qualifier, referto reversibleprocesses.
Heal and Work at Constant Temperature or Constant Pressure
HEAT AND WORK AT CONSTANT TEMPERATUREOR CONSTANTPRESSURE
Isothermal work. We show ihai the ioia! work performed on a system in
a reversible isoiherma! process is equai io Ihe increasein the Heimlioltz free
energy F ~ V \342\200\224za of ihe system. For a reversible process tfQ= ida ~ d{ia),
because dx = 0, so that
dW = dV ~ dU - d(za) \302\253dF. C6!
Thus in such processes the Helmhoitz free energy is ihe natural encrgc:;';
ftiuciion, more appropriate than the energy V. When we treat an isotEicrltui
process in terms of (he Hclrnholiz free energy, we automatically include ihe
additional work that is required to make up for the heat transfer from the
system to the reservoir. Often the heat transfer is ihe major part of the work;for the ideal gas the energy V does noj change in an isothermal process,andthe work done is equal lo the heat transfer.
Isobaricheat and work. Many energy transfer processes\342\200\224isothermal or not\342\200\224
lake place at constant pressure, particularly those processes that take place
in systems open to the atmosphere.A process at constant pressure is said to bean isobaricprocess.A simple example is the boiling of a liquid as in Figure 8.10,
n
F=/<\342\200\236<<
Figure 8.10 When a liquid boils under aimospheru:pressure,the vapor
displacing the aimosplierc does work againsi ihe atmospheric pressure.
ChapterS: Heal and Work
where the pressure on the piston is the external atmospheric pressure. If (hesystem changes its volume by dV, the work -pdV = \342\200\224
d{pV) is part of thetotal work doneon the system. If positive, this work is provided by the environ-
environment and is in this sense \"free.\" if negative, the work is delivered to the
environment and is not extractablefrom the system for other purposes. For
thjs reason it is often appropriate io subtract ~d(pV) from the total work. We
thus obtain the eflccthe work performed on the system, defined as
ilW = itW + d{PV)= itU + d{pV) -
itQ= till ~ HQ , C7)
where we have defineda new function
H = V + pV , C8)
calledthe enthalpy which plays the role in processes at constant pressurethai
the energy V plays isi processes at constant volume.The term/it7in C8) is the
work required to displace the surroundingatmospherehi order to vacate
the space to be occupiedby the system. Implicit in these definitions is the ideathat there are other kinds of work besides ihut due to volume changes.
Two classes of the constant pressure processesare particularly important:
(a) Processes in whichiiodTcctive work is done.Thehcattransfcrist7Q~ <///,
from C7). The evaporation of a liquid {Chapter 10) from an open vessel is
such a process, becauseno effective work is done, The heal of vaporizationis ihe enthalpy difference between the vapor phase and the liquid phase-
(b) Processesat constant temperaiure and constant pressure. Then iJQ =*
if/a ~ d{xa), and the effective work performedon the system is, from C6) and
C7),
d\\V => dF + d{pV) => dG , C9)
wherewe have definedanother new function
G ~ F + pV \302\253U + pV- xa , D0)
the Gibbs free energy. The effective work performed in a reversibleprocessat constant temperature and pressure is equal to the change in the Gibbs
free energy of the system. This is particularly useful in chemical reactions
where the volume changes as the reaction proceeds at a constant pressure.The Gibbs free energy ts used extensively in Chapter 9, and Ihe enthalpy is
used in Chapter 10.
Heat and Work at Constant Temperature or ConstantPressure
isobaricConsider an ciccHoiylc of dilulc sutfurie acid iti which arc immersed philiiuiin
electrodes ilia t do noi renet w&in lnc acid [r igtirc s. 11J. Tnc ^titfiir tc sci^i dissocjaics in i o
H+ ant) SO4~~ ions:
H2SOa \302\2532H+ + SO.,\"\" D!)
When a current is passed through the ceil lhe hydrogen ions move to lhe negaihe clecirodewhere llicy lake up electrons and form molecular hydrogen g;is:
-+ H2. D2)
The sulfatc ions move to the positive electrodes where they decompose water with therelease of molecular oxygen gas and clocirons: _.
SO4~ + H;O - H3SO4+'iOj + 2c!^. D3)
The sum of the above liirccstepsis the net rcucikm ctttuiluin in lite cell;
H2O-\302\273 H2 + 1O3. {44}
Whch carried out slowly in a vessci open to lhc almosplicrt!, ihc prticessis at conslain
prcsburcaiKicoitMiiiil temperature. A negligible pjft of the olcciricul injml power goes iulo
Figure 8.11 An electrolysis cell An electrical current passes through an electrolyte, such
as dilute sulfurtc acid. The overall result is the decomposition ofwaterinto gaseoushydrogen
and oxygen. The process is an example of work being done at constant
temperature and constant pressure. . .
resistance heating of the electrolyte. The effectivework required to decompose1moleof
water is related to the molar Gibbs free energies of Use reactants:
W => AG =G(H2O)
- G(Hj) - \\G{02). D5)
Chemical tables list the Gibbs free energy differenceAG as - 237 kJ permole at room
temperature.
In electrolysis tins work is performed by a current / that flows under an external voltageVo. If I is the time requiredto decomposeone male of water. Q \302\273I x I is lhe total charge(not the heat!) flowing tlirough the cell, and we have
W ~QV0. D6)
According to D3), there are two electrons involved in decomposing one water molecule,
llcnce
Q = ~2/v> = -1.93 x 10scoulomb. D7)
We equate D6) to D5) 10obiain the condition for electrolysis lo take place.This requiresaminimum voltage
. Vo= ~AC/2NAe , D8}
or 1.229volts. A voltage larger lhan Vo must be applied to obtain a finite current flow,
because VQ alone merely reduces to zero the poierciuit barrier between the systems on lhc
two sides of the reaction equationD4).When. V > Va, the excess power (V-
VQ) x /will
be dissipaicd as heat in trie electrolyte.
If V < Vo, the reaction D4) will proceed from right to Icfl provided gaseoushydrogen is
available at the positive electrodeand gaseous oxygen at lhc negative electrode. In lhesimplesetupof Figure 8.11 the gases are permitted to escape,and for V < Vo nothing will
happen al all. Ev is possible, however, lo construct the electrodesas poroussponges,wiih
hydrogen and oxygen foccedthrough under pcessure (Figure 8-12). Such adevice producesa
voltage f'o between the electrodes and, tf the electrodes arc connected, eucnwl current will
{law. This arrangement is called a hydrogeci-osygen fuel cell. Fuel cells were usedas power
sources on board the Gemini and Apoilo* ipacccta.fiand incidentally produced dtinking
water for the astronauts.
The principal technological limitation of fuel cells is their low current, per unit electrode
area. In the Apollo cell the current density was only a few hundred mA/cmJ; hence largeelecirode areas are required lo produce reasonable currents. The current-voltagecharac-characterise of an clearochcmi'cal ceil in Us two operating ranges as fuel cell and as electrolyticcell are shown in Figure K.I 3.
\342\200\242The Apollo End criU used Ni and NiO uniict ibn Pt as decltoiici. ami K.OH racier sVian H .SO\302\253
to J. O. M. Bockrisand S. Sriili^asan, fwf cells: Their electrochemistry, McGraw-Hill, New York,1969.
Porous electrodes
Figure 8.12 A fuel ecl! is an electrolysis celloperated in
reverse, with hydrogen and oxygen supplied as fuels. The
fuels arc forced under prow tin: through porous dmroOes
sqwratcJ by an electrolyte The hydrogen and oxygen reato form waicr; the excess Gibbs Treeenergy is deliveredoutsideas electrical energy. Water forms at the positiveelectrodeand is removed there.
Elcnrolysis\342\200\224
Figure 8.13 Tliecui
celt or fuel cell, indie
ic of aa electroiyiic
anges.
ChaptcrS; Heal and Work
Chemical Work
Work performed by the transfer of particles to a system is calledchemicalwork, because it is associated with the chemical potential.
When particles are transferred, the numberof particlesin the system is one
of the independentvariableson which the energy U depends. If U = U(atV,N),then for a reversible process
dU = xda -pdV + fidN , D9}
by tlie thermodynamic identity of Chapter5.Herewe have replaced the partialderivatives by their familiar equivalents (Table 5.1). By our definition of heat,the zdo lerm represents Ihe transfer of heal and the ~pdV and fidN terms
represent the performance of work, ali understood to be reversible:
<tW ~ -pJV+ pJN. E0}
The -pdV.termismechanical work; the pdN term is the chemicalwork:
dWc= ftdN. . E1}
If there is no volume change, dV = 0. All the Work is chemical,In particletransfer there arc usually two systems involved, both in contact
with a heat reservoir, and the total chemicalwork is the sum of the contributionsfrom both systems. In the arrangement of Figure8.14a pump transfers particles
from system &x to system .&;. The chemicalpotentialsare /it and p2. If
dN =i dN2 \342\204\242\342\200\224dN, is the number of particles transferred, the total chemical
work performed is
dWc = OWcl + d\\Vcl= fadNt + pldN2 = (p2 -
nt)dN. {52}
The work that must be supplied to the pump is a\\Vr if there is no volume work(dV1
*= dVi = 0), and if all processes arc reversible
The result E2) givesart additional meaning of the chemical potential. We
summarize the propcriics of ihe chemical potential:
{a} The chemical potential of a system is the work required to transferoneparticle into the system, from a reservoir at zero chemical potential.
(b) The difference in chemicalpotentialbetween two systems is equal to thenet work requiredto movea particlefrom one system to the other.
Figure 8.t4 Chemical work is the work performed when particles arc moved
rcs'ersibly from one system to another, with the two systems having different
chemical potentials. ITilie two volumes do not change, ihe work is purechemical work ; the amount per pariicie is the difference in chemical potentials.
{c} If ihe two systems are in diffusive equilibrium ihey have the same chemicalpotential;no work is required io move a particle from one syslem 10
the other,
{d} The difference in internalchemicalpotential {Chapter 5} between two
systems is equal but opposite to the potential barrier thai maintains thesystems in diffusive equilibrium.
Example i Chemical work for an ideal gas. We considerlhc work per particle required tomove revcrsibly ins atoms of a monatomic idea! gas from -Sj wuh concentration t\\^. to -S.with concentritiion n2 > ii,, both sysiems being al the same leinpcrntnre tFtgure 8.151 IfdV \302\2530, the work contains only a chemical work term, which cati be calculated from ihedifference in chemical potential, no matter how the process is actually performed. Thechemical potential difference between two ideal gassystems with different concentrations is
02 -/fl
=t[10gOl2/\302\273iQ)
- logOlj/Hfl)] = Tlog0l2/Jl,}. E3)
This rcs\\ih is cqtial io lite n\\cch:tnit;tl work per particle required io compress the gas
isothciniiiily from the concentration \302\253jto the concern mi ion ux. Tltc work required iocompressN particles of an ideal g;ts from an initial voluitw i\\
to a littal volume ''. is
W --\302\247pdV
=-Nx^tiVjV^ NTlog{IV^} = WtlogfHj/ii,}. {5A}
Hence the mechanical work per particleis iSogtoj/iij),identical to the result E3). The
ChapterSt Heat and Work
Reservoir'
-ilnergy exchanges-
Figure 8.15 Isothermal diemiciit work. The amount of chemical work per panicledoes not change if the process is performed isoihermallywith both sysiems inthermal equilibrium wiih a common targe reservoir.
ideniiiy of the chemical work wiih the isoihcrmal compresslence or convcriibiiiiyof dilferem kinds of work.
illustrates the equi-
MagneticWork and Superconductors ,-
An important form of work is magnetic work.Themostimportant application
of magnetic work is to superconductors, and this application is treated here.
Below some critical temperature Tc that is usually less than 20K, manyelectricalconductorsundergoa transition from their normal state wiih a finite
electrical conductivity to a superconducting slate wiih an apparently infinite
conductivity.
Superconductors expel magnetic fields from their interior. If the super-
superconductor is first cooled below the critical temperatureand ihen inserted into a
magnetic field, we might expect that the infinite conductivity would shield the
interior from the penetration by a magnetic field. However, iheexpulsionoccurs
even ir i|ie superconductor is cooled below Tcwhile in a magneticfield (Figure
8.16). This active expulsion, called ilic Metssncreffect,shows that supefcon-
duciivity is more ilian an infinite conductivity.TheMcissncreffect is caused by
shielding currents lhat are spontaneouslygeneratednear the surface, in a layerabout lG~5cm iliick. The magnetic field expulsion is not always complete.Superconductors aresaidtobeof type
It if the expulsion is incomplete, but stilt
nonzero, in a range of fields above some low field. We shall restrict ourselves
ark and Superconductors
(
)
Hgure 8-16 Mcissner effect in a supercon6tiding sphere cooledin a constant applied magnetic field; on passingbelow ihe
transition temperature tlic iiiics of induciion B arc ejectedfrom
the sphere.
In
TiX
Figure 8,17 Thresholdcurves
fidd versus lempcratui;: for scvconductors. A specimen \342\226\240i;jpe
below ihe curve and noiii..;! iib
Temperature, in K
,-->I[-lOUl fill 111
Magnetic I Curt and Superconductors
Figure 8.19 A superconductor
arcu A in n superconducting sol
produces a magnetic Ikld B.
transition it is the electronic system rather than the crystai structureof the metai
that undergoes a phase transition.The superconductingstaleis a distinct .<hermodynamic phase, as confirmed
by differences in the heat capacity of the normai and the supcrconduciingstales.Theiicai capactiy (Figure 8.1 S) cxiiibhs a pronounced discontinuity at iheonsetof superconducifvhy -di x = xt; when superconductivity is desiroyed by a
magnetic field, ihe discontinuiiy disappears. The stablephasewill be the phase
with the iower free energy. Beiow t = tc in zero magnetic fieid the free energyof liie superconductingphaseis lower than that of the normal phase. The freeenergy of the superconduciingphaseincreasesin the magnetic field, as we showbeiow. The free energy of the normal phase is approximately independent ofthe field. Eventually, as the fieid is increased, the free energy of the super-superconducting phase will exceed that of the normal phase.The normaiphase is
ihen the stable phase, and superconductivity is destroyed.The increase of the free energy of a superconductor in a magneticfield is
calculated as the work required to reduce themagneticfield to zero in the interior
of the superconductor; the zero value is required to account for the Meissnereffect.Considera superconductorin the form of a long rod of uniform cross-sectioninsidea long solenoid that produces a uniform field B,as in Figure 8.19.
The work required to reduce the field to zeroinsidethe superconductor is
equai to the work required to createwithin the superconductor a counteracting
field - B thai exactlycancelsthe solenoid fteld. We know from electromagnetic
theory that the work per unit volume required to create a field B is given by
(SI) B2/2{t0\\ E5a)
(CGS) E5b)
Chapter Si Heal and Work
Figure 8.20 Tlic free energy dcnsiiy F* of anoi\\ti\\^i syndic t\\otii\\3.t meiat is appcox.iiv\\u.lcty
independent of [he intensity of the appliedmagnetic fkUi $\342\200\236.Al a temperature x < xt the
melai is a superconductor in zero magnate
field, so dial Fsti,O)is lower than F^i.O). An
applied magnetic field increases Fs by Bj/l/i,,,in SI unite land by BBJ/8n in CGS uniis), so
lhai Fs(r,/iJ =\342\226\240F5(r,0) + Ba3/2j@. If Bu is
larger liian ihe criikjil field BM ihe free energydeiisiiy is lower in ihe normal siaic than in
the siiperconduciing siaic, and no* ihc nornmt
si ale is ide siabic st,iie.The origin of i tic
vertical se;ilc in ilic drawiiij; is ai f Jr-O)-The
ftgute equally applies 10 U5 und Us ii\\ t = 0.Applied magnelic field Ba
Tiiis is the amount by which the free energy density in the bulk superconductor
is raised by application of an externalmagnetic field, in an experiment at con-
constant temperature.
There is no comparable free energy increase for the normal conductor,
because there is no screeningof the applied field. Thus
(SI) E6a)
{CGS) E6b)
En a plot of the free energy density of both phasesversus the magnetic field
{Figure 8.20), tne free energy of the superconductingphasewill ultimately rise
above that of the normal phase,sothat in high fields the specimen will be in thenormal phase,and the superconducting phase is no longer the stable phase.This is the explanationof the destruction of superconductivity by a critical
magnetic field Bf.
Wiih increasing temperature the free energy differencebeI ween normal and
superconducting phase decreases as t --\302\273rr, and the critical magnetic field
decreases. Everything else being equal, a high stabilization energy in a type I
superconductor will lead to boilia high critical temperature and a high critical
field. The highestcriticalfields are found amongst the superconductors with the
highest crilical temperatures, and \\ice versa.
SUMMARY
1. Heat is the transfer of energy by thermal contact with a reservoir. In areversible process dQ = xda,
2. Work is the transfer of energy by a change in theexEerna! parameters that
describe the system. The entropy transfer in a reversibleprocessiszero when
only work is performed and no heat is transferred
3. The Carnot energy conversion efficiency, j;c =(zH
\342\200\224tJ/tj, is the upper
limit to t!ie ratio WjQh of the work generated to the heat added.
4. The Carnot coefficient of refrigerator performance, -,'c ~ t,/(ta -r,), is the
upper limit to the ratio QijW of the heat extracted to the work consumed
5. Thetotalwork performed on & system at constant temperature in a reversible
process is equal to the changein the Hclmholtz free energy F se U -ro-
roof the system.
6. The effective work performed on a system at constant temperature and
pressure in a reversible processis equal to the change in the Gibbs free
energy C = U - w + pK.7. The chemical work performed on a system in the reversible transfer of t.lN
particles to the system is pdN.
8. Tlicchangein the free energy density of a superconductor {oftype {) caused
by an external magnetic field B is B2/2{i0in SI and S2,8;i in CGS.
PROBLEMS
/. Heat pump, (a) Show that fora reversible heal pump tile energy requiredperunit of heat delivered inside the building isgiven by the Carnoi efficiency F):
Chapter S: Heat and Work
2. Absorption refrigerator. In absorption refrigerators the energy driving theprocessis supplied not as work, but as heal from a gas flame at a tempenisurei(,h > xh. Mobile home and cabin refrigeratorsmay be of this type, with propane
fuel, (a) Give an energy-entropy flow diagram similar to Figures 8.2 and 8.4 for
such a refrigerator, involving no work at all, but with energyand entropy Hows
at the three temperatures rfch > ta > ij. (b) Calculate the ratio QJQhh, for ihe
heat extracted at r \302\253r,, where QM is the heat input at r =
xhh. Assume reversible
operation.
3, Photon Camot engine. Considera Carnotengine that uses as the workingsubstance a photon gas. (a) Given i,, and r, as well as Vl and V2, delerwne
Vi and VA. (b) What is the heat Qh taken up and the work done by thegas during
the first isothermal expansion? Are they equal to eachother,as for the ideal gas?
(c) Do the two isentropic stagescanceleach other, as for the ideal gas? (d)Calcu-Calculate the total work done by the gas during onecycle.Compare it with the heat
taken up at rh and show that the energy conversion efficiency is the Carnot
efficiency.
4. Heat engine\342\200\224refrigerator cascade. The efficiency of a heat engine is to beimproved by lowering the temperature of its tow-temperature reservoirto avaluer,.,below the environmental temperature r,, by means of a refrigerator.Therefrigeratorconsumes part of the work produced by the heat engine.Assume
that both tlie heat engine and the refrigerator operatereversibly.Calculatethe
ratio of the net (available) work to the heat Qh supplied to the heat engine attemperature ift. Is it possible to obtain a higher net energyconversionefficiency
in this way?
5. Thermal pollution. A river with a water temperature T} = 2CTC is to be
used as the low temperaturereservoirof a large power plant, with a steamtemperature of Th
= 500JC. If ecological considerations limit the amount ofheat that canbedumped into the rtver to \\ 500 MW, what is the largest electricaloutput that the plant can deliver?If improvements in hot-steam technologywould permit raising Th by lOQ'Q what effect would this have on the plantcapacity?
6. Room ah conditioner. A room air conditioner operatesas a Carnot cycle
refrigerator between an outside temperature Th and a room at a lower tempera-temperature7\"j. The room gains heat from the outdoors at a rate A[Th
\342\200\224T,); this heat
is removed by the air condilioner.The power supplied to the cooling unit is P.
(a) Showthat the steady state temperature of the room is
T, \302\253(Th + PjlA)-
[<Tfc4- P/2AJ
- T,,2]1'2.
(b) ir the outdoorsis at 3VC and the room is maintained at 17\302\260Cby a cooling
power of 2kW, find the heat losscoefficient A of the room in W K\021.A good
discussion of room air conditioners is given by H. S. LefT and W. D. Teeters,Amer.J. Physics 46, 19 A978). In a realistic unit the cooling coils may be at
2S2 K and the outdoor heatexchangerat 378 K.
7. Light bulb in a refrigerator, A 100 W light bulb is left burning inside a
Carnoi refrigerator that draws tOOW. Can the refrigerator cool below roomtempera!ure?
S. Gcoihermal energy. A very large mass M of porous hot rock is to beuiilized to generate electricity by injecting water and utilizingthe resulting hot
steam to drive a turbine. As a result of heat extraction, the temperature of iherockdrops,according to dQh
= -~MCdTh, where C is the specificheat of the
rock, assumed to be temperature independent.If theplainoperatesat the Carnoi
limit, calculate ihe tola! amount W of electrical energy estractable from therock, if the temperature of the rock was initially Th
= T,, and if the plant is tobe shut down when the temperature has dropped to Th
=Tf. Assume that the
lowerreservoirtemperatureT,stays constant. -.
At the end of the calculation, give a numerical value, in kWh. for M - 10u kg(about 30km3). C=Hg\"!K\"\"', T,-=600C, 77
- 110 C, 7\",\302\27320 C.
Watch the units and explain all steps!Forcomparison:The total electricity
produced in the world in 1976was between1 and 2 times 10!4 kWh.
9. Cooling of nonmetatlic solid to T~Q. We saw in Chapter 4 that the heatcapacity
of nonmetallic solids at sufficiently low temperatures is proportionalto Ti, as C
~ aT3. Assume it were possible to cool a piece of such a solid toT \302\2530 by means of a reversible refrigerator that uses the solid specimen as its
(varying.) low-temperaturereservoir,and for which tile high-temperaturereservoir has a fixed temperature 1\\equal to the initial temperature Tj of the
solid. Find an expressionfor the electrical energy required.
10, ltrerersibte expansionofa Fermtgas. Considera gas of N noninteractiilg,
spin \\ fermions of mass A/, initially in a volume V;at temperature if
~ 0. Lei the
gas expandirreversibly into a vacuum, wiihout doing work, to a final volume
V}. What is the temperature of the gasafterexpansionif Vfis sufficiently large
for the classicallimit to apply? Estimate the factor by which the gasshouldbeexpanded for its temperature to settle to a constant final value. Give numerical
values for ihe final temperature in kelvin for two cases: (a)a particlemassequal
to the electron mass, and NjV = 10\"cm\023,as in metals; (b) a particle mass
equal to a nucleon,and N/V = 10JO,as in white dwarf stars.
Chapter 9
Gibbs Free Energyand Chemical Reactions
GIBBS FREE ENERGY 262
Example: Comparison of G whh F 265
EQUILIBRIUM IN REACTIONS 266
Equilibriumfor IdealGases 267
Example: Equilibrium of Atomie and Molecular Hydrogen 269Example;pH and the Ionszatton of Water 265
Example: Kinetic Modelof Mass Action 270
SUMMARY 272
PROBLEMS 272
1. ThermalExpansionNearAbsolute Zero 272
2. Thermal lonization of Hydrogen 2733. lonizationofDonorImpurities in Semiconductors 2 73
4. Hiopolynicr Growth 2/35. Patticlu-Antiparticle Equilibrium
274
Chapter 9: Gibbs Free Energy and Chemical Reactioi
GIBBS FREE ENERGY
The Helmholtzfree energy F introduced in Chapier 3 describes a system atconstant volume and temperature. But many experiments, and in particularmany chemical reactions, are performed at constant pressure, often one aimospherc, h is useful to introduce another function to treat the equilibrium
configuration at eonslant pressure ;tnd temperature. As in Chupicr 8, wedefine the Gibbsfree energy G as
G a U - xa + pV. A)
Chemists often call this the free energy, and physicists often call tt the
thermodynamie potential.The mostimportantproperty of the Gibbs free energy is thai it is a minimum
for a system S in equilibriumat constant pressurewhen in thermal contact
with a reservoir (H.The differential of G is
dG = dU ~ ida -cdx + pdV + Vdp.
Consider a system {Figure9.1)in thermal contact with a heat reservoir <Rl
at temperature t and in mechanical contact wilh a pressure reservoir<R2 that
maintains the pressure p, but cannot exchangeheat. Now dt = 0 and dp = 0,so that the differential dG oflhe system in the equilibrium configurationbecomes
dGi =dUs
- xdax + pdV*
The thermodyiiamicidentity E.39) is
zdds = dUi -ttdNi + pdVs,
so that B) becomes dG^ \302\253ndNx. But dNs = 0, wh ence
B)
C)
dGj \302\2730 ,
i eser
m,
\342\226\240oir
. System
\342\226\240\342\226\240\342\226\240v\"'Jif
Hea rese rvoir
Pressure
NPSu
qualiz
cserv
-
ger loe pressure
v.hh a heat reservoir and in mechanical
equilibrium with a barysiai or pressurereservoir whkh maintains a constant picon Hie system. The barysial is thermally
insuSalcd.
which is tlic condition for Gj to be ;m extrcmum wilh respect to system varia-variations at constant pressure, temperature, and particle number. Theseare,there-
therefore, the natural variables for G(N,z,p).That ihe extremumof G3 musi be a minimum, raiher than a maximum,
follows direciiy from the minus sign associated w'Hh the eniropy in (i); Any
irreversible change laking place entirely wiihin & will increase a and ihusdecrease Gs.
With B),
- adx + E)
The difierentiai E) may be writicn as
Comparison of{5)and F) gives the relations
0)
(8)
Chapter 9: Gibbs Free Energy and Chemical Reactions
, = V. E)
ThreeMaxwell relations may be obtained from these by cross-differentiation;seeProblem 1.
In the Gibbs free energy G = V \342\200\224za + pKthevariabfesr and pare intensive
quantities: they do not change value when two identical systemsare pui together.
But U, a, V, and G are linear in the number of particlesA':their value doubles
when two identical systems are put togeiher,apart from interface effects,. We
say that V, a, V, A7 and G are extensive quantities. Assume that only one particle
species is present. If G is directlyproportionalto N, we must be able to write
G =Nip(p,x) , A0)
where ep is independent of N because it is a function only of the intensive
quantitiespand r. If two identical volumes of gas at equal pressure and tempera-temperature,c;idi wjih j.V molecules, iire |>ui together, the Gibbs free energy
does not change in the process. It follows from this argument that
We saw in G) that
(II)
A2)
so that <p must be identical with ;j, and (iO)becomes
G(,V,p,t) A3)
Thus the chemical poiemiaf for s single-componentsystemis equal to 'h
Gibbs free energy per particle,G'/A'.I or G for an ideal gas, sec B1)below.
If more than one chemical species is present, A3) is replacedby a sum ov
all species:
A4)
GibbsFreeEnergy
The ihermodynamic identity becomes
xda = dV + pdV-
^e/INy, A5)
and E) becomes
dG \302\253\302\243njdNj
~ adz + Vdp. (!6)
We shall develop the theory of chemical equilibria by exploiting the property
that G ss:YJ^jfif
ts a minimum with respect to changes in the distributionofreacting molecules at constant t, p. No new atoms comeinto the system in a
reaction; the atoms that are presentredistributethemselves from one molecular
snucics to another molecular species.
iiiutupfri Omipurhtw \302\273fti with F. Let its sttf wluit is Jitiacut :ihuuJ \\\\vi iwo rd.ttions
{cFfdN)tty = p(iVrT,K) A7)
and
$G/dN)tiP*= fi{T,p). (IS)
We found in F.18) lM for an ideal gas
MN.r.F) -Tlog(WKnQ) , A9)
so that ti(N,z,l') is not independent of N and therefore we cannot write F = JV;i(r,f) .isSheiaiegra!of{S7).
Thai is, f is not direciiy proportional lo N if ilie system is kept at cotislanl volume as ihc
number of particles is increased.Instead,from F.24),
F(t,V,N)^ Ni[\\og(NJVnQ)- I]. B0J
But she Gibbs free energy for t-he ideaJ gas is
C(r,/\302\273,N)\302\273F + PK
= Arr[!og(p/rny)- l] + -Vr
\302\273.NrIog(;j/THQ) , PO
by use of ihe ideal gas Saw in the form N/V ==p/r. We readily identify in B1) the chemical
potential us
/((t,p) = ilo\302\243(pJxnQ) , B2)
Chapter 9; Gibbs Free Energy and Chemical Reactions
by reference to the result G \302\253iVjih.pV We see [hat N appears unavoidably in sit.V) in
A9), but not in /j(t,p) in B2). The chemical potential is the Gibbs free energy per particic,but il isnol ihcHctnihoHz free energy pet panicle. Of course, we are free lo wrile p its cilher
A9) or 122},as is convenient.
EQUILIBRIUM IN REACTIONS
We may write the equation of a chemicalreactionas
v,A, + v2A2 + \342\226\240- \342\226\240+ v,A, = 0 , B3)
B4)
where the Aj denote the chemical species, and theVj
are the coefficients of the
species in the reaction equation. Here v is the Greek lettermi.For the reaction
Hj + Clj = 2HC1 we lave
Ai= H,; A, - Clj; Aj
= HC1; Vi=l; fj=l; \302\273j=-2.
B5)
The discussion of chemical equilibria is usually presented for reactions under
conditions of constant pressureand temperature. In equilibrium the Gibbs free
energyisa minimumwith respect to changes in the proportions of the reaclants.The differential of G is
dG - Z fj^i -adz + Vi!p. B6)
Here//j is the chemical potentialofspeciesj, asdefined by \\is=
{BGJdNj)s-p. At
constant pressure dp = 0 and at constant temperaturefa = 0;then B6) reduces
to
rfC-1/,/W,. 07)j
The changein the Gibbs free energy in a reaction depends on the chemicalpotentials of the re\302\243tctbnts\302\273Xti comlit^rtufii G is &i cxtrcmum miu uu musl tjo
zero.
Equilibrium for Ideal Gases
The change dN} in the number of molecules of species/ is proportionalto the
coefficient v,m the chemical equation ]>>/.; - 0.We may Write dN} in the form
JNj \302\253rjd$ ,
\"
B8)
where dfif indicates how many timesthe reactionB4) takes place. The change dGin B7) becomes
dG \302\273I vjfi dft- B9)
In equilibriumdG = 0, so that
C0)
This is the conditionfor equilibrium in a transformation of matter at constantpressureand temperature.*
Equilibrium for Ideal Gases
We obtain a simple and useful form of the genera! equilibrium condition
Y^Vjf-j= 0 when we assume that each of the constituents acts as an ideal gas.
We utilize F.48) lo write the chemical potential ofspeciesj as
fij-
rflogMj- logc;) , C1)
wherenj
is the concentration of species j and
cj a\302\253QJZ/int) , C2)
which depends on ttie temperalure but not on ttie concentration.HereZ/int)is the internal partition function, F.44). Then C0) can be rearrangedas
5\302\273gnj= $>,!(*<:, ,
\342\226\240
C3a)
* Uul ihcmull is moregencrat; onceequilibrium is reached, ihe rcaclion does nol proceedfurihi:r,
when p and t arc specified..
Chapter9: Gibbs Free Energy and Chemical Rcactia
The left-hand sidecan berewritten as
and the right-hand side can be expressedas
C3b)
C3c)
C3d)
Here K(t}, called the equilibrium constant, is a function only of the temperature.
Wiih{32)\\vehave
K{t) snil\302\253/' C4)
because the internal free energy is Ffim) = -TiogZj{int).From C3c,d) and
C4) we have
Fk-Vj C5)
known as the law of massaction.Theresult says that the indicated product of theconcentrations of the reactanlsis a function of the temperature alone. A change
in the conceniraison of any one reaciam will force a change in the equilibriumconcentration of one or moreofthe other reaclants.
To calculate the equilibrium constant K(x}in C4), it is essential to choose in
a consistent way the zero of the internal energy of each reactant.We need
consistency here because the value of each partition functionZji'mi) depends
on our choice of the zero of the energy eigenstales. The different zeros for thedifferent reactants must be related to give properly the energyor free energy
difference in ihe reaction. It is not diilicuh to arrange ibis, but it docs not
happen without a conscious effort on our part. For a dissociation reaction suchas l\\2 ^ 2H, ilie simplest procedure is to clioosethe zero of the internal energyof each compositeparticle(here the H2 molecule} to coincide with the energyof the dissociatedpanicles(here 2H) a! res!. Accordingly, we piacc the energyof the ground state of the composiie particle at ~\302\243fl, where \302\2433is the energy
Equilibrium for IdealGases
requiredin the reaction to dissociate the composite particle into its constituentsand is laken to be positive.
Examplei Equilibrium of atomic and molecular hydrogen. The siatcmcnl of ihe law
mass aciioil for the rcacuon Hj = 2H or ttz~ 2H = 0 for the dissociaiion of molecul
hydrogen imo atomic hydrogen is
C6)
Here [llj] denotes ihc concenmuion of nlolecutar hydrogen, and fjt] i
of atomic h>drogen. li foltov-sihai
[h^tpo17^'l37)
lhai is, the relume concentration of atomic hydrogen at a given temperature is inversely
proportional lo the square rooi of ihe concent ration of molecular hydrogen.The equilib-
equilibriumcon:.lani K is given by
logK=
log^tHj)- 2IognQ(H) - F(U2)/i, C8)
in terms of the internal freeenergyof H,. per molecule. Spin factors are absorbed in F(H,}.
Here [he zero of energy is laken for an H atom at rest. The more lightly bound is H,, the
more negative is F{Ht),and ihe higher is Kt leading 10 a higher proportion of Hj in ihe
mixture. The energy to dissociate Ht is 4-476 eV per molecule, al absolute zero.It may be said lhai ihe dissociationof molecular hydrogen into atomic hydrogen is an
example of entropy dissociaiion: The gain En entropy associated wiih ihe decompositionof Hj into two independent particles compensates [he loss in binding energy. It is believedthat most of the hydrogen in intergalactic space is present as H and not Ht: The reaction
equilibrium is thrown in the direction of H by the low values of ilie concentrationof Hj.
Hjdrogcn is very dilute in intergalactic space.
Example: pi! and ihe limitation of water. In liquid waicr ihe ioni^lion process
H2O*~>H+ + OH\" 09}
eeds to a slight txicnt. Al room tcinpcraluic the reaciion cquilibfium i
o.ximalcly by she coiiccmraikm product
D0}
Chapter 9; Gibbs Free finersy and ChemicalReactions
where ihc jonac concentrations arc given in moles per liicr. inpurcualcr\302\243H*l\342\200\224
fOH \021 ^=
!0\021molr''. An acid is said lo aa as a prolon donor. The concentration of H* ions is^nefcasco oy aGGin& z\\t\\ iictu to llic wai^r \302\273^ntJihe concentration of C3i1 tons wtll decreaseas required lo mainiain the product [H*][O!!'] constant. Similarly, ihe concentrationof OH\"
ions can be increased by adding a hase lo ihe water, and ihe H*
concentration will
decrease accordingly. The physical siaie oi wafer is more complicated than ihe equationof ihe ionizaiion process suggests\342\200\224ihc H* ions ate not bare protons, but are associated
with groups* of H2Omolecules.This does not sigiltficantry affect ihe vulidily of ihe reaction
equation.it is ofien convenient lo express theacidily oraikalinii) otasolulionin letmsof ihe pti,
defined as
pH s -log10[H+]. D!)
The pH ota solution is ihe negative of the togarilhm base ten of iht; hydrogen ion concentra-
concentrationin moles per liier oCsolution.ThepH of pure water is 7 because [H*] = !CT7mair'.The slrongesi acidic sotuiions have pH near 0 oi even negative; an apple may have pH - 3.Human blood plasma has a pH of 73 lo 7-5; it is slightly basic.
Kxati\\ple2 Kttictic modelof/nms action. Suppose that atoms A and B combine to form a
moieculc AB. We suppose that AB is formed in a biaiomic collision of A and B. Let ha,
nB, nAB denote t!ie concentrations of A, B, and AB respectively. The rale of change of nAB is
where the me constant C describes the formation o[AB in a collision of A with B, and ihe
rale constant D describesthe reverse process, the ihemiat decay of AB into its componentatoms A and B. in thermal equilibrium the concermaiions of all consiiiutents ace constant,
so that du.a:di= Oand
nAB= D/C , D3)
a function of temperature only. This result is consistent with ihe law of mass action that we
derived earlierby standard thermodynamics.
Suppose AB is noi formed principally by the bimolecular collision of A and B, but is
formed by some catalytic process such as
\"The dominant species present is most likely it*1 4HjO, acomplc*of4 water molecules surround-
surroundingone ptolon. A [cvie* is given by M. Eigcn and L. Dc Macycr, P(oc_ Roy. Soc tLondon) A147.505A958). , \342\226\240
.
Equilibrium for IdealCases
Here E is ihe catalystwhich is returned to its original slate ai ihe end of the seeond step.So long as ttie intermediate product AE ts so short lived that no significant quantity of A
is lied up as AE, ihc ratio iiaiiu/iUb in equilibrium must be the same us if AB were formed in
the direct process A + B<--AB treated above. No rnaiier by what route Hie reaction
actually proceeds, ihc equilibrium must be the same. The rates, however, may differ.
The equality in equilibrium of the direct and inverse reaction rate's is culled the principle
of detailed balance.
Comment: Reactionrates. The law of mass action expresses ihc condition satisfied by ihe
concentrations once a reaction lias gone to equilibrium, li iclis us nothing aboui how fasi
ihc reaction proceeds. A reaciion A + B = C may evolve energy AH as it proceeds, bat
before the reaction can occur A and B may have 10 negotiatea potential barrier, as in
Figure 9.2. The barrier heigh! is called the activation energy. Only moleculeson the high
energy end of their energy distribution will be able to read; others will not be able to getover the potential hill. A catalyst speeds up a reaction by offeritig an alternate reaction path
with a lower energy of activation, but it does not change the equilibrium concentrations.
Schematic coordinate
Figure 9.2 The quantityAH measures ihe energy evolved in the reaction
nnd determines the equilibrium concentration ratio [A][B]/[C]. The
activation energy is the height of the potential barrier to be negotiated
before the reaction can proceed, and it determines the rate at which the
reaction takes place.
Chapter 9: GibbtFreeEnergy and Chemical Reactions
SUMMARY
1. The Gibbs freeenergy
G 3 U - iff + pV
is a minimum in thermal equilibrium at constant temperature and pressure.
2. (cG/3r)H,= -a; tfG/cph.,= V; (SG/SN),.P= p.
3. C(r,p,W) = W,,(r,ri4 The law ofmass action for a chemical reaction is that
IK'- -^w.
a function of the temperature alone.
PROBLEMS
/. Thermal expansionnear absolutezero, (a) Prove the three Maxwell rela-relations
, (dV/di)P= -{dts/dp), , D5a)
(aVldN)p = +(ap/ap)^ , D5b)
(Qj/cr)jV= ~Ea/dN)x. D5c)
Strictly speaking, D5a) should be written
and two subscripts should appear similarly in D5b) and D5c).It is commontoomit those subscripts that occur on both sides of theseequalities,(b) Show wilh
(lie help of D5a) and the third law of thermodynamics tjiat the volume Coeffi-Coefficient of tjicrniiti expansion
approaches zero as t -* 0.
2. Thermal ionization of hydrogen. Consider the; formation of atomic hy-
hydrogen in the reaction c + H*ftH, wheree isad eicutron, as the adsorptionof an electron on a proton H*. (a) Show that the equilibrium concentrationsof the reactants satisfy the relation I
[e][H+J/[HJs\302\253acxp(-//T), D7)
where / is (he energyrequited to tontze atomic hydrogen, and tiQ~ {im/2zh2K11
refers to the electron.Negiecithe spins of the particles; this assumption doesnot aSTed the final result. The result is known as the Saha equation.Ifall the
electrons and protons arise from the ionization of hydrogen atoms, then the
concentration of protons is equal to that of the Electrons, and the electronconcentration is given by ]
0]~[H]\"V'3\302\253p(-//iT). D8)
A similar problem arises in semiconductor physics in connection u-jth the
thermal ionization of impurity atoms that are donors of electrons.Nolice lhat: I
A) The exponent involves ^1 and not /, which shows that this ts not a simple\"Boltzmannfactor\" problem. Here / is the ionization energy,
B) The elccjron concentration is proportional to the square root of thehydrogen atom concentration. !
C) If we add excess electrons to the system,then (He concentration of protons
will decrease.
(b) Let [H(exc)]denotethe equilibrium concentration of H atoms in thefirst excited electronic state, which is \\l above the ground state. Compare
[H(exc)] with [e] for conditions atthe surface of the Sun,with [H] = 1023 cm~ 3
andT - 5000K. |I
3, hnization of donor impurities in semiconductors. \\ A pentavalent impurity
(called a donor) introduced in placeofa tetravalent silicon atom in crystalline
silicon acts likea hydrogen atom in free space, but with e2/e playing the role of
e2 and an effective mass m* playing the role of the electron mass m in the
description of *hc kmizalion energy and radiuso( the ground state of the
impurity atom, and alsofor the free electron. For silicon the diclctltic consta.t\\t
e ~ 11.7 and, 0pproxima!e!y, m* =0.3 m. U there are 10\" donors p^r cmJ,
estimate the con cent ration of conduction electronsat 100K.
4. Siopotymergrowth. Consider the chemical equilibrium of a sotutionof linear polymers made up of identical units. The basic reaction step ismonomer 4- A'mer = (A' -f l)mer. Let K^ denoteliieequilibrium constant for
Chapter 9: Gibbs free Energy and Chemical Reactions
this reaction, (a) Show from the law of mass action lhat the concentrations[\342\226\240
\342\200\242
\342\226\240]
satisfy
[N + 1] = [If \"/K.KjKj \342\200\242\342\200\242\342\226\240K.v D9)
(b) Show from the Iheory of reactionsthai for ideal gas conditions (an idealsolution):
wQ(W) ^Bnti2/M,^rm , E1)
where MN is the mass of the Nmer molecule,and FN is the free energy of oneiVmer molecule, (c) Assume N \302\273j, so thai nQ{N) =
Hq{N+ 1), Find the
concentration ratio [N 4- t]/[N]at room temperature if there is zero free
energy change in the basic reaction siep: that is, if AF = FKi.l ~ Fs ~ fj =-0.Assume [I]
= 10aocm~3, as for ammo acidmoleculesin a bacteria! ceil. The
molecular weight of the monomer is 200.(d) Show that for the reaction to go in
the direction of long molecules we need AF < ~0.4cV, approximately. Thiscondition is not satisfied in Nature,but an ingenious pathway is followed thatsimulates the condition. An elementary discussion is given by C. KiUel, Am. J.
Phys. 40,60A972).
5. Pavtkte-antipartkk equilibrium, (a) Find a quantitative expression for the
thermal equilibrium concentration n = n+ = n~ in the particle-antiparticlereactionA+ 4- A\" = 0. The reactants may be electronsand positrons; protons
and antiprotons; or electrons and hoies in a semiconductor. Let the mass ofeither particlebe M; neglect
the spins of the particles. The minimum energy
release when A* combines with A\" is A-Take the zero of the energyscaleasthe
energy with no particles present, (b) Estimate n in cm\023 for an electron (or a
hoie) in a semiconductor T \302\253*300 K. with a A such that A/t = 20. The hole isviewedas the antiparticic to the electron. Assume that the electronconcentrationis equaito the hoie concentration; assume aiso titat the particles are in the
classical regime,(c)CorrectIheresult of (a) to let each particle have a spin of 3.
Particles that have amiparticfes are usually fermions with spins of \\.
Chapter 10
Phase Transformations
VAPOR PRESSURE EQUATION 276
Derivation of the CoexistenceCurve,p Versus t 278
Triple Point 284Latent Heat and Enthalpy 284
Example: Model System for Gas-Solid Equilibrium 285
VAN DER WAALS EQUATION OF STATE- 287
Mean FieldMethod 288CriticalPointsfor the van der Waals Gas 2S9Gibbs Free Energy
of the van der Waals Gas 291Nucieation 294Fefrom agnet ism 295
LANDAU THEORY OF PHASE TRANSITIONS 298
Example: Ferromagnets 302
First Order Transitions 302
PROBLEMS 305
1. Entropy, Energy, and Enthalpy of van der Waals Gas 3052. Calculationof
dTjdp for Water 305
3. Heat of Vaporization of Ice 305
4. Gas-Soltd Equilibrium 305
5. Gas-Solid Equilibrium 3056. Thermodynamicsof the Superconducting Transition 306
7. Simplified Model of the SuperconductingTransition 307
8. First Order Crystal Transformation 307
Note: tn the first section s denotes c/iV, the entropy per atom. In the section on fcr
Chapter 10: Phaie Transformations
VAPOR PRESSURE EQUATION-
The curve of pressure versus volume for a quantity of matter at constanttemperature is determinedby the free energy of the substance. The curve iscalledan isotherm.We consider the isoihcrmsof a real gas in which the atomsor moleculesinteract with one another and under appropriate conditions canassociatetogether in a liquid or solid phase, A phase is a portion of a systemthat is uniform in composition.
Twophasesmay coexist, with a definite boundary between them.An isotherm
ofa real gas may show a region in the p-V plane in which liquid and gascoexistin equilibrium with each other. As in Figure 10.1, part of the volume containsatoms in the gas phase. There are isotherms at low icmperaturcsfor which
solid and liquid coexist and isothermsfor which soiid and gas coexist. Everythingwe say for the liquid-gas equilibrium holds also for the solid-gas equilibrium
and the solid-liquid equilibrium.Liquid and vapor* may coexist on a section of an isotherm only if the
temperature of the isotherm lies below a criticaltemperaturerc. Above the
critica! temperaiure only a single phase\342\200\224thefluid phase\342\200\224exists, no matter
how great the pressure. There is no more reason to cali this phase a gas than
a liquid, so we avoid the issueandcallit a fluid. Values of the critical temperaturefor severalgasesare given in Table iO.t.
Liquid and gas will never coexist along the entire extent of an isothermfrom zero pressure to infinite pressure; they coexist at most only along a
section of the isotherm. For a fixed temperature and fixed number of atoms,
there will be a volume above which all atoms present are in the gas phase.
A small drop ofwater placedin an evacuated sealed bell jar at room temperaturewill evaporate entirely, leaving the bell jar filled with H2O gas at some pressure.A drop of water exposed to air not already saturated with moisture may
evaporate entirely. There is a concentration of water, however,above which
the atoms from the vapor will bind themselves into a liquid drop. The volumerelationsare suggested by Figure JO.i.
The thermodynamic conditions for the coexistence of two phases are the
conditions for the equilibrium of two systems that are in thermal, diffusive,
Vapor Pressure Equatio
Liquid Liquid + gas Gas
Figure IO.I Pressure-volume isotherm of a reyj gas at a
temperature such that liquid and gas phases may coexist,that
is, t < tc. in the two-phase region of Hquid 4- gas the pressureis constant, but the volume may change. At a given lempcratureEfiei^ is only <t s^nclc v*iluc of tltc ptcssurc for Vvh^cri a Jjtjuiu
and its vapor are in equilibrium. Jf at this pressure we move the
piston down, some of the gas iscondensedto liquid, but the
pressure remains unchanged as Jong as any gas remains.
Table 10.1 Critical temperatures of gases
T,. in K Tt, in K
HeNcAr
Kr
Xc
5.2
414151210289 7
H2N.
Oj
CO,
33.2126.01543647.1
.W.2
Chapter 10: Phase Transformations
and mechanical contact.Thesecondilionsare that tj = t2: fit= p2; pl =
or, for liquid and gas,
*j= 18\\ Ml
=/Jj,; Pi = Pg . A)
where the subscripts / and g denotethe liquid and gas phases. Note that thechemical potentials of the same chemical species in the two phases must beequalif the phases coexist. The chemical potentials are evaluatedat the commonpressure and common temperature of the liquid and gas,so that
/l,(p,T)=
/ijl B)
At a general point in the p-x plane the two phases do not coexist: If /i, <ng
the liquid phase alone is stable, and if ns< Pi the gas phase alone is stable.
Metastablephasesmay occur, by supercooling or superheating. A metastable
phase may have a transient existence, sometimesbrief,sometimeslong,at a
temperature for which another and ntore stablephaseof the same subslance
has a lower chemical potential.
Derivationof the CoexistenceCurve, p Versus r
Let pQ be the pressure for which two phases, liquid and gas, coexist at thetemperaturez0.Supposethat the two phases also coexist at the nearby pointPa + dp;iQ + rfi.Thecurve in thep.T plane along which the two phasescocxist
divides the p, t plane into a phase diagram,as given in Figure 10.2 for H2O.It is a condition of coexistencethat
dt).
Equations C) and D) give a relationship betweendp and dx.
We make a series expansion ofeachsideofD) to obtain
C)
D)
-. E)
ion ojthc Cot
Figure 10.2 Phase diagram of H.O. The
relationships of the chemical poienmls /t,. ;i,.and jiB in the solid, liquid, and gas phasesateshown.The phase boundary here bciwcen iceand water is not cxacily vertical; the slope is
actually negative,although very large. After
Iniemalionat Critical Tobies.Vol. 3. and P. \\V
Qridgman, Proc. Am. Acad. Sci.47, 4-i| A912)for the several forms of ice. see Zemamky,
p. 375.
-100 0 100 200 300 400
Temperature,in \302\260C
In the limit as dp and dz approach zero,
by C) and E). This result may be rearrangedto give
F)
which is the differential equation of the coexistencecurve or vapor pressure
curve.
The derivatives of the chemicalpotentialwhich occur in G) may be expressedin termsofquantities accessible to measurement. In the treatment of the Gibbs
Then G) for dp/dt becomes
Chapter 10: Phase Transformations
free energy in Chapter 9 we found the relations
With the definitions
v s V/N, 5 =o/W (9)
for the volume and entropy per moleculein each phase, we have
1 (cG\\ V(dp\\
)JJ
01)
Heresa
~st is the increase of entropy of the system when we transfer one
moleculefrom the liquid to the gas, andvg
\342\200\224t\\ is ihc increase of volume
of the system when we transfer one molecule from the liquid to the gas.It is essential to understand thai the derivative dp/dz in (I!) ts not simply
taken from the equation of state of the gas. The derivative refers to the very
special interdependent change of p and i in which the gas and liquid continueto coexist.The number of molecules in each phase will vary as the volume is
varied, subject only to N, + Na=
A\\ a constant. Here Ns and(Va
are the
numbers of molecules in the liquid and gas phases, respectively.The quantity sa
~ s, is related directly to She quantity of heat that must
be added to the sysfern to transfer one moiecule reYersibly from the liquidto the gas, while keepingthe temperature of the system constant. (If heat isnot addedto the system from outside in the process, the temperature will
decrease when the molecule is transferred to the gas.)The quantityof heat
added in the transfer is
l1Q ~TE*
\"5'}'
A2)
by virtue of the connectionbetween heat and the change of entropy in a
reversible process.Thequantity
L = tEs- A3)
Dcritaiion of the Coexistence Cur
defines She latent heat of vaporization,and is easily measured by elementary
calonmetry.We let
A*\302\253 i-g-v, A4}
denote the change of volumewhen one molecule is transferred from the liquidto she gas. We combine A1), U3), and A4) to obtain
A5)
This is known as the Clausius-Clapcyron equationor the vapor pressure equation
The derivation of this equation was a remarkableearly accomplishment of
thermodynamics. Bo!li sides of (!5) are easily determined experimentally, and
the equation has been verified to high precision.
We obtain a particularly useful form of A5} if we make two approximations:(a) We assume that vg \302\273I'j: the volume occupied by an atom in the gas
phaseis very much larger than in the liquid (or solid)phase,so that we may
replace Av by vg:
Avsva= vt/yr {16)
At atmospheric pressure i'9/i;( ^ !03, and the approximation is very good.
(b) We assume that the ideal gas lawpVg
=Ngz applies to the gas phase,
so that A6) may be written as
Av S zip. A7)
With these approximations the vapor pressureequationbecomes
where L is the latent heat per molecule.Given L as a function of temperature,
this equation may be integrated to find the coexistence curve.
if, in addition, the latent heat L is independent of temperature over the
temperature range of interest, we may take L =l.,Q outside the integral. Thus
when we integrate (IS)we obtain
/dp
r (lv
A9)
Chapter 10: Phase Traasfortnat
logp =\342\200\224Z-0/i+ constant; ) = p0CXp(-L0/T) ,
where p0 is a constant.We defined LQ as the latent heat of vaporizationofonenioiecuie. If Lo refers instead to one mole, then
whereR is the gas constant, R = A'cta, where No is t!ie Avogadro constant. Forwater the latent heat at the iiqufd-gas transition is 2485J g~' at O'Cand 2260
J g~' at lOO'C.a substantial variation with tcmpcraiure.
The vapor pressure of water and of ice is plottedtn Figure !0.3 as !ogp
versus 1/T. The curve is linear over substantial regions, consistent with the
X 103
Vapor pressure ofwaier and of ice \342\226\240-102
is 1/T. The vertical scale is \"
The dashed line b a straiglif iine. S
I 10
\\
\342\200\2241
-Crit ca! p
atm
\\
Jin.
id wa
\\
er
Si*\"
\\ s
vc
V \342\200\242Ice
1
\\
1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
lO'/T, in K~\302\273
2 3 4
Tcmperamre, in K
Figure 10.4 Vapor pressure versus iemperaiurcfor 4He. After H. van Dijkeial.. Journal of Research offheNauonal Bureau of Standards 63A, \\2
A959)-
approxmiate result B0). The vapor pressure of 4He,plottedin Figure 10-4,
is widely used in the measurement of temperatures between I and 5 K.The phasediagram of 4He at low temperatures was shown in Figure7.14.
Notice tUat the liquid-soHd eoexistence curve is closely horizontal below 1.4K.We infer from this and (I!) [hat the entropy of the liquid is very nearly equalto the entropy of the solidin this region. It is remarkable that the entropiesshouldbeso similar, because a normal liquid is much more disordered than asolid, so that the entropy of a normal liquid is considerablyhigherthan that
Chapter 10: Phase Transformations
of a normal solid. But 4He is a quanhim liquid. For another quantum liquid,3He, the slope of the liquid-solid curve is negative at low temperatures(Figure 7.15), and in this region the entropy of the liquidis lessthan the entropy
of the solid. The solid has moreaccessiblestatesthan the liquid! Liquid 3He
has a relatively low entropy for a liquid because it approximates a Fermi gas,which generally has a low enSropy when t \302\253zF because a large proportionof the atoms have Sheir momenta ordered into the Fermi sphere of Chapter7.
Triple point. The triple poin! t of a substance is that point p,, t, in She p~z
plane at which all three phases, vapor, liquid,and solid, are in equilibrium.
Hereng
= /i( = /js. Consider an equilibrium mixtureofliquidand solid phases
enclosed in a volume somewhat larger than that occupied by the mixture
alone. The remainingvolume will contain only the vapor, in equilibrium wishbo!h condensedphases,and at a pressure equal to the common equilibriumvapor pressure of both phases. This pressure is the triplepointpressure.
The iriple point temperature is not identical wish the melting temperatureof She substance at atmospheric pressure. Melting temperatures dependsomewhaton pressure;the triple point lemperaSure is the meiSing temperatureunder the common equilibrium vapor pressure of the two condensedphases.
For water the triple point temperature is 0.01 K above the atmosphericpressure melting temperature; T, = O.Oi\302\260C = 273.16K. The Kelvin scale isdefinedsuch that the triple point, of water is exactly273,16K;seeAppendix
B.
Latent heat and enthalpy. Tile latent heat ofa phase transformation, as from
the liquid phase to she gas phase,is equal to t times the entropy differenceof the two phases at constant pressure. The latent heat is also equalto the
difference of H s V -f pV between the two phases, where H is calledthe
enthalpy. The differential is dH = dU -f pdV + Vtlp. When we cross the
coexistence curve,the thermodynamicidentity applies:
Tito = dV -f- pdV -(}tg
- jt,)dN , B2)
On the coexistencecurvejig
~ //,. Thus at constant pressure
L \302\253tAa \302\273At/ + pAV \302\273Ml = Ha- llt. B3}
Values of// are tabulated; ihey are found by integrationof the heat capacity
at coiisnint pressure:
T\342\200\224- = Hr- 1
Derivation of the Coexistence Curve, p Ver
jc,, B5)
Example: Model systemfor gas-sotidequilibrium. We construct a simple model io de-describe a solid in equilibrium with a gas, as in Figure 10-5. We can easily derive ihe vaporpressurecurve for ihis model. Roughly the same model would apply to a liquid.
Imagine the solid to consist of N atoms, each bound as a harmonic oscillator of fre-
frequency u to 3 fixed center ol force. The binding energy oleach atom in ihe ground siaicis\302\243fl;that is, ihe energy of an atom in its ground state is \342\200\224
co referred to a free atom at rest.
The energy states of a singleoscillatorarc ntioi \342\200\224r.o, where h is a positive integer or zero
(Figure 10.6). For the sake of simplicity we suppose itial each ntom can oscillaiconly in o:.e
dimension. The result for oscillators in [hrce dimensions is left as a problem.The pariiiion Junction ol a single oscillaior in llie solid is
Z, =\302\243\302\253p[-(n/10>
-eo)/r] = expOWt) \302\243\302\253p(--,,Aa./T)
=
The Ziee energy F, is
F* = u* ~Tff,
- -tlogZ,.
The Gibbs free energy in the solid is, per atom,
Gs\302\253
Vt- to-, + pt', =>
Fs + pvx = //,.
B6)
B7)
Figure 10.S Aloms in a solid in cqmSibn
wiih aionis in the gas phase.The equilibripressureis a function ol temperature. The
energyol the atoms in [he solid phase is lothan in ihe gas phase, bm the cmiopy of \\
aioms lends lo be higher in the gas phase.
equilibrium configmaiion is dfiietiiiiacd
cotmtcrpkiy of the iwo ciTccls.A! low
lanpemSurc most of iIk atoms ;src in ll
at high temperature all ot most of ihe a
may be in ihe gas.
. They ihe
Chapter 10: Phase Transformation
lor of frequency (u. The
ssumed 10 be % below that o(a
in!he gas phase.
Ground Male <
bound aloui
The pressure in the solid is equal to that o[ [he gas with which il is in contact, but Uie
volume i>i per atom in ihe solid phaseis much smaller than the volume vt per atom in [he
gas phase:c, \302\253vr
\342\226\240
If we neglect the term pv, we have (or the chemical potential of [hesolidp, S f \342\200\236whence
the absolute activity is
,/t) = exp(-logZJ
<stp(-to/i)[l - B9)
We make the ideal gas approximation to describethe gas phase, and we take Ihe spin ofthe atom to be zero. Then, [rom Chapter 6,
C0)
C1)
C2)
'\302\253q '\"a t\\
The gas is in equilibrium with the solid when).f
=).\342\200\236or
p = inQexp(-E0/t)[l- exp(-/1<0/t)].
K we insert nQ from C.63J:
V=
(j^j
Van Der Waats Equation of State
VAN DER WAALS EQUATION OF STATE
The simplest mode! of a liquid-gas phase transition is that ofvander Waals,\\vho
modified the ideal gas equation pV = Nx to take into accounlapproximately
the interactions between atoms or molecules. By the argument that we give
below, he was led to a modifiedequation ofstateof the form
(p + N2a/V2)(V - Nb) = Nx , C33
known as the van der Waals equation of state. Tliis is written for N atoms in
volume V. The a, b are interactionconstantsto bedefined;the constant a is
a measure of the long range attractivepart of the mieraeiion between two
molecules, and the constant /' is a measureof their short range repulsion
(Figure 10.7). We shall derive C3) \\vitli..tlie help of tlic general relation p =
~-(SF/DV)liN.We shall then trcal the ihcrmodynamic properlics of the modelin order to exhibitthe liquid-gas transition.
For an ideal gas we have, from F.24),
F(idealgas) = -NT[log(na/n) + i]. C4|
The hard corerepulsion
at short distances can be treated approximately asif the gas had available not the volume V, but the free volume V ~ Sb, when
b is the volume permolecule.We therefore replace the concentration n = N/V
in C4) by N/{V ~ Nb).Thus, insteadofC4),we have
F = -NT{log[)fQ(V- Nb)/N] + 1}. C5}
To this we now add a correction for the intermolecular attractive forces.
Figure 10.7 The iiucraaion energy between
repulsion plus a long range aiiracHou. The
short range repulsion can be describedapproximately by saying that each moleculehas a hard, impeneirable coie.
Chapter 10: Phase Transformation*
Mean Field Method
Thereexistsasimple approximate method, called the mean field method, for
taking inio account the effect of weak long range interactions among the
particles of a system.Themostwidely known applications of the method are to
gases and to ferromagnets.Lettp(r) denote the potential energy of interaction
of two atomsseparatedby a distance r. When the concentration of atoms in
the gas is n, the average value of ihe total interactionof all other atoms on
the atom at r = Ois
C6)
where -2a denotesthe value of ihe integral \\dVip{r). The factor of two is a
useful convention. We exclude the hard core sphereof voiume b from the
volume of integraiion. In writing C6) we assume that the concentration n is
constant ihroughout ihe volume accessible to the moleculesof the gas. That is,
we use the mean value of n. Tins assumption is the essence of the mean field
approximation.By assuming uniform concentration we ignore the increaseof concentrationin regions of strong attractive potential energy. In modernlanguage
we say thaS the mean field method neglectscorrelationsbetween
interacting molecules.\" \342\226\240
From C6) it follows that the interactions change the energy and the free
energy of a gas of N moleculesin volume V by
AF s MJ = -\\BNna) *= -N2a/V. C7)
The factor j is common toself-energy problems; it arranges that an interaction
\"bond\"between two molecules is counted only once in the total energy.Theexact number of bonds is |-N(N - I),which we approximate as |N2.
We add C7) to C5) to obiain the v;m der Waais approximation for the
Heimholu free energy of a yas:
F(vilW) =-.Vi{log[\302\273u(l'r
-Sh)/N~\\ + 1}
- Nza/V. 0$)
The pressure is
'\342\226\240-^\302\253'\027^5-FC9)
Critical Pawlsfor the
Figure 10.8 Direciions of intermolecularforces that ad on molecules near ihe boundaryof a volume Y. The van dci Waats argument
suggests lhal ihcse forces contribute art internal
pressure Nxa!Vl which Is lo be addedlo ihe
be used as ihe pressure in the gas taw.
o
V
0
\302\251
o
o
o
Q
O
O
1
o
Q
Figure 10.9 Hie coniainer of volume Khas NmolcculcSi each of volume b. The volume not
occupied by molecules is V - S'b. Intuition
suggests that iHis fece volume should be used in
llic gas law in place of the coiiiaincr Volume V.
(p + N2a/V2){V ~ Nb) \302\253A'l . D0)
der Waals equatum of staSe. The terms in a and b arc interpreted in
10.8 and 10.9.
Points for She van der Waals Gas
ne the quantities
pc ^ a!21b2\\ Vcs 3Nb; xc
^ 8a/27b. D1)
Chapter 10: Phase Trans/or,
P/Pc
0
. - 0.95tt
Figure 10.10 Tile van der Waals equation of stal
the critical temperature. Courtesy of R.Cahn.
In termsof these quantities the van der Waals equation becomes
3 \\(Vl\\ 8t
f7HJAH 3/ 3t/D2)
This equation is plotted in Figure 10-10 for several temperatures near the
temperature tc. The equation may be written in terms of the dimensionlessvariables
p s pjPc\\ 9sVfV.\\ t e t/tc , D3)
D4)
This result ts known as the law of correspondingstates. In termsofp, V, t,
si! gases !ook alike\342\200\224if they obey the van der Waals equation. Valuesof a
Gibbs Free Energy of the van (let M'aah Gas
and b are usually obtained by fitting to the observed pc and tc. States of two
substances at the same p, Vy x are called corresponding states of ihe subsiances.Realgasesdonot obey the equation to high accuracy.
At one point, the critical point, the curve of p versus V at constant f has ahorizontal point of inflection. Here the local maximum and minimum of theP~Vcurve coincide, and there is no separation between the vapor and liquid
phases. At a horizontal point of inflection
a- @i-
These conditions are satisfied by D4) if p = 1; 9 = 1;i = 1. We call pc, Vet
and tc the critical pressure, critical volume,and criticaltemperature,respectively.
Above tt no phase separation exists. \342\226\240-
Gibbs Free Energy of the van der Waals Gas
The Gibbs free energyof the van der Waals gas exhibits the characteristics ofthe !iquici-gasphase transition at constant pressure. With G = F + pV, we
have from C8) and C9) the result
G(x,VtN)=
-\342\200\224^
--^
-Nt{log[\302\273u(K
- Nb)/N] + 1], D6)
Thisequationgives G as a function of V, t, N; the natural variables for G arep, t, N. Unfortunately we cannot conveniently put G into an analytic form as a
function of pressure instead ofvolume.We want G(z,p,N) because we can thenobtain ;i(t,p)as G(z,p,N)/N by (9.13). It is /i that determines the phasecoexistencerelation
ft,=
fig. The results of numerical calculationsofG versus pare plottedin Figure 10.!1 for temperatures below and at ihe critical temperature. At any
temperature the lowest branch represents the stable phase;the otherbranches
represent unstable phases. The pressure at which the branchescrossdetermines
the transition between gas and liquid; this pressureis calledthe equilibrium
vapor pressure. Results for G versust areplottedin Figure !0.12.
Figure 10.13 shows, on a p-V diagram,the region V < Vs In which only ihe
liquid phaseexistsand the region V > V2in which only the gas phase exists.
The phasescoexist between Vx and V^. The value of Kj or V2 is determined bythe condition that /i](r,p)==
}ia{x,p) along the horizontal line betweenVx and
V2- This will occur if the shaded areabelowthe Jine is equal to the shaded area
Chapter 10; Phase Ttansfo
-0.40 t\342\200\224
--
/
\\
,-Vapor[ rasure
Figure 10.11 (a) Gibbs free energy versus pressure for van
der Waals equaiion of stale: i = 0.95tc. Courtesy of R. Cahn.
(b) Gibbs freeenergy versus pressure for van der Waals equationofstale;i = %..
above the line. To see ihis, consider
dG \302\253~adx + Vdp + pdN. D7)
We have dG = Vdp at constant i and constanttola!number of particles. The
difference of G between V\\ and V1 is
G,-G,= fWp, D8)
G/Nrc
j
LiqutaS^
-
i t
1
~p- 0.95
nqb = !
\302\273bic
k
i
0.984 0.986'
0.988 0.990
't/t,\342\226\240
(a)
Figure 10.12a Gibbs free energy versus temperature for van der Waals equalioof slalea!p
= 0.95 pe. Courlesy of A. Manoliu.
-2.70
0.90
-
-
\\
\\
p= 1.0
nqb = 1
\\oas
I.OO
t/t,
(b)
1.05 1.10
Figure 10.12b Gibbs free energy versus temperature for van der Waais cquatiO!of slate a! the crilical pressure pc.
Chapter Id: Phase Transformations
T
-
\\
~ conslan
^Liquid
i<\302\243
-Cocx Stetice fir
X
|\\
c
a as
s.
v.t
Figune 10.13 Isotherm of van der Waalsgasat alemperafure belowliie critical lempcralure. For volumes
less than Vt only !hc liquid phase exisls; for volumes
above Kj only lliegas phase exisls. Between Vx and V2
Ihe system in stable equilibrium lies along the coexistence
line and is an inhomogencousmixture of two phases. The
liquid and gas phases coexist. The proportion of liic
liquid and gas phases must be such lhai ihe sum of fheir
volumes equals the volume V that is available.
bul ihe integral isjust ihesumofthe shaded areas, one negalive and one positive.When Ihe magnitudes of the areas are equal, G?(t,/>)
- Gj(t,p) and [is{x,p}=
/^{r,p)alongHie horizontal coexistence line drawn in the figure. In equilibriumwe require j.tg
= /*,.
Nuclcatiou. LetAji
\342\200\224ftg
\342\200\224;i, be ihe chemical potential difTerence between
the vaporsurroundinga smallliquid droplet and the liquid in bulk (an infinitely
iarye drop), if A/i is positive, Ihe buik liquid will have a lower free energy thanthe gas and thus the liquid will be more stable than the gas. However,the
surface free energy of a liquid drop is positiveand tends lo increase the free
energy of'he liquid. Al small drop radii the surface can be dominantand the
drop can be unstable with respect to the gas.We calculate the change in Gibbs
freecnergy when a drop of radius R forms. If\302\273,is the concentration of molecules
in the liquid,
AG = G,-
G9= -{ + 4nR2y D9)
Ferromagtutism
where y is the surface free energy per unit area, or surface tension. The liquiddrop will grow when Gt < Gr An unstable maximum of AG is attained when
ti&G/dR= 0 = -4nR2iii&n + SnRy , E0)
Rc \302\2532y/H,A/i. E1)
This is the critical radius for nuclcation of a drop- At smaller R the drop will
fend to evaporaie spontaneously because that will Jower the free energy. At
larger R tlie drop will tend to grow spontaneously because that, too, will lower
the free energy.The free energy barrier(Figure10.14)that must be overcome by a ihcrni.i!
fluctuation in order fora nucleusto grow beyond R,. is found by substitution of
E1) in D9):
(&G)C=
(lenPW/nMl*I}. E2)
If we assume [hat the vapor behaves like an ideal gas, we can useChapter5
to express Aji as
Aft-r
tlog{p/peq) ,
where p is the vapor pressuretn the gas phase and pe<lthe equilibrium vapor
pressure of the bulk liquid (R~* co). We use y
~ 72erg cm\022 to estimate Rtfor water at 300 K and p = l.J^tobel x !0~6cm.
Kcrro magnet ism
A ferromagnet has a spontaneous magneticmoment,which means a magnetic
moment even in zero applied magneticfield.We develop the mean field approxi-approximation SO the temperature dependence of ihe magnetization, definedas the
magnetic moment per unit volume. Tlie centra! assumptionis that each niagnelic
alom experiences an effective fieid BEproportional to shemagnetization:
BE\302\253AM , E3)
where I is a conslans. We take the external applied field as zero.
Chapter 10: Phase Tram/or
Critical barn*\" or cortdcnsut
t for growH,3n nuclei
1
L
fx
1'
ondensation
| nuclei__
evaporate
, \\
\342\226\240'\342\226\240\342\226\240-
1
Figure 10.14 Excessfree energy of drop relative lo gas,asfunction of drop radius R, both m reducedunits.The gas is
supersaturated because the iiquid has the lower free energy forthis curve as drawn, but the surface energy ofsmaii dropscreates
an energy barrier ihat inhibits the growth of nuclei of the liquid
phase. Thermal fluctuations eventually may carry nuclei over ihe
barrier.
Consider a system with a concentration h ofmagneticatoms, each of spin j
and of magnetic moment fi. In Chapter3 we found an exact result for the
magnetization in a field B:
M = HjitanhOiB/t). . E4)
In the mean field approximation E3) this becomes, for a ferromagnet,
. .\342\226\240\"\342\226\240_\342\226\240_\342\226\240
. Me n/nanh(/dMA) , . '.. '. E5)
Fcrromu^cth,
0 0.2 0.4 0.6 0.8 1.0
Figure 10.15 Graphical soiuiion of \302\243q.E6) for the
reduced magnetization ,,i as a function of temperature.The reduced magnetization is definedasm = M/n/t. Theleft-hand side of Hq. E6) is piolicd as a straight line in
with unit slope. The right-hand side is tanh(m/0 and is
plotted versus in for three difierenl values of the reduced
temperature t = r/n;i'^ *=i/tt. The three curves
correspond to the temperatures2x,,zt, and Q.5rt. Thecurve for i = 2 intersects the straight iine in only al
m s= 0, as appropriate for ihe paramagnetic region (ihercis no externaiapplied magneiic fieid). The curve for / = i
(or t = it) is tangent lo the suatght Sine m at ihc origin;this ictnpetalure murks ilic onset of fcrroiiiagnetisiu.TheCurve for i = 0.5 is in iiie ferromagnciic region andtmersccts ihc straight line in at about \302\273is= 0.94 n/i. As1 -\342\200\2420 the intercept moves up to in - l.so that all
magneiic moments are iined up at absolute zero.
a transcendental equation for M. We shall see that solutions of this equationwith nonzero M exist in the temperature range between0 and v To solve E5)we write it tn terms of the reduced magnetization ??i \302\253M/h/i and the reduced
temperature f s z/n[i2X, whence
hi = ianh{Hi/r). E6)
We plot the right and left sides of this equation separately ;is functions of \302\273t,
as in Figure 10.15. The intercept of the two curvesgives the value of m at Ihe
temperature of interest.The critical temperature ts f = I, or zt = h/i^L ThecurvesofM versus x obtained in ihis way reproduce roughly the featuresofthe
Chapter 10; Phase Transformation
Figure 10.16 Saiuration magnetization of
nickel as a function of temperature,together
with ihc theoretical curve fci spin \\ on ihe
mean fieid theory.
experimental resuiis, as shown in Figure 10.16 for nickel As t increases themagnetizationdecreasessmoothly to zero at z = rc,called theCurietemperature.
LANDAU THEORY OF PHASER TRANSITIONS
Landau gave a systematic formulation of the mean field theory of phasetransi-transitions applicable to a large variety of systems exhibitingsuch transitions.We
consider systems at constant volume and temperature, so that their Heimholtz
free energy F = U \342\200\224ta is a minimum in equilibrium. The bigquestionis,a
minimum with respect to what variables? It isnot helpful to consider all possible
variables. We suppose here that the system can be described by a single orderparameter\302\243,the Greek xi, which might be the magnetization in a ferromagneticsystem, the dielectric polarization in a ferroelectric system, the fraction ofsuperconducting electrons in a superconductor, or the fraction ofneighborA-Bbondstotola!bonds in ait alloy AB. In thermal equilibrium the order parameterwill have a certain value c,
= cc{c). In the Landau iheory we imagineIhat \302\243
can be indepaideally specified, and we consider the LandauTree energy (\"unction
FL(\302\243.x)= E7)
where the energy and entropy are taken when the order parameter has the
specified value \302\243no! necessarily c0. The equilibrium value \302\2430{t)is the value of
Landau Theory of Phase Transitions
c, that makes FL a minimum,at a given t, and the actual Hdmholtz free energyf(r) of the system at i is equal lo thai minimum:
F(t) \302\273Fl($0,t) S FL(\302\243r) if \302\243* Co- E8)
Plotted as a function of \302\243for constant r, ilic Landau free energymay have more
than one minimum. The lowest of thesedetermines the equilibrium state. In afirst order phase transitionanotherminimum becomes the lowest minimum as
i is increased.We restrict ourselves to systems for which Ihe Landau (\"unction is an even
function of \302\243in the absence of applied fields. Most ferromagneticand ferro-ferroelectric systems are exam pies of this. We also assume that F 1(^,1) is a sufficientlywell-behaved function of \302\243that it can be expanded in a power series in \302\243\342\200\224
something that should not be taken for granted. For an even function of \302\243,as
assumed,
The entire temperature dependence of FL(\302\243,x) is contained in the expansioncoefficients ga\\ g2,gx, g6- These coefficients are matters for experiment or theory.
The simplest example of a phase transitionoccurswhen ^(x) changes sign at
a temperature i0, with y4 positive and the higher terms negligible.For simplicity
we take g2(t) linear in i:
<72(r)~ (r - to)* . F0)
over the temperature range of interest, and we take g4 as constant in that range.Witli these idealizations,
The form F0) cannot be accurateovera very wide temperature range, and it
certainly fails at low temperatures because such a lineardependenceon tem-
temperature is not consistent with the third law.
The equilibrium value of \302\243is found at the minimum of FL{$ ;t) with respect
which has the roots
f~Q and ? = (to - r)(a/g4). F3)
Chapter iff; Phase Transformations
With a and ga positive, the root c,= 0 corresponds to the minimum of the free
energy function F1) at temperatures above i0; here the HelinhoHzfreeenergy is
J=Xt) = 0g(l). F4)
The other root, c,2~ (a/gj(to
- t) corresponds to the minimum of the free
energy function at temperatures below t0; here the Helmhoitz free energy is
The variation of F(r)with temperature is shown in Figure 10.17.The variationof FdZ'J-) as a function of \302\243:for ihree representative temperatures is shown in
Figure 10.18,and ihe temperature dependence of the equilibrium value of \302\243is
shown in Figure 10.19.Our modeldescribesa phasetransition in which the value of the order paratn-
elergoescontinuously to zero as the temperature is increasedto t0.The entropy
Figure 10.17 Temperature dependence of (he free energy
for an ideatized phase t ransition of the second order.
Figure 10.18 Landau free energy function versus \302\2432at
reprcscniaLivc temperatures. As the temperature drops below t0
the equilibrium value of \302\243gradually increases, ;is tic fined by the
posiiion of the minimum of the tree energy.
Figure 10.19 Spontaneous polarization versus
temperature,for a second-order phase transition. Thecurve is not realisticat low temperatures because of theuse OfEq.F0): the third law of thermodynamics requiresthatdf/rfr ->0asi-*0.
Chapter 10: Phase Transformations
\342\200\224dFfdz is continuous at t = rQ,so that there is no latent heat at the transition
temperaturef0. Such a transition is by definition a second order transition.Transitionswith a nonzero latent heat are called first order transitions; we
discuss them presently. The real world contains a remarkable diversity ofsecond order transitions; the best examples arc ferromagnels and super-superconductors.
Example: Ferromagnets. In the mean field approximation, ferromagnets satisfy the
Landau theory. To showthis, consideran atom of magnetic moment pin a magnetic field 3,
which we shall set equalto ihe tijean field >M as in E3). The interaction energy density is
V(M) =-\302\261 F6)
where ihe factor j iscommon to self-energy problems. The entropy density is given approxi-
g{M) = constant -M2J2nn2 , F7)
in Ihe regime in which M \302\253n/j. Thus the free energy fursciion per unit volume is
FL{M) ~ constant - \\M2(/.'~\342\200\224A+ lermsof higher order. F8)
At Ihe transition temperature the coefficient of M* vanishes, so ihat
i0 -\302\273}i2>., F9}
in agreement with the discussion following E6).
First Order Transitions
A latent heat characterizes a first order phase transition. Theliquid-gastransi-
lion at constant pressure is a first order transition. In ihe physics of solids first
order transitions are common in ferroelectric crystals and in phase transforma-
transformationsin metals and alloys. The Landau function describes a first order iransition
when the expansion coefficientg*is negative and gb Is positive. We consider
first Order Transitions
Figure l(K20 Landau free energy function versus i1 m a
first order transition, at representative temperatures. At xc
the Landau function has equal minima at \302\243= 0 and at a
finite \302\243as shown. For t below rf the absolute minimum is
iatger values of ^; as r passes through tc there is adisconimuouschcinge in the position of the absoluteminimum. The artows mark the minima.
The extrema of this function are given by the roots oFigureJ0.20:
Either \302\243- 0 or
G1)
G2)
At the transition temperature rc the free energies will be equal for the phaseswith c,
~ 0 and with the root c^O. The value of xc will not be equal to r0,
Chapter 10; Phas
_60 -40 -20
T~ rrt in K
and the order parameter\302\243(Figure 10.21} does not go continuously to zero at
xc. These results differ from those in the second order phase transition treatedearlier, where \302\243weui to zero continuously at t0 - tf. A first order transfor-
transformation may show hysteresis, as in supercooling or supcrsaturation, but no
hysteresis exists in a second order transition.
1. The coexistencecurve in the p-x planebetween two phases must satisfy tiic
Clausius-Clapeyron equation:
dp _L
where L is the latent heat and An is the volume difference per atom betweenthe two phases.
2. The latent heat L ~ H, - Hltwhere H ~ V + pV is the enthalpy.
3. Thevan der Waais equation of state is
.'
(P + N2a!V2){V- Nb) = Nx.
4. In the Landau free energy function
the energy and entropy are taken when the order parameter has thespecifiedvalue \302\243,,not necessarily the thermal equilibrium value \302\2430.The function Fl
is a minimum with respect to \302\243when the system is in thermal equilibrium.
5, A first order phase transition is characterizedby a Intent heal and byhysteresis.
PROBLEMS
/, Entropy^ energy, and enthalpy of van der IVaah gas. (a) Show that the en-
entropy ofthe van dcr Waats gas is
o -N[log[nQ(V
-Nb)JN] + 1}. G3)
(b) Showthat the energy is
U = INi.- N2.ii!V. G4)
(c) Show thai the enthalpy// = U + pV i%
H{i,V)= |Nt + N2bx/V -
2N2ajV\\ G5)
//(r./>) - -jNr + NhP~ 2NuPfx. G6)
All results arc given to first order in the van der \\V;ials correction terms o, h.
2. Calculationof (IT{dp for water. Calculate from the vapor pressure equa-equation the value of elT/dp near p ~ 1 atm for the Hquid-vypor equilibrium of
wmcr. The heat of vaporizational 100Xis2260Jg\021. Express the result in
keivin/atm.
3. Heat of vaporization of ice. The pressure of water vapor overiceis3,SSmm
Hgal ~2=C and 4.58 mniHgai OX. Estimatein Jmol\"' the heat of vaporiza-vaporizationof ice at ~l\302\260C.
4. Gas-solid equilibrium. Consider a version of theexampleB6}-C2)in which
we let the osciSiators in the soiid move in three dimensions, (a) Show that in thehigh temperatureregime(t \302\273haS) the vapor pressure is
(b) Explain why the latent heat per atom is tQ\342\200\224
\\i.
5. GaS'SoU'd equilibrium. Consider the gas-solid equilibrium under the ex-
extreme assumption that the entropy of the solidmay be neglected over the tem-
temperature range of interest. Let -e0 be the cohesiveenergyofthesolid,per atom.
Chapter 10: Phase Transformatio
Treat the gas as idea!and monatomic. Make the approximation that the volumeaccessible to the gas is the volume V of the container, independent of the muchSmaller Volume occupied by the Solid, (a) Show thai the total HeimhoHzfree
energy of ihe system is
F = F, + F, \" -Ufa + N,r[!og(N,/l'il0)- I] , G8)
where the total number of atoms, N = N, + Ngis constant, (b) Find the mini-
minimum of ihe free energy with respect to N^show that in ihe equilibrium condition
(c) Find the equilibriumvapor pressure.
6. Thermodynamics of the superconducting transition, (a)Show that
2/i0 th . [i0 ih
09)
(SO)
in SI units for Be. Because Bcdecreases with increasing temperature, the rightside is negative.Thesuperconductingphase has the lower entropy: it is the more
ordered phase. As t ~+ 0, the entropy'in both phases will go to zero, consistent
with the third law. What docs this imply for the sliapc of the curveof Bt versus r?
(b) At r =xtt
we have Bt = 0 and hence a^ = aN.SIiow llutt this result hits ihe
following consequences: A) The two free energy curves do not cross ;if tt but
merge, as shown in Figure 10.22.B) The two energies are the same: Usfr,.) =
U.vW- C) There is no Intent heat associated with the transition at r \342\200\224tt.
What is the latent heat of the transitionwhen carried out in a magnetic field,at r < i{7 (c) Show that Cs and CN, the heat capacitiesper unit volume, are
related by
(81)
Figure S.iS is a plotof Cj'Tvs T1 and shows that Cs decreases much faster
than linearly with decreasing r, while Cs decreasesas yz. For t \302\253tc> AC is
dominated by Cs. Show that Hiis implies
_- -0.2
.1
Superconductor
X. Normal
*STC=1.180K
0.5
Temper; lure. K
Figure 10.22 Experimental values of the Tree energy as a function of
temperature Tor aluminum in the superconducting state and in the normal
stale. Below the transition temperature T, = 1.180 K (he free energy is lowerin !he supcrcondtiding slate. The (wo curves merge at the transition
tempcra(ure, so thai the phase transition is second order {(hereis no laient heat
of transiiion a! Tc).The curve Fs is measured in zero magnetic field, and FN is
measured in a magnetic field suftkien! to pu! the specimen in (he normal slaie.
Courtesyof N. E. Phillips.
7, Simplified model of the superconductingtransition. TheBc(i}curves of most
superconductors iutve shapes close to simpleparaboias.Supposethat
Bt(i)= Bt0[l
- d/r(}2]. (83}
Assume that Cs vanishes faster than linearly as t -*0.Assume also that Cs is
linear in r, as for a Fermi gas (Chapter 7}. Draw on the resultsof Problem 6 to
calculate and plot the i dependencesofI he two entropies, the two heat capacities,and ihe latent heat of the transition. Show that Cs(rc)/Cv(r(}= 3.
8. First order crystal transformation. Consider.'! crystal that can existin eilher
of two.structures, denoted by a and /?. We suppose 'hat the a slrucmre is thesiablelow temperalure form and the /J structure is the stable high semperalure
form of the substance. If the zero of the energy scale is taken as the stale of
separated atomsat infinity, then the energy density 1/@) at r == 0 uill be
Chapter 10: Pha:
negative. The phasestable at t \342\200\2240 will have the lower value of U{0);thus
Ux@) < U?@). If the velocity of soundve
in the /J phase is lower than vx in the
a phase, corresponding to lower values of the clastic moduli for /?, then thethermal excitationsin the $ phase will have larger amplitudes than in the aphase.The larger the thermal excitation, the larger the entropy and the lower!he free energy. Soft systems tend to be stabie at high tempera!ures,hardsystems at low. (a) Show from Chapter 4 that the free energy density contributed
by the phonons in a solid at a temperature much icssthan the Debyetemperatureis given by -~rc2r'i/3Oi;3/i3, in the Debye approximation with v taken as the
velocity of all phonons.(b) Show that at the transformation temperature
>-v.-'). (84)
Therewill be a finite real solution ift'p
< iv This example is a simplified modelofa classofactual phase transformations in solids, (c) The latent heat of trans-transformation is defined as the thermal energy that must be supplied to carry the
system throughthe transformation.Show that the latent heal for this model is
L = 4[U,@)- t/,@
In (84) and (85), U refers to unii volume.
(85)
Chapter 11
Binary Mixtures
SOLUBILITY GAPS 310
ENERGY AND ENTROPY OF MIXING 31-1
Example: Binary Alloy wrth Nearest-Neighbor Interactions 318
Example: Mixture of Two Solidswith Different Crystal Structures 319
Example: Liquid 3Hc-\"HeMixturesat Low Temperatures 320
Phase Diagrams for Simple Solubility Gaps 321
PHASE EQUILIBRIA BETWEEN LIQUID
AND SOLID MIXTURES 322
Advanced Treatment:Eutectics 325
SUMMARY 330
PROBLEMS 330
1. Chemical Potentials in Two-PhaseEquilibrium330
2. Mixing Energy in 3He-4He and Pb-Sn Mixtures 3303. Segregation Coefficient of Impurities 331
4. Solidification Range of Binary Alloy 331
5. Alloying of Gold into Silicon 33i
Chapter 11: Binary Mixtures
Many applications of materials science,and large parts of chemistry and
biophysics, are concerned with the properties of multicomponentsystems that
have two or more phases in coexistence.Beautiful,unexpected,and important
physical effects occur in such systems. We treat the fundamentals of the subjectin this chapter,with examples drawn from simple situations.
SOLUBILITY GAPS
Mixturesarc systems of two or more different chemical species.Birjury mixtures
have only two constituents. Mixtures with three and four constituents are called
ternary and quaternary mixtures.If the constituents arc atoms, and not mole-molecules, the mixture is called an ailoy.
A mixture is homogeneous when its constituents are intermixedon an atomic
scale to form a single phase, as in a solution. A mixture is heterogeneous when it
contains two or more distinct phases, such as oil and water. The everyday
expression \"oii and water do not niix\" means [hat their mixture does not form
a single homogeneousphase.Thepropertiesofmixtures differ from the properties of pure substances.The
melting and solidification properties ofmixturesareofspeciai interest. Hetero-
Heterogeneous mixtures may melt at lower temperatures than their constituents.
Consider a gold-silicon alioy: pure Au melisat lO63cCand pure Si at I404X,but an ailoy of 69pet Au and 31 pet Si melts (and solidifies}at 37OCC. This is not
ihc result of ihe formationof any low-melting Au-Si compound: microscopic
investigation of the solidified mixtureshows a two phase mixture of almost
pure Au side by side with almost pure Si (Figure I I.I}.Mixtures with such
properties are common, and they are of practicalimportancepreciselybecause
of their lowered melting points.What determines whether two substances form a homogeneousor a hetero-
heterogeneous mixture? What is ihe composition of ihe phaseslhat are in equilibrium
with each osher in a heterogeneous mixture?Thepropertiesof mixtures can be
underslood from the principle that any system at a fixed semperaiure will
evolve to the configuration of minimum free energy. Two subsianceswiil
dissoive in each oilier and form a homogeneousmixtureif that is the configura-
configurationof iowest free energy accessible to the components. The subsianceswill
Solubility Gaps
SO/tm
Figure II.I Heterogeneousgold-siliconalloy. When a mixture of 69 pci Au and 31 petSi is melted and then solidified, the mixture segregaies into a phase of almost pure Au
(Sight phase) coexistent \302\253iiha phase of almost pure Si{darkphase).Magnified aboui
800 times. The composinon given is that of the lowest-melting Au-Si mixture, the
so-called eutectic mixture, a conceptexplainedlater in she text. Photograph courtesyofStephan Justi.
form a heterogeneous tnixlure if [he combined free energy of the Uvo separate
phases side by side is tower ihaii the freeenergy of the homogeneous mixture:
then we say Ihat the mixture exhibits a solubility gap.A hclerogeneous miMure will melt at a lower [cmperalurethan the separate
substances if I he free energy of the homogeneousmeltis lower than the com-
combined free energies of the two separate solid phases.Throughout ihis chapter we assume for simplicity that the external pressure
may be neglected, and we sel pV = 0. Then volumechangesdo not involve
work, and the appropriate free energy is tire HelmhoUz free energy F rather
Ilian the Gibbs free energy G. We will usually simply speak of the free energy.
We discuss binary mixtures of constituents Ihat do no! form well-defined
compounds with each other. Our principal interestis in binary alloys. Consider
Chapter H: Binary Altitun
Many applications of materials science, and large parts of chemistry and
biophysics, are concerned with the properties of multicomponent systems that
have two or more phases in coexistence. Beauiiftil, unexpected, and important
physical effects occur in such systems. We treat Ihe fundamentals of the subject
in this chapter, with examples drawn from simplesituations.
SOLUBILITYGAPS
Mixtures are systems of two or more different chemical species. Binary mixtures
have only two constituents. Mixtureswith three and four constituents are called
ternary and quaternary mixtures.If the constituents are atoms, and not mole-
molecules, the mixture is called an alloy.A mixture is homogeneous when its constituents arc intermixedon an atomic
scale to form a Single phase, as in a solution. A mixture is heterogeneous when it
contains two or more distinct phases, such as oil and water. The everyday
expression \"oil and water do no! mix\" means that their mixture does iiot form
a single homogeneous phase.The properties of mixtures differ from the properties of pure substances. The
meltingand solidification properiies of mixtures are of special interest.Hetero-Heterogeneous mixtures may melt at lower temperatures than their constituents.Considera gold-silicon alloy: pure Au melts at 1063\302\260C and pure Si at 1404\302\260C,
but an alloy of 69 pet Au and 31 pet Si melts (and solidifies}at 370\302\260C This is not
the result of the formationof any iow-me!ting Au-Si compound: microscopic
investigation of the solidified mixture shows a two phase mixture of almost
pure Au side by side with almost pure Si (Figure 11.1).Mixtures with such
properties are common, and they are of practical importance precisely becauseof their loweredmelfing points.
What determines whether two substances form a homogeneousor 3 hetero-
heterogeneous mixture? What is the composition of the phasesthat are in equilibrium
with each other in a heterogeneousmixture?Thepropertiesofmixtures can be
understood from the principle that any system at a fixed temperature wiil
evolve to the configuration of minimum free energy. Two substanceswill
dissolve in each other and form a homogeneousmixtureif that is the configura-configurationof lowest free energy accessible to the components. The substanceswill
Solubility
10/mi
jure H.I Heterogeneous gold-siliconalloy.When a mixture of 69 pet Au and 31 petis melted and ihen solidified,!hemixiure segregates into a phase of almost pure Au
Figure 11.1
Siisn
(lighi phase) codxisieniwiih a phase of almost pure Si (dark phase). Magnified aboui
800 limes. The composition given is !hal of ihe towesi-melting Aii-Simixiure, ihc
so-called eulectk: mixture, a concept explainedtater in ihe texi. Photograph courtesyofSiephanJusii.
form a heterogeneous mixture if the combined free energy of the two separate
phases side by side is lower than the free energy of the homogeneous mixture:then we say that the mixture exhibiis a solubility gap.
A hcierogeneous mixture will melt at a tower temperature than the separatesubstances if the free energy of the homogeneous melt is lowerthan ilie com-
combined free energies of the two separate solid phases.Throughout Ihis chapter we assume for simplicity that the externalpressure
may be neglected, and we set pV ~ 0. Thenvolumechangesdonot involve
work, and the appropriate free eneryy is the Hetmlioltzfree energy F rather
than the Gibbs free energyG.We will usually simply speak of the free energy.We discuss binary mixtures of constituents that do not form weil-defined
compounds wiih each other. Our principal interest is in binary alloys. Consider
Chapter Hi Binary Mixtures
a mixture of JVA aloms ofsubsiance A and NB atoms of substance B.Thetotalnumber of atoms is
We express the compositionof the system in iermsof the fraciion x of B aioms;
x \302\273A'b/N; 1 - x \302\273
jVa/N. B}
Suppose the sysiem forms a homogeneoussolution,wiih an average free energy
per atom given by
/ = F/N. C)
Supposefurther that/(.\\) lias the functional form shown in Figure 11.2. Because
this curve contains a range in which ihe second derivative d2f/dx2 is negative,we can draw a line tangent to the curve at two points, at x = xx and x \342\204\242
x^.
Free energy curves of this shape are common,and we wilt see later what maycause this shape. Any homogeneous mixture in the composition range
x, < x < xp D)
is unstable with respect to two separate phases of composition x, and x^. We
shall show that ihe average free energy per atomof the segregated mixture is
given by the point i\"on the straight line connecting the points a and [}. Thus in ihe
entire composition range D) the segregatedsystem has a lower free energy thanthe homogeneoussystem.
Proofi The free energy of a segregated mixture of the two phasesa and j$ is
F => NJix,) + NfJ\\xfi) . E)
where ,V, and Nfi are the total numbers of atoms in phasesa and ji, respectively.
These numbers satisfy the relations
which may be solved for .V, and Ny.
0)
Solubility Gaps
Figure tl.2 Free energy per alom as a function of composition, for a
system with a solubility gap. tf the free energy per aiom of a
homogeneous mixture has a shape such that a tangent can be drawn
that touches the curve at two diftereiit points x and /?, (lie compositionrange between the two points is unstable. Any mixture with a
composition in this range will decompose into two phases with the
composition _v, arid ,\\f. The free energy of the two phase mixture is
given by the point / on the straight line, below the point It.
From E) we obtain
fjix)JV
(S)
for the free energy of the two phase system. This result is linear in x and is a
straight line in thef-x plane. If we set .v. = A'3 or.v^, v,c see (hat the line docsgothrough the points a and /?. Thus/ in (he interval between .y4 and
xfiis given by
the point i on the straight lineconnectinga and p.
Chapter II: Binary Mixtures
We have not yet made useof (he assumption that (he straight line is tangenttof{x)at the points 3 and /J, and therefore our result holds for any straight line
that has two points 2 and/Jin commonwith/(.v). Bui fora given vaiue of x, (he
lowest free energy is obtainedby drawing (he lowest possible straight line thathas (wopointsin common wiih/(-v), on opposite sides of a-. The lowest possible
straight line is the (wo-pointtangent shown. The compositions x3 and x? arethe limits of the solubility gap of (he system.
Once (he system has reachedits lowest free energy, (he (wo phases must be in
diffusive equilibrium with respect to both atomic species,so thai their chemical
potentials satisfy
/*a> =/*\302\253; Pb\302\273
= Pb*- (9)
We show in Problem i that jja and/jB
are given by (he intercepts of the two-point tangentwith the two vertical edges of thc/(x) plot at x *= 0 and a ~ 1,as in Figure 11.2.
ENERGY AND ENTROPY OF MIXING
The Heimhoitz free energy F ~ U ~ to has contributions from Iheenergyandfrom the entropy. We treat the effect of mixing two components A and B on
both terms. Let uA and \302\273abe the energy per atom of the pure substancesA and B,
referred to separated atoms at infinity. Tlie average energy per atom of the
constituents is
u = (uANA + vsyn)/N -nA 4- (\302\253B
- uA)x , A0)
which definesa straightline in the u~x plane. Figure 11.3. The average energyper atom of the homogeneous mixture may be larger or smallerthan for the
separate constituents. In (he exampleof Figure11.3,(he energy of the homoge-
homogeneous niixture is larger than the energy of the separate constituents.Theenergy excess is called the energy of mixing.
If (he \342\200\224re term in the free energy is negligible,asat 1 = 0,a positive mixing
energy means that a homogeneous mixture is not stable.Any such mixture will
then separate into two phases. But at a finite temperature the \342\200\224iaterm in the
free energy of the homogeneousmixturealways tends to lower the free energy.
Theentropy ofa mixturecontainsa contribution, called the entropy of mixing,that is not present in (he entropies of the separate components. The mixingentropy arises when atoms of the different species are interchangedin position;
this operation generates a different state of (hesystem.Becauseof such inter-
Energy and Entropy of Mixing
Figure 11,3 Energy per atom as a function of composition
in a sysicm with a positive mixing energy. A simple
example for which a solubiliiy gap may occur is thai of a
system in which the energy per atom of the homogeneous
mixilire is greater than Him of ihe separate phases, so that
1 0 for att cdiffer
mposi s. The ing e rgy i
e bci een the u[x) curve and the straight line.
changes a mixture has moreaccessiblestates than tlie two separate substances,and hence the mixturehas the higher entropy.
In C.80) we calculated the mixing entropy erM of a homogeneous alloy
A, ^B,. to find
(ID
as plotted in Figure 11.4. The curve of aM versus x has the important propertythat the slope at the endsof the composition range is vertical. We have
N dx-
X)- bgx - ioj
which goes to + co as x -+ 0 and to \342\200\224co as x -* I.
A2)
ChapterII: Binary Mixtures
da^/elx = \342\200\224X
Ffgure 11.4 Mixing entropy. Tn any mixture of two constii uenis an
interchange of two atoms of different species leads to a new slate of the
system. The logarithm of liienumber of slates related in this way is the
mixing entropy.
Consider now the quantity
u[x)- (a -
ffjtf A3)
which is the free energy per atom without the mixing entropy contribution. The
non-mixing part of the entropy,<j\342\200\224an, is usually nearly the same for (he
mixture as for (he separate components, so that {er\342\200\224
cM)r is nearly a linear
function of the compositionx. If we assume this, the /0(.v) cuive has the same
shapeas the u{x) curve, but offset vertically.If we add the mixing entropy contribution ~-zaM/N to fo(x), we obtain at
various temperatures the f{x) curves shown in Figure 11.5. In drawing the
figure we have ignored the temperaturedependence of/0(.v) itself, because for
Energy und Entropy of Mixing
Figure 11.5 Free energy per atom versus composition, at threetemperatures.The curve fQ is the free energy per atom without the
mixing entropy contribution. For ilHistraiion a parabolic composition
dependence is assumed, and the temperaturedependenceof/0 is
neglected. The tliree solid curves represcnl the free energy includingthe mixing entropy, for the temperatures 0.8 rM, 1.0 tM, and 1.2 rM,
where rw is the maximum temperature for which there is a solubility
gap. The phase separation at 0.8 rw is apparent.
our argument this is irrelevant. Three importantdeductionsfollow from the
construction of the/(x) curves:
(a) At all finite temperatures f{x) turns up at both ends of the composition
range, because of the infinite slope of the mixing entropy contribution.
(b) Below a certain temperature rM there is a com position range within which
the negative second derivative of the fo[x) curve is stronger than ihe
positive second derivative of the -taM contribution,thereby making it
possible to draw a common tangent to f(x) at two different values of x.
(c) Above Ty the curve has a positive second derivative at al!composilions.
Chapter 11: Binary Mixtures
We conclude that the A-B system with positive mixing energy will exhibii a
solubility gap below the temperature tM. The composition range of the gapwidens with
decreasing temperature, but the gap can reach the edgesof the
composition range only as t -* 0. At any finite temperature there is a finite
solubility of A in B and of B in A, a result obtained earlier in Chapter 3. The
new result is that the mutual solubility is limited only below tw. Positive
mixing energies arise in different ways. We now discuss three examples.
Example: Binary alloy with nearest-neighbor interact'ionSi Consider an alloy A^jB, in
which (he attractive interaction between unlike atoms is weaker than the attractive inter-
interaction between like atoms. For simplicity wespeakof the interactions as bonds. There arethree different bonds: A-A, A-B, and B-B. Let uAA) uAB and uBB be ihe potential energies ofeachbond.These binding energies will usually be negativewith respect to separated aloms.
We assume the atoms are randomly distributed among the lattice sites.The average
energy of ihe bonds surrounding an A alom is
uA= A
-x)uAA + xuAa , A4)
where (t - x)is ihe propoition of A and x is the proportion ofB.This result is wiiucn in ihe
mean field approximationof Chapter tO. Similarly, for B atoms,
\"a= A -
X)HAD + N\"UD. A5)
The total energy is obtained by summing over both atom types. Ifeach atom has p nearest
neighbors, the average energy per atom is
- ip[(l -xJUAA + 2jcA
- *Kb + *3\302\253db]- A6)
The factor^ aiises because eachbond issharedby the two atoms it connects. The result A6)can be written as
u = ip[(l-
x)uAA + xum] + uM. A7)
is the mixing energy. On this model the mixing energy as a function ofx is a parabola, as in
Figure II.5.
Energy and Entropy of Mixing
A solubility gap occurs whenever (/'//dx1 < 0, that is, when
^r= -2P[fAB -
i(\302\253AA + O3- B0)
From A2),
N dx2 x{l -x)
The equal sign holds for ,x = $. Wilh these results{19)yields
T*i-
M\302\253ab- iO'AA + ^a)] B2)
as die lower limit of the temperature for a solubility gap.
There are many reasons why mixed bonds may be weaker than ihe bonds of the sepafaicconstituents.If the constituent atom* of an alloy differ in radius, the difference introducesclasticstrains that raise the energy. Water and oil \"do not mix\"' because water moleculescarry a large electric dipole moment that leads to a strong electrostaticattraction between
water molecules. This attraction is absent in water-oil bonds, which are only about as
strong as the weaker oil-oil bonds.
Example: Mixture of two solids with different crystal structures. Consider a homoge-homogeneouscrystalline mixture of gold and silicon. Thestable crystal structure of gold is the face-centeredcubic structure in which every atom is surrounded by twelve equidistant nearest
neighbors. The stable crystal structure of silicon is the diamond structure in which every
aiom is surrounded by only four equidistant nearest neighbors, if in pure Au we replace asmall fraction xof the atoms by Si, we obtain a homogeneousmixture Au^.Si., wiih the
fee crystal structure of Au. Similarly, if in pure Si we replacea small fraction 1 - x of theaioms by Au, we obtain a homogeneous mixture Au, -,5^, but with the diamond crystalsiructure of Si.There are two different free energies, one for each crystal structure (Figure
11.6).The two curves must cross somewhere in the composition range, or else pure Au and
Si would not crystallize in different structures. The equilibrium curve consists of the lower
of the two curves, with a kink at the crossover point. Such a system exhibits a sotubility
gap on eithersideof the crossover composition. The curves shown in the figure are sche-
schematic; in the actual Au~Si system the unstable rangeextends so close to the edges of the
diagram that it cannot be representedon a fult-scale plot extending from x = 0 to x = 1.
Chapter 11; Binary Mixture
\\/
Figure 11.6 Tree energy versus composition for cryslallinchomogeneousmisiuresfor which [he [wo constituents of the
mixture crystallize in ^ilfcrem crystal structures. Two differe
free energy curves are involved, one for each crystal structure
Different crystal structures for the pureconstiiucmsarean important cause or solubility
gaps in crystalline solid mixiures. Our a/gument applies to mixtures of ihis kind, providedthe two structures do not transform coniiriuously into each other wilh changing composi-composition.This is a tacit assumption in our discussion, an assumption not always satisfied when
the two crystal structures are closely similar. The other assumption we make throughoutthis chapter is that no stable compound formation should occur, in the presence of com-compound formation the behavior of the mixture may be more complex,
vs^~7cz:\":\"s'.\".r:'~ \342\226\240\342\200\242\342\226\240-\342\226\240->\342\226\240\342\226\240-\342\226\240\342\226\240\342\226\240-\342\226\240\342\200\224-\342\226\240\342\226\240\342\226\240---\342\200\224.-.\342\200\224--...-.. ... -..,.-,... ~ ;-.-,,
Exampk: Liquid SIU-* He m*.Uuivi at W tanpcraiures. The moat interesting liquid mix-mixture with a solubility gap is the miMuniof ilie two iiefium isotopes JHeand JHe, atoms of
tiie Toruicr ocmii fcimjon^ unti of the 'aHer bosons, 1 lie re js a soluoiltty u\302\273iosn the mtx turc
oclow 0.S7rCj ii1/ in i igure 11.\027. 1 Ins property ss utilised m the Iicliuitt cJj'tiiion refftccr^tor
{Chapter 12).Tito mi.\\ing energy must be positi\\e to have a solubility gap. The origin of the
positive mixing encray is tht! folio\" ing: 4Hc aloiiisarc bosons.At suliieiently low tempera-temperaturesalmost jli \342\226\240*!le afoins occurs) the ground state orbii;tl of the sysicm, vvherc they have
Phaie Diagrams for SimpleSolubility Gaps
10 20 30 40 50 60 .70 80 90 100Atomic percent \342\226\240'He
Figure 11.7 Liquid mixtures of JHe and 4Hc.
pure\342\226\240\"He
zero kinetic energy. Almost trie entire kinetic energy of the mixture is contribuicd by t!ie
3He atoms, which are fermions.The energy per atom of a degenerate Fermi gas increases
v,ilh concentration as n1'*,as in Chapter 7. This energy has a negativesecondderivative
Pltase Diagrams Tor Simple Solubility Gaps
A phase diagram represents the temperature dependence of solubiiilygaps,as in
Figure 11.8. The two compositions xx and xf arc plaited horizontally, the
corresponding temperature vertically. The .v^ and xf branches merge at the
maximum temperaturet,m for which a solubility gap exists. At a given tempera-
temperature,any mixture whose overall composition falls within the raogeenclosedby
the curve is unstable as a homogeneous mixture.Thephasediagramsof actual
mixtures with solubility gups may be more complex,according to the aclual
form of (he free energy relation/(.v),
but the underlying principles are ihesamc.
Chapter II; Binary Mlxtur
/ Dec
1
1
1
1
Uns
Slabk
omposilion \\
Figure 11.8 Phase diagram for a binary system with a solubility
gap. A homogeneous mixture oCcomposition x will be unstable aitemperature i if the point (*,i) Tails below tlic stability boundarycurve. The mixture will then form two separate phasesof the
compositions given by the intersections of the stability boundary
curve with the horteontal line for temperature r. The stability
boundary curve shown here was calculated quantitatively for the
system of Figure lt.5, with a parabolic fo{x).
PHASE EQUIUBRIA BETWEEN LIQUIDAND SOLID MIXTURES
When a small fraction of a homogeneous liquidmixturefreezes,the composition
of the solid that forms is almost alwaysdifferent from that of the liquid. The
phenomenonis readily understood from the free energies for liquid and solidmixtures. We consider a simple model, under two assumptions; (a)Neitherthe
Phase Equilibria Between Liquid and Solid Mixtures
solid nor sheliquid has a solubility gap. (b) The melting temperature ta or pureconstituenlA is lower Jhan the melsing Semperature tb of pure constituentB.We consider a SemperaSure between ta and ra.
The freeenergiesper atom, fs{x) Tor the solid and fL(x) Tor the liquid, are
shown qualitatively in Figure 11.9a.The two curves intersect at some com-
posision. LeSus draw a jangenScommonlo boSh curves, touching/j aS .\\ ~ xs
and fL a! x = xL.We can define three composision ranges, each with differcnS
internal equilibria:
(a) When x < xL, the system in equilibrium is a homogeneous liquid.(b) When xL< x < xs, the system in equilibriumconsistsof two phases,
a
solid phase of composition xs and a liquid phaseofcompositionxL.
(c) When x > xs the system in equilibriumjs a homogeneoussolid.The compositions xs and xL of a so!id and a liquid phase in equilibrium arc
temperature dependent. As ihc temperature decreasesthe free energy of thesolid decreases more rapidly than tlKll of the liquid. The Ungctitiaipoints inFigure11.9amove to the Icfi, Tliis behavior is rcprcscnScdby a phase diagram
stinihlr to the earlier representationof the equilibrium composition curves for
mixtures with phase separation. In Figure 11.9b the curve for xL is called the
Hquidus curve; the curve for xs is the soltdus curve.
The phase diagrams have been determinedexperimentallyfor vast numbers
of binary mixtures. Those for most of the possible binary alloys are known.*For most metal alloys the phase diagrams are more complicatedthan Figure
11.9b, which was drawn for a simple system, germanium-silicon.
When the temperature is lowered in a binary liquid mixturewith the phase
diagram of Figure lj.9b, solidification takesplaceover a finite temperature
range, not just at a fixed temperature.To see this, consider a liquid with the
initial composition xiL shown in Figure 11.10.As the temperature is lowered,
solidification begins at t \342\200\224x,. The composition of the solid formed is given
by xtsi so 'hat the composition of the remaining liquid is changed. In the
example xiS > xiL, so that the liquid moves towards lowervaluesofx,where the
solidification temperature is lower. The temperature has to be lowered if
solidification is to continue. The composition of the liquid moves alongthe
liquidus curve until the solidification is compleiedat t =tA. The solid formed
is nonuniform in composition and is not in equilibrium. The solid may homoge-homogenizeafterward by atomic diffusion, particularly if lhe temperature remains
high for a long time. But for many solids atomic diffusion is too slow, and thein homogeneity remains ''frozen in
\"indcnniieiy.
\342\200\242The slandard iabutatlons arc by M. Hansen, Coitsilxatlon of binary allays, McGraw-Hill. 1958;R. P. Edioti, Constitution of binary alloys, firsi supplement, McGraw-Hill, 1965; R A. Shunfc,
Constitution of binary alloys, second supplement, McGraw-Hill, 1969. .
Figure 11.9 Phase equilibrium btiwccn liquid and solid mixtures. In
ihis example neither phasecxhibhs a solubiliSy gap. We assume
tA < x < xlt. The upper figure (a) shows the free energies for i|ie two plxiscs;
ihc lower figure (b) shows ihc correspondingphase diagram. The curves xLand xs in She phase diagram are called ihe liquid us and She solid us curves.The phasediagram is She Gc-Si phase diagram, wish TCt
= 940cC and
7\342\204\2425i- I412\"C
324
Phase Equilibria Between Liquid and Solid Mixtm
Figure 11.10 Mosi liquid mixiuresdo not solidify at a sharpScmperalurc,but over a finite temperature range from t, 'o ta. The
higher-nwiliiig consiiluenl precipilaSes first, thereby enriching thelower-mching consliSucnt in ihc liquid phase and thus loweringShe
solidification lemperalure of ific liquid.
Advanced Treatment; Eutecltcs. Therearemany binary systems in which the
liquid phase remainsa liquid down to temperatures significantly below thetowermeltingtemperatureof the constituents. The go!d-si!icon alloy is such a
system:a mixtureof69pet An and 31 pet Si starts to solidify at 370\302\260C. At other
compositions solidification starts at a higher temperature.When we plot the
temperature oflhe onset of solidificationas a Function of alloy composition,
we obtain the two-branch liqutdus curve in Figure 11.11. Mixtures with two
liquidusbranchesarecalledcutectics.Theminimum solidification temperature
is the eutectic SemperaSure,where She composition is She eutectic composition.The solidifiedsolidat the eutecticcompositionis a two phase solid, wiih
nearly pure gold sideby side wiih nearly pure silicon, as in Figure 11.1.In thesolidAu-Si mixture shcre h a very wide solubility gap.The low mching point
occurs for the eutectic composisionbecausethe freeenergy oFthe homogeneous
melt is lower than the free energy of the two phase solid, for temperatures at
or above She cutectie temperature.
Such behavior is common among systems thai exhibita solubilitygapillthe
solid but not in the liquid. The behavior ofeutccjj'cscan be understood from the
free energy plotsin Figure11.12a.Weassuine_/^(.\\\") for the solid as in Figure 11.6,
Chapter 11: Binary Mixtures
1600
1400
1200U
E 1000
1063\302\260
\\
\\
/
1/-31
//
0\302\260
404\302\260
0 10 20 30 40 50 60 10 80 90 100Pure Au Atomic percent silicon Pure Si
Figure 11.11 Euieciic phase diagram of go!d-siljconalloys.The Hquidus consisisof iwo
branches ihai come iogeihcr ai ihe euieciic iempctaiure T, = 37O;C.The horizonial
line and ihe experimcnial daia poinls ai 37OX indicaie ihai ihroughoul the eniirc
composiiion range ihe mixiurc docs noi complcic iis solidificalion unlit ihe euieciic
corresponding to difTcrcnt crystal structures a and ^ for the two pure con-constituents. Figure II.12a is constructed for a temperature above the cutectic
temperature but below the melting iemperature of citherconsihuent, so that
tUe free energy of the liquid reachesbelow the common tangent to the solid
phase curves.We can draw two new two-point tangents tltat give even lower free
energies. We now distinguishfive different composition ranges:
(a) and (e). For x < xaS or x > x^, the equilibrium state of the systemis a
Homogeneous solid. In the first range the solid will have the crystal structurea;in the second range the structure is ($.
(c). For xlL < x < XpL, the equilibrium state is a homogeneousliquid.(b) and (d). For x^ < x <
xaL or x^L < x <Xp$,
a liquid phase is in
equilibrium with a solid phase.As the temperature is lowered, faS and fa decreasemorerapidly than/L,
and the range of the homogeneous liquid becomesnarrower. Figure H.12b
shows the corresponding phase diagram, includingthe two solidus curves.
Figure 11.12 Free energies(a)and phase diagram (bj in a
sysiem.
At theeulecltc temperature t, Jhe free energy of Jhe liquid phaseis tangential
to the common tangent to f^ and fps, as in Figure 11.13. The composition at
which fL touches the tangent is the eutcctic composition. At x < xt, the free
energy fL iies above (he tangent, although fL may be beiowthe freeenergy of
a homogeneous solid.
A mixture of composition equal to the eutectic composition solidifies and
meits at a single temperature,just like a pure substance. The solidification of
Chapter11: Binary Mixtures
Figurel!.!3 FiCe energies in a euseclic system at t , andati < xr.
compositionsaway From the euieciic composition starls at a higher temperatureand ends at the eutecttc temperature. Melting starts at the eulectictemperatureand ends at a higher temperature.
The minimum properly of the melting temperature of eutectics is widelyutilized. The Au-Si eutectic plays a large role in semiconductor device tech-
technology: the cutectie permits low temperature welding of electrical contactwiresmadeofgoldtosilicondevices. Lead-tin alloys exhibit a euieciic (Figure11.14)at i83\302\260C to give solder a melting temperature below that of pure tin,
232;C. According to whether a sharp melting temperature or a melting range is
desired, citlicr the exact cutectie compositionB6pet lead) or a different com-
composition is employed. Salt sprinkled on ice melts the icebecauseof the low
eutectic temperature -2L2\"Cof the H2O~NaC! eutecttcat 8.17moipet NaO.
The solfdus curves of eutectic systems vary greatly in character, for the
Pb-Sti system (Figure Il.M) die solid phases in equilibrium with die ioclt
contain :tn appreciable fraction of tltc minority const [merit, and this fraction
increases with decreasing temperature, in other systems this fraction may besmallor may decrease with decreasing temperature, or both. The Au-Si system
is an example: The relative concentration of Au in solid Si ill equilibrium with
an Au-Si melt reaches a ma\\imum value of only 2 x !G~6s.-ound i 300\" C, and
it drops off rapidly at lower temperature.In our discussionof the free energy curves of Figures 31.12 and 11.13we
assumed tltat lite composition at which the liquidphasefree energy touches the
\020 10 20 30 40 50 60 70
pure Sn Atomic percent lead
90 100
pure Pb
10/tm
Figure 11.14 {a} Ptiasc diagram of the Pb-Sn s> stem, after Hatlicn.{b) Microphotngrapf;of [he Pb-Sn eutmic, magnified about S0Otimes.Courtesy of J. D. Hunt and K. A.
Jackson.S29
Chapter 11: Binary Mixtures
tangent to the solid phase curves liesbetweenthe compositions xlS and xfiS.In some systems this point liesoutsidethe interval, as if either/aS and/t or/flSand fL were interchanged iti Figure i i .\\2a. Such systems arc caiied peritecticsystems.
SUMMARY
1. A mixture exhibits a solubility gap when the combinedfreeenergy of two
separate phases side by side is lowerthan the free energy of the homogeneousmixture.
2, The mixing entropy arises when atoms of different speciesare interchangedin position. For the alloy A} _IBI, we have
3. The mixing energy for nearest-neighbor interactions is
uM= px(l
-x)[uAB
- j(uAA + uBB)] ,
for p nearest neighbors.
4. The Hquidus is the composition curve .xL versus t for a liquid phase in
equilibrium with a solid. The solidusis the compositioncurve Xs versus i
for a solid phase in equilibriumwith a liquid.
5. Mixtures with two branches to the liquidus curve are called eutectics. The
minimum solidification temperature is called the eutectic temperature.
PROBLEMS
L Chemicalpotentialsm two-phase equilibrium. Show that the chemical po-[eniials ;iA and /jB of die two atomic species A and B of an equilibrium two
phase mixturearegiven by the intercepts oFthe two-point tangent in Figure 11-2
with the vertical edges oFthe diagramat x = 0 and x \342\200\2241.
1. Mixing energy in 3He~*He andPb-Sn mixtures. ThephasediagramoF liq-
liquid 3He-4He mixtures in Figure 11.8 shows that the solubility of 3He in 4He
remains finite (about 6 pet)as r ->0.Similarly, the Pb-Sn phase diagram of
Figure.11.14 shows a finite residual solubility oFPb in solid Sn with decreasing
t. What do such finite residual solubilities inipty about the Form of the Functionu(.x)?
3. Segregation coefficient of impurities. Let B be an impurity to A, wish A' \302\2531-
In this limit the non-mixing parts oF the Free energy can be expressed as linearfunctions of x, as fQ(x) = /0@) + x/0'@),for both liquid and solid phases.Assume thai the liquidmixtureis in equilibrium with the solid mixture. Calculatethe equilibrium concentration ratio k ~ xs/xL, called thesegregationcoefficient.
For many systems k \302\253I, and then a substance may be purified by melting
and partial resolidificatioti, discarding a smallFractionoFthemeit.Thisprinciple
is widely used in the purification of materials,as m the zone refining of semi-
semiconductors. Give a numerical value for \302\243for/os'\342\200\224
/Dt'= ! eV and T = 1000K.
4. Solidification range of a binary alloy. Consider the solidificationofa binary
alloy wish the phase diagram of Figure.'11.10.Show that, regardless of the
initial composition, the melt will always become fully depleted in component B
by the time the last remnant of the meit solidifies. That is, the solid i Seas ion
will not be complete until the temperature has dropped to TA.
5. Alloying of gold hto silicon, (a) Suppose a 1000A layer of Au is evaporatedonto a Si crystal, and subsequently heated to 400\302\260C- From the Au-Si phase
diagram, Figure 11.11,estimate how deep she gold will penetrate into thesiliconcrystal. The densities of Au and Si are 19.3and 2.33gcm\"\023. (b) Redo
the estimate for 800\302\260C.
Chapter 12
Cryogenics
COOLING BY EXTERNAL WORK
IN AN EXPANSION ENGINE 334
Gas Liquefactionby the Joule-Thomson Effect 337
Example: Joule-Thomson Effect for van der Waals Gas 333LirideCycle 339Evaporation Cooling: Pumped Helium, to 0.3 K 341HeliumDilutionRefrigerator;Miilidegrees
342
ISENTROPIC DEMAGNETIZATION:
QUEST FOR ABSOLUTE ZERO 346
NuclearDemagnetization 348
SUMMARY 350
PROBLEMS 350
1. Helium as a van der Waals Gas 359
2. Ideal Carnot Liquefier 35i3. ClaudeCycle Helium Liquefier 35!
4. Evaporation Cooling Limit 3525. InitialTemperaturefor Demagnetization Cooling 352
Cryogenics is the physics and techiioiogyoftheproductionoftow temperatures.
We discuss the physical principles of the most important coolingmethods,down to the lowest temperatures.
The dominant principleoflowtemperaturegeneration down to lOmK is the
cooling of a gas by kiting it do work against a force during an expansion. Thegas employed may be a conventional gas; the free electron gas in a semicon-semiconductor; or the virtual gas of 3Hc atoms dissolvedin liquid 4He. The force
against which work is done may be external or internal to the gas. Below10mK the dominant cooling principle is the iscntropic demagnetizationof aparamagnetic substance.
We discuss ihe cooling methods in the order in which they occur in a
laboratory cooling chain lhat starts by liquefying helium and proceeds from
there to the lowest laboratory temperatures,usually lOmK, sometimes 1 ;iK.
Household cooling appliances and automobile air conditioners utilize thesameevaporation cooling method that is used in the laboratory for cooling
liquid helium below its boiling temperature, to about 1 K.
COOLING BY EXTERNAL WORK
IN AN EXPANSION ENGINE
In the isentropic expansion of a monatomicidealgas from pressure pi to a
lower pressurep2, the temperaturedropsaccording to
(i)
by F.64). Suppose p, = 32atm; p2= iatm; and Ti = 300K; then the tem-
temperature will drop to T; \342\200\22475 K. We are chiefly interested in helium as the
working gas, and for helium A) is an excellentapproximationif the cooling
process is reversible.The problems in implementingexpansioncoolingarise from the partial
irreversibility of actual expansion processes.The problems are compounded
by the nonexistence of good low temperature lubricants. Actual expansion
cooling cycles follow Figure 12.1.The compression and expansion parts of
itisng by External Work in on {Expansion Ens
Heal ejection
Expansionngine
Working\342\200\224
volume
FEgure I2.I Simple expansion refrtgeraior.A working gas ts
compressed; the heat of compressionis ejected into the
environment. The compressed room temperature gas is
precooled further in the counleriiow heat exchanger. It then
does work in an expansion engine, where it cools to a
temperature below that of the working volume. Afkr extracting
hea{ from the working volume, the gas returns to the compressorvia {he heat exchanger.
Chapter 12: Cryogenics
the cycle arc separated.The compressionis performed at or above room
temperature. The hot compressedgas is cooledto near room temperature
by ejecting heat into the environment. Thegasis further precooled in a counter-
flow heat exchanger by contact with the cold return gas stream at the lowtemperatureofthe cooling load. The gas is then cooled to itslowesttemperaturein the expansion engine, usually a low friction turbine.The coldgas extracts
heat from the cooling load and then returns to the compressor via the heat
exchanger. The heat exchanger greatly reduces the cooling requirements im-
imposed on the expansion engine. The design of Sheheat exchangerisasimportant
as the design of the expansion engine.The work extracted by the expansion engine is the enthalpy difference
between the input and output gas: The iota!energy flowing into the expansion
engine is the iruernat energy U^ of the gas plus the displacement work p{Vldone by the compressor, where boih fij and Vx refer to a given mass of gas.The total energy leaving She engine with the gas is the energy U2 of the gas
plus the work p2 V% required to move the gas against the pressurep2- The work
extracted by ihe engine is the difference
W = {Ul 4- PlVt) - {Uz + PlV1)= Hs
~~ H2. B)
For a monatomic ideal gas U j= |Nr and pV = Nr, hence H =\\Nx. The
work performed on the engine by the gas is
W \302\273|N{t,
- r2). C)
The countefHowheat exchangeris an enthalpy exchange device: it is an
expansion engine which extractsno externalwork.
Most gas Hquefiers use expansion engines to prccool the gas closeto itsliquefaction temperature. It is impractical to carry She expansion coolingtothe point of liquefaction:the formation of a liquid phase inside expansionenginescausesmechanical operating difficulties. The final liquefaction stageis usually a Joule-Thomson stage, discussed below. Helium and hydrogenliquefiersusually eontain two or more expansion engines at successive tem-temperatures, with multiple heat exchangers.
The principle of cooling by isentropic expansion of an ideal gas is applicableto the electron gas in semiconductors. When electrons How from a semi-
semiconductor wuh high electron concentration into a semiconductorwith a lower
electron concentration, the electron gas expands and does work againstshe
potential barrier between the two substances that equalizesthe two chemical
potentials. The resulting electronic cooling, called the Peltiereffect, is used
Gas Liquefaction by the Joule-Thomson Effect
down to about 195K quite routinely;in multistage units temperatures downSo 135K hnvc been achieved.
Gas Liquefaction by the Joule-Thomson Effect
Intcrinolccular attractive interactionscausethe condensation of al! gases. At
icmpcratures slightly above the condensationtemperature the interactions are
strong enough that work against them during expansion causes significant
cooling of the gas. If the coolingis sufficient, part of the gas will condense.
This process is Joule-Thomsonliquefaction.Thepractical implementation is simple. Gas at pressure p, is forcedthrough
a constriction called an expansion valve into space with a lower pressure p2,as in Figure 12.2.The work is {he difference between the displacement work~plt(Vj doneon the gas in pushing it through {he expansion valve and the
displacement work +p2(\"/2 recovered from the gas on the downstreamside.Here dVt is negative and dV2 is positive.
The overallprocessis at constant enthalpy. To sec this, notice that the
expansion valve acts as an expansion engine that extracts zero work. With
If = 0 in B), we have H\\ ~ li2 in the Joule-Thomson effect. For an ideal gas11 \342\204\242
\\Nr, so that ts \342\200\224r2 in the expansion. There is zero coolingeffect for an
ideal gas.
in real gases a small temperaturechange occurs because of the internal
work done by the molecules duriiig expansion. The sign of the temperature
Expansionvalve
Figure |2,2 The Joule-Thomson eflccl.A gas is pushed
through an expansion value. If the gas is notlflieal. ihere will
be a temperature change during the expansion because of work
done against the intermolecuhtr forces. If the temperature is
initially below a certain inversion temperature, riB,,the gas
will cool on Joule-Thomsonexpansion-
Chapter 12: Cry
Table U.I Liquefaclion dala for lo
(jas
CO,cmo,N,H,
\342\200\242He
JHe
n.K
19511290277.320.4
4.SS
3.20
Tt,
K
30419115512633.35.253.35
T,,., AH,
K U/mol
B050) 25.2
A290) 8.18893 6.82
621 5.57
205 0.9051 0-082
B3) 0.025
V,.
em'/mol
22334.428.134.628.632050.8
wall Mite
314
6667458.70.7!0.14
the liquid. The las umn, Atl/V, tndkai n walls thai can be la'ken up for
Jrti T( ano\" oot measured 'Carbon dioxide solidifies mosphcric pressure. because its lrjple poinl c
of natural gas, which is liquefied in huge quamilics for shipping us LNG fuel. Liquid
and niirogen are separaied iu lhe liquefaction of air. For helium, we give daia boih for ihe
ei isotope 4Hc and for 3He.
change during a Joule-Thomsonexpansiondepends on the initial temperature.
All gases have an inversiontemperatureTIn, below which such an expansion
cools, above which it heals (he gas. inversion temperatures for common gasesare listed in Table 12.1.
Example: Joule-Thomson effectfor ran far li'aab gas. We found in A0.75) that
H \302\253=JWt + {S2fV){bx
- 2a) D)
for a van der Waais gas, where a and b are positive constants. The last two terms arc the
corrections caused by the short range repulsion and the long range atiraction. The correc-corrections have opposite signs. The tola! correciionchanges sign at the temperature
tinv== lab = 2/rt, E)
where xc is the critical tempera!urc, defined by A0.46).
The temperature iln, is the inversion lemperature. For t < iin, the enthalpy at fixed
temperature increasesas the volume increases; here in expansion the work done against the
attraciive interactions between molecules is dominani. In a process at consiant enihafpy
this increase is compensated by a decrease of the \\Nt ierm, that is, by cooling the gas. For
Gas Liquefactionby the Joutc-Thot
i a fixed lemperamre
ioIccuIcspenetrate farther inio lhe repulsive regio
s becausenow the work done by theanl: ai lhc higher lempcraluie the
Linde cycle. In gas litjueficrs the Joule-Thomsonexpansioniscombined wilh
a counlerflow heat exchanger, as shown in Figure J2.3.The combination is
called a Lindc cycle, aficr Carl von Linde who used such a cycle in 1895 to
liquefy air starting from room temperature.In our discussionwe assume that
the expanded gas returning from the heat exchanger is at ihe same temperatureas the compressed gas entering it. We neglect any pressure differencebetweenthe output of the heat exchanger and the pressure above the liquid.
Toand from comprcs
Figure 12.3 The Lindecycle.Gasby combining Joule-Thomson expaa countcrflow Iieatexchanger.
JT expansionvaJv.
Liquefied gas
Figure 12.4 Performance of helium litjueficrs operating by the Liude cycle,as a fund ion ofihe inpui pressure, for an ouiput pressure of 1 aim andfor various valuesof ihc input temperature. The solid curvesgive
ihe
liquefaciion coefficient The broken cunes give QiM=
tfDUl- Hia, ihe
inierna! refrigeration load available at 4.2 K if (he toad is placed inside iheii^uciicrand ihc still coid helium gas boiledoff by lhc load is relumed
through ihe heal exchanger rather than boiled oiT into liie atmosphere.See Problem 3. Afier A. J. Croft in A&mWL-dcryo&mcs (C. A. Baiiey, ed.),Plenum, 1971. p. 1S7.
Evaporation Coaling; PumpedHelium, to 0.3 K
The comhimtiion hc;\302\273exchanger-expansion valve is ;i consimu enthalpyarrangement. Let one moleof gas enter Hie combination; suppose ihat thefraction X is liquefied. Constant cniluilpy requires lhat
//;\342\200\236=
-IWii, + \302\243!-
J)H9Ui' \302\2436}
Here Hla= H(TiMp-a) and llou, = H(Tin.pBJare the enlhalpies per mo!c of
gas at lhe input and output pressures, both at lhe common upper temperatureof the heat exchanger- tfHl) is lhe enthalpy per moleof liquid at its boiling lcm-
peraturc under lhe pressurepoal. From F) we obtain the fraction
17)
calledthe liquefaction coefficient.
Liquefaction lakes place when //\342\200\236\342\200\236,> Hia; thai is, when
HiT^J > H{Tia,p-J. (8)
Only the enthalpies at the input lemperalure of lhe heat exchangermaHer. If
the Joule-Thomson expansion at this temperature cools the gas, liquefaction
will take place.The three enthalpiesin G) are known experimentally. Figure 12.4 shows lhe
liquefaction coefficient calculated from them Tor helium. The liquefactioncoefficientdrops rapidly with increasing Tiat because of the decrease of thenumeratorin G) and the increase of the denominator. To obtain useful lique-
liquefaction, say ;. > 0.!, input temperatures below one-thirdof the inversion
temperature are usually required. For many gases this requiresprecedingofthe gas by an expansion engine. The combination of an expansionengine and
a Linde cycle is called a Claude cycle.The expansion engine is invariably
preceded by another heat excitauger,as in Figure 12.1-
Evaporatfon Coofing: Pumped Helium, lo 0.3 K
Starting from liquid helium, the simplest route lo lower temperaturesis fy
evaporation cooling of !hc liquid helium, by pumping away lieiium vapor, j'
latent heat of vaporization of the liquid Iteltum is extracted along with the vap,-..The heatextractioncauses ihc further cooling: work is done against the inter-:itonu'cforces that cnuscJ ihc helium io liquefy in the first pi;>ce. hi JouL-
Tltomson cooling tlte initial staie is a gas, while in evaporation cooling the
initial state is a liquid.
Chapter 17: Cryogenics
Table 12.2 Tempera turds, in kctvifi.at which 1 he vapor pressures of 4He and3Hc reach specified values
p(lorr)
0.660.28
0.790.36
0.980.47
1.27
0.66
1.741.03
2.641-79
The lowest tempcralure accessible by evaporation cooling of liquid heliumisa problemin vacuum technology (Chapter 14). As the ternperaluredrops,the
equilibrium vapor pressure drops (Table 12.2}and so docsthe raie ai which
helium gas and its heat ofvaporization can be extracted from the liquid heliumbath.
Evaporation cooling ts the dominant cooling principle in everyday cooling
devices such a5 household refrigerators and freezersant! in ait conditioners.
The only difference is in the workingsubstance.
Helium Dilution Refrigerator: Militdegrees
Once the equilibriumvapor pressure of liquid 3He has dropped to I0\023 torr,
classical refrigeration principles lose their utility. The temperaiurerange from
0.6 K io 0.0! K. is dominated by the helium dilution refrigerator, which is an
evaporation refrigeratorin a very clever quantum disguise.*We saw in Chapter 7 that *He atoms are bosons, while 3He atomsare fer-
mions. This distinction is not important at temperatures appreciably higherthan the superfluid transition temperature of \"fie, 2.17K. However,the two
isotopes behave as altogether different substances at lowertemperatures.Below
0.87 K. liquid 3He and 4He are immiscibleovera wide composition range, like
oil and water. This was discussed in Chapter 11 and is shown in the phase
diagram of 3He-4He mixtures in Figure 11.7. A mixture with composition in
the range labeled unstable wil! decomposeinto two separate phases whose
compositions are given by the two brandies of the curveenclosingthat area.
The concentrated 3He phase floats on top of the dilute 3He phase.
As T -\302\2730, the 3He concentration of the phase dilute in 3He drops to about
6 pet, and the phaserich in 3He becomes essentially pure 3He. Consider a liquid
*For good reviews, sec D.S.Belts.Contemporary Physics 9.97 {1968): IC. Wheatley. Am-1 Phys.
36, 181A968);for a general review or cooling techniques below 1 K see W. J. Huiskamp and O. V.Lounasmaa, Repts. Prog, Phys. 36, 423A973); O. V. Lounasmaa, Experimental principles and
methods below t K, AcademicPress,Hew York, 1974. A very elementary accoun! Is O. V. Lounasmaa,Scientific American 221,26 (t%9). . \342\226\240 .
Figure 12.5 Cooling principle of ilm helium dilution refrigerator. LiquidHlc is in equilibrium wiih a JHc-4He nmiure. When 4He is added io the
mixiure, sHe evaporaics from ihc pure ]He fluid and absorbs heat in ihc
3He-4He mixture wiih more than 6 pet 3Hea* a temperature in the millidcgree
range, near the bottomofFigure11.7.At these temperatures almost all the 4Heatoms have condensedinto the ground state orbital. Their entropy is negligiblecomparedto that of the remaining 3He atoms, which then behaveas if they were
present alone, as a gas occupyingthe volutne of the mixture. If the 3He concen-concentration exceeds 6pct, the excess condenses into concentrated liquid 3Heandlatent heat is liberated. If concentrated liquid 3He is evaporatedimo the 4He
rich phase, the latent heat is consumed.The principle of evaporation cooling
can again be applied: this is the basts of the heliumdilution refrigerator.
To see how the solution of 3He can beemployed to obtain refrigeration,
consider the equilibrium between the concentrated3Heliquidphase and the
dilute iHc gas-like plliise (Figure 12.5).Supposelliai tile JHc:4lic nilio of
the dilute phase is decreased,as by dilution with pure *He. In order to restoretheequilibriumconcentration, 3He aiomswil! evaporate from the concentrated3He liquid.Coolingwill result.
To obtain a cyclic process the 3He-4He mixturemust be separated again.
Tile most common method is by distillation, using tile different equilibrium
vapor pressures of 3He and *Hc (Table 12.2). Figure 12.6 shows a schematic
diagram of a refrigerator built on these principles. The diagram is highly
oversimplified. In particular, in actual refrigerators titehcat exchangerbetweenthe mixing chamber and the still has an elaboratemultistagedesign.An alternate
method* to separate the. 3He--4He mixture utilizes tile superfluidity of 4He
below 2.17 K. For a variety, of practical reasons it is less commonly used,althoughUs performance is excellent. . \342\226\240
.
Chapter 12: Cryogenics
3He pump loop
Key:Liquid
\342\226\240I 'lie
Helium Dilution Refrigerator: AtitliJegre
Figure 12.6 Hdium dliulion refrigerator. Prccooledliquid 3He enters a mixingchamber a( (he tower cud of the assembly, wlicrc cooling takes place by ihe quasi-
cvaporaiion of the 3He atoms into the denser JHc-Jf1cmixed phase underneath.
The quast-gas of JH atoms dissolvedin liquid *He then diffuses through a countcrfiow
heat exchanger into 3 still. There the JHe is disiilledfrom the 3Hc-4Hc mixture
selectively, and is pumped olf.To obtain a useful 3He evaporation and circulationrate, heat must be added to the still, 10raiseUs temperature to about 0.7 K, at which
temperature the *He vapor pressure is ssiH much smaller. Thus, the 4He does not
circulate lo any appreciable extent; ihe *Hcmoves riirough a nearly stationarybackground of4Hc.The pumped-off JHe is returned to ihe system and is condensedin a condenser that is cooled to about I K by contact with a pumped 4He bath. Theconstrictionbelow the condenser takes up the excess pressuregenerated by the
circulation pump over ihe pressure in the still. The liquified JHe is cooled further,
first in rhe siill. ihcn in the counter/low heat exchanger, beforere-entering tlic miung
chamber.
The helium dilution refrigerator has a low temperaturelitnft. In the conven-
conventional evaporaiioii refrigerator this limit arose because of the disappearance ofthe gas phase, but the quasi-gas phase of 3Hepersistsdown to t = 0, However,ihe heat of quasi-vaporizationof JHe vanishes proportionally to x2, and as a
result, Ihe heat removal rate from the mixing chamber vanishes as i1. TI'Spractical
low temperature limit is about 10 mK, In onerepresentativedevice;*
a temperature of 8.3 mK has been achieved:ihe same device was capable of
removing 40/AVat 80mK.Temperatures
below SmK can be ncltievcu by single shot operation. If, in
ihe design of Figure I 2.6, we shut off the 3He supplyafter some time of opeiation,
there is no needto cooi the incoming 3He itself, and ihe temperature of the
mixingchamberdrops below its sleady state value, until all 3He has been
removed from the chamber.
The dilution refrigerator is not the oniy cooiingmethodin the inillikelvin
range that utilizes the peculiar propertiesof JHe.An alternate method, known
as Pomcranchsik cooling, utilizes the phasediagramof3He,as shown in Fig-
Figure 7.15, with its negative slope of the phase boundary between liquid and
solid 3He, The interested reader is referredto the reviews by Huiskamp ant!
Lounasmaa, and by Lounastnaa, citedearlier.
'N. H. Pcnnings, R. de Bruyn Ouboicr, K. \\V. Tacoois. Phjiica 8 SI. !0!A976). and Physiea B
84, 102A976}.
ISENTROPIC DEMAGNETIZATION:
QUEST FOR ABSOLUTEZEROBelow 0.01 K the doniimim cooling process is the isciitropic(adiabaiic) dcm;ig-
iictizatioii of a paramagnetic substance. By this process,temperaturesof I niK
have been attained with electronic paramagnetic systemsand j /(K with nuclear
paramagnetic systems. The method dependson tlie fact that at a fixed tempera-temperaturethe entropy of a system of magneticmomentsisloweredby application of a
magnetic field\342\200\224essentially because fewer slates are accessible to the systemwhen ilic level splitting is large than when the level splitting is small. Examplesof the dependence of the
entropyon tlie magnetic field were given in Chapters 2
and 3.
We first apply a magnetic field Bt at constant temperature ij. The spin excesswill attain a value appropriate to the value of Bj/tj. If the magnetic field is thenreducedto B2without changing the entropy of the spin system,the spin excesswill remain unchanged, which means that B2/z2 will equa' #i/ri- HBz \302\253Bi,
then t2 <\342\226\240<tj-
When the specimen is demagnetized isentropically,entropy can
flow into the spin system only from the system of lattice vibrations, as in Fig-Figure 12.7- At the temperatures of interest the entropy of the lattice vibrations is
usually negligible; thus the entropy ofthe spin system will be essentially constant
during isentropie demagnetization of the specimen.
w Total
Spin
Lattice
Before
1
Time\342\200\224-
New equilibrium Before
\\Latttce
Time\342\200\224\342\200\242
New equilibr:
Time at which
magnetic field
is removed
Time at which
magnetic field
js removed
Figure 12.7 During isentropie demagnetization the total entropy of the S
specimen is constant. The initial entropy of the lattice should be small in
comparison with the entropy of the spin system in order to obtain significant
cooling of the lattice. - - .
lsentropic Demagnetization: Quest for Absolute Zer
Figure 12.8 Entropy fora spin \\ sysiem as a function of Icmpcralure.assum
an inicrna! random magnetic field Bx of 100 gauss. The specimenis magnetiz
isothermaiiy along ah, and is then insulated ihcrmaMy. Thu cxlcrna! ntagnctit
field is 1 timed off along/>c. Ill order to keep the figure on a reasonable sculcllic initial temperature Tj and tlie external magnetic field are lower than woi
used in practice.
The steps carried out in the cooling processarc shown in Figure 12.8. The
field is applied at temperaturetx with the specimen in good thermal contactwith the surroundings, giving the isothermal path ab. The specimenis ihen
insulated {At? ~ 0) and the field removed; the specimenfollows the constant
entropy path 6c, ending up at temperature t2. The thermal contact ai t, is
provided by helium gas, and the thermal contact is broken by removing the
gas with a pump.
The population of a magnetic subievelis a function only of ntB/x, where m
is the magneticmomentof a spin. The spin-system entropy is a function only
ofthe population distribution; hence the spin entropy isa function only oimBjx.
If SA is theeffeclive field that corresponds to thediverselocalinteractions among
the spins or ofthe spins with the lattice, the final temperature r2 reached in an
isenlropic demagnetizationexperimentis
(9}
where B is the initial field and rt the initial temperature.Results are shown in
Figure IZ9 for the paramagnetic salt known as CMN, which denotes cerous
magnesium nitrate, . .
0 0.1 0.2 0.3 0.4 0.5 0.6
Final temperature, in K
Figure 12.9 Final magaclic field Bf versus final
nil rale. In fhese experiments ihe magnetic field was not
removed cniirciy, bul only m tfic indicated values.Theiiiiiial fields and icmpcraiurcs were idcnliait in all inns.
After unpublished results 61\" J. S. Still and J. H. Milncr.
as cilcd by N, Kuril, Kuovo Cimcnio(Supplement)6,1109A957).
Theprocessdescribedsofar is a single shot process. It is easily convertedinto a cyclicprocessby thermally disconnecting, in one way or another, the
demagnetized working substance from the load, reconnectingit to the reservoir
at t,, and repeating the process.*
NuclearDcmagnelizalion
Because nuclear magnetic moments arc weak, nuclear magneticinteractionsare much weaker lhan similar electronic interactions. We expect to reach a
temperature 100 limes lower with a nuclear paramagnet lhan with an electron
paramagnet. The initial temperature of the nuclearstage in a nuclear spin-
\342\200\242C. V. Hctr. C. B. Barnes,and J. G. Daunt. Rev. Scj, insi. 25. IGS8 j|954); W. p. PraH, S. S.
Rosenblum, W. A. Slcyerl. and i. A. Barclay. OHgcnics 17, 3S! A977).
t\\'uc!ear Demagnetization
Iniiial magnetic Held in KG
0.6 1
Initial B/T'm \\QS G/K
Figure I2J0 Nuclear demagnctizaflons ofcoppernuclei in the metal, skirting from 0.012 K and various
fields. After M. V. Hobdcn imd N. Kuril. Phil. Mag.-1.1902!1959).
coolingexperimen! must be lower than in an electron spin-cooling experiment.lfwestartatB= SOkGandT,= 0.01
K,then/fiB/*87\\* 0.5, and the entropy
decrease on magnetizationis over 10 percent of the maximum spin entropy.This is sufficient to overwhelm ihe lattice and from (9) we estimate a final
temperature T2 ss 10\027K.The first nuclear cooling experiment was carried
oui by Kurli and coworkerson Cu nuclei in the metal, starting from a first
stage at about 0.02 K as attained by electron demagneltzallon cooling. The
lowest temperature reached in this experimen!was 1.2 x I0~6K. The results
in Figure 12-10fil a line of Ihe form of (9): 7\\ = T|C.1/B)wilh Bin ^auss, so
llial B\302\261= 3.1 gauss. This is the ciTcctive interaction field of the magnetic mo-
moments of the Cu nueiei. The motivation for using nuclei in a metal rather than
in an insulatoris that conduction electrons help ensure rapid thermal contactof latticeand nuclei at the temperature of the first stage.
Temperatures below 1//K have been achieved in experiments in which the
cooling load was the system of nuclearspinsitself, particulatly in experiments
that were combinations of coolingexperimentsand nuclear magnetic resonance
experiments.*
SUMMARY
1. The two dominant principles of the production of low temperaturesarc ihe
cooling of a gas by letting it do work against a force during an expansionand the iscntropicdcmagncii/atioiiof a paramagnetic substance.
2. Joule-Thomson cooling is an irreversible process in which work is done
against interatomic attractive forces in a gas. It is used as the last coolingstagein liquefying low-boiling gases.
3. In evaporation cooling the work is alsodoneagainst the interatomic forces,
but starting from the liquid phaserather than the gas phase. Using different
working substances, evaporationcoolingforms the basis of household
cooling devices, automobile airconditioners,and laboratory cooling devices
(in the range 4 K down to SGmK).
4. The helium dilution refrigerator is an evaporation cooling device in which
the gas is the virtual gas of 3Heatomsdissolvedin 4He.
5. Isentropic demagnetization utilizes the lowering of the temperature of a
system of magnetic moments,when an external magnetic field is reduced in
strength. The magnetic moments may be electronic or nuclearmoments.
By using nuclear moments, temperatures in the microkelvin range may be
achieved.
PROBLEMS
L Helium as a van der H'aafcgas. (a) Estimate tlte liquefaction coefficient X
for helium by treating it as a van der Waals gas. Select the van der Waalscoefficientsa and 6 in such a way that Tor one mole 2Nb is the actualmolarvolume of liquid helium and that 2a/b is the actual inversion temperature.Use the data in Table 12.!. Approximate the denominator in G) by setting
Hout-
Hi(q- AH 4- f(rin -
xliH), A0)
\342\200\242See,for example, M. Chapcllier, M. Goldman, V. H. Chau and A. Abragara, Appl. Phys-41,
849A970). . -.-._...\342\226\240
where All is the latent heat ofvaporizationof liquid helium. (Explain how this
approximation arises if one treats the expanded gas as an ideal gas).Theresulting expression gives /. as a function of the molar volumes Yin and Vatll.Convert to pressures by approximating the l\"svia the ideal gas law. (b) Inaertnumerical values for T = 15 K and compare with Figure 12.4.
2. Ideal Carnot liquefier. (a)Calculatethe work \\VL thai would be required toliquefy one mole of a monstomic ideal gas if the iiqticfier operated rcversibiy.Assume that the gas is suppliedat roomtemperature To. and under the same
pressure p0 at whidi the liquefied gas is removed, typically 1 atmosphere. Let
7\\ be the boiling temperature of the gas at this pressure, and A// the latent heatof vaporization.Show that under these conditions
A1)
To derive A1)assumethat the gas is first cooled at fixed pressure p6 from To
to Tfc, by means of a reversible refrigeratorthat operates between the fixed
upper temperature Tb~ To attd a variable lower temperatureequalto the gas
temperature. Initially T, = To, and at the end T, ~ 1\\.After reaching Tb the
refrigerator extracts the latent heatof vaporization at the fixed lower tempera-temperatureTb. (b) Insert To = 300 K and values for Tb and AH characteristic of
helium. Re-express the result as kilowatt-hours per liter of liquid helium.Actua! helium liquefiersconsume5 to lOkWh. liter.
3. Claude cycle helium Hqucfier. Considera heliumliquefier in which 1 mols\021
of gas enters the Lrnde stageat T(o= 15 K and at a pressure pla
= 30 aim.
(a) Calculate the rate of liquefaction, in liter hr\"'. Suppose that all the liquefied
helium is withdrawn to cool an externalexperimentalapparatus, releasing the
boiled-off helium vapor into the atmosphere. Calculatethe cooling load in
watts sufficient to evaporate the heliumat the rate it is liquefied. Compare thiswith the cooling load obtainable if the liquefier is operatedas a closed-cycle
refrigerator by placing the apparatus into the liquid collectionvesselof theliquefier, so lhat the still cold boiled-off helium gas is returnedthrough the heat
exchangers, (b) Assume that the heat exchangerbetween compressor and ex-
expansion engine (Figure 12.1)is sufficiently ideal that the expanded return gasthat leaves it with pressure pout is at essentially the same temperature Tc as thecompressedgasenteringit with pressure pc. Show that under ordinary liquefieroperationthe expansion engine must extract the work
Te - TJ , A2)
Chapter 12: Cryogenics
per mole of compressed gas.HereTin, pin, pBUt, and X have the same meaningas in ihe Undo cyclesectionof this chapier. Assume the expansion engineoperates isemropically between ihe pressure-temperaturepairs {pc,Tc) and
(Pia>Ti(l). From A2) and the given values of (pia,Tia),calculate(pc,Te).(c) Estimate
the minimum compressor power required to operate the iiquefier,by assuming
that the compression is isothermal from poai to pc at temperature Tc ~ 50\"C.Combinethe result with {hecooling loads calculated under (a) into a coefficientof refrigerator performance, for both modes of operation. Compare with the
Carnot iimit.
4. Evaporation cooling limit. Estimate the lowesttemperatureTmia that can
be achieved by evaporation cooling of liquid 4He if the cooling load is 0.1 W
and the vacuum pump has a pump speedS = I02filers\021. Assume (hat the
helium vapor pressure above the boiling helium is equal to the equilibriumvaporpressurecorresponding lo TBliJ1, and assume that ilic helium gas warmstip to roorn temperature and expands accordingly before it enters the ptunp.
Nota: Tlte molar volume of an ideul gas at room icmpcramrc and atmosphericpressureG60torr) is about 24 liters. Repeat the calculation for a mtjch smaller
heat load (I0~3 W) and a faster puinpA0J liter s\"\021). Puntp speed is defined in
Chapter 14.
5. Initial temperature far demagnetizationcoaling, Considera paramagnetic
salt with a Dcbyc temperature {Chapter 4) of 100K.A magnetic field ofiOOkG
or lOtcsia is available in the laboratory. Estiniate the temperature to which thesalt must be prccoolcd by other means in order that significant magnetic cooling
may subsequently be obtained by !he isentropic demagnetization process. Take
the magnetic momentofa paramagneticion to be I Bohr magneton. By signifi-significant cooling we may understand cooling to 0.1 of the initial temperature.
Chapter 13
Semiconductor Statistics
ENERGY BANDS; FERMI LEVEL;ELECTRONS AND HOLES 355
ClassicalRegime 358Law of Mass Action 362
intrinsic Fermi Level 362
/r-TVPE AND P'TYVE SEMICONDUCTORS363
Donors and Acceptors 363FermiLevel in Extrinsic Semiconductors 364
Degenerate Semiconductors 365Impunly Levels 368OccupationofDonorLevels 369
Example: Semi-Insulating Gallium Arsenide 372
p-n JUNCTIONS 373
Reverse-BiasedAbrupt p-n Junction 377
NONEQUIUBRIUM SEMICONDUCrORS 379
Quasi-FermiLevels 379
Current Flow: Drift and Diffusion 379Example:InjectionLaser 381
Example: Carrier Recombination Through an Impurity Level 383
SUMMARY 385
PROBLEMS 387
1. Weakly Doped .Semiconductor 3872. Intrinsic Conductivity and Minimum Conductivity 387
3. Resistivity and Impurity Concemraiiou 3874. Mass Action Law for High Electron Concentrations 3875. Electronand Hole Concentrations in InSb 387
6. Incomplete lonizationof Deep Impurities 3S7
7. Built-in Field for Exponential DopingProfile 3888. Einstein Relation for High Electron Concentrations 388
Chapter 13: Semiconductor Statistics
9. Injection Laser10. Minority Carrier Lifetime
11. Electron-Hole Pair Generation
38838S388
iiiiiiui^zny v\302\273'ilticonduct [tin xind valence b jfiu\302\243\\ elects oeis snd hoics^ donors *inu 3cccplovs. I he
Ha^ conccnlration of holes ^
ht= cffcciivc c^uantiiEn conccnlnU/on for condudion ctccirons;
n,= effective quamum coiKcntralion for holes.
In the semieonductoi tileralurc n,. and % ate called ihe effective densities of states for the conduction
and valence bands. Notice iKal we use ;i fo( tin; chemical potential or Fcimi level, and we use Ji foi
cai-iicr mobilities.
ENERGY BANDS; FERMI LEVEL;ELECTRONSAND HOLES
The application of the Fermi-Dirac disiributionto eiecironsin semiconductors
is central to the design and operationof all semiconductor devices, and thus
to much of modern electronics.We treat below those aspects of the physicsof semiconductorsand semiconductor devices that are parts of thermal physics.We assume that the reader is familiar wjlh the basic ideas of the physics ofeleclrons in crysialline soiids, as treated 'n the texts on solidslatephysics
and on semiconductor devices cited in the general references. We assume the
concept ofenergybandsand of conduction by electrons and hotcs. Our principalaim is to understand the dependence of the alt-important concentrations ofconduction electronsand of holesupon the impurity concentration and the
temperature.A semiconductor is a system with electron orbitats grouped into two energy
bands separatedby an energy gap (Figure 13.1). The lowerband is the valence
band and the upper band is the conductionband.* In a puresemiconduciorat
t = 0 al! valence band orbitats are occupiedand alt conduction band orbitals
are empty. A full band cannot carry any current, so that a pure semiconductor
at r = 0 is an insulator.Finiteconductivity in a semiconductor follows eitherfrom the presence of electrons,catled conduction electrons, in the conduction
band or from unoccupied orbitats in the valence band, called notes.Two different mechanisms give rise to conduction electrons and holes:
Thermalexcitation of electrons from the vatencc band to the conductionband,or the presence of impurities that change the balance between the number
of orbitats in the valence band and the number of electrons available to fill them.
We denote the energy of the top of the valence band by \302\243,.,and the energy
of the bottom of the conductionband by e{. The differedrence
is the energygapof the semiconductor.Fortypical semiconductors Eg is between
0.1 and 2.5electronvolts.In silicon,e, ^ 1.1eV. Because t ^ 1/40 eV at room
\342\200\242We tieai both bands as single bands; for out (imposes it does noi mailer thai boih may be groups
of bands wtih additional gaps wiihtn each gioup.
Chapter 13: Semiconductor Statistics
Conduction band
Energy gap
Empiyatr =0
Filledair =0
Figure t3.1 Energy band structure of a puic semiconductor or insulator.
The electron orbilajs occur in bands \\vhjch exlerid through the crystal.Air = Gallorbitaisuplothe top of the valence band are filled,and iheconduct ion bantl is empty. The energy interval between the bands is calledthe energy gap.
temperature, we usually havec,, \302\273t. Substances with a gap of more than about
2.5 eV are usually insulators. Table 13.1 gives the energy gaps for selected
semiconductors, together with other properties needed later.
Let nt denote ihe concentration of conduction electrons and nhthe con-
concentration of holes. In a pure semiconductorthe two wiH be equal:
ii, =\302\253*. B)
if the crystal is electrically neutral.Most semiconductorsas usedn\\ devices have been inteniionaiiy doped with
impurities that may become thermally ionized In the semiconductor at room
\342\226\240temperature. Impurities tiiat give an electron to the crystal (and become
positively charged in the process) are calleddonors.Impurities that accept
Energy Bands; Fermi Lml; Elec
Table 13.1 Band structure data
SiCcGaAs,,p
InSb
Energy
gaps at
300 K
eV
t.[40.671.431.350.18
Q
c
c
2.71.04.64.94.6
antun
Idea
\",.
x 10x !Ox 10x 10X 10
i concentr
ions and li
1300K
c
' 1.19
5-2' 1.5' 6.9' 6.2
itions
olcs
x 10\"x to'ax 10\"x 10'*x 10\"
Dcnslty- f-statcs
|effective masses,
in units of the
free cicctr
in/ftn
1.06
0.56
0.070.0730.015
on mass
0.580.350.710.420.39
Dielectric
constants
relative to
vacuum
ilia
11.715.813.1312.3717.S8
an electron from the valence band (and become negatively charged in the
process) are called acceptors.Let \302\253j+be !he concentration of positively charged donors and na~ the
concentration of negatively charged acceptors. The difference
An=
/[/-
na~ C)
ts called the net ionized donor concentration.Theelectricalneutralitycondition
becomes
= An = Hj*\342\200\224
no~~t D)
which specifies the difference betweenelectronand hole concentrations.
The electron concentration may be calculated from the Fermi-Diracdis-distribution function of Chapter 6:
exp[(e -/
E)
where [i is the chemical potentialof the electrons. The subscript e refers to
electrons. In semiconductortheory the electron chemical potential is always
called the Fermilevel.Further, in semiconductor theory the character fi is
almost always reserved for ihe electron and hole mobilities,and the Fermi
level is designated by ef orby \302\243.To avoid confusion with the Fermi energy
of a meia!which we designated as cf and which stands for tile Fermi level
in ihe limil r -* 0, we shall maintain our previous usage of the letter /j for the
chemicalpotentialat any temperature.
Given ;i and t, the number of conductionelectronsis obtained by summing
the distribution function /,(e) over all conduction band orbitals:
The number of holes is
*\\ =\302\243[l
- /.(*)] = I ftU). V)
VB VI!
where the summation is overall valenceband orbitals. Here we have introducedthe quantity
y.
which is the probability that an orbital at energy e is unoccupied.We say
that the unoccupied orbital is \"occupiedby a hole\"; [hen/h(e) is the distributionfunction for holes just as
f\302\243t) is the distribution function for electrons. Com-Comparison of (8) with E) shows that the hole occupation probability involves
y.\342\200\224e where the electron occupation probability involves c - p.The concentrations nt
= NJV and nh= NJV depend on the Ferm't level.
But what is the value of the Fermi level? It ts determined by the electrical
neutrality requirement D), now written as nt{y)\342\200\224
nh{y)= An. This is an
implicit equationfor y.; to solve the equation we must determinethe functional
dependences ne{y) and nh{y}.
Classical Regime
We assume that both electron and hole concentrations are in the classical
regime defined by the requirements that fr \302\2531 and fh \302\253I, as in Chapter 6.
This will be true if, as in Figure 13.2, the Fermi level lies insidethe energy gap
and ts separated from both band edgesby energies large enough that
exp[-(\302\243c-
/i)/t] \302\2531; exp[-0*- ej/t] \302\2531. (9)
To satisfy (9) both (gc\342\200\224
n) and (/t-
eu)have to be positive and al least a
few times larger than t. Such a semiconductor is callednondegenerate.Theinequalities (9) place upper limits on the electron and holeconcentrationsand
are satisfied in many applications. With (9) the two occupation probabilities
/J.E.)and /h(g)reduceto classical distributions:
Classical Regi
':.-\"-\\~'\\ I IConduciion
band
Figure 13.2 Occupancy oforbiials as a finite temperaiure, according to the Fermi-Dirac disSribution function. The conduclion and valence bands may be represented
in terms of temperature-dependenteffedive numbers Nc, Nc of degenerate orbiialslocated aS the iwo band edges e,, \302\243\342\200\236.The n(, n( arc ihe corresponding quantum
= exp[-(e- => cxp[-(,,
- A0)
We use F) and A0) to write the total number of conduction electrons in the
form
N, A1)
Chapter 13: Semiconductor Statistics
where we define
N, -\302\243exp[~(s
-ej/r]. A2)
Cfl
Here \302\243\342\200\224\302\243cis the energy of a conduction electron referred to the conduction
band edge ec as origin.The expression for Nc lias the maihematical fomt of a partition function
for one electron in the conduction band. In Chapter3 we evaluated a similar
sum denoted there by Zlt and we can adapt that rcsuil lo the prescuJproblemwith an approximate modificalion for band siructure effects. Because of the
rapid decreaseofcxp[-{\302\243
-e()/i] as c increases above its minimum value
at ee,only ihe distribution of orbitals within a range of a few x above cc really
matters in ihe evaluation of the sum in A2). The orbitalshigh in the band
make a negligible contribution. Theimportant point is that near the band edge
the electronsbehavevery much like free particles. Not only arc the electronsmobile,which causes the conductivity of the semiconductor, but the energydistributionofiheorbitalsnear the band edge usually differs from that of free
particles only by a proportionality factor in the energy and eventually in the
sum for Z%.We can arrange for a suitableproponionaliiyfactor by use of a device
called the densify-of-sfateseffective mass. For free particles we calculated the
partition function Z\\ in C.62), but for zero spin. For particlesof spin | the
result is larger by a factor of 2, so that A2) becomes
Nc = Zj =2nQV= 2(mx/2nh2)i!iV. A3)
Numerically, this gives
NJV =- 2.509 x 1019x G/300KK'2cm\023 . (H)
where Tis in kelvin.
The quantity Nt for actual semiconductors exhibits ihe same temperaturedependenceas A3), but differs in magnitude by a proportionality facior. We
express this formally by wriiing, in analogy io A3),
Nc *= 2(mSz/2nti2K'2V , A5)
where me* is called the denshy-of-btates effective mass for electrons. Experi-
Experimental values arc given in Table 13.1. The introduction of effective masses
is more than a formality. In thetheory
of electrons in crystals it is shown that
Classical Regime
Ihe dynamical behaviorofelectronsand holes, under the influence of externalforces such as electricfields, is that of particles wilh effective masses different
from the free electron mass. The dynamical massesusually are different from
Ihe density-of-staJes masses, however.We define the quantum coticetilralion ??\302\243for conduction eleclrons as
nc = NJV -
By A0 the conductionelectronconcentration ne = NJV becomes
A6)
A7)
The earlier assumption (9) is equivaleni Jo the assumption that n,. \302\253nc, so
that the conduction electrons act as an ideal gas. As an aid to memory, we may
think of Ne as arising from N( orbilalsat \302\243\302\243,wiili the Fermi level at ;i. IVanriiuj:
In the semiconduclor literature nt is invariably called the effective density ofstatesof the conduction band.
Similar reasoning gives the number of holesin ihe valence band:
- e)/t]
wiih the definition
A9)
We define the quantum concentrationnv for holes as
e NJV s 2{in^z/2jihi)m.
where wk* is llns dcnsity-of-sJaJcs effective mass for holes. By (IS) ihe hole
concentration/^ s NJV is
nh\342\200\224
aBexp[\342\200\224(/(\342\200\224
e,-) B1)
Like A7), this gives the carrier concentrationin terms of (he quantum con-ceniraiion and the positionof the Fermi level relative to the valeuce bandedge. In (he semiconductor literature nu is called the effective density of statesof the valeuce band.
Law of Mass Action
The productn^nh is independent of the Fermi level so longas the concentrations
are in the classical regime. Then
'V'k = 'Wexpf-fE,.- O/r] s=ncnuexp(~ eJz) , B2a)
where the energy gap \302\2439s= ec
-eff. In a pure semiconductor we have ut.
\342\200\224>ibt
and the common value of the two concentrations is called the intrinsic carrier
concentration t^ of the semiconductor. By B2a),
B2b)
The Fermi level independence of the product n^iij, means that this product
retains its value even when ne <\302\243nh, as in the presence of electricallychargedimpurity atoms, provided both concentrations remain in the classical regime.We may then write B2a) as
B2c)
The value of the product depends only on the temperature.This result is the
mass action law of semiconductors, similar to the chemical mass action law
(Chapter 9).
IntrinsicFermiLevel
For an intrinsic semiconductor ne =nh and we may equate the right-hand
sidesof A7) and B2b):
neexp[-(ec- ^)/r] =
(vO\022exp(-\302\243/>*). B3)
Insert eB= e^ \342\200\224
eB and divide by n,.exp( \342\200\224\302\243(/r):
ej/2r].
Donors and Acceptors
We lake logarithms to obtain
/i= {U, + \302\243,.)+ |t Iog(u,/u,) = l(cc + e,) + 3rlog(\302\273ifc7\302\273i,*), B4)
by use of A6) and B0).TheFermilevel for an intrinsic semiconductor lies near(he middle of lhe forbidden gap, but displaced from the exact middle by anamount that is usually small.
w-TYPE AND p-TYPE SEMICONDUCTORS
Donors and Acceptors
Pure semiconductors are an idealization of Httle practical interest. Semicon-
Semiconductors used in devices usually have impurities intentionally added in order to
increase the concentration of either conductionelectronsorholes.A semicon-
semiconductor with more conduction electrons than holes is called\302\273i-type;
a semi-
semiconductor with more holes than electrons is calledp-type.The letters n and p
signify negative and positive majority carriers.Considera silicon crystal in
which some of the Si atoms have been substituted by phosphorus atoms.
Phosphorus is just to the right of Si in the periodic table, hence each P hasexactly one electron more than the Si it replaces. These extra electrons do notfit into the filled valence band; hence a Si crysial with some P atoms wiil containmore conductionelectronsand, by the law of muss action, fewer holes than \302\253
pure Si crystal Next consider aluminum atoms. Aluminum is just to the left
of Si in the periodic table, hence Al lias exactly one electronfewer than the Si
it replaces. As a result, Al atoms increase the number of holes and decreasethenumber of conduction electrons.
Most impurities in the same columnsof the periodic table as P and Al will
behave in St just as P and Al behave. What matters is the number of valenceelectronsrelative to Si and not the total number of electronson the atom.
Impurities from other columns of the periodic table will not behave so simply.
Similar reasoning can be appliedto other semiconductors, for example GaAs.
For the presentwe assume lhat each donor atom contributes one electronwhich
may enter the conduction band or fill one hole in the valence band. We also
assume that each acceptor atom removes oneelectron,either from the valence
band or from the conduction band.Theseassumptionsare called the approxi-
approximation of fully ionized impurities: all impurities when ionizedareeither posi-
positively charged donors D+ or negatively charged acceptors A\".The electrical neutrality condition D) told us that
An = nt\342\200\224
nfc~ nj+ \342\200\224
na~. . . . B5)
Chapter J3; Semiconductor Statistics
Becausenh=
m,V\302\273*from the mass aciioti law, we secthat B5) leads to a quadraticequation for nc\\
V-\".^!-^1. B6)
The positive root is
\302\273*= i{[{A'O2 + V]1/J + An} , B7a)
and because nh~
^ - An we have
\302\273*\302\253
l([(AnJ + V]1''2- An], B7b)
Most often the doping concentration is Urge compared to the intrinsic con-concentration, so that either nr or ft* is much larger than n,:
\\An\\\302\273nt. B8)
This Condition defines an extrinsicsemiconductor.Thesquarerootsin B7)
can then be expanded:
[(AnJ + An*]\"*=
|^[i + (hj&nJ]1'2
- m +ln*l\\bt\\. B9)
In an n-type semiconductor &n is positive and B7) becomes
nt =x A/i + nffAn-
An; nh ** n^/An \302\253n,. C0)
In a p-typc semicondiiclor An is negative and B7) becomes
Ji. ^n?/\\An\\
\302\253n,; nk ^ {A/i| +h;V1A*i1
^jAffj. C1)
The majority carrier concent ration in the extrinsic limit B8) is nearly equal to
the magnitudeof An, while the mmoriiy carrier concentration is inversely
proportionalto jAnj.
Fcrmf Level in Extrinsic Semiconductor
By use of the massaction law we calculated the carrier concentrations without
havingto calculatethe Fermi level first. Tlic Fermi level is obtained from n, or
/ij,by solving A7) or B1) for/c
Degenerate Scum
Figure 13.3 The Fermi icvet in silicon as a function of lempcraturc, for
various doping concentrations. The Fernifleveis are expressedrelative Ic
the band edges. A sniaii decrease of liic energy gap wiih icmperaiure has
been negieaed.
We may now use B7) to find ft as a function of temperature and dopinglevel
An. Figure 13,3 gives numerical results for Si. With decreasing temperature ths
Fermi level in an extrinsic semiconductorapproacheseiiher the conduction cr
She valence band edge.
DegenerateSemiconductors
When one of the carrier concentrations is increasedand approachesthe quan-
quantum concentration, we may no longer use the classicaldistribution A0) for iliiit
carrier. The calculation of the carriorconcenir.iiionnow follows the treatment
of the Fermi gas in Chapter 7. The sum over all occupied orbitais, which n
equal to [he numberofelectrons,iswritten as an iiiteura! over the density yf
states times the distributionfunction:
N
Chapter IS: Semiconductor Statistics
where for free panicles ofmass\302\273jllie densily of stales is
C4)
Thai is, \302\251(\302\243)(&is ihe number of orbitals in the energy interval (\302\243,e+ ck). To
make ihe transition to conductionelectronsin semiconductors we replace iV
by n,V; in by m,*; and \302\243by e \342\200\224ec. We oblain
Lei x s (e -er)/r and ij
=(fi
~et)/t. We use the definition A6) of J(c to obtain
C6)
The integral /(;;)in C6) is known as the Ferml-Dirac integral.When ee
-p. \302\273x we have \342\200\224ij\302\2731, so that cxp(x -
jj) \302\2731. In this limit
C7)
the familiar result for the ideal gas.In semiconductors the electron concentrationrarely exceeds several limes
the quantum concentration nc.The deviation between the value of/i from C5)and the approximation C7) then can be expanded into a rapidly converging
power series of the ratio r = njn^ calledthe Joyce-Dixon approximation:*
\302\253-\342\200\224i-S-f)
\342\226\240\",/\"\342\200\236 C8)
Figure t3.4 compares the exact relation C6) witll the approximations C7) and
C8).
\342\200\242W. B. Joyce and B.W. Dixon. App!. Phys. Lclt. 31,354A977). If the right side ofC8) is wrine
1.483S6 x lO\"*:^, - -4A2561x 1Q-6.'
Dtgcncnitt Slmicomlitct
7 -6 -5 -4 -3 -2 -I 0 1 2 3 4 5 6
i) = (|.- i,)h
above conduciion band edge Er. The dashed curvereprcsenisihc firsi icrm of ihc
Joycc-Dixon approximation C8).
Chapter 13: Semiconductor Sta
When neisno longer small comparedton(, Ihe expression of the mass action
law must be modified.In Problem4 we ask ihe reader io show ihat
C9)
If the gap Itself depends on the carrier concentrations, the value of n, to be usedhere will depend on concentration.
Impurity Levels
The addition of impuritiesto a semiconductor moves some orbitals from theconduction or valenceband into Ihe energy gap, where the orbiials now appearas localized bound states. We consider phosphorous in a silicon crystal.If the
P atom has released iis extra electron to the Si conductionband, the atom
appears as a positively charged ion. The positiveion attractsthe electrons in
the conduction band, and the ton can bind an electron just as a proton canbind an electronin a hydrogen atom. However, the binding energy in thesemiconductorisseveralordersofmagnitude lower, mostly because the binding
energy is to be divided by the square of the static dielectric constant, and
paniy because of mass effects. Table 13-2gives the ionizationenergies for
column V donors in Si and Ge. The lowest orbital of an electronbound to a
donor corresponds to an energy level Asd= st
\342\200\224st below the edge of the
conduction band (Figure13.5).There is one set of bound orbitals for every
donor.
A parallel argument applies to holes and acceptors.Orbiialsare split off
from 'he valence band, as in Figure 13.5. For each acceptor atom there is onesetofbotmdorbitalswiih an ionization energy Asa ~
Ea\342\200\224
\302\243,,,ofthesameorder
as Aej. Ionization energies for column III acctipcorsin Si are listed in Table 13.2.
In GaAs the ionizationenergies for all column VI donors except oxygen arecloseto 6rrteV. For zinc, the most important acceptor, &\302\243a
= 24meV. Some
T:.Mc 13.2 !uiiu..iio.i encr^e*ofcuhmm V <j
column il! acceptors in Si arui Ge, in mcV
4912.7
11
4510.4
Ace
AI
57
10?
cp OISGa
65iOS
In
1611.2
Occupation of Donor Letch
'At, !' \"'
\302\261
Figure 13.5 Donor and acceptor impurity levels in Uic energy
impurities generate orbitals deep inside the forbidden gap, sometimes with
multiple orbitals corresponding to different ionization states.
Occupation of Donor Levels
A donor level can be occupied by an electronwith either spin up or spin down.Hence there are [wo different orbitals with the same energy. However, the
occupationsof thesetwo orbitals are not independe nt of each other; Once the
level is occupiedby one electron, the donor cannot bind a second electronwith
opposite spin. As a result, the occupation probability for a donor level is not
given by the simple Fcrmi-Diracdistribution function, but by a function
treated in Chapter 5. We write the probability that the donor orbital is vacant,so that the donor is ionized, in a form slightly different from E.73):
Here Ed is the energy of a singly occupied donor orbital relative to the origin
of the energy. The probabilitythat the donor orbital is occupied by an electron,
so thitt the donor is neutral, is given by E.74):
Chapter 13; Semiconductor Statistics
Acceptors require extra thought. In the ionizedconditionA\" of the acceptor,
each of the chemical bonds between ihe acceptor atom and the surroundingsemiconductoratomscontainsa pairofelectronswith antiparaliel spins. There
is only one such state,hencethe ionized condition contributes only one term,exp[(/i - cJ/t],to the Gibbs sum for the acceptor, lit the neutral condition A
of the acceptor, one electron is missing from the surrounding bonds. Because
the missing electron may haveeithcrspin up or spin down, the neutral conditionis representedtwice in the Gtbbs sum for the acceptor, by a term 2 x J ~ 2.
Hence the thermalaverageoccupancy is
A '2 + exp[{^ ~
Efl)/i] \"\021+2 exp[(Efl- $x]\"
{~ \"}
The neutral condition A, with the acceptor orbityl unoccupied, occurs with
probability
______ _ __________ D3)
The value of An == nd+ \342\200\224na~ is the difference of concentrations of D* arid A\342\204\242.
From D0) or D2) we have
D4)
D5)
The neutrality condition D) may be rewritten as
This expressionmay be visualized by a logarithmic plot of n\" and n* as func-
functions of the position of the Fermi level (Figure 13.6}. The four dashed lines
represent the four terms in D6); the two solid lines representthe sum of aii
positive and all negative charges.Theactual Fermi level occurs where the total
posicivechargesequal the total negative charges.
For nd+- jio~ \302\273nh as in Figure 13.6, the holes can be neglected;for
nu~~
\024*-^ \"i 'he electrons can be neglected. If one of the two impurity
species can be neglected, the majority carrier concentrationcan be calculated
in closed form- Consider an it-type semiconductor with no acceptors. The
Occupation of DonorLetch
Figure 13.6 Graphical determinaiion of Fermi teve!and eteciroi
an n-1ypc sciniconducior coniaining both donors and acceptors.
neutrality point in Figure 13.6 is now given by the intersection point of the n*curve with the ?!e curve, U ihe donor concentration is not too high, the inter-
intersection will be on the straight portion of the incurve, alongwhich the approxi-
approximation A7) holds. We rewrite this as
cxp(/</T)- (\302\273A)\302\253PfcA); \302\2537>
exp[(^-
\302\243i)/t]=
(\302\273>Jexp[(\302\243,-
tj/t] = njn* , D8)
nc* s neexp[~(et-
Eli)/i] = ;i,exp(-A^/t) . {49}
Is the electron concentration that would be present in the conduction band if
the Fermi level coincided with the donor level. Here Aed~
ec~ Ej is the donor
iontzactonenergy.We insert D8) into D4) and set nt
~ nd* to obtain
E0)
nf3 + \\n,nt* =i\302\273rfn,*. E1)
llus is a quadratic equation in n,; ihe positive solution is
\302\273\342\200\236= V([l + (S\",,//)/)]1'3- I}- E2)
Forshallow donor levels, nt* is large and closeto nc.ffthe doping is sufficiently
weak that 8\302\273j\302\253\302\273/, the square root may be expanded by useof
A + xI1* =? 1 + \\x -ix2 + \342\226\240\342\226\240\342\226\240, E3)
for -v \302\253i. With x ~8\302\273d/He* we obtain
\302\273\342\200\236^\302\253\342\200\236
- 2it//ne* \302\253;tj(l
- 2<td/ite*). E4)
The secondtermin the parentheses gives the first order departure fromcompleteionization.For
example, for P in Si at 300 K, we have Ae,j ^ t.74r from
Table 13.2, so that ne* =s 0M5nc from D9). If \302\253j= 0.0!nf, Eq. E4) predicts
that il.4 pet of the donorsremain un-ionized. The limic of weak ionization isthe subjectofProblem 6.
Example: StuM-itiiHtuting gallium anaiM*.: Could pure GaAs be prepared, it would have
an intrinsic catikt concciUfdtioita! room temperature of\302\273,< 10'cm\023. Wjih such a tow
conceuiraiion of carriers, @~\" less than a nicial, the conductivity would bs; closer !O art
insulator than to a conveniiona! sciniconductor. [<itn'iisk G;iAs would be useful as an
\302\273isulat:n\302\243substiatc on whidi to prepare lEiin layers of doped GaAs as neeJed for devices.
There Joes no! exist a technology to purify any substance to 101 wnpuriircs per em1.
p-n Junctions
However, ic is possible to achieve near tnerinste carrier concentrations in GaAs by dopingwith high concentra lions (tOi5-IO17 Cm-3) of oxygen and chromium together, two impu-impurities ttiat have iticir impuriiy levels near the middle of ihe energy gap. Oxygen enters an
As site and is a donor in GaAs, as expecicd from the posiiion of O in the periodic Cablerelative to As; the energy level* is about 0.7 eV belowts.Chromium is an acceptor v> i'h an
energy level about Q.S4eVbelowet.Consider a GaAs crysiai doped with boih oxygen and chromium. The ratio of liiS two
conceniraiions is not critical; anything with an O:Cr raiio between abouc 1:10and 10:1
will do. If the conccitiraiions of all olher impurities are small compared with those of O andCr, (he positionof ihc Fermi level will be governed by ihe equilibrium betweenelm -ons
on O and holes on Cr. The construction of Figure t3.6 applied io tliis system shows thatover the indicated concentration raiio rangedie Fermi level is pinned io a range between
t.5i above the O fevel and l.5r below (he Cr level. With ihe Fermi level pinned near themiddle of the energy gap, the crysiai must act as nearly imrinsic
Gallium arsenide doped in this way is called semi-insulating GaAs and is used extensivelyas a tiijjh-resisiivity [!0a to to10 il cm\\ substrate for GuAs devices. A similar doping pro-procedure is possible in inP, with iron taking ihe place of chromium.
p-n JUNCTIONS
Semiconductorsused m devices are almost never uniformly doped. An under-
understanding of devices requires an understanding of nonuniformiy doped semi-
semiconductors, particularly of structures called p-n junctionsin which the doping
changes with posi! ton from /vtype to n-type within 'he samecrystal.We consider
a semiconductor crystal inside which the doping changesabruptlyat .v = 0
from a uniform donor concentration nd to a uniform acceptor concentrationna, as in Figure i3.7a. This is an exampleof a p~i\\ junction. More complicateddevice structures are made up from simple junctions: a bipolar transistor h.-isiwo closelyspacedp-njunctions,ofthe sequence p-i\\~p or n~p-n.
p~n junctions contain a built-in electrostaticpotential step Vbi, even in the
absence of an externallyappliedvoltage (Figure 13.7b). With no externally
applied voltage, the electronson the two sides of the junction are in diffusive
equilibrium, which means that the chemical potentials(Fermilevels) of tlic two
sides are the same. Becauseihe posiiionof the Fenni level within ihe band
siructurc depends on the localdoping,constancy of the Fermi level forces :i
shift in the electron energy bands in crossing ihe junction (Figure 13.7c). The
shift is eVN. The potential stop of height eVbii% an example of the potential step
required to equalizethe total chemical potential of two systems when ihc
intrinsicchemicalpotentialsare unequal, as discussed in Chapter 5.
Figure 13.7 A p-n junclion. (a) Dopingdislribulion.!l is assumed
that the doping changes abruptly from n-type to p-type. The two
doping levels arc usually different, (b) Electrosiaiic poteniiat. Thebuitl-in voltage Vbl ealabtishes diffusive equilibrium between ihe
two sides wilh differenl electron concenlralions as wet! as hole
concentrations, (c) Energy bands. Becauseihe Fermi level must be
shifted relaiivc to each other, (d) Spacecharge dipolc required to
generate the buitl-in voltage and to shift the energy bands.
We assume that the two doping concentrationsnd, tta lie in the extrinsic but
nondegenerate range,as defined by
Hf \302\253nd \302\253\302\273c; \302\273(\302\253na \302\253\302\273t. E5}
If the donors are fully ionized on the n side and the acceptorsfully ionized on
the p side, then the electron and ho!econcentrationssatisfy
ne=z vd\\ nh
s: na, E6)
one on the n side and the other on thep side.(Wehavedroppedthe superscripts
\302\261from Jij, \302\273\342\200\236.)The conduction band energies on the i\\ and p sides follow from
A7):
\302\253\302\253-/*-xlbgl^/nj; E7)
ccp = H~ xlo&{nJnc) = ;1- tlogK1/\"^). E8)
by B2c). Hence
eVtf -\302\243CJ.
-e\302\253
= Tlog(W>,2) . E9)
For dopingconcentrationsnd~
0.0inrand Hfl=s0.0inL.,we find eHi = es -~9.2t,which is 0.91 eV in silicon at room temperature.
A step in electrostatic potential is required to shift the band edge energies on
the two sides of the junction relative to each other. The electrostatic potential<p{x) must satisfy the Poisson equation
(SI) ~ = ~~ , F1)
where p is the space charge density and e thepermittivity
of the semiconductor.
Space charge must be presentwhenever <p varies. In the vicinity of the junctionthe charge carriers no Songer neutralize the impurities as in the bulk material.
The space charge must bepositive
on the h side and negative on the p side(Figure13.7d}.Positive space charge on the n side means that the electron
concentration is less than the donor concentration, indeed, as the conduction
Chapter 13: Semiconductor Statistics
band edge ts raised relative to the fixed Fermi level, A7) predicts an exponentialdecreaseof the electronconcentrationie.
Take the origin ofthe electrostatic potential at x = -co.soihai <p(-00}= 0.Then ec(x)
=\302\243c(~-a))
\342\200\224e<p{x), and A7) becomes
ne{x) = ntcxp[e<p[x)/T]. F2)
ThePoissonequation F1) is
^ ] ^ F3)
Multiply by Jdtpfdx to obtain
_</WV </AM2 ^ J
fr / , ,1 i6d\\
d.x i/x^ (/.x\\il\\-J
t dx { e J
Integrate uiih the initial condition<p{\342\200\224oc)= 0:
] F5)
F6)
\\dxj
At the interface x = 0 we assume that'
where Vn is that part of the built-in electrostaticpotentialdropthat occurs on
the n side. The exponential on the right-handsideofF5)can be neglected, and
we obtain
E - [peii,/c)(K,- */e)Vn F7)
for the -v component ofthe electric field E = -dipjdx. at the interface.Similarly,
\302\243=[{2eMB/\342\202\254)(^-t/e)]\022,FS)
where I-; is that part of ilie built-inelectrostaticpotential drop that occurs on
ihepstdcTlie two \302\243fields must be the same; from this and from Vn -VVp
=Vbi
we find
(\302\243^')\022 F9)
rse-BiasedAbrupt p~n Junction
The field E is the Sameas if On the ii-type side all electrons had been depletedfrom lite junction to a distance
-(Vw-2r/f) , G0)
with no depletion at\\x\\
> \\vK. The distance wa is used in semiconductor devicetheory as a measureofthe depth of penetration of the space charge transitionlayerinto the n side.
Similarly, on the p side,
The totaldepletionwidth wK
1 f we assume \302\273o= nd = i0licm\023;e ~ l(ko;aiid Vbi
- 2z/e == 1 volt, we lind\302\243- 4.25 x 104VcmwI and w = 4.70 x iO^5cm.
Reverse-Biased Abrupt p-n Junction
Let a voltage V be applied to a p-n junction, of such sign that the p side is at a
negative vohagerelativeto the n side, which means that V raises the potential
energy of the electronson the p side.This voltage will drive cond uction electrons
from the p side to the \302\273side, and holes from the ;i side to the p side.But the
p side in bulk contains avery low concentration of conduction electrons, and ifie
n side contains a very low concentration of holes,consistentwith the mass
action law. As a result, very little current flows. The distributions of electrons,holes,and potential are approximately ihe same as if the built-in voltage were
increased by the applied voltage,Figure 13.S. The field at the interface is now
given by
G3)
Chapter 13: Se
\342\200\224
I
r. \342\200\236._
it
\342\200\224'/
/
1mi .->
..-\342\226\240
\342\200\2247\342\200\224
//
/
Figure 13.8 Reverse-biased p-?i junaion, showing tfic cjuast-Fcrmilevels fi, and//P.
and the junctionthicknessis given by
G4)
In the semiconductor device liierature we often find G3) and G4) without the
term 2t/e, becausecertain approximations have beea made about the spacechargeand field distribution; we have solved the Poisson equation with the
correct electron distribution F2).
Currtrni Flow: Drift and Diffus
NONEQUILIBRIUM SEMICONDUCTORS
Quasl-Fcrmi Levels
When a semiconductor is illuminated with light of quantum energy greaterthan !he energygap,electronsarc raised from the valence txind to the conduc-conductionband. The electron and the hole concentrations created by iliumination
arc larger than their equilibrium concentrations. Similarnonequiiibrium con-
concentrations arise when a forward-biased p-n junctioninjectselectrons inio a
ptype semiconductor or holes into an ii-type semiconductor. The eiectric
charge associated with the injected carrier lype allracis oppositely charged
carriers from the external electrodesof the semiconductor so thai bolh carrier
concentrations increase.Theexcesscarriers eventually recombine with each otlier. The recombination
timesvary greatly with the semiconductor, from less than lo~9s to longerthmi
10\" 3s. Recombination tiniesin high purity Si are near 10~35.Even the shortest
recombination times are much longer than the times (-^ lG~l2s) required at
room temperature for the conduction electronsto reachthermal equilibrium
with each other tn the conduction band, and for the holes to reach thermalequilibriumwith each other in the valence band. Thus the orbitaloccupancydistributionsofelectrons and of holes are very close to equilibrium Fermi-Diracdistributionsin each band separately, but the tola! number of holesis not in
equilibrium with the total nuniber of electrons.We can express this steady state or quasi-equilibrium conditionby saying
at there are di fierent Fermi levels;<c and ;iv for the two bands, called quasi-emiilevels:
that t
ii levels:
7\021\"I +exp[(t-\342\200\236\342\200\236)/,]\342\200\242
Quasi-Fermi levels are used extensively in the analysis of semiconductor devices.
Current Flow: Drift and Diffusion
If the conduction band quasi-Fermi level is at a constantenergy throughout a
semiconductor crystal, the conduction electrons throughout the crysta!are in
thermal and diffusive equilibrium, and no electron current wjl! flow. Any
conduction electron flow in a semiconductor at a uniform temperature must be
caused by a position-dependence of. the conductionband quasi-Fermi level.
Chapter 15; Semiconductor Statistics
If the gradient of this levelis sufficiently weak, we may assume that the con-contribution of conduction electrons to ihe total electrical current density is
proportional to this gradient:
J, cc grad/*(. G6)
HereJeis an electrical current density, not a particle flux density. Because each
electron carries Ihe charge \342\200\224e,we have
5(\342\204\242e)x {electron flux density), G7)
where the electroniluxdensity is defined as the number of conduction electrons
crossingunit area in unit time. The ciose connection of G6) to Ohm'sSaw is
treated in Chapter 14. Because the flow of particies is from high to low chemicalpotential, the conduction electron flux is opposite to grad nc, but because
electrons carry a negative charge, the associated electricalcurrentdensityis
in the direction of grad fic. We view grad jj\302\243as the driving force for this current.Fora given driving force, the current density is proportional to the concentrationnt of conduction electrons. Thus we write
Jt_ ~. G8)
where the proportionality constant fi't is the electron mobility. The symbol ptshould not be confused with the conduction band quasi-Fermi level,/jc.
If the electron concentration is in the extrinsic but nondegenerate range,
n, \302\253nf \302\253nc , G9)
the cottduclion band quasi-Fermi level is given by A5), which can be written
in lerms of the eleclronconcentration as
ut -\302\243c+ Tlog(it7\302\253A (80)
Thus G8) become:
(81)
A gradient in the conduction band edge arises from a gradient in the electrostatic
potential and thus from an electricfield:
Current Flaw: Drift and Diffusion
We introduce an electron diffusion coefficient Dc by the Einstein relation
(83)
(84)
Dt -pex/e ,
discussed in Chapter 14. We now write (?S)or (8i) in the final form
J, - eptntE 4- eDe grad n
There arc two different contributions to the current:onecaused by an electric
field and one caused by a concentration gradient.
Analogous results apply to holes,with one difference. The valence band quasi-Fermi levelis not ihcchemicalpotential for holes, but is the chemical potentialfor the electrous in the valence b;ind. Holes arc missingelectrons;a hole current
io the right is really an electron current to the left. But holes carry a positive
rattier than a negativecharge.Thetwo sign reversals cancel, and we may view
grad j.iv as the driving force for the contribution Jh of holesto the total electrical
current density. We write, analogously to G8),
Jh\302\273
jvifcgradjv (85)
Carrying through the rest of the argumentleadsto
as the analog of (84), with the Einstein relation Df,= ji^fe- Note the different
sign in the diffusion term: Holes, like electrons, diffuse from high to low con-concentrations, but hole diffusion makes the opposite contribution to the electriccurrent, because holes carry lhe opposite charge.
Example:Injection laser. The highest nonequilibrium carrier concenlrations in semi-
semiconductors occur in injection lasers. When by efectron injection ifie occiipaiion f&c) ofIhc towesi conduciion band orbital becomes higher than lhe occupalioii/XeJofthe higheslvalence band orbiial, the population is said lo be invened. Laserilieory iclls us ihai lighi
Vvilh a quantum energy ec \342\200\224\302\243\342\200\236=
&f can ihtn be amplified by siimulylcd emission. The
condiiion for populaiion inversion is ihat
/Me) > fM (S?)
Wiih ilie quasi-Fernii distributions G5) this condition is expressed us
(88)
Figure 13.9 Double-heterostructtirc injection laser. Electrons Row from the
right into the active layer,where they form a degenerate eieciron gas. Thepotential barrier provided by t!ic wide energy gap on the p side prevents theelectrons from escaping to the left. Holes flow from ihc Icfi iiiio ihe aciive
layer,but cannot escape to the right. When (83) is attained, laser aciionbecomespossible.
For laser action the quasi-Fermi levels must be separated by more ihan ihe energy gap.The condition (88) requires that at least one of the quasi-Fermi levels lie inside ihe band
to which it refers. This is a necessary,bui not a sufficient condiiion for laser operation.An important additional condition is that the energy gap is a directgap rather than an
indirect gap. The distinction is treated in solid state physics texts. The most important
semiconductors with a direct gap are GaAs and inP.
The population inversion is most easily achieved in the double hcterostructtire of Fig-Figure 13.9; here ihe lasing semiconductor isembeddedbetween two wider-gap semiconductor
regions of opposite doping.An example is GaAs embedded in A!As. In such a structurethere is a potential barrier that prevents the outflow of electrons to the p-type region,and
an opposiic poieniial barrier ihat prevents ihe outflow of hoies to then-type region.Except for the current caused by tlic recombination itself, the electrons in the waive liiy\302\253r
arc in diffusive equilibrium wilh tlic electrons in the n contact, ana\" the electron quasi-Fermi level in the active layer linesup with the Fermi level in then contact. Similarly, the
valence band quasi-Fermi level linesup with the Fermi level in the p coniact. inversioncan be achieved if we apply a bias voltagelarger than ihe votlage equivalent of the activelayer energy gap. Most injection lasers utilize this double heierostf uciure principle.
rent Flow: Drift and Diffusion
Example; Canter recombination throush an iaipsiriiy level, Electrons and holes can re-
combine eiiher by an electron falling dircciiy inio a hole with itic emission of a photon,
or ihey can recombine ihrough an impurity level in ihe energy gap. The impuciiy process
is dominant in silicon. We discuss ihe process as an insiruciivc example ofquasi-equilibf iumsemiconducior statistics. Consider an impurity recombination orbiiai ut energy t, in
Figure 13.10. Four transition processes are indicated in the figure. We assume that ihe
rate Rc, at which conduction electrons fail into the recombination orbiiais is describedby
a law of the form
*\342\200\236-(! ~/>A \342\200\242 (89)
w here fr is the fraction of recombinationorbitais already occupied by an electron (andItcncenot available), and t. is a characteristic time constant for the capture process. Weiissume the reverseprocessproceeds ai tiie rate
Kc = /,'lj'e' , (90)
\302\253hcrcit'is the time constant forttie reverse process.We tnke R,, independent of the con-
concentration of conduUion ekurons, becausewe assume that i\\e < nt 1 he time consiaitts r.
Figure 13.10 Electron-hole recombination ihrough
impurity recombination orbitais at t, inside ihe energy
\342\226\240gap- .
Chapter 13; Semiconductor Statistics
and r,' arc related,becausein equilibrium ilie two raies Rrt and Rcr musi cancel Thus
wiihf, and n, evaluated in thermal equilibrium, which means we use A7) for h,. We ignore
ihc spin muitipiiciiy of the rccombinaiion levels, Wiih ihe equilibrium Fcnni-Diracdistribution for/, we have
A -/,)//, = exp[~{^~e,)/r]. (92)
Thus (91) becomes
^=
^exp[~fc-\342\200\236)/!]\302\273\"\302\243.
(93)
where ne' is defined as ihe conduction electron conceniralion lhal would be present if ihc
equilibrium Fermi level ft in A7) coincided wjlh the recombination level, if (92) and (93)arc insenedinto (89) and (90), ihe net electron recombinalion rale becomes
Re =Rcr
~R,c - ~ [A'- f,)nt
-/,\302\273/]. (94)
The analogous recombinaiion rate Tor holes is obtained by ihe subsiituiions
Re, %,\302\273/, te -* Rh, flh, nh*, !h;
and
/-I ~Jr\\ 1 -/,-/,. (95)
Here fk is the lifetime of holes in ihe limit thai ail recombination ceniersare occupiedby
eiecirons, and n,,* is, by definiiion,
ah* snBexp[-(\302\243,
- ej/i] -;i,V\302\273.*- C96>
With these substiiutioiis the net hole rccombiiiationrale is
Iiisteady staieUicuvorecombinaUonfaiesmusibcequaS:R, =Hft
= R.Equations 194)
and (97) arc iwo equal ions for ihe two unknowns f, and /?. W'i. eiiminalc /, to find
(9S)
This is the basic rcsuli of ihe Hail-Siiockley-Readrecombination iheory.' Applications
are developed In Problems 10 and I).
SUMMARY
!. In semiconductorsthe electron orbilals are grouped into a valence band(completelyoccupiedat r
\342\200\2240 in a pure semiconductor) and a conduction
band (complelelyemptyal x = 0 in a pure semiconductor), separated by
an energy gap. Electrons in the conduction band are calledconduct:-:1'electrons;empty orbitals in the valence band are called holes.
2. The probability of occupancy of a band orbital with energy e is governedby ihe Fermi-Dirac distribution function
Here \\i is the chemical potential of ihe electrons, called the Fermi ievei.
3. The energetic location of the Fermilevel in an eleclrically neutral semi-semiconductor is governed by the neutrality condition
Here ne and i\\h are the concentrations of condiiction eiecirons and holes,and An is the excess Concentration of positively charged impuritiesovernegatively charged impuriiies.
4. A semiconductor is said to be in the classical regime when ne \302\253nt and
nh \302\253nv. Here
are the quantum concentratioi)s for electrons and holes; hi/ and \302\253?/arc
efTecItve masses for eiecjrons and Ijoles. In the semiconductorHteralure,nc and \302\273\342\200\236are called the effective densities of states for the conduction;:ndvalence bands.
* R. N. Hall, Phys. Rev. 87, 387{i952); \\V. Shockley and W. T. Read.Jr.. Phys. Rev.S7. S35{!952).
. In ihe classical regime
where \302\243\302\243and eu are the energies of ihe edgesof the conduction and valence
bands.
6. The mass action law stales that in the classical regime the product
\"\302\273\302\273*= 'I.-1 -
'M'uexp(-\302\243a/i)
is independent of the impurity concentration. The intrinsicconcentrationn,
is the common value of n{ and nh in an intrinsic ( \342\200\224pure) semiconductor.
The quantity
is the energygap.
7. A semiconductor is called Ji-type when negative charge carriers{- con-conduction electrons) dominate; it is called p-type when positive chargecarriers (wholes) dominate. The sign of the dominant chargecarriersisoppositeto the sign of the dominant ionized impurities.
S, A p-n junction is a rectifying semiconductor structure with nn internal
transition from p-typc to n-typi. A p-n junction contains internal electricfields even in the absenceof an appliedvoltage.Foran abrupt junction the
field at the p-n interfaceis
Heree is the permittivity, nu and Mj are ionized acceptor and donor con-concentrations, and
| V\\and Vbi are the applied and the built-in reverse bias.
9. Theelectric current densities due to electron and hole flow are given by
JB = e/yi,E + eDegmdne,
Jp ~ephnkE
- ePsgrad nh,
. Here % and\302\243\342\200\236are ihe electron and hole mobilities, and \342\226\240
are ihe electron and hole diffusion coefficients.
PROBLEMS
1. Weakly doped semiconductor. Caiculale the electron and hole concenlra-tjons when the net donor concentration is small coinpared to the intrinsic
concentration, jAnj \302\253nt.
2, Intrinsic conductivity and ininimuni conductivity. The electrical conduc-
conductivity is\342\226\240. --
a = e{ntpe + njh) , (99)
where Jit and ph are the electron and hole mobilities.For most semiconductors
fl( > Jfj,. (a) Find the net ionized impurity conccntral/on An ~ na*\342\200\224
/ia~ for
which the conductivity is a minimum.Give a mathematics!expressionfor this
minimum conductivity, (b) By what factor is it lower lhan the conductivity of anintrinsicsemiconductor?(c) Give numerical values at 300 K for Si for which the
mobilities are jie = 1350and pk= 4S0cm3 V~*s~l, and for InSb, for which
the mobilities are p# = 77000 and jih= 750cm2V\"Is\021. Calculate missing
d;itu from Table 13.1.
3. Resistivity and impurity concentration. A manufacturer specifies the re-resistivity p
= I/a of a Ge crystal as 20ohm cm. Take pc = 3900 cm2 V\021 s\021
and /Zk=s 1900cm1 V\021 s~l. What is the net impurity concentration a) if the
crystal is n-type; b) if the crystal is p-type?
4. Mass action law for high electron concentrations. DeriveC9),which is the
form of Ihe law of mass aclionwhen ne is no longer small compared to ne.
5. Electron and hole concentrations in InSb. Calculate nc, nh, and /i\342\200\224
Et for
n-type InSb at 300K, assuming nd+\342\200\224
4.6 x 1016cm\"\"J = ne. Because of the
high ratio njn,. and the narrow energy gap, the hole concentration is not
negligibleunder theseconditions,nor is the nondegsnerate approximation
ne \302\253nc applicable. Use the generalized mass action Saw C9). Solve Ihe tran-
transcendental equation for nc by iteration or graphically.
6. incompleteiont'zation o/ deep imparities. Find the fraction of ionizeddonor impurities if the donor tontzalton energy fs large enough lhat Aed is larger
than t }og(nJ8nd) by several times t. The result explains why substances with
large impurity tonization energies remain insulators, even if impure.
doi
Chapter 13 s Semiconductor Statistics
7. Built-in field for exponentialdoping profile. Suppose that in a .P-typesemiconductor the ionizedacceptorConcentration at x = xt is \302\273\342\200\236\"
<-\302\273%\302\253\302\273\342\200\236
and falls off exponentially to a value na~= n2 \302\273ns at X = x2. What is the
built-in eiecuic field in the interval (xtlJCj)? Give nurnerica!values for /!s//ii~
103 and Xj \342\200\224X[
= I0~ 5 cm. Assume T = 300K.Impurity distributions such
as this occur in the base regionof many n-p-n transistors. The built-in fieldaids in driving the injected electrons across the base.
8. Einstein relation for high electron concentrations. Use the Joyce-DixonapproximationC8)to give a series expansion of the ratio DJfic for electronconcentrationsapproaching
or exceeding itc,
9. Injection laser. Use the Joyce-Dixonapproximationto calculate at T =
300 K the electron-holepair concentrationin GaAs that satisfies the inversioncondition (88).assumingno ionizedimpurities.
10. Minority carrier lifetime. Assume both electron and holeconcentrationsin a semiconductor are raised by 6n above their equilibriumvalues.Definea netminority carrier lifetime f by R =* Sn/i. Give expressions for i in terms of thecarriereonccniraiions ne and i\\h; the energy of I lie recombination level, as
expressed by ne* and nh*; and the timeconstantsit and tk, in the limits of verysmall and very large values of Sn. Uiider what doping conditionsis f indepen-
independentof ditt
11. Electron-hole pair generation. Inside a reversebiasedp-n junctionbothelectrons and lides have been swept.out. (a) Calculate the elcciron-holepairgeneration rate under these conditions, assuming ne* =
nfc* and te ~th
= t.
(b) Find ihe factor by which tin's generation rate is higher than the generationrate in an \302\253-type semiconductor from which the holes have been swept out,but in which the electron concentration remains equal to itd+ \302\273'i,-. (c) Give a
numerical value for this ratio for Si with ji/ = 10Ificm~3.
Chapter 14
Kinetic Theory
KINETIC THEORY OF THE IDEAL GAS LAW 391
MaxwellDistribution of Velocities 392Experimental Verification 394
Collision Cross Sections and Mean FreePaths 395
TRANSPORT PROCESSES 397
Particle Diffusion 399Thermal Conductivity 40!Viscosity 402
Generalized Forces 404
Einstein Relation 406
KINETICSOF DETAILED BALANCE 407
ADVANCED TREATMENT;
BOLTZMANN TRANSPORT EQUATION 408
ParticleDiffusion 409
Classical Distribution 4!0
Fermi-Dirac Distribution 4! I
Electrical Conductivity 413
LAWS OF RARF.FIED GASES 413
Flowof Molecules Through a Hole 414
Example: Flow Through a LongTube 4!6Speed
of a Pump 417
SUMMARY
I'KOIiUCMS 419
1. Mean Speeds in a Maxwcllian Distribution 4192. Mean KineticEnergy in a Beam 420
3. Ratio of Thermal to ElectricalConductivity 420
4. Thermal Conductivity of Metals 4205. Boitzmann Equation and Thermal Conductivity 421
Chapter14: Kinetic Theory
6. Flow Through a Tube7. Speedof a Tube
421421
I ant conscious of being only an individual struggling weaklyagainstthe stream
of time. But if still remains in my power to contribute in such a way thai, when
the theory of gases is againreviled,not too much will have to be rediscovered.
L, Betiynann
Kinetic Theory of the Ideal Ga>
!n this chapter we give a kinetic derivationof dieideal gas law, the distribution
of velocities of gas molecules,and transport processes in gases: diffusion,thermal conductivity, and viscosity. The Bohzmaim transport equation isdiscussed. We also treat gases at very low pressures, with referenceto vacuum
pumps. The chapter is essentially classical physicsbecauseIhequantum theory
of transport is difficult.
KINETIC THEORY OF THE IDEAL GAS LAW
We apply the kinetic method to obtainan elementary derivation of the ideal gaslaw, pV
~ Nx. Consider molecules that strike a unit area of the wall of acontainer. Let v, denote the velocity component normal to the planeof the wall.as in Figure 14.1. If a molecule of mass M is reflected specularly (mirror-likcjfrom the wall, the changeofmomciitumofthe molecule is
\342\200\2242A/|i-x|. A)
This gives an impulse 2M|i>.|to the wall, by Newton's second law of motion. The
pressureon the wall is
_ /momentum cliangc\\/number of molecules strikmgNpj
\\ per molecule )\\ unit area per unit lime /
. Let a(vI)ilv1 be the number of molecules per unit volumewith the 2 component
of the velocity between i>, and v2 + dii2. Here \302\247a{v.)dvz\342\200\224
NjV= n. The
number in this velocity range that strike a unit area of the wall in unit time is
a(vJ)vIdDx. The momentum change of these moleculesis -2Mi-.fl(i'.)iyf[i;,sothat the totai pressure is
p = {a>2Mv1la{t\\)dv1= M f\" v.la{vMv.. C)JO J-a
The integralon the right is the thermal average of v22 times the concentration,
so that p ~Afn<yI2>, The average value of JjMi-e2 is |t, by equipartition of
change of momentum of aily v which is reftecied from
ntaincris~-2M\\vz\\.
energy (Chapter 3). Thus the pressure is
p =iiAf<ps2>
= 'it = (NfV)z; D)
This is the idealgas law.
The assum ption of specular reflection isconvenient,but it is immaterial to the
result. What comes into t!:e surface must go back, with the same distribution,
if thermal equilibrium is to bemaintained.
Maxwell Distribution of Velocities
We now transform the energy distribution function of an ideal gas into aclassical
velocity distribution function. Often when we mean \"speed\"we shallsay \"velocity\", as this is the tradition in physics when no confusion is caused.In Chapter 6 we found the distribution function of an ideal gas to be
Ati = E)
where/(cn)is tltc probability of occupancy of an orbital of energy
\342\200\242
ui\\l
in a cube of volume V = Li, The average number of atoms with quantum
number between it and n + tin is (tlie number of orbilals in this range) x (the
probability such an orbital is occupied).Tltenumberoforbitals in the positive
MaxwellDistribution of t'ekctlies
octant of a sphericalshellof thickness tin is ^Dnn2)(Int whence the desiredproduct is
{{nn2dn)f{zK)= &/.n*exp(-\302\243jz)dn. G)
We take the spin of the atom as zero.To obtain the probability distribution of the classical velocity, we must
find a connection between the quantum number n and the classical velocnyof a particle in the orbital \302\243\342\200\236.The classical kineticenergy \\Mvz is related to the
quantum energyF) by
We consider a system of N particles in volumeV.LetNP{i)ili}be the number of
atoms with velocity magnitude, or speed,in the range dv at v. This is evaluatedfromG) and (8) by setting ,!>: - (iln/ilijib-
(\\lL/hn)il\302\273. We have
W(*/i> ={-n;ji!exp(-\302\243./T)*i/\302\273
=
ii.;/^Yi.2cxp(-A/o:1/2i),/i..(9)
From Chapter 6 we know that }. =n/\302\273Q
= (.V/L^^lnt^fMTK'2, so that the
fiidor staiidiiig to the left of v1 becomes
Thus
\342\226\240!n(A//2j[i)lV<:.\\p(-A/rV2r). (ll)
Tin's is the Maxwell velocity iIis(ribulion(Figure14.2).The quantity P(i Mr is (lie
probability that a particie has its speedin i/r at i\". Numerical values of the root
mean square thermal velocity and the mean speed are given in Table !4.I,
using the results \302\273,\342\200\236= Ci/.W)\022 andf - (Sr/irU)\022from Problem !.
Figure t-l.2 Mawveil velocity distribution as a
function of the speed in units of the most
probable speed tm(, * {2t/A/I\". Also shown
arc the mean speed c and ttie loot mean squarevelocity i-AK1.
1/
A
/
i
_
i
\\
\\
\\
\\
-
Table I
Gas
H,HeH,nNeN,
4.1 M
\".\342\200\236,
18.4
13.1
6.25.84.9
ofcculat \342\226\240
?
16.9
12.1
5.7534.5
*e!ocilies31273 K, i
Gas
OiArKrXe
Free electron
in 10*01115\"'
\342\226\240\342\226\240,.,
4.6
4.3
2.862.27
1100.
1
4.2402.632.09
1013.
Experimental verification. The velocity distribution of atoms of potassiumwhich exit from the slit of an oven has beenstudied by Marcus and McFee.*
The curve in Figure 14.3 compares the experimental results with the predictionof A2) below; theagreement is excehent. We need an expressionfor the velocity
distribution of atoms that exit from a smatf hole1 in an oven. This distributionisdifferent from the velocity distribution within the oven, because the flux through
the ho^e involves an extra factor, thevelocity component normal to the watt.
The exit beam is weightedin favor of atoins of high velocity at the expenseofthoseat low velocity. In proportion to thetr concentration in the oven,fast atoms
\342\200\242P. M. Marcus and J. H.McFee,Recent research in molecular fceanw, ed. }. Esterman, Academic
Press, 1959.1 In such experimenls a round bole is said io be small if ihe diameicr is less than a mean free paihof an aiom in ihe oven, tf the bole ti not smalt In tbis sense, [be flow of gas from it will be governed
\342\226\240\"\342\226\240by lhetawsofbydrodynamic Bow and not by gas kinetics.
i SecUtms and Me
109
>?7
15
321
zjn
-?*
V
5 6 ?
Tiansit time
10 U 12 13
figure 14.3 Measured traiistnissio\302\273 points and calculated
Maxwell transmission curve for potassium atoms that exit froman oven al a temperature E?\302\260C.The homontal axis is Ihe transit
lime of the utoms iransmiitcd. The inicnsity is in arbitrary units;the curve attd fhe points are normalized to the same maximum
value. After Marcus and McFce.
stitke the walismoreoften ihan slow atoms strike Ihe walls. The weightfactoris the velocity component ucosf? normal to the plane of ihe hole.Theaverage
of cos 0 over ihe forward hemisphere is jusl a numericalfactor, namely J. The
probability thai an atom which leavesthe hole will have a velocity between
v and v + dv deftnes ihe quantity PbciJ,v)iSv, where
wiih P,,(,Iwett given by (H). The distributionA2) of ihe transmission through a
hole is calledihe Maxwell transmission distribution
Collision Cross Sections and Mean FreePaths
We can estimate ihe collision rates of gas atoms viewed as rigid spheres. Two
aioms of diameter d will collide if their centers pass wiihin ihe distance4 ofeachother.From Figure 14.4 we see thai one collision will occur when an atom
has traversed an averagedistance
A3)
Chapter 14: K'tnaic Theory
Figure 14.4 (a) Two rigid spheres will collide
if their centers pass within a distance d of eachcacliother, (b) An arom of diameter d which
travels a long distance L will sweep oul avolume n^L, in the sense that it will collide
with any atom whose centerlieswithin the
volume. If n is the concentration of atoms, theaveragenumber of ajoms in this volume isi\\r,dlL. This is the number of collisions.Tlieaverage dkwnCt between collisions is
L 1
where >i is the number of atoms per unit volume. Tiie length /is called the meanfreepath:it isthe average distance traveled by an atom betweencollisions.Ourresult neglects the velocity of the target atoms.
We estimate the order of magnitude of the mean free path. If tlie atomicdiameter d is 2.2 A as for [ielium,then diecollisioncrosssectionac is
at= nit2 = C.14)B.2 x KT'crnJ \302\253=15.3 x lO-i6cm2. A4)
The coiicenEration of moleculesofan idea! gas atO:Cand 1 atm is given by the
Losclimidt number
m0= 2.69 x IOi9atumscnrJ. A5)
definedas the Avogadro number divided by the molar volume at O'C and 1aim.The Avogadro number is the number of molecules in one mole; tUe molar
volume is the volume occupied by one mole.We combine (M) a ad A5) to obtain
Transport Processes
the mean free path under standard conditions:
/ =\342\200\2243\342\200\224
= tttz]7pT6 2\\rT9 lO*7\"\"\"\"^
= 2M * 1\302\260\"iCm' ^16a'
This length is about 1000 times larger than the diameter of an atom. Theassociatedcollisionrateis
At a pressure of I0~6atm or Idynecm\022, the concentration of atorr.: sreducedby iG\026 and the mean free path is increasedto 25cm.At 10\026aim the
mean free path may not be smallin comparison with the dimensions of anyparticularexperimentalapparatus.Then we are in wliat is called the high
\\ acui:m
region, also called the Knudscn region. We assume below that the meaii free
path is small in comparison with the relevant dimension of the apparauis,
except in the section on lawsof rarefiedgases.
TRANSPORT PROCESSES
Consider a system not in thermal equilibrium,but in a nonequilibrium steadystate with a constant How from one end of tiic system to the otUer. For example,we may create a steady state noncquilibrium condition in a system by placingopposite endsin thermal contact with large reservoirs at two different tempera'
tures. If reservoir 1 is at the higher temperature, energy will How through the
system from reservoir 1 to reservoir2. Energyflow in this direction will increase
the total entropy of reservoir 1 -f reservoir 2 + system. The temperaturegradientin the system is the driving force; the physical quantity that is trans-
transported through the specimen in this process is energy.Thereis a linear region in most transport processes in which the flux is
directly proportional to the driving force;
flux s= (coefficient) x (driving force) , A7}
provided the forceis not toolarge.Such a relation is called a linear phenome-
noiogicailaw, such as Ohm's law for the conduction ofelectricity.Thedefinition
of the flux density of a quantity A is:
3A = flux density of A = net quantity oM transported acrossunit area in unit time. (IS)
Table 14.2 Summary of phcnomenologica! transport laws
Flux of Approximateparticle expression
Effect property Gradient Coefficient Law Name of law for coefficient
Diffusion Number ~DifTusmty/) J, = -
\302\243>grad\302\273 Pick's law D =\\~cl
dz
dv.Viscosity Transverse M ~~
Viscosity i;
Thermal Energy \342\200\224= Ct Thermal JB = -K gradr Fourier's |a<
conductivity^ \" \"z
conductiviiy K
ictricaiconductivity
Electrical Charge -~ = E, Conductiviiy s J. = a
a:
bols: n xx number of panicles per unit volume ip *=\302\273elccirosuiicpolcniial
Z = mtan thermal speed = <Ji'J> E = electricfield intensity
/ \342\226\240=mean free path q = elcciticchdrgt;C|-= heat capacity per unit volume M ^= mass of paruclepu \302\253=thermal energy per unit volume p \342\200\224mass per unit volume
Fx/A *2- shear force per unit area - p ^ momentum
The net transport is the transport in one direction minus the transport in the
opposiie direction.Varioustransportlaws are summarized in Table 14.2.
Particle Diffusion
In Figure 14.5 we consider a system with one end in diffusive contact with a
reservoir at chemical potential /ij; the other end is in diffusive contact with a
reservoir at chemical potentialft2. The temperature is constant. If reservoir Iis at the higher chemical potential, then particles wili flow through the systemfrom reservoir 1 to reservoir 2. Particle flow in this direction will increase the
total entropy of reservoir 1 H- reservoir 2 -f system.
Consider particle diffusion, first when the difference of chemical potential iscaused by a difference in particle concentration. Tile flux density Jn is the
number of particles p;isshigthrough a'uait urea in unit time. Tltc driving forceof isothermaldiffusion is usually taken as thc\"gradicnt of the particle concentra-concentrationalong the system;
-,)n = -Dgradti. A9)
The relation ts called Fick's law; here D ts the putt tele diffusion constant or
diffusivity.
Particles travel freely over distances of the order of the mean free path /
before they collide. We assume that tn a collision at position z the particlescome into a local equilibrium condition at the local chemicalpotentialjt{z) and
local concentration >\\{:). Let /. be the z component of the aiean free path. Acrossthe plaaeat z there is a particle flux density to the positive z direction equal to
4>j(z \342\200\224/;)\302\243(and a flux density in the negative z direction equal to ~-{n{z+
L]cI.Here n(z~ L) means the particle concentration at z \342\200\224
/,. The net particle
Figure 14.5 \342\226\240Opposite ends of the system are in diffusive
contact with reservoirs at chemicalpotentials /it and/ij. The
-. temperature is constant everywhere.
Chapter 14i Kinetic Theory
flux density is the averageoverail directions on a hemisphere of
B0)
We want to express the average value of cjg tn terms of?/. Here it = IcosOts the projection of the meanfree path, and c. \342\200\224?cos<? is the projection of thespeed on the z axis.The average is taken over the surface of a hemisphere,becauseall forward directions are equally likely. The eletnent of surface areabecais2xs'm0i}9.Thus
B1)
B2)
On comparison with A9) we see that the diiTusivtty is given by
The particle diffusion problemis the model for other transport problems. In
particle dilTusioii we are coneerncd with the transport of particles;in thermal
conductivity with the transport of energy by particles; in viscosity with the
transport of momentum by particles; and in electrical conductivity with the
transport of charge by particles. The linear transportcoefficients that describe
tile processes are proportional to the particledilTusivity D,
Let pA denote the concentration of ihe physical quantityA. If A is a quantity
like chargeor :r.:issthat has tile same value for all the molecules, then the flux
density of A in the z direction is
j; =e,<\302\273,>. (-\342\200\242')
ulierc (i';) is the mean drift velocity of the p;irtidcs in tlie z directioii.'! lie drift
velocity is zero in thermal equilibrium.
If /{ is a quantity like energyor momentum that depends on the velocity of
a molecule, then we always find a similar expression;
\342\226\240>/= IaPa<v.) , B5)
Thermal Conductivity
where fA is a factor with magnitude of the orderofunity. The exact value o(fA
depends on the velocity dependence of A and may be calculated by the method
of the Boltztnann transport equation treated at the end of this chapter. For
simplicity we set fA~lin this discussion. By analogy with A9) for particlediffusion, the phenomenologt'cal law for the transport of A is
126)
with the particle diffusjvity D given by B2).
ThermalConductivity
Fourier's faw
Ju = \342\200\224^Cg B7)
describes the energy llux density Ju in terms of the thermal conductivity K andthe temperaturegradient(Figure14.6).This form assumes that there is a nci
transportof energytbut not of particles. Another term must be addedif addi-
additional energy is transported by means of particle flow, as v,hen electrons flow
under the influence of an eiectricfield.The energy flux density in the z direction is
JJ =s />\342\200\236<!\342\200\242->, 128)
where ^o,i> is the mean drift velocity; pu is the energy density. This result isvalid within a factor of the order of unity, as discussed.By analogy with the
diffusion equation, the right-hand side isequalto
-DdpJAx= -D(tV,/cr)(i/t/i/x). B9)
Reservoir t Reservoir 2
Figure 1-1.6 Opposiie endsof the system arc in thermal
conijci wiih reservoirs at lempcratures r, and ij.
Chapter14: Kinetic Theory
This describes the diffusion of energy. Now dpjet is just the heat capacityper unit volume, denoted by Cy. Thus
-Ju- -.DCYgradr;
on comparison with B7) the thermalconductivity is
C0)
C1)
The thermal conductivity of a gas is independentof pressureuntil at very low
pressures the mean free path becomeslimitedby
the dimensions of the appara-
apparatus, rather than by intermolecular collisions. Until very low pressures are
attained there is no advantageto evacuating a Dewar vessel, because the heat
lossesare independentofpressureas long as C1) applies.
Viscosity
Viscosity is a measureof the diffusion of momentum parallel to the Howvelocityand transverseto the gradient of the flow velocity. Consider a gas with flow
velocity in the .x direction, with the Slow velocity gradient in the z direction. The
viscosity coefficient tj is defined by
C2)
Here vx is the x component of the flow velocity of the gas; px denotes the .v
component of momentum; and X. is the x componentoftheshearforce exerted
by the gas on a unit urea of thexj' plane normal to tlte z direction. By Newton's
second law of motion a shear stress X. acts on the xy plane if the plane receives
a net flux density of x momentum J:{ps), because this flux density measures the
rate of change of the momentum of the plane,per unit area.
In diffusion the particle flux density in the z direction is given by the number
density n times the mean drift velocity <i--> in the z direction, so that Jn: =
h<Uj>=
\342\200\224Ddnjdz. In the viscosity equation the transverse momentum density
is nMvx; its flux density in the z direction Is (hMdJO-)- By analogy with B6)
this flux density equals \342\200\224Dit(nMvxl'tlz, within a factor of the Order of unity.
With p = nM as the mass density,
-Dp daJds = -n C3)
Thus, with D given by B3),
rj= Dp = y-c! C4)
gives the coefficientof viscosity. The CGS unit of viscosity is called the poise.Themean free path is / =
\\jnd2n from A3), where d is the moleculardiameierand n is the con cent rat ion. Thus the viscosity may be expressed as
rt= htcfiml1 , C5)
which is independentof the gas pressurp. Tlie independence fails at very high
pressures when the molecules arc nearly always in contact or at very low
pressures when the mean free path is longer than the dimensions of the
apparatus.Robert Boyle In 1660 reported an early observation on the pressureinde-
independence of the damping of a pendulum in air:
Experiment 26 We observd also that when the Receiver was full of Air,
the included Pendulum continud its Recursions about fifteen minutes (or aquarterof an hour) before U left off swinging; and ikat after the t'xsnciionofthe Air, ihe Vibration of the same Pendulum {being fresh put into motion)appeared not (by a minutes Walch) to last sensiblylonger.Sothat the event of
this experiment being other than we expected, scarce afforded us any oihersatisfaction,than that of our not haviiuj umitleil to try it.
Although at first glance implausible, this result is readily understood. With
decreasing pressure the rate of momentum-transfer collisionsdecreases,btit
each colliding particle comes from farther away.The largerthe distance, the
larger the momentum difference; the increasingmomentumtransfer per collision
cancels the decreasing collision rate.It iseasiertomeasurethe viscosity than the diffusivity. HD =
r\\jp as predicted
by C4), then K is rclaiedto jj by
K - nCyfp. C6)
The observedvalues of the ratio Kp/rjCy given in Table 14,3 are somewhathigher than the value unity predicted by our approximate calculations.Im-Improved calculations of the kinetic coefficients K, D, i] take account of minor,
Chapter 14: Kinetic Theory
Tabie 14,3 Experimental values of K, D,ij, and Kp/tjCy at 0rC and l atm
Gas
HeAr
Hi
N,
Oj
K, in mWcm\"' K~l
1.50
0.181.820.260.27
D, incm*sM 1
0.158
1.28
_
/, in ;\302\273poise
186.
210.
84.
167.189.
we.
2.402.491.911.911.90
but difficult, effects we have neglected; see the works cited in the general
references.
Comment, The dilTusivity of gas atoms is directly proportional to iheir viscosity. The
diffusivily of a parliclc suspended in a liquid or gas is a different probian: !hc viscosiiy of
tile sojvent opposes !hc diffusion of ihe suspended particle. We find D oc l/i;, where Drefersto ihe particles and r\\ refers to the liquid. The Stokes-Einstcinrebiion for suspended
particles is D = r/6n^R, where R is Ihe radius of !tiespherein suspension.
Comment, Tlie quaniily v = tj/p is caJlcdrlIie kinemaiic viscosiiy;if C4) iiotds, v should
be equal to ihe diffusii-ify D. The raiio i\\/p eiiicrs into iiydrodynamic iheory and into the
Reynold'snumber criierion for taminar flow.
Generalized Forces
The transfer of entropy from one pars of a system to another is a consequenceof any transport process. We can relate the rate of changeof entropy to ihe
flux density of particles and of energy.By analogy with the thcrmodynaniic
identity at constant volume,
da = -c\\V- }~ (IN , C7)
we write the entropy current density Ja as
C8)
Let a denote the entropy density; let cafdt denote the net rate of change o(
entropy density at a fixed position r. Then, by the equation of continuity,
cajdt divjo. C9)
In a unit volume element the net rate of appearanceof entropy is equal to the
rate of productiongaminusthe loss - div Jo attributed to the transportcurrent.
In a transfer process U and N are conserved.The equation of continuity for
the energy density u is
~ \342\226\240=-divju;
the equation of continuity for the particleconcentrationti is
D0)
{41)
Let us take the divergenceofJo in C8);
divJa = idivJ. + Ju-grad{l/t)
-(tt/x) div Ja
- Jn\342\226\240
grader). D2)
Let C7) refer to unit volume; we take a partial derivative with respect to timeto obtain the net rate of entropy change:
dd 1cu i\\ en
ct~~
t ci tot'D3)
We use D0H43) to rearrange C9) in a form suggestive of the ohntic power
dissipation:
3. = J. -grailUM + J,
\342\226\240gradi-;</r) , W)
Here Fu and Fa are generalizedforcesdefined by
F. s grad(Vr); F, = grad(-;</r). D6)
Chapter 14: Kinetic Theory
In an isothermal processF, iii D6) may be written as Fn== {-I/t)grad/i or,
lit terms of ihe internal and externalpartsof the chemical potential, as
Fn = -\"{i/t D7)
For an ideal gas n,Rt==
xlog{n/nQ), so that grad/jjni = {r'n)gradn; for an
electrostatic potential gfad^\302\2431,<= q gradtp \302\273\342\200\224
qE. Thus
Fn- -UAJO^gradn -gE] D8)
How the particle flux density also has two terms, written as
3\302\253>= -i>ngrad\302\253 4- uflE , D9)
where D, is thedtfTusjvity and p. is the mobility, which is the drift velocity per
unit eleciricfield.The raiioof the coefficients of gradn to E is Djnp. in D9)
and rfnq in D8). These ratios must be equal, so that
E0)
which is calledihe Einsiein relation between ihe diffusivity and the mobilityfor a classical gas.
Comment. We gain an advantage,for reasons relalcd to the thermodynamics of irrevers-
irreversibleprocesses, if we use FM and FH in D6) as the driving forces for the linear transportprocesses. We write
Ju =\342\200\242Z,aiF, + L12FB; Jn = t2tF,,+L22F, , E1)
The Onsager relation of irreversible thermodynamics is that
L^B)~
Lj,i-B) E2)
where 8 is the magneiic field intensity. If B = 0, then L;J=
Ljf always. For E2) to hold,the driving forces F must be defined as in D6). Other definitions of the forces are perfcclly
valid, such as the pair grad i and grad n, but do not necessarily lead lo coefficients L that
saiisfy the Onsagcr relation.For a derivation see the book by Landau and Lifshiiz cited
in the general references.
of Detailed Bohr.
KINETICS OF DETAILED BALANCE
Consider a system whh two states, one at energy A and one at energy -A. Inan ensemble of N suchsystems,N* are at A and N\" are al \342\200\224A,with \\ ==
N* + N\". To establish thermal equilibrium theremust exbt sonic mechanism
whereby syslems can pass belwecnihe two stales. Consider the rate equaiionfor transitions into and out of the upperstate:
dN+/(lt= uN~ - PK+ , E3)
wherea, /J may be functions of the temperature. The transitionrate from \342\200\224to
+ ts directly proportional to the number of systems in the \342\200\224state. The transi-
transition rate from -1- to \342\200\224is directly proportional to the number of systemsin the4- state.
In thermal equilibrium (i/JV*/i/i) = 0, which can be satisfied only if
afp = <N+>/<N\">
= exp(~2A/r) , E4)
the Boltzmann factor. This result expresses a relation betweena(rj and /J(t) that
must be satisfied by any and every mechanism that assists in the transitions. As
an example, supposethat the transition + -+ \342\200\224proceeds with the excitation
of a harmonic oscillatorfrom a state of energy se to a state of energy (s + I)e;
in the inverse process * + the oscillator goes from se to (s \342\200\224i)e. In the
quantum mechanical theory of the oscillatorit is shown that
0 __Prob{s -+ s + 1) __
s + 1
for the excitation and de-excitation of the oscillator,a result derived in most
texts on quantum 'heory.The valueof(s) isfound from the Planck distribution;
<*\342\200\242>\302\253 J \342\200\242<sy + i = exp(E/T)
W W<*> \302\253\342\200\224J _ <sy + i =Wexp(e/r)- P W
exp(e/t)- 1
'
so thai, with \302\243= 2A to conserve energy,
aft -<5>/<s + 1) = exp(-2A/r). E5)
This satisfies the conditionE4).The principle of detailed balance emerges as a generalization ofthisargument:
in thermal equilibrium the rate of any process that leads to a given state must
Chapter14tKinetic Theory
equal exactly the rate of the inverse process that leads from the state. Onecommon application of the principleis to the Kirchhoff law for the absorptionand emissionof radiationby a solid, already discussed in Chapter 4: radiation
of a wavelength that is absorbedstrongly by a solid will also be emitted
strongly\342\200\224othenvise the specimen would heat up because it could not come into
thermal equilibrium wi!h the radiation.
ADVANCED TREATMENT:BOLTZMANN TRANSPORT EQUATION
The classical theory of transport processesis based on the Boltzmann transport
equation. We work in the six-dimensional space of Cartesian coordinatesr mul
velocity v. Tiie classical distribution function /{r,v) is defined by ihe relation
f{t,\\)thdv = number of particles in drdv. E6)
TlieBoltzmann equation is derived by the following argument. We consider
the effect of a time displacement dt on the distribution function. The Liouville
theoremofclassicalmechanics rells us that if we follow a volumeelementalonga flowline the distribution is conserved:
f(t + i!t,r -+ tk.v + d\\) = /(f.r.v) , E7)
in the absence of collisions. With collisions
fit + dt.T + tle,\\ + dv) - f(t,r.v) =dt(ff/ct\\mijoni, E8)
Thus
dt(cf/c!) + dr- gradr/+(/v gradv/ =(/t(f//^)Mu- E9)
Let a denote the acceleration dv/tli; ihen
cflct + v\342\226\240
grad, /+ a \342\226\240gradv / \302\253
{df/ct)tM. F0)
This is the Bohzmann transport equation.In many problems the collision lerm (cf/ct)eolJ may be treatedby the introduc-
introductionof a relaxation time rc{r,v\\ defined by the equation
ParticleDiffusion
Here/0 is the distribution function in thermal equilibrium. Do not confusercfor relaxation time with t for temperature. Suppose thai a noncquilibriumdistribution of velocities is set up by externalforceswhieh are suddenly removed.
The decay of the distribution towards equilibriumis then obtained (torn F1) as
- f0) f ~fo F2)
if we note that dfo/dt = 0 by definition of the equilibrium distribution. This
equation has the solution
F3)
ft is not exeluded that xc may be a function of r and v.
We combine E6), \302\24360),and F1) to obtain the Boitzmanu transport equaiu the relaxation time approximation:
F4)
In the steady state cf/ct = 0 by definition.
Particle Diffusion
Consider an isotliermalsystem with a gradient of the particle concentration.The steady-stateBoltzmann transport equation in the relaxation time approxi-approximation becomes
v.dffdx* -(/-/0)/i{, F5)
where the nonequiltbritim distributionfunction/ varies along the x direciion.
We may write {65)to first order as
/\302\273̂fo-vsMJx, F6)
where we have replaced cf/'dxby dfojJx. We can iterate to obtain higher ordersolutionswhen desired. Thus the second order solution is
fi= fo- o^M'lx =/\342\200\236
- vj^lfjdx -f vxlr*tllfalilx\\ F7)
The iterationis necessary for the treatment of nonlinear effects.
Chapter 14: Kinetic Theory
Classical Distribution
Let/0 be the distribution function in the classical ijmil:
/0 = exp[(/*- e)/i], F8)
as in Chapter 6. We art; ai Iibeny io take whatever normalization for the
distribution function is most convenientbecausethe iransport equation is
linear in / and f0. We can take the normalization as in \302\24368)rather than as in
E6). Then
dfoftlx -(ilfofduWtifdx)
= ifoh){dn/dx) , F9)
and the first order solution F6) for the nonequilibrium distribution becomes
/ = /o -(^J0/x)(dti/dx). G0)
The particle flux density in the x directionis
J/-$vxff>ie)de, G1)
where \342\226\240Dfc)is the density of orbitals per unit volume per unit energy range:
\302\273, G2,
as in G.65) for a particle of spin zero. Thus
J/ =JuI/01D(\302\243)f/E
- idn/dx) jiuJxJohM^k G3)
The first integral vanishes because v3 is an odd function and fQ is an even func-
function of vx. This confirms that the net particle flux vanishes for the equilibrium
distribution fa. The secondintegralwill not vanish.
Before evaluating the second integral, we have an opportunity to makeuseof what we may know about the velocity dependence of the reiaxatton time ic-
Only for the sake of examplewe assume that r{ is constant, independent of
velocity; rc may then be taken out of the integral:
Fermi'DiracDistribution
The integral may be written as
i JV/oO(E)<fe=
~j J(iMi'!)/0O(e)\302\253fe
= ,n/M , G5)
because ihe integralisjust the kinetic energy density |nt of the particles. HereJ/0\342\202\254){\302\243)(/e
= n is the concentration. The particleflux density is
J' *= -{nxJKtyiflnldx)\302\253
-{z(z/M)(dn/(lx) , G6)
because fi= xlagn + constant. The result G6)is of the form of the diffusion
equation with the diffustvity
P = Vr/A/.= K\022>V G7)
Another possibleassumptionabout the relaxation time is that it is inversely
proportional to the velocity, as in rc ~ l/v, where the meanfreepalh1isconstant.Instead of G4) we have
J* =~(dft/dx)(l/T)j(vx2/v)focD(\302\243)dB
, G8)
and now the integral may be written as
J i>ic , G9)
where c is the average speed. Thus
J.x \302\273-i(^\302\273A)W/V^)
= ~\\mdn/dx) , (80)
and t!:e diffusivity is
\302\243>= lie. (81)
Fermt-Dlrac Distribution
The distributionfunction is
^_ (82);XP[(\302\243
-WAJ + ' \342\226\240\342\200\242
Chapter 14: Kinetic Theory
To form dfo/dx as in F9) we need the derivative dfo/dfi. We argue bciow lhat
'tfJdH x 6{e-'y) , (83)
at low temperaturesr \302\253ft. Here 5 is the Dirac delta function, which has the
property for a general function F[c) that
Now consider the integral j\302\243F{c){dfOl'dii)j\302\243. At low temperatures dfQ/dfi is verylarge for s =^ n and ts smail elsewhere. Unlessthe function F(\302\243)is very rapidly
varying near n we may take F{e)outside the integral, with the value Fin):
(85)
where we have used dfo/dn =*~df0/dz. We have also used f0 = 0 for \302\243= co.
At low temperatures /@) = 1; thus the right-hand side of (85) is just F[}i),consistentwith the delta function approximation. Thus
dfoftlx= S(e - v)d[i/dx. (86)
Theparticle fiux density is, from Giy
^ (87)
wheret{ is the relaxation time at the surface e = /( of the Fermisphere.The
integral has the Vaiue
M3n/2cf) -nlm , (88)
by use of Q([t) ~ 3;i/2efat absolulczero,from G.!7), where zF =\\mvF2
defines tile velocity vf ort the Fermi surface.Thus
JM'\302\253
-(m,;i\302\273)Jii!Jx. (S9)
At absoluie zero ii@) ={h2j2in)Cji2n}2 J, whence
Jlt/ilx = {ilftV-mKJii')''1/\"\021}''\"/*
=5(\302\243r/\302\273),/ii.V/.v, (90)
Laws of Rartfitd Gaits
so that (S7)becomes
//= ~Bxt/3m)t:Fdn/dx= ~\\ve\\dnfdx. (91)
The diffusivity is the coefficientotdn/dx:
D - \\vF\\ , (92)
closelysimilarin form to the resuh {77}for the classicaldistributionofvelocities.
In {92} the relaxation time is to be takenat the Fermi energy.
We see we can solve transportproblemswhere the Fermi-Dirac distribution
applies, as in metals, as easily as where the classical approximation applies.
Electrical Conductivity
The isothermalelectricalconductivity a follows from the result for ihe panicledifiusivity when we muliiply the particle flux density by the particle chargeq
and replace the gradient dji/dx of the chemical potential by the gradient
cfdq>fdx =~-<j\302\243i of ihe external potential, where Ex is the .v component of the
electric field intensity. The electriccurrentdensity follows from {76}:
J, -H\\/'\302\273)E; a = uqhjni , (93)
for a classical gas with relaxation time xc. For the Fermi-Dirac distribution,from (89},
J,= (nq2xc/m)E; a *=
nq'xjm. (94)
LAWS OF RAREFIED GASES
Thus far in this chapter the discussion of transport has assumed that the
molecular mean free path is short in comparisonwith the dimensions of the
apparatus. At a gas pressure of i(T6atm at room temperature,the mean free
path ofa molecule is of the order of25cm. The diameter of a laboratory vacuum
system conneciionmay be of the order of 25 cm, thus of the order of ihe mean
free path. We may usefully draw a line here and denotepressureslower than
1 x 10~6atm as high vacuum. This pressure is approximately0.1Nra\021 or
1 x 10~6kgcm~2 or 7.6 x 30\"\"*mmHg or 7.6 x I(T4torr. The Knudsen
region of pressures is understood to be the regionin which the mean free pathis much grealer than the dimensions of the apparatus. A knowledge of the
behavior of gases in this pressure region is important in the use of high vacuum
pumps and allied equipment.Theterminology recommended by the American Vacuum Society isexpressed
in terms of torr, where 3 torr ;= 1 mm Hg = 1.333 x 3O\"~3bar=133.3Nnf2 =1333dynecm~3; here 1 bar ~ !06dynecm~2 = 0.9S7 standard atmospheres.Then:
high vacuum iO~3-i(T6 torr
very high vacuum 3O~6-3O~9 torr
ultra high vacuum below 3G~9 torr.
Flow of Molecules Through a Hole
in ilic Knudsen regime we do not need to solve a hydrodynamic flow problem
in order to get the rate of efflux of gas molecules through a hole, because themoleculesdo not sec each other. We have merely to calculate the rale Jfl at
which molecules strike unit area of surface per unit time. We find for tlie flux
density
(95)
where n is the concentration and c is the mean speedof a gas molecule.Toprove (95), consider a unit cube containing n molecules. Each molecule strikes
the + z faceofthe cubejct times per unit time, so thattn unit time^nc, moleculesstrike unit area.
We solve for ?, in terms of c. Becausec. *= c cos 0, we require the average ofcos#over a hemisphere:
2*J^2
cos 8 sin 1(8 j=
~2n [\"\\\\aedf
\"\"
1'(96)
Therefore ?, \302\253\\c, and (95) is obtained. The expression (95)for the flux forms
the basis for many calculations of gas flow in vacuum physics in the Knudsenregime.
IfA is the area of the hole, the iota! particle flux, which is the number ofmoleculesperunit time, is
nS , (97)
Flow of Molecules Through a Hols
S = l-ic. (98)
Theconductancesofthe hole is defined as the volume ofgasper unit lime flowing
through the hole, with the volume taken at the actual pressure p of the gas. The
conductance is usually expressed in liters per second. For the average airmoleculeat T => 300 K we have c x 4.7 x lQ4cms~';for a circular hole of
10cm diameter, (98) leads to a conductanceof 917liter/sec, roughly 1000
liter/sec.
For a hole with a given conductance the total particle flux is proportional to
the concentration n or, because p ~ m, to the pressurep:
Here-we have defined the quantity
Q~pS, A00)
) A01)
Theconditionfor zero net flux is
using the proportionality of c io t1'2.In the Knudsen regime equal pressuresdo not imply zero net flux tf the temperatures on the two sides are different.
At equal pressures gas will flow from the cold side to the hot side; zerogasflow requires a higher pressure on the hot side. ...
Chapter !4: Kinetic Theory
Eft, - t2, Eq.A01)can be \\
- -(p, - p2)S=~AQ , C03)
where
Example; Flow through a hns tube. We assume that the molecules which strike the innerwail of the tube are rc-emittcd in ali directions; that is, the reflection at [he surface is
assumed to be diffuse. Thus when there is a net flow there is a net momentum transfer tothe tube, and we must provide a pressure head to supply the momentum transfer. Let u be
the velocity component of the gas moiecuicsparallelto the waii before striking the wall.
We estimate the momentum transfer to the waii on the assumption that every collisionwiih the wall transfers monicmurn Af <n). The rate of ttow down the tube is (M<\302\273>. where
A is the area ofilieopening.The rafe at which molecules strike ihe wall is, from (95)
\\nld)ic , . A05)
whered is the diameter and' L the length of the lube. The momentum transfer to the tube
must equal the force due to the pressure differential &p:
{nLMcM(u} =* AAp. A06)
We solve for tile flow velocity (u) to obtain
<u>--\302\243w:ln\302\260'\302\245mrA07)
The net flux is
AO - n(u)A -Ap
~ = \342\200\224S , A0$)
wher
S =tA\302\253/Ap=
\342\200\224j-
A09)
is the conductance of the lube, defined analogously to the conductance of a hole,Eq.(97).
Speed of a Pump
A more detailed calculation, with averages over the velocity distribution take
arefully, leads lo a conductance differing from A09) by a factor S/3*:
8 xAi 2t\302\253/J
The conductance of a tube cannot be larger than that of a bole with the same area. From
(98),A10), and A21)below.
This ratio will be larger than unity for 1L < 4d. In writing A06) we assumed implicitly
that every molecule hits ihc tube wall. This will tioi be true for a short tube.For our result
to be valid we must suppose that the tube is long enough to make the ratio A11] be small
compared to uniiy, which means
L \302\273id. A12)
Using our earlier example for the conductanceof a hole, we find ihat the conductance ofa tube 1 meter Jong and !0 cm in diameter is about 122 Jiler/sec, for air at 300 K.
Speed of a Pump
Thespeedofapump is defined similarly to thecOnductancoofaholeorofa tube;it is defined as the volume pumped per unit time, with the volume taken at theintake pressureof the pump. The same symbol S is used as for conductance;
(H3)
just as for i!ie conductance of two dectriaii conductorsin series.
Proof: Let pE denote the pressure al the input cad of the lube, and let p2
denote [lie puntp intake pressure at the output end of the tube. Continuity of
flux requires that
Chapter14: Kinetic Theory
SO that
^i =W^ +
),(ll5)
Pi ScS[ S, \\Sp SJ
equivalent to A13).
The relation A13) for Scft explains why in high vacuum systems ihc connec-connections between the pump and the vessel to be evacuatedmust be as short and ofas largd a diameteraspossible.A long and narrow connecting lube makes pooruseofa high speed pump. Further, the speed of the pump itself cannot be larger
than the conductance of its awn aperture.
How rapidly does a pump with effective speed S evacuate a volume VI Fromthe idealgas law p V = Nx, and from the definition of pump speedanalogoustoEq.(99) we find
iP^^E^Q^Pl A16)dt V di V V
If the pump speedis independent of pressure, this differential equation has the
solution
W0-^0)cxp(-(/io); to\302\253V/S. A17)
For a volume of! 00 litersconnectedto a pump with a speed of 100 liter/sec,the
pressureshoulddecreaseby 1 e per second.
Any user of vacuum technologysoon discoversthat the pumpdown of a
vacuum system proceeds much more slowly in the high and ullrahigh vacuum
regions than expected on the basis ofpumping speedand system volume. The
desorption ofsurface gas predominates\342\200\224often by many orders of magnitude\342\200\224
over volume gas. The surface emits adsorbed moleculesas fast as the pump
evacuates molecules from Uievolume.
SUMMARY
1. Theprobability that an atom has velocity in do at v is
P(v)dv \302\2734;:{.Vf/27rrK'Vexp{~Mi>V2
the Maxwell velocity distribution.
2. Diffusion is describedby
Jn = -Dgradtr, D = %cl,
wiiere t is the mean speed and I is the mean free path.
3. Thermalconductivity
is described by
Ju =* -Kgradr; K = iC,,ff/ ,
where (V refers to unit volume.
4.- The coefficientof viscosity is given by
where p is the mass density.
5. According to the principle of detailed balance, in thermal equilibriumthe
rate of any process that leads to a givenstatemust equal exactly the rate o( theinverse processthai leadsfrom [he state.
6. The Boltzmann transport equation in the relaxationtimeapproximation is
jf+ f gradY /+ v \342\226\240
gmd, /- -Lzh.
7. The electricalconductivity of a Fermi gas is
a \342\200\224nqixjm ,
where xc is the relaxation time.
PROBLEMS
/. Mean speeds In a Maxwellian distribution, fa) Show that the root mean
square velocity vtaa is
\302\253n\302\253-
<\302\273*>m\302\273
f3t/M)!'2. (MS)
Because <>3> -<Pjt2> + <w,:> + <\302\273/> and <^a> -
<\302\273/>- <fI2), it fol-
follows that
(_Vx3ylS-
(t/M)\023\302\273V^JV'1. A19)
Chapter 14: Kinetic Theory
The results can alsobe obtained directly from the expression in Chapter 3 forihe averagekinetic
energy of an ideal gas. (b) Show that the most probablevalue of the speed vmP
is
vmp- BT/MI'*. A20)
By most probable vaiue of ihe speed we mean the maximumof the Maxwell
distribution as a function of v. Notice thati>mp
< v,mi. (c) Show lhal the meanspeedc is
c \302\253
j^dvvP(v)= (8T/rcAf)m. A21)
The mean speedmay also be written as <juj>. The ratio
v,mJZ~ J.0S6. A22)
fd) Show that ?., the mean of the absolutevalue of the z component of the
velocity of an atom, is
\"
?, s (\\v:{> =$\302\243
= {2t/jiA/)\"\\ A23)
2. Mean kinetic energy in a beam.' (a) Find [he mean kinetic energy jji a beamofmoicculcsthat exits from a small hole in an oven at temperature i. (b) Assumenow lhal the moleculesare collimated by a second hole farther down the beam,so thai the moiccuies that pass through ihe second hole have oniy a small
velocity component normal to ihe axisofemission.What is ihe mean kinetic
energy? Continent: The moiecuiesin the beam do not collide and are not in real
thermal equilibrium after they have exited from ihc oven. The gas left in the
oven is depletedwith respect lo fast moiecuies, and Ehe residua! gas will cooi
down iCH is not reheatedby heat flowing s'u (iirough the walls of the oven.
3. Ratio of thermal to electrical conductivity. Show for a classical gas of
particles of charge q that
K/ra - 3/2q2, or K/Ta- 3/:s2/2^2
\342\226\240A24)
in conventional units for Kand T.Thisis known as the Wiedemann-Franz ratio.
4. Thermal conductivity of metals. The thermal conductivity or copper at
room temperature is largely carried by the conduciion electrons, one per atom.Themean free path of the electrons at 300K is of the orderof400x 10~8cm.
The conduction electron concentration is 8 x 102Ipercm3.Esiimaiefa) the
electron contribution to the heat capacity; (b) the electroniccontributionto the
thermal conductivity; (c) the electrical conductivity. Specify units.
5. Boltzinaun equation and thermal conductivity. Consider a medium with
temperature gradient dx/dx. The particle concentration is constant,(a) Employ
the Boltzmann transport equation in the relaxation time approximationtofind the first order nonequiHbrium classical distribution:
A25)
wherevx2= 2e/3m. fc) Evaluate the integral to obtainfor the thermal conduc-
conductivity K = Sinijm.
6. Flow through a tube. Show that when a liquid flows through a narrow tubeunder a pressure difference p between the ends, the total volumeflowing ihrough
the lube in unit time is
A27)
(b)Show that the energy flux in the x direction is
where)/is the viscosity; L is the length; a is the radius.Assume that the flow is
laminar and that the flow velocity at the walls of the tube is zero.
7. Speedofa tube. Show that for air at 20eC the speedofa tube in liters per
second is given by, approximately,
L + id(I2S)
where ihe length L and diameicr J are in centimeters; we have tried to correct
for end effects on a tube of finite length by treating the ends as two halves of ahole in series with ihe tube.
Chapter 15
Propagation
HEAT CONDUCTION EQUATION 424
Dispersion Relation, (n Versus k 425
Penetration of Temperature Oscillation 426Developmentof a Pulse 427
Diffusion with a Fixed BoundaryConditionat x ~ 0 429Time-Independent Distribution
\342\200\242' 429
PROPAGATION OF SOUND WAVES IN GASES 430
ThermalRelaxation 432Example: Heat Transfer in a Sound Wave 434
SUMMARY 435
PROBLEMS 436
1. Fourier Analysis of Pulse 4362. Diffusion in Two and Three Dimensions 4363. TemperatureVariations in Soil 437
4. Cooling of a Slab 437
5. p~n junction: Diffusion from a Fixed SurfaceConcentration 437
6. Heat Diffusion with Internal Sources 4377. CriticalSize of Nuclear Reactor 437
The purpose of this terminalchapteris to bring within the compass of the textthe most important problems in the propagation of heat and the propagation
ofsound, both dassicai subjects that are part ofan educationin thermal physics,
HEAT CONDUCTION EQUATION
Consider first the derivation of the diffusion equation, which in found from the
Pick law A4.19) for !hc particle flux density:
J, a-/?ngrad\302\273 , (I)
where A, is the particle diffusivjty and n the particle concentration. The equationof continuity,
*-'+ divJ, - 0 , B)
assuresthat the number of particles is conserved. Because div grad ss V2,
substitution of A) in B) gives
y=
DnV2n. C)
This partial differential equation describes tlie time-dependent diffusion of the
particle concentration n.
The thermal conductivity equationis derivedsimilarly.By A4.27-14.30) we
have in a homogeneous medium
J_ = -/igradr. D)
The equation ofcontinuily for the energy density is
^ + div.l.= 0. M
Dispersion Relation, w Versus k
where C is the heat capacityper unit volume. We combine D) and E) to obtainthe heatconductionequation
~ \302\253Z),V2r; Ds = K/C. F)
This equation describesthe time-dependent diffusion of the temperature. The
equation is of tbc form of ihe particle diffusion equation C). Thequantity Ds
is called the thermal diffusivity; fora gasii isapproximatelyequalto the particle
diffusivity, as in (i 4.23}.
Comment. The eddy current equation of electromagnetic theory* has l\\\\c
C) and F). If 0 is (he magnetic field intensity, then
G)
The constant DB may be called the magnetic diffusivity and in SI is equal to I/ff/i; in CGS,
is % \342\200\224c j^JZtJ^i, it itiiS Inc (if n^Cfisiotts (jcnctli) (ttoiej af^d is dirccfjy j^ro^orijoDiiJ to ttic
(skin depihK limes the frequency. When we have solved one equaiion, say C), we have
sohed lhree problems.
Dispersion Relation, to Versus k
We look for solutions of the diffusiviiy equation
DV20 = cOJct {8}
that have the wavclikeform
0 ^ 0oexp[i{k-r- tor)] , (9)
with oj as the angular frequency and kas ]hc vvavevector. Plane wave analysis is
an excellent approach 10 this problem,even though it will turn out that the
diffusion waves or healwaves are so highly damped ihai they arc hardly waves
Chapter IS: Propagation
at all Substitute (9}in (S) to obtain [he relation between k and to:
Dk2 - ioi. A0)
A relation io{k) for a plane wave is calleda dispersionrelation.
PenetrationofTemperatureOscillalion
Consider the variation of temperature in the semi-infinitemedium z > 0 when
the temperature of the plane z = 0 isvaried periodically with lime as
0(<U) = 0ocosw; , A1)
which is the real part of 0oexp(-iw/), for real 6Q. Then in ihe medium z > 0the temperatureis
O(z,t)*=
e0Rc[aip[i(kz- at)]}
where Re denotes real part and i3'2 - {i - l}/%/2.Thus, with S s BDJqiI11,
6(z,i) = 0oRe{exp(-_-/a)exp[i(i/<5)-iwO)
= 0ocxp(-z/5)cos(w(- z/5). A3)
The quantity 5 \342\226\240=BZ)/u)}\022 has the dimensions of a lengihand representsthe
characteristic penetration depth of the lemperature variation: at this depth the
amplitudeof the oscillations of 0 is reduced by e\"J.The characteristic depth iscalled the skin depth if we are dealing with the eddy current equation. The
wave is highly damped in ihe medium\342\200\224the wave amplitude decreases by e\" '
in a distance equal to a \\vave!ength/2jr.
If the thermal diffusely of soil is taken as D s= 1 x 10\023cm2s'\"', then the
penetration depth of the diurnal cycieof heatingof the ground by the sun and
cooling of the ground by the night sky (w s= 0.73 x 10\"\024s\"\"!)is
L(diumal)- B0/cu}1'1 as 5cm.
For the annualcycie,
L(aniiual) k lm.
Development of a Pulse
A layer of tOcm of can h on top of a cciiar will lend 10averageout day night
variations of surface lontperalure, but liic summer/winterwffxitiofi ;i! ihc topof lhc cdi;tr requires severalMiclcrs of earth, Actual values of the tliertir.ti
dtlTustvity arc sensitive to the composition and conditionof lhc soii or rock.
Not'tcc that a figure of merit for cellar construction involves the thermal
dilTusivity, anrf not ihe conductivity alone.
Development of a Pulse
In additionto the wavelike solutions of the form (9), the diffusion equalion hasseveralother useful forms of solutions. We confirm by insertion in {8}thai
0(x,t) \302\253DnDi)~inexp{-x2J4Dt) A4)
is a soiution. The proportionality factorhas beenchosen so that
j*y{x,t)dx*~ I. A5)
The soiutionA4} corresponds to the time developmenl ofa pulse whichat t = 0
has the form ofa Dirac delta function S(x}t sharply localized at x = 0, and zeroelsewhere.
The pulse might be a temperature pulse, as when a pulsed laseror pulsed
electron beam heats a surface briefly. Let Q be the quantity of heat deposited on
the surface, per unit area. The temperature distribution is then given by
0{x,t) A6)
where Cv is the heat capacity per unit volume of the material. The function is
plottedin Figure15.1.The factor 2 arises because all heat is assumedto flow
inwards from the surface, while for the solution A4) symmetrical flow was
assumed. Another example of the applicationofA4) is the diffusion of impurities
deposited on the surfaceofa semiconductor, to form a p-~ujunction inside ihesemiconductor.
The pulsespreadsout with increasing time. The mean square value of x is
given by/-
X)dx ~2Dit A7)
after evaluating the Gaussian integrals. The root meansquarevalue is
. . . A-Imi(/)- (x1I'1 * (IDty1. A8)
Chapter I Si Propegai
s.b
0.8
0.6
0.4
= 1.0
,^\\
\"^
/ =4.0
\342\200\224^
= o.s
Figure 15.1 Piol ofspieadof temperature pulse whh lime. Tor 4nD = 1,from Eq. A6). At i = 0 (he pulse is a delta function.
Comment, This result shows thai the width of tile distribution increases as t\022t which is
a general characteristic of diffusion and random walk problems in one dimension. It is
quile unlike the molion of a wave pulse in a nondisperstve medium, which is a medium
for which <o = rfc, where v is the constant velocity.The connection with Brownian motion
or the random walk problem follows if we let t0 be die duration of each step of a random
walk; then r = Ni0. Mscre .\\' is the number of iteps. it follows that
XrBU(i) = {2Dto)llZN1'2 , A9)
so that the rnts dtsplacemcni ts propodional to the square toot of the number of steps.This is the result observed iri sludics of il.c Bfowiiian inolion, the random motion of
suspensions of small particles in liquids.
Time-Independent Diitribution
Diffusion with a Fixed Boundary Condition at x ~ 0
If a solution of (8) is differentiated or integrated with respect to any of its
independent variables,theresultmay again be a solution. An important exampleis obtainedby integrating A4} with respect to x;
=*-= [\"dscxpf-s^icrfu, B0)
where u = x/DDt)U2. Here we have introducedthe error function defined by
->\302\253u
erf\302\253 = \342\200\224
jo<feexp(-s2). B1}
Tables of the error function are readilyavailable.The error function has the
properties
erf{0) = 0; lim erf(x)= 1. B2)
Of particular practical interest is the diffusion of heat or ofparticles into an
infinite solid from a surfaceat x \342\200\2240, with the fixed boundary condition 0 = 0o
at x = 0 and 0 ~ 0 at x =* co. (For [ < 0 we assume 0=0 everywhere.)Thesolutionis
0(x,t)- 0o[l
- zrf(x/DDt)U2)l B3)
Again we see that the distance at which 0(x,t)reachesa specified value is propor-
proportional to DDiI11. The application of this solution to the dilfusion of impurities
into a semiconductor is discussed in a problem.
Time-Independent Distribution
Let us look at a solutionof (S) that is independent of the time. The diffusivily
equation reduces to lite Laplace equation
V20 =-- 0. B4)
Consider a semi-infinite medium bounded by the plane 2 = 0 and extending
along the positivez axis.Let the temperature vary sintisoidaliy in the boundary
plane;
0(x,y\\0) - 0os B5)
Chapter ]5: Propagation
The solution ofB4) in the medium is
0{x,y,z)= 0o5inkxexp[~kz). B6)
The temperature variation is damped exponentially with the distance from the
boundary plane. The temperaturedistributionin the time-independent problem
must be maintained by constant healsourceson the boundary plane z = 0.
PROPAGATION OF SOUND WAVES IN GASES
Results developed earlier in ifiis book can be applied to the studyofsoundwaves
in gases. Thermal effects are important in this problem. Let dp(x,t) denote the
pressureassociatedwith the sound wave; the form of the wave may be written
as
dp = 6poenp[Hkx - on)}, B7)
where k is the wsvevecior and w is the angularfrequency. The wave propagates
in the .V direction.
We suppose the equasion of state is that ofan idealgas:
pV= Nz , or p =
pr/A-f , B8)
where p ~ NM/V is the mass density, and M is the mass of a molecule.Theforceequation referred to unit volume is
Here u is the x component of the velocityofa volume element. The motion is
subject to the equation of continuity
dp/dt + div(py) = 0 , C0)
or, in one dimension,
dp/dt + S{pu)/dx - 0. C!)
Thethermodynamic identity is
dU + pdV = jda , C2a)
Propagation of Sound Waves in Gases
which can also be written
du iV da-_ + p_ =Ty. C2b)
If we assume (pending discussion beiow) that there is no entropy exchangeduring the passage of a sound wave, Eq. C2b) becomes
Cy(cr/ui) + (p/V)lcV/ct) = 0 , C3)
where Cv is the heat capacity at constant volume, per unit volume. We can re-
rewrite the second term in terms of dp/ct becausep -NMfV &\\\\d(l/V)[dV/ct)
=
-~(\\/p)(dp/dt). Now the thermodynamic identity appearsas
Cv(dx/di)- {p/p)(ep/ct) - 0.
C4)
Let us define the fractional deviations s, 0 by
P= Pod + s); t ~ T0(l + 0) , C5)
where p0, t0 are the density and temperature in the absence of the sound wave.We assume that u, s, Shave the form of a travelingwave;exp[t(Jcx- wO].The
three equatioas B9), C1), C4) that govern the motion now become
-iwpu 4- ik{(?po/M)s + (pTofM)O] = 0; C6)
-ia)pos + ik{pous + pn) = 0; C7)
-iwiQt,y0+ icu{pfp)p0sk 0. C8)
We assume that at sufficiently small wave amplitudes it is a good approxima-approximationto neglect in these equations terms in the squares and cross products of
u.s, and 0. For example, pu ~po{\\ + s)u becomes po\\\\ if the cross product su is
ncgiecicd.The equations thus reduce to, with the subscripts dropped from
p and r,
cott - (kx/M)s- {kxjMH= 0; C9)
as ~ ku = 0; D0)
rCyO- ps = 0; or CVB
- us = 0 , D1)
Chapter IS; Propagation
where n is the concentration. Theseequationshave a solution only if
to = fyt/M)llik , .\"
where y = (Cy + n)jCv =\302\243,/\302\243(,
in our units. The velocity of sound is
it,= den/dk *
D2)
D3)
Tiiis result applies to nionatomic gases from the lowest frequenciesup to high
frequencies limned only by the requirementthai the acoustic wavelength should
be much larger than the mean free path of the atoms. This requirement is the
criterion for the appHcabiiiiy of she hydtodynamic approach embodied m theforceequation B9).
Thermal Relaxation
Wiih polyatomic gases {43}is valid at low frequencies, but as the frequency isincreasedthere is a transition frequency region above which (he velocity of
\342\226\240sound increases. The transition region belwcen low
frequency and high fre-
frequency propagation is associated with relaxation effects.Thermal relaxationdescribesthe establishment of thermal equilibrium in a
system. Energy dissipation results when al!partsofa system are not at ihesame
temperature; the dissipation is strongest when the period of the heating and
coolinghalf-ejclein the sound wave is comparable with the lime required for
heat exchange between the different degrees of freedom of the system. In
polyatomicgasesunder standard conditions there are lime delays of the orderof 10~5sin Hie transfer of energy between the internal vibrational stalesof amoleculeand the external translation states.
LetthehealcapacityCi and temperature tj =to(\\ -f 0,) refer to the internal
states, while Cy and t =: to(l + 0) refer to the translations! states.Then C4)
becomes
Ct(dxt/dt) \\ (cz/ct)-
(p/p)(dp/ci) = 0 , D4)
or, in pSacc of QS),
-kotoCiQi-
ia>x0CvO + ioj{p/p)pQs - 0. D5)
Supposethai the transfer of energy between ihe internal and externalslateshas
the characteristic tune delay tQ such that
t)/t0 D6)
koO, = @ - 0,)/!Q. D7)
Here t0 is called tiie relaxation time.Therewill be separate relaxation times forthe rotational-iranslational transfer and the vibrationaf-iranslaiionaltransfer.
We combine C9), D0), D5) and D7) to obtain the dispersionrelalion
,48,
where CV, Cp refer to the translational states alone. In the low frequency limit
(otQ \302\253I and
k2 = ftj*(A//r) -^-i\342\200\224i = w^Af/vor) \342\226\240 D9)
where y0 is the low frequency limit of the total heal capacity ratio [C^ + C\\)/
(O 4- Ct). The low frequencylimit of the velocity of sound is
11,@)= (Vqt/A/)\022. E0)
In the high frequency limit cur0 \302\2731 and
J = w2(A//yxt). E1)
Here yM refers only to the translations! stales; at high frequencies the internalstalesare not excited by the sound wave. The high frequency limit of the velocity
of sound is
*,(cc)= G*t/Af)\022. E2)
Values of y0 are given in Table 15.1;if no internal stales at all are excited,y\302\273
= I
The wave is ai(enua(ed when k is complex; the imaginary part of k gives the
pressure attenuation coefficient a. From D8) it is found that the maximum
absorptionperwavelength occurs when cu = 2n,'t0 and is given approximatelyby
H C. ~ Cy C, ,,,.2
Cp Cv -f d
Chapter IS: Propm
Table 15.1 Ralio Cp Cy ~ \342\226\240;for gascS
Gas Temperature, CC
Air 17 1.403
HjO 100 1.324H2 15 1.410Oa -LSI 1.450
15 1-401200 1.396
2000 1.303
CO, 15 1.304Ar 15 1.668
He -180 1.660
kois: For imoniiomic Idea! gai,C,/Cr - S/J= 1.667, a!
tof ftf 3nd tic. roi a qj^equuc q&$ sJ 2 I c in peruiurc ni^ii
as for Oj and Hj at room (cmpe^aturci ai levnperalufcssuAicicr\302\273tly high lo excite also the v?branooal motion,
C,!Ct- = 9/7 - 1.2S6, as for O, aj 2000=C.The values given<lrc of v^i spplicsblc to slslic processesmiti Eo sound waivesin ihe ijmil of low frequencies. For very high frequency
\342\226\240/^\302\2535.3 is appftcabk.
For CO2 gas a( the relaxation frequency of 20kHz under standard conditionsthe intensity is observedto decreaseby 1/e in about A wavelengths\342\200\224a massive
absorption, in agreement with theory.
Example: Heat transfer in a sound Have. Equation {33)expressesthe iserttropic assump-assumption:ihe equation ncglecis ihe thermal conduciivity which gives rise to some transfer of
thefniat energy Within ihc sound wsvc betweensuccessivewsrm and cool half cycles. The
assumpiion thai da \302\253=0 musi be modified lo lake account of heal (low. The heat conductionequaiion F) may be wtiuen as
Kch/cx2 = x edict , E4)
where 8 is (he entropy density. ThenC4)becomes
Cytftjdt)-
lpfp)idpjdt) - Ktfh/dx2) ,
icaps= -Ktk28. . E5)
When we use this in place of {4\\\\, the dispersion relation lj{1.)becomes
With W = K/ai. At low frequencies Wk* is much Smaller lhan Cy, so thai the sound velocity
essentiallythe condiiion / \302\253}., where I is the molecular mean fiec path and X is the wave-
length of the sound wave. The attenuation of the pressure oscillation is given by the imag-
imaginary pan of the wavcveclor k and is denoted by a. The result from E6) in the low frequency
region Wk1 \302\253Cr is that
* = f/o ~l)pK<a2j2i\\3Cp , E7)
where Ce refers lo unit volume.
SUMMARY
1. The heat conduction equation is the partial differential equation that follows
when the phenomenologicaitransport equation(here the Fourier iavv) is
combined with the equation of continuity. We obtain
^=
D,V2r; D, s K/C.
2. The time-dependent diffusion equation and the eddy current equation havethe same form, so that their solutions may be translated from the solutions
of the heat conduction equation, these being often more familiar in the
Hterature.
3. Frequently H is useful to construct solutions in the form of superpositionsof plane waves of the form
0^ 0oexp[/(k-r -tot)}.
The differential equation then gives the relation between <y and k, called the
dispersion relation of the problem.
4. The propagation of sound waves in gases depends on the rateofexchange of
energy between the translation^!, rotaUonal, and vibrationaimotionsof a
moiecuie. A low frequency sound wave is describedby isentropic, and not
isothermal, parameters\342\200\224a result that seems paradoxical at first sight.
Chapter IS; Propagation
PROBLEMS
/\342\226\240Fourier analysts of pulse. Consider a distribution ihat at the initial time1= 0 has ihc form of a Dirac delta function iSf.v). A delta function can be
represented by a Fourier integral:
0{x,0)= Six)^~j^<!kcxp{ikxl
E8)
At later times the pulsebecomes
0{x,t)= ~ f\" dktxp[i(kx
- a)!)}, E9)Ik \342\200\242)-\"\342\226\240\"
or, by use of A0),
D[x,t) = \342\200\224
J^dk exp{ikx
- Dk2t). F0)
Evaluate the integral to obtain the result A4). The melhodcan be extended to
describe the time development of any distribution given at r ~ 0. If the distribu-distributionis f{x,0), then by the definition of [he delta function
/(x,0) -Jdx'f(x',0)S(x
- x'). F1)
The time development of 5{x - x') is
0(x - x',t)\302\273D7i/)f)^+
exp[-(.v- x'J/4Dt] , F2)
by A4}.TSms at time t the distribution f{x,0) lias evolved to
fix,!) ={4nDi)~{
fdx'f{x\\0)cxp[-(x
- x'J/4Dtl F3)
This is a powerful general solution-
2. Diffusion in ma and three dimensions, (a) Show that tlte diffusion equa [ion
in two dimensions admits [hesolurion
02{t)- (C2/!)exp{-~r2/4Dt) F4)
and in tlirce dimensions
o,F - {C3/iin)tsp{~r/4Di); F5)
(b) Evaluate the constants C2 and C3. Thesesolutionsare analogous to C4)and describe ihc evoltiuon of a delta function at t = 0.
3. Temperature variations in soil. Consider a hypothetical climate in which
boih ihe duiiy and the annual variationsofthetemperatureare purely sinusoidal,wiih amplitudes 0d =* IOC. The mean annual temperature0a
s= 1OX. Take
[lieihcrmaidifTusivityofthesoiltobel x IO~3 cm1 s\021. What is Uic minimumdepih at which water pipesshouldbe buried in this climate?
4* Cooling of a slab. Supposea hot slabof thickness 2a and initial uniform
temperature 8t is suddenly immersed into water of temperature 0o < 0uthereby reducingthe temperatureat the surfaceof the slab abruptly to 80 and
keeping it iherc. Expandthe temperature in the slab in a Fourier series. After
some time all but the longest wavelength Fouriercomponentofihetemperature
will have decayed, and then the temperature distribution becomessinusoidal.After what lime will the temperature difference between the centerof Uie slab
and its surface decay to 0.0! of the initial difference 0s~ 0O?
5, p~-n junction: diffusion from a fixed surface concentration. Supposea sili-silicon crystal is /j-type doped with a concentration of ua
\342\200\2241016cm~3 of boron
atoms. If the crystal slab is heated in an atmosphere containing phosphorusatoms,the latter will diffuse as donors with a concentration nd(x)into the semi-
semiconductor. They will form a p-n junction at [hat depth at which na= nd.
Assume that the diffusion conditions are such that the phosphorus concenu:;-
lion at the surface is mainiained at nd@)= 10l7cm'\023. Take the diffusion
coefficient of donors to beO s= iO~13cm2s\"l,What is the value of ihe constant
C in the equation ,\\\" =* Ct1'2, where .x is ihe depth of the p-njunctionand i is
ihe lime?
i5. Heatdiffusion with internal sources. When internal heat sources are pres-
present, the continuity equation E) must be modified to read
C~ + divju=gu , Ffi)
ct
where guis the heat generation rate per unit volume. ExamplesincludeJoule
heat generated in a wire; heat from the radioactivedecayof trace elements
inside ihe Eanli or ihe Moon. Givean expressionfor the temperature rise ai die
center of (a)a cylindrical wire and (b) iiic spherical Earth, on the assumptionthat ga is independent of position and is constant with time.
7, Critical size of nuclear reactor. Extendthe considerations of the preceding
problem to particle diffusion, and assume thai there is a net particle generalionrate i/B that is proportional to the local parsicleconcentration,gn
\342\200\224n/t0, \302\253here
f0 is a characteristic time constant. Suchbehaviordescribesthe neutron genera-
generationin a nuclear reactor. The value off0 depends on dieconcentraiionof\"SU
Chapter IS: Propagal
nuclei; if no surface losses took place,the neutron concentration would growas exp(f/f0}. Consider a reactorin the shape of a cube of volume I? and assumethat surface losses pin the neutron surface concentration at zero. Show that
Eq. C), if augmented by a generation term g, = n/t0, has solutionsof the form
n(x,y,z,!} c F7)
where kxL, kyL and k:L areinteger multiples of n. Give the functional depen-
dependence of the net time constant i\"; on klt ky, k. and r0, and show that for at least
one of the solutions of the form F7) the neutron concentration grows with time
if L exceeds a criltcal value Ltli(. ExpressLc,-n as a function of Dn and @. In
actual nuclear reactors (his increaseis ultimately halted because the neutron
generation rate gn decreases with increasing temperature.
Appendix A
Some Integrals Containing
Exponentials
THE GAUSSINTEGRAL
Let
J - <x> JO(i)
The following trick is used to evaluate /o. Write (i) in terms of a different
integration variable:
/\342\200\236-
j'~mp(-y')cly.B)
Multiply (i) and B) and convert ihe result to a double integral:
V - J_>^exp(-xa)*cJ_*\"exp(-.y2)<fy=
j\" J_*Jexp- (x1 + y')ilxdy.
C)
This is an integral over the entire *->\342\226\240plane. Convert to polar coordinates r and
tp, as shown in Figure A.S. Then, x1 \342\200\242}\342\200\242y1= r1, and the area element dA =
dx dy becomes dA = ritnltp:
'\302\2602=
Jo [j\020<=^-r')r'lr\\dif= 2i
Becauseof d[exp{~r2)]= - 2exp(-r2)rjr, the integral over r is elementary:
V - -n f4exp(-r2)]\302\273-fnexpl-r1)]'
\"- n.J
L J'-\302\260
D)
C iFitc\302\243r@'5 LOfl
area element iU = rrfrtltp.
GENERALIZED GAUSS INTEGRALS,
AND GAMMA FUNCTION INTEGRALS
Integralsof the form
Im - 2JoVexp(-x2)(k, (m
> -1) , E)
where m need not be an integer, may be reducedJo the widely tabuiatcd gamrna
function V{:), by the substitutionsx2 ~y, 2dx
~y~idy:
Im = >dy -V(n f I), (m
- l F)
Theintegralin F) may be viewed as the definitionof f(z) for nonititeger positive
\\alues of r.
The gammafunction satisfies the recursion rdalion
It is easily obtained for n > 0 from F) by integration by parts, and it is used to
extend the definition F) of T{z) to negative values of z. By using (?) repeatedlyit is always possible to reduce F{z) for arbitrary argument lo a value in the
interval 0 < z < !.
The Stirling Approximation
Form = 0,11 = -J; from D):
Urn is an even integer, m \342\200\22421 > 0, n is a half-integer, and i!~ 1\342\200\224}, then we find
by repeated application of G), with the aid of (8), that
h,
(9)= ry + j) = (/- i) x (i - j) x
\342\226\240\342\200\242\342\200\242x \302\247x i x u\"!.
Form - l.ii -0:
/, = 2Jo*.xexp(-*!)ifc
=JoV'<0'
= HI) ~ 1. A0)
If in is an odd integer, in = 2i H- I >: 1, n is an integer, n = / 2: 0, and we find
similarly, with the aid of A0),
'lit i - 2JoV*lexp(-x')J.\\-= HI + 1) = I x (I- 1) (HI
The gammafunction for positive integer argument is simply ihe factorialofthe integer preceding the argumeni.
THE STIRLING APPROXIMATION
For largevalues of\302\253,n! can be approximated by
A2a)
logn! * lj\\og2n + (n + J)logH - n +j\342\200\224
Here the term 1/12/? is the first term of nn expansion by powers of !.'\302\273,and
0{i/\302\2732) slands for omitted higher order terms in this expansion, of order l/\302\273
Some Integrals Containing Exponential*
or higher. In practice,even the lerm I/I2n is usually omitted. Its principalroleis to check on the accuracy of the approximation. If the effect of the I/12n-
correction introduces only a change below the desired accuracy, ihe entire
expression has the desiredaccuracy.Toderive A2) we write, in accordance with A1),
\"! =jo\302\260x\"e\"llx
\"J0\"exP[/(x>]'h
\342\200\242 A3>
/(x)=)ilogx-x A4)
We make the substituiion
x = n + yii* = n(l + j'lr1) , dx - rfdy. A5)
Then
J{X) = I! lOgl! - It + S(j') ,
With these,
9(y) = npog
exP[/(-v)]= irV-'e
A7)
(IS)
A9)
The function g(y) has its maximumat y= 0; g@) = 0. Using the Taylor
expansionof the logarithm,
log(l + s) = s - is' + Js1- |s\"
with s = (.y2/\"I'2. we expand g{y):
B0)
B0
The Stirling Approximation
In tile limit n -\342\226\272a}, s ~ 0, and all but llie first term in B1) vanish, and the
integral in A9) becomes
/ J\"exp(-j.72)rfi.= On)'11 , B2)
with the aid of D). If B2) is insertedintoA9) the result is identical to A2a) exceptfor the correction term 1/I2n. Its derivation is a bit tedious. We work with
Iogii! and wrile
log/ii = {login+ (\302\253+ iltog'i - ii + \342\200\224+\302\260(i)-
B3a)
If we replace h by n - 1,
Iog(n- 1)! - ^Iog2n + (n
-
A A
M- 1
\\n
<23b>
We sublracl B3b) from B3a):
ii I
logn! - log(n - 1I -log
jj-^-jT;
- loS\"
= (n + J)logn -(ii
- 5)log(il - 1) - 1
where all omitted terms are now at least of order I/a3.The two terms in A
can be combined:
If this is inserted into B4), we find
4 = (n-i)iog B6)
Some Integrals Containing Exponentials
For large n the logarithm may be expanded according io B0),with s~ \342\200\224
\\/n:
r + i + i +\302\260i B7)
B8)
If this is inserted in B6) wesee that A -\\/\\2.
We are often interested not in n\\ but oniy in Iog/i!, and only to an accuracysuch that the relative error between an approximate value of Iog/i! and the
true value decreases with increasing \302\273.Such an approximation is obtained by
neglecting all terms in (I2b) that increase less rapidly than linearly with n:
Iogu! =s if log ii B9)
Appendix B
Temperature Scales
DEFINITION OF THE KELVIN SCALE
Numerical values of temperature*-1 are not expressedin practice in terms of thefundamental temperaturer, whose unit is the unit of energy, but on the(absolute)thermodynamic temperature scale T, the Kelvin scale, whose unit is the kclvin,
symbol K. The kelvin was defined in 1954 by international agreement as ihefraction 1/273.16of the temperature T, of the iriple point of pure water.Henr;,by this definition, T, s 273.16 K, exactly.Thistemperatureis O.GI K above the
atmospheric-pressure freezing point of water (the ice poini), To ~ 273.15 K.The triplepoint is more easily and accurately reproducible than the ice poini.
The triple point establishes itself automatically tn any clean evacuaied vessel
that is partially backfilledwith pure water and cooled until part but not all of
the water is frozen, leadingto an equilibrium between solid ice, liquid water,and the water vapor above the ice-watermixture.
The Celsius temperature scale / is defined in terms of the Kelvin scale, by
t a T - 273.15K. (I)
Temperatureson this scale are expressed in degrees Celsius, symbol 'C. Tem-Temperature differences have liie same value on both Kelvin and Celsius scales.
The conversion factor kB between the fundamental temperature i and theKelvin temperature,
t =kBT , B)
*We appreciate ihe assisiance of Norman E. Phillips in ihe preparation of this appendix.
1The ullimale survey of the slale of development of precise lempcralure measurement is Ihe
try. Tfic proceedings arc published urider the SiSn.C
ihoU. 19-10. This volume is largely obiolcie. Vol. 2: E. C.
ger reflecting the State of ihe art, this volwnc is stills of p/tEittpfci ofvanous methods of tc
liiic. Vol. 1:C O. Fairchild, editor: Reir
W.Jfc. editor; Koicltotd, \\<jS5. Although4..tfful for ils thorough inlioductory i!isi
edili r; Rcitihold, (962- Pc/liapsthcipks of various methods of tcm
nfcU*ii c becauseof its tiitroduLioty JisfUiiion uf
letiis. Vol. 4 C pans): H. H. Plumb, edilor;
is called the Boiumann constant. Its numerical value must be determined
experimentally; [lie best current value* is
ka= A.350662 \302\2610.000044) x 10~t6ergK\"\\ C)
The value of the BolUniannconstant is determined with the aid of certainmode!syslcmswhose structures are sufficiently simple that one can calculatethe energy distributionof !hequantum states, and from it the entropy as afunclion of the energy, a = a(li).The fundamental lemperalure as a functionof Ihe energyis i(U) =\342\226\240
(ca/cU)~l. Examples of mode! syslems used in !he
determination of k8 are the following:
(a) Idealgas. iti the limit of low particle concentration all gases behaveasidea!gases,satisfying pV
= NkBT. One obiahis/cB by measuring the pV product
of a known amount of gas al a known kelvin tcmperalure T, extrapolated to
vanishing pressure.The determination of the number of particles N invariablyinvolves the Avogadro constanl NA, independently known.
(A) Black bodyradiation. We can obtain k,, by filling Utc measured spectraldislributionof a black body of known Kelvin temperature T to the P!;mck
radialion law (Cliapler 4). Because this law involves t through iheralioIioj/x~
tici>/kBT, this dcterminalion requires the independent knowledgeof Planck'sconstant.
(c) Spinpammagnetistn. In the limit of vanishing interaction the magnetic
moment M of a syslem of .V spins in a magnetic field B, at temperature t, isgiven by Eq. C.46). Various paramagnetic salts, such as cerousmagnesium
nitrate (CMN) are good approximations to nomnieractmg spin systems if the
temperature is not too low. By filling measured values of M as a function of BjT
to C.46) we can determine Ihe ratiomfkB, where in is the intrinsic magneticmoment of the eleclron, known independently. Usually only the low-field
portionis used,in which case ihe number of spins must also be known, which
tnvoives again A^. Precision results require correctionfor weak residual spin
interactions, similar to corrections for particle interactions in a gas.The kB value given in C) is a weighted averageof several deterrninations.
With an uncertainly ofaboui 32 parts permillion,it isoneof the least accurately
known fundamenial constants. Most of this uncertainty is due to the difficultyof the measurements and to the nonideaHty of the systems used for thesemeasurements.About 5 parts per million are due to the limited accuracywith
which h and NA are known.
\342\200\242E. R. Cohen and B. N. Taylor, J. Phys. Chem. Reference Dam 2, No. 4 A973).
Primary and Secondary Thermometers
When expressing temperature as conventionalKeivin temperature 7\" rather
than fundamental temperature t, it is customary to absorb the Bohzmannconsiam into the definhionof a conventional entropy 5,
S s ksa. D)
The relation$Q = xda between reversible heat transfer and entropy transferthen becomes
dQ\302\253=TtlS. E}
PRIMARY AND SECONDARY THERMOMETERS
Any accurately measurable physical property A' whose value is an accuratelyknown function ofthe temperature, A' *= X{T], may be used as a therniomelricparameter10measure the temperature of ihc System possessing the property.V and of any system in thermal equilibrium with it. Used til this way. thesystem with the property X is a thermometer. The principles underlying
the niost commonly used thermometers are listed in Tables 13.1 and 13.2. The
thermometers listed in Table 13.2arecalled secondary thermometers, defined
as thermometers whose temperature dependenceX(T) must be calibrated
empirically, by comparison with another thermometer whosecalibrationisalready known. The calibration of all secondary ihcrmonieicrs must ultimately
be traceable to a primary thermometer. But once calibrated, secondary ther-
thermometers arc easier to use and arc more reproduciblethan the primary ther-
thermometersavailable at the sametemperature.Any calculable model system that can be used to determinethe value of the
Boltzmann constant kB ean be used as a primary thermometer, and the threemodelsystems diseussed above are the most important primary thermometers(TableB.I).
The precision and accuracy of thermometers vary greatly. Precision is
expressedby the variation AT observed when the same temperatureis measured
at different times with the same instrument. Accuracy is expressed by the
uncertainty AT with which the thermometer reproduces the true Kelvin scale.
Secondarythermometersbased on electrical resistance measurements mayachieve a precision of 1 part in I05. The precision of thermometers based onmechaniea!pressuremeasurements is much poorer, particularly at low pres-pressures. For example, helium vapor pressure thermometers at the iower end oftheir useful range have a precision of about 1 part in 103.The accuracy of
secondary thermometers is limited by the accuracyoftheprimary
thermometers
Table B.I Principles of the mosl important primary thermomete
Model sysicm and
propeny utilised
Idea! gas
a| Static pressure ai constam voliim
b) Speed of sound
Magnetic susccplib'iljtyof noninteractm,
spin sjsii;ma) Eiccironic spinsb) Nuclear.spins
Black body radiaiion
note: The coiumn \"Defining cquaiion\" indicates which equation underlies ilic ba&ic Wca. plus lest reference.In pracl'rcc, c
various nuniiicaiiiics uiay be nccilccl. The temperature rjmecsqtiotcd ;iro the nui^es used >n pr;icikc for primary iliL'niuimc
and diluted CMNii
Thcrtt'oJyttanuc Thet
Table B,2 Principles of the most important secondarythermometers
Physical property
Thermoelectric votlageof thermocouples*Thermal expansion ofliquid in glass
Electrical resistancemetals*semiconductors (germaniumIcommeEciai carbon resistors'
Vapor pressure of Jiqucfied gas4
He
Useful rangetaK
400-1400200-400
14-7000.05-770.05-20
1-5.20.3-3.2
p
ltimate limits.\342\200\242
Used as interpolating instrument in the 1PTS.'
Widely used as cryogenic laboratory ihernion
must be individually calibrated to deliver usable
used lo calibrateIhem.The accuracy of primary thermometers is limited hy
their relatively poor precision and by residual variations between different
thermometers. As a rough estimate, the present-day accuracyofprimary
thermometers is about 1 part in 104 above 100 K, about 1 part in 103 around
1K, and about 1 part in 102 near 0.01 K,
THERMODYNAMIC THERMOMETRY
It is possiblelo perform primary thermometry without relying on the theoreti-
theoretically known properties of simple mode! systems, by somehow utilizing therelationE).We give three examples.
(a) Carnot cycle. Consider a Carnot cycleoperating between a known tem-
poralure '!\\ and the unknown temperature7\\, Because of entropy conservation
the heat transfers at the two temperatures satisfy Qt/Tv\302\253
Q2/T2. The un-
unknown temperature can be determined by measuring the ratio of the \\\\\\o hem
transfers. The: method is hot very practical.
(b) Magneticcabriinetry. Supposea paramagnetic substance is initially at a
known temperature7\\, in a magnetic field B{. Let the substance be cooledby
isentropic demagnetization to the unknown temperature T2. If now a known
Temperature Scales
smaii amount dQ of hem is added to the substance, its entropy is raised bydS = dQjT2. The substanceis then isenlropically re-magnetized, and ihc
magnetic field B2 is determinedat which the temperature has returned exacilyto T,. The field B2 will be found siightiy different than Bt:B2 = #i + <l&.
Entropy conservation requires
cIS = dQ/T2 \302\253SG,,Bi)- 5G,,^,) = &S/cB)TdB. F)
Fromthe thermodynamic identity for the Hclmholtz free energy for a magnetiz-
magnetizablesubstance,
dF = -SdT - MilB , G)
oneobtains,by the usual cross-differentiation, the Maxwell relation
(cS/dB)T= {dM/3T)B. (8)
We insert (8) into F) to find the expression for the unknown temperature:
72 -UTQ/<lB)(dM/dT)B. (9)
The quantities <fQ at T \342\200\224T2anddBzlT - Tt are known,and the temperature
derivative of M at 7\" = T, and JS = B, is easily measured. The method makes
no assumptions about the ideality of [he paramagnetic substance, and it has
therefore been used extensively at low temperatures.
(c) Clausim-Clapeyronthermometry. The melting temperature Tm of a sub-substance varies with pressure p according to the Clausius-ClapeyronequationofChapter10:
dTJdp=TmAV/&H , A0)
where AV is the volume change during melting, and AH the latent heat offusion.If both quantitieshave been measured as functions of pressure, A0) canbe integrated;
7*2/7,=
cxpJPl(AK/AH)rfp.(U)
If 7, andp, are known, a measure men! of the pressure p% at which the unknown
temperature T2 is the equilibriummelting temperature permits calculation of
International Practical Temperature Scale {IPTS)
T2 from (U). By utilization of the strong temperature dependence of thesolidificationpressureof liquid ^He, the method lias been used as an alternativeto magneticthermometry at low temperatures.
INTERNATIONAL PRACTICAL TEMPERATURESCALE (IPTS)
Many known phase equilibrium temperatures can be reproduced far more
precisely than the accuracy with which their exact location on the Kelvin scalecanbedetermined by primary thermometry. To facilitate practical thermometry,a numberof easily reproducible phase equilibrium temperatures have beendetermined as accuratelyas possibleand have been assigned best values todefine an InternationalPracticalTemperature Scale (IPTS). On the IPTS theselectedequilibriumpoints are treated as if their temperatures were known to beexactlyequalto theirassignedvalues.Intermediatetemperatures are determined
by a precisely specified interpolation procedurethat is chosen to reproduce the
true Kelvin scale as accuratelyas possible.The present version of the scale isIPTS68, adopted in 1968 by international agreement, covering temperaturesfrom the triple point of hydrogen {13.81K) upward.*Table B.3gives tlte
assigned temperatures for IPTS68.In the range between 13.81K and 903.89 K, which is the melting point of
antimony, a platinum resistance thermometer is used as the interpolatinginstrument.In the range from 903.89 K. to 1337.58 K.,the meltingpoint of gold,
a platinum-piatinum/rhodiuni thermocouple is used. Above 1337.58 K black
body radiation is used.Below 13.81K. no precisely defined procedure has been agreed, In the range
between 5.2 K. and 13.S1K. various scales based on the vapor pressure of
hydrogen are in practical use. Below 5.21 K, the critical point of 4Hc( down to
about 0.3 K, the 1958and 1962helium scales* are widely used as de factoextensions of 1PTS68.The 1958 4He scale relates the vapor pressure of 4Hetothe temperature T; the 1962 3He scale uses the vapor pressureof 3He.
As the accuracy of primary temperature measurements improves, errors in
practical scales such as IPTS become uncovered, leadingeventually to revision
of the practical scales. Table B.3 listssomeerrors now believed to exist in
IPTS68.
\342\200\242Sec, for eiampte, American Iniiiiuie of Physics handbook, 3rd cd., McGraw-Hill,1972;Sectio
Hcaf, M, W\". Zcmansky, ciiiiot. Contains complete original references-
Tempera e Scales
Table B.3 Assigned temperatures of lhe lnternalionalPracjical TemperatureScaleOf I96S
in K in KEquilibrium poinl
Substance Type
6 B50lorr)b
b
\342\200\242>
i
b
ff
13.8117.04220.2827.40254.36190.188
273.16373-15
505-tISI
692.43
1235.081337.58
exact0.0250.0440.066
noi-e: Except for the triple poinisand the 17.042 K point, all
fG^mlEoriu *irc tHosc at a pressure of one st-mdarti airrio^pncre\302\273
p0 - l0J,325Nmwi <= 760iorr). The I7.O42K poinl isihe
boiling point ot hydrogen at 25OtOff^ llic notations f, bt and
f tti lhe second colurnn refer to uiplc poinis* boiling points,and freezing points.* The lasi column contains esiimatcs oferrors known to exisr, from Physics Today 29, No. 12,p. 19(Dec.1976).\342\200\242Alt data from lhe American luxituie of Phyws handbook,3rded., McGraw-Hill.1972; Seciion 4: Heat, M. W. Zcmansky,ediior.
Appendix C
Poisson Distribution
ThePoissondistribution law is a famous result of probability ihcory.Theresult
Is useful in tUe design and analysis ofcountingexperimentsin physics, biology,
operations research, and engineering. The statisticalmethodswe have developed
tend themselves to an elegant derivationof the Poisson law, which is concerned
with the occurrence of small numbers of objects in randomsampting processes,
it is also called the law of smallnumbers. If on the average there is one badpenny in a thousand, what is ihe probability that N bad pennies will be found in
a given sample of one hundred pennies? The problem was first considered andsolved in a remarkablestudy of the rote of luck, in criminal and civil law trials
In France in the early nineteenth century.We derive ihe Poisson distribution taw with the aid of a mode*system that
consists of a large number R of independentlatticesitesin thermal and diffusive
contact wilh a gas. The gasservesas a reservoir. Each lattice site may adsorbzero or oneatom.We want to find Ihe probabilities
that a total of 0, !, 2,.... N,.... atoms are adsorbed ou the R sites, if v/e are
given the average number <N> of adsorbedatomsover an ensemble of similar
systems.Consider a systemcomposedofa singlesiie.Itisconvenient to set the binding
energy of an atom to the site as zero. The identical form for the distribution is
found if a binding energy is included in the calculation. The Gibbs sum is
where the term A is proportional to the probability the site is occupied,and
the term ! is proportional to the probabilliy the sheisvacant.Thus the absolute
probability ihat ihe site Is occupiedis
J ~1 -r /.\"
{>
The actual value of /, is delermined byihe condition of the gas in the reservoir,
Polsson Distribution
because for diffusive comae! beivveen i\\\\c lattice and the reservoir we must have
by the argument of Chapier5.The evaluation of /.(gas) for an ideal gas was
given in Chapier 6.
We now extend [he treatment10R independent siics. Then
(<*)
By the argument used in Chapter 1 we know lhal the binomial expansion of{O + \302\251)Bor (! + /.)* counts once and only once every stale of Ihe system ofR sites. Each site has two alternative states, namely O for vacant or \302\251for
occupied, which corresponds in the Gibbs sum to the term ! for X\302\260and the
term / for )}.In the low-occupancylimitof/ \302\2531 we have / s /.,whence
fR- },R E)
is the average total numberof adsorbed atoms. The Poisson distribution is
concerned with this low-occupancylimit.We can now write D) as
F)
Next we let the number of sites R increase without limit, while holding the
average number of occupied sites <N> constant. The Poissondistribution is
concerned with infrequent events! By the definition of the exponential function
we have
exp<N> , G)
(}R)NJM = exp<N>=
\302\253p(;j!)-
I\342\200\224- (8)
Ttie last step here is the expansionof the exponential funclion in a power series.
The term in '/!\" in 3\"^, is proportional lo the probability F(A') that .V sites
are occupied. With the Gibbs sum as the normalisation Factor we ha\\c in the
limit of large l(:
C)
or, because/.K = <A'> from E),
A0)
This is the Poissondistribution law.'
Particular interest attaches to the probability P@) that none of the sites
is occupied. From A0) we find, with <N>\302\260= 1 and 0! = I,
P@) =exp(-<N\302\273; logP(O)
- -< (II)
Thus the probability of zero occupancy is simply related to ihc averagenumber
(N) of occupied sites. This suggests a simpleexperimentalprocedure for the
determination of <N>: just count thesystems that have no adsorbed atoms.
Values of P(A') for several values of <N> are given in Table CM. Plots are
given in Figure C.I for <N> = 0.5,1, 2,and 3.
Table
f@)
PW
FB)
FC)W)P{5)
PG)
F(9)
C.l Values of the Poisson disi
0.1
0.90480.0905
0.0045
0.0002
1
0.3
0.740S
0.2222
0.03330.00330.0003
0.5
0.60650.3033
0.0758
0.0126
0.0016
0.0002
Iribution ft
0.7
0.49660.34760.12170.02840.00500.0007
0.0001
inction PIN)
0.9
0.40660.36590.16470.0494
0.0111
0.0020
0.0003
<N>
1
0.36790.36790.18390.06130.01530.00310.00050.0001
\302\253p(-W
N!
2
0.13530.27070.27070.18040.09020.03610.01200.0034
0.0O09
0.0002
>)
3
0.0498
0.1494
0.22400.22400.16S00.100S005040.02160.00810.0027
0.0008
4
0.01830.07330.14650.19540.19540.15630JO42
0.0595
0.0298
O.OI32
0.0053
5
0.00670.03370.08420.14040.17550.17550.14620.10440.06530.03630.0S8!
0.6
gO.40.2
0
Tisttibuti
(N> =-
J) 1
0.5
J -2 3
W _
t 1
4 5
0.4
0.2
<,V> = I
)s , .
\302\2430.2
LLjl
Example:Incorrectand correct counting of states, (a) The Gibbssum for the R sites is nai
Jwi = 1 + X + )} + -I3 4- - ' ' + )*. {12}
Why noi?
(b) Thccorrcci suns is
A3)
Ji!
[R - W N!
islilc binomial cocflicicm. Nolc that tjill.V) is Ihc number of independent states of Ihe
system for a given number of adsorbed alor.is .V. The Gibbs sum is a sum over ail stales.
Example: Elementary demotion of P@). I.ei a Iota! of R bacteria be distribulcd at
randoln among L dishes.Each dish is viewt j as a syslen:of many siles 10 which a bacterium
PohsonDhtributior.
may aiiach.TlieL dishes represent an ensemble ofLidcnticalsyslems.The average numberof bacieria per dish is
<N> \342\200\224R/L. {14}
Each time a bacleriumis distributed, iheprobability that a given dish will receive thai
bacterium is t/L The probability [he given dish will not receive the bacterium is
A5)
Theprobability in R tries lhat the given dish will receive no baderia is
/><0)=
(l~~Y A6)
because the faaor (i 5) eniers on eachtry.
Wemay\302\253ri(e(l6)as
lW-(l -<\302\243>)'.(.7,
by use of<.V> = R/L. We know iliai in Hie lirail of large R,
rap(-<N\302\273= limfi -\342\200\224), A8)
!>y llie Jcrmilion of Ihc cxpoocnlu! funclron. Thus for R \302\273I and L \302\273I we have
P@) =cxp(-<N\302\273 A9)
in aercemcni wjih (U).
PROBLEMS
/. Random pulses. A radioactive source emus alpha particles which arccounteda! an average rate of Oiie per second, (a) What is the probability of
counting exactly iOalpha particlesin 5s?(b)Ofcounting2m 1 s?{c) Of countingnone in 5 s? The answers to (a) and {b}arenot identical.
2. Approach to Gaussian distribution. Showthat the Poisson function P{N) ~
\342\226\240(N>'\"'exp{ \342\200\224<W)}/A'! closely approaches a Gaussian function in form, for
large (A'>. That is, show when N k closeto <;V> thai
PIN) * Aexpt-tyS - (N}I] ,
where A, B are quantities to be determinedby you. Hint: Work with !oeP{A?};use the Stirling approximation.In the Gaussian form both A and B are functions
of <N>; in the development of the Poissonfunction you may find A, B arefunctions ofN, but the two forms of A, B are closelyequivalentover the range in
which the exponential factor has significant values.
Appendix D
Pressure
Let a pressureps be applied normal to ihe faces of a cube filled wilh a gas or
liquid in quantum state s. By elementary mechanics {Chapter 3} the pressureis equalto
Pl~ -dUJdV , {1}
where Vs is tlie energy of the system in the states.We can also write the pressureas
Pl- ~(dUJdV\\ , B)
where (dU/,tV), denotes ihe expcciaiton value\" of dUfdV over the siatc s at
volume y. Ii is important that we can calculate p by {2} which is at a fixed
volume with no ambiguity about the identity of the selected state 5. whereas
(!) involves followingthe state through two volumes. Kand V + dVt with some
possible doubt whether the state remains ihe same.The ensemble average
pressure p is the average of p, over the states represented in the ensemble:
f>W--(Wl,> C)
Becausethe number of states in the ensemble isconstant, the entropy is constant,
so that the derivative is at constantentropy.We may therefore write
The result D) uses the energyof the system expressed as V(a,V,...}; that is,as a function of the volume V and the entropy a\342\200\224not the temperature r. It is
the entropy and not the temperaturethat is to be held constant in the
differentiation.
\342\200\242The equivalence of (t) and B) is an example of ihe Hellmana-Feynman theorem of quantur
resped to a parameter X are related by dVfdX=>
<|A*7<*A|). The derivation may be found o
p. 1192of C. Cohen-Tannoudji, B. Diu, and F. Laloc. Quamum mechanic*, Wiley, [977; see alsE. Merzbacher,Quantum mechanics, 2nd ed., Wiley, 1970;p. 442.
Appendix E
Negative Temperature
The result of Problem 2.2 for the entropy of a spin system as a function of the
energy in a magnetic field is plotted here in Figure E.I. Noticethe region in
which Ca/cV)s is negative (Figure E.2).Negativer means that the population
of the upper state ts greater than the population of the lower state. When this
condition obtains we say that the population is inverted,as illustrated in
Figure E.3.
The concept of negative lemperaturets physically meaningful for a systemthat satisfiesthe followingrestrictions:(a) There must be a finite upper limit to
the spectrum of energy states, for otherwise a system at a negative temperaturewould have an infinite energy. A freely moving particle or a harmonic oscillatorcannothave negative temperatures, for there is no upper boundon theirenergies.Thus only certain degrees of freedom of a particle canbe at a negative tempera-
jr= +0
\342\226\240
\\
\\
\\r = -0\\0 0.2 0.4 0.6 0.8 1.0
Figure E.1 Entropy as funciion of energy for a iwo siaic sysicm.The scparaiion of the siaiea is c ~ I in ibis example. In ihe
Icfi-liandsideof ihe figure da/cU is posiiive, so ihal r is posiiivc.On ihe riglii-hand side ca/cU is negaiive and i is negaiive.
Negmlrt Temp,,
+6
+ 5
+ 4
+ 3
+ 2
K -f 1
\302\247\302\2610
I -\342\226\240
-1
-3
-4
_5
.\342\200\224\342\200\224
5 o 2 0
1
/
4Migy 0\342\200\224\342\200\224
0.6 01 ^1
/
!/
Figure E.2 Temperature versus energy for Ihe twostale system. Here
Noiice tiiai ihe energy is not a maximum at x ~ + x,but is 3 maximum al i = -0.
ture: the nuclear spiu orientalion in a magnetic field is ihe degreeof freedom
most commonly considered in experiments at negative temperatures,(b) The
system must be in internal thermal equilibrium.Thismeansihestates must have
occupancies in accord with the Bolumann factor taken for the appropriate
negative temperature, (c) The states that are at a negative temperature must be
isolated and inaccessible to those states of the body that arc at a positivetemperature.
The ordinary translations! and vibrational degrees of freedom of a bodyhave an entropy that increases without limit as the energy increases,in contrast
to the two stale or spin system of Figure E.I. If a incieases without limit, then iis always positive. The exchange of energy between a system at a negativetemperatureand a system that can only have a positive temperature(becauseof
Negative Temperature
I I I I I I I I I I
Figure E.3 Possible spirt distributions for various positive and
negative tempera!ures.The magneiic fietd is direcied upward. Thenegative spin temperaturescannot Jasi indefinitely because of weak
coupting betweenspinsand the lattice. The lattice can oniy be at a
positive temperature becauseils enefgy level spectrum is unboundedon top. The downward-directed spins, as at i =
\342\200\224r^,turn over one
by one, I hereby releasing energy to the lattice and approaching
equilibrium with the lattice a! a common posiiivctemperature. A
nuclear spin system at negative temperature may reiax qutie slowly,over a time of minutes or hours; during this time experiments at
negative temperaturesmay be carried out.
an unbounded spectrum) will iead always to an equilibrium configuration inwhich both systems
are at a positive temperature.
Negative temperatures correspondto higher energies than positive tempera-
temperatures. When a system at a negative temperature is brought into contactwith a
system at a positive temperature, energy will be transfused from the negative
temperature to the positive temperature.Negative temperatures are hotter
than positive temperatures.The temperaturescalefrom coid to hot runs + 0 K,.... 4- 300 K,..., H- go K,
-03K -300 K,..., -OK. Notethat if a system at -300K b broughtinto thermalcontactwith an idem teal system at 300 K, the final equilibrium
temperature is not 0 K, but is \302\261co K.
Nuclear and electron spin systemscanbepromoted to negative temperatures
by suitable radio frequency techniques.If a spin resonance experiment is
carried out on a spin system at negati\\e temperature, resonant emission of
energy is obtainedinsteadof resonant absorption.* A negative temperature
system is useful as an rf amplifier in radio astronomy where weak signalsmust
be amplified.
Abragam and Proctor* have carried out an elegantseriesofexperiments on
calorimetry with systems at negative temperatures.Working with a LiF crystal,
they established one temperaturein the system of Li nuclear spins and anothertemperaturein the system of F nuclear spins. In a strong staiicmagnetic field
the two thermal systems are essentially isolated,but in the Earth's magneiic
field the energylevelsoverlapand the two systems rapidly approach equilibriumamong themselves(mixing). It is possible to determine the temperature of ihcsystems before and after the systems 3re allowedto mix.Abragam and Proctor
found that if both systems were initially at positive temperatures they attained
a common positivetemperatureon being brought tnto thermal contact. If boih
systems were preparedinitially at negative temperatures, they attained a
common negative temperatureon being brought into thermal contact, if
prepared one at a positive temperatureand the other at a negative temperature,then an intermediate temperaturewas attained on mixing, warmer than the
initial positive temperature and coolerthan the initial negative tempcraiure.
FURTHER REFERENCESON NEGATIVE TEMPERATURE
N. F. Ramsey, \"Thermodynamics and statistical mechanics at negative absolutetemperature,\"Physical Review 103, 20A956).
M. J. Klein, \"Negativeabsolutetemperature,\" Physical Review 104, 5S9 (i956).
\342\200\242E. M. Puicelland R. V. Pound, Physical Review 81,279 A951).1 A. Abragam and W. G. Proctor,Physical Review 106.16Q(l<>57); 109,1441 A958).
Index
Abragam, H.. 350, 463
Abraham, B. M.. 209
Absolute activity, 139
Absorption refrigerator, 25SAbsorptivity, 97
Acceptor, 357, 363Accessible state, 29
Activation energy, reactions, 2Active transport, 145
Age, Sun, HiAir conditioners, 235
room, 258
Alloy, binary, !6, 310. 331
gold in silicon, 331
mixing energy, 3IS.33Osolidification ranee,33!system, 16
Anderson, A. C, 219Amonini, E.. 142
Atmosphere, 126,145,179Atoms, in a box, 74
velocity distribution, 394
Average value, 22
Avogadro constant, 2S2
Barclay, I A., 345Barnes,C.B.,345
Barometric pressure, 125
equation, 126Battery, electrochemical, 129
Bertram, B., 210Belts,D.S.. 342
Binary alloy, 16,310.331
Binary model systems, 10Binomial distribution. 22
expansion. 14BiopoJynwr growth, 273
Hiack body radiation, 98Bockris,J. O.M.,248
Bolununn constant, 4i, 45, 446Boiizmannfactor. 58,61Bolutiminn transport equation. 403, 42!Born, M.. tOfi
Bose-EinStcin Condensate, 202Bose-Einstein distribution, 157, 159Boson,152Boson gas, 199
condensation, 205degenerate, 221fluctuation, 222
one dimension, 222Boson system, 223
Bridgman, P. W., 279
CMN, 348, 448Carbonmonoxidepoisoning,i46Camot coefficient,
refrigerator performance, 234
Camot cycle, 236ideal gas,237Ihermodynamic thcrmomctry, 449
Camot efficiency, 230Carnot engine,photon,258Camot inequality, 228, 232
Carnot Jiqucfier, 351Carrier concentration,intrinsic. 362
Carrier lifetime. 3SSCarrier recombination, 383Catalyst. 271Celsius temperature scate, 4-15
Centrifuge, 145Cerium magnesium nitrate, 348, 448
Chapeiiier, M, 350Chauikasekhar limit, 222
Characteristic height, atmosphere, \\26
Cliau, V. H., 350
Chemical equilibria, 266 Critical point, 291
Chemical potential. 118, JJ9, J43, J61 van dcr WaaJs gas, 2%9and entropy, J3I Critical radius, nucleation, 295
equivalent definition, J4S Critical size, nuclear reactor, 437external, J49 Crilical temperature, 276
ideal gas, J20, J69 gases,277
internal, J22, 12-4 Croft, A. J., 340mobile magneticparticles,J27 Cryogenics, 333
near absolute zero, 199Crystal transformation, 307
loiaj. 122. J24 Crystalline mixture, 319
two phase equilibrium, 330 Curie temperature, 298
Chemical reaction, 266 Cycle, Carnot, 236
Chemical work, 250
idea! gas, 25! DNA molecule, S3
Classical distribution, 410 Daunt, J. G, 348
function, J6J de Bruyn, R., 345Classicallimit, 160 De Maeyer, L., 270Classicalregime,74, 153,159,358 Debye T1 Saw, 106Claude cycle, 341 Dcbyetemperature, 105
helium liqucfier, 351 Debye theory, 102Ciausius-Clapeyronequation, 281 Degenerate Fermi gas, 219Clayton, D. D.,222
Degenerate gas, !82
Closed system, 29 Degeneratesemiconductors, 355. 365
Coefficient, refrigerator performance, 234 Demagnetization, cooling, 352
Coefficient, viscosity, 402 isenttopic,346
Coexistence curve, 278 nuclear, 348Cohen, E. R.,446 . Density of orbitals, 187,218Cohen-Tannoudji, C.,459 Density testates, 186
Collision cross sections, 395 effective, in semiconductors, 360Collision rates, 395 effective mass, 360
Concentration fluctuations, 147 Detailedbalance,kinetics, 407
Condensed phase, 303 principle of, 271
Conductance, hole, 415 Deviation, integrated, 54
tube, 416 Diatomic molecules, rotation, 84Conduction band, 355 Diesel engine, 180
Conduction electrons, Differential relations, 70semiconductors,355 Diffusion
Conductivity, electrical, 4!3, 421 current flow, 379
intrinsic, 387 equation, 437thermal, 401,421 fixed boundary, 429
Configuration, most probable, 33, 35 heat, 437
Convectivc isentropic equilibrium, 179 internal heat sources, 437
Cooling, demagnetization, 352 particle, 399,409
evaporation, 34t . p-n junction, 437
external work, 334 Diffusive equilibrium, 120
nonmctallic solid, 259 DtlTustvity, 399, 428, 437
of slab, 437 Dilution refrigerator, helium, 342
Cooper pair, 250, 257 Dispersionrelation, 425
Corresponding states, law of, 290 Distribution, classical,161,410Cosmicbackground radiation, 98 Base-Einstein, 158Countcrflow heat exchanger,336 Fennt-Dirac, 154,411
,. Critical magnetic field, 253 . Dju, B.,459
Dixon,R. W., 366
Donor, 356, 363levels,'369
Donor impurities, tonization, 273
Doping concentrations, 375Doping profile, 383
Dry adiabatic lapse rate, 270
Earth, distance from Sun, 111Eddy current equation, 425Effective density of stales, 361
valence band. 362
Effective mass, 360
EfTcctivc work, 246Efficiency, Carnot, 230
Eigen, M., 270Einstein condensation, 199
temperature, 205Einstein relation, 406
high electron concentrations, 388Einstein temperature, solids, 84
Elasticity of polymers,86Electrical conductivity, 413, 421Electrical noise, 98Electrochemical battery, 129
Electrolysis, 247Electron-hole pair generation, 388
Electron mobility, 380
Elementary excitations, 212EHiott, R. P., 323
Etnissivity, 97Energy, conversion, 240
conversion efficiency, 230
degenerate boson gas, 221equiparijtion, 77
Fermi gas, 185fluctuations, 83, 113
geothermal, 259ideal gas, 76magnetic, 252
mean kinetic, 420mixing, 314, 330thermal average, 140
transfer, 227two state system, 62
van der Waals gas, 305
Energy gap, 355
Ensemble, average, 31,62construction, 32
systems, 31
Enthalpy, 246, 284van der Waals gas, 305
Entropy, 42, 45, 52accumulation, 229
and chemical poicniiai. 131conventional, 45degeneratebosongas,221degenerateFermi gas, 2j9
fr\302\253energy. 165
heat now, 44law of increase, 54as logarithm, 50of mixing, 78, 178, 314and occupancy, 114and temperature, 52
transfer, 227
van der Waals gas, 305
Equation of continuity. 424
Equation of stale,van dcr Waais, 287, 289Equilibria, chemical. 266\"
phase, 322
Equilibrium, hydrogen, 269gas-solid,285, 305
panidc-anttpartide, 274
reactions, 266two phase, 330vapor pressure, 291
Equilibrium constant, 268Equipartition of energy, 77
Error function, 429Euteclic, 325Evaporation cooiing, 341
limit, 352
Expansion, cooling, 334
engine, 334
Fertitt gas, 259irreversible,175isothermal.171
Extensive quantities, 264
Extreme rdativtstic particles, 117Extrinsicsemiconductor,364
Feher, G., 148Fermi energy, 155, 183Fermi gas, 183
fluctuations, 222ground slate, 185Irreversible expansion. 259
liquid hdium-3, 219metals. 194relativistic, 218
Fermi level, 155,357intrinsic, 362extrinsicsemiconductor,364
Fetmi-Dirac distribution (unction, Gibbs free energy,246, 262
153, 177, 411 van dcr WaaU gas, 291Fcrnfi-Dirac integral, 366 Gibbs sum, 134, 138,146FcrmtOil, 152 ideal gas, 169, 180Fer tomagnet ism, 295, 302 two level system, 146Pick's law, 399 GifTard, R. R., 103First law, 49 Goldman, M., 350First order transition, 302 Grand canonical distribution, 138
Flow, through hole, 415 Grand partition function, 138
through tube, 416, 421 Grand sum, 138
speed, 422 Greenhouse eflect,115Fluctuations, Bose gas. 222 Guyer, R. A., 210
concentration, 147
energy, 83, 113Fermi gas, 222
time of, 178 Half-cell potentials, 131Flux density, 397 Hall, R. N.. 385Fourier analysis,436 Ha|I-Shockley-Readtheory, 383
Fourier's law, 401 Hansen. M., 323
Free energy, 163 Harmonic oscillator, 52, 82
Gibbs, 246, 262 free energy, 82harmonic oscillator,82 multiplicity function, 24
Helmholu.68 Harwit, M, 219paramagnetic system, 69 Heat, 44,68, 237, 240photon gas, i 12 definition, 227two state system, 8i isobartc, 245
Free energy function, Landau, 298 path dependence,240
Fmlon, J. S., 142 ... vaporization of ice, 305
Fuel cell, 247, 248 Heat capacity, 63, 165Fundamental assumption,29 '
degenerate boson gas, 221
Fundamental temperature, 41 electron gas, 189
intergalactic space, N3Gallium arsenide, semt-tnsulating, 372 liquid helium-4, 113Gamma function integral, 440 photons and phononS, U3Gasconstant,166 solids, 113
Gas, critical temperatures, 277 two state system, 62
degenerate, 182 Heat conduction equation, 424
degenerate boson, 221 Heat engine, 228, 230
degenerate Fermi, 219 refrigeratorcascade,258
ideal, 72 Heat exchanger, counter Row, 336
liquefaction, 337 Heat flow, 44
one-dimensional, 86 Heat transfer, sound wave,434potential energy, 145 Heat pump, 235, 257quantum, 182 Heat shield, 112
rarefied, 413 reflective, 115
sounii waves, 430 Hecr, C. V., 34S
Gas-solid equilibrium Helium dilution refrigerator, 342
Gauss integral, 439 Helium liquerler, 351Gaussiandistribution, 20 Helmholu dec eneray, 6SGeneralizedforces,404, 405,453 Heinegroup, 140
Generation, dectron-hoic pair, 3S8 Hemoglobin.141Geothermal energy, 259 Heitshaw, D. G., 216Giauque,W., t67 High, vacuum region, 397Gibbs factor, 134, I3S Hill, J. S., 34S
Hobdcn.M. V.,349
Hots, 177,355
conductance, 415quantum concentration, 361
Hook,J. R.,217Huiskamp, W. }., 342
Hydrogen, equilibrium, 269
IPTS,451Ice,heat of vaporization, 305Ideal gas, 72, 74, 160,169
calculations, ISO
Carnot cycle, 237chemical potential, 120,169chemical work, 251
energy, 76
Gibbssum, 180internal degree of freedom, 179
iscntropic relations, 179Kelvin, 446
law, 77
law, kinetic theory, 391
one-dimensional, 86
sudden expansion, 243therm odynamic identity, 177two dimensions, 180
Impurity atom ionization, 143
Impurily level, 368, 383carrier recombination, 383
Increase of entropy, law of, 45
Inequalily, Carnot, 232Injection laser, 381,38S
Integrals comaining exponentials, 439Intensive quantises, 264
Intergalaciic space, heat capacity, 113Internal chemicaipotential, 122, 124
International Practical TemperatureScale,451
Intrinsic conductivily, 387
Intrinsic Fermi level, 362Inversion temperature, 336
lonization, deep impurities, 3S8donor impurities, 273
impurily alum, 143
thermal, 273water, 269
Irreversibilily, sources, 232Irreversible thermodynamics,406Irreversiblework, 242Isemropicdemagnetization, 346
Iscntropic expansion, 114, 148Isentfopicprocess,173Iscntropic rclalioos, ideal gas. 179isobaric process,245
Isotherm, 276
Langmuir adsorption, 143Isothermal work, 245
James,H.M, 86Jfohnson, J. B., 98
Johnson noise, 98Johnston, H. L., 167Joule-Thomson effect, 337
van dcr Waais gas, 338Joyce, W. 8-, 366
Joyce-Dkon approximation, 366JQttner, F.,219
Kelvin temperature, 41
scale, 445Kinematic viscosity,404
Kinetic model, mass action, 270Kinetictheory, ideal gas law, 391
Kinetics, detailedbalance,407
Kirchhoir law, 96, 115Klein, M.J., 463
Knudsen regime, 397,413Kramers, H. C, 114Kuril\", N-, 348, 349
LaioS, F-, 459Lambda poinl.heIium-4, 210
Laudau free energy function, 298
Landau function, 69, 298Landau theory, phase transitions, 298
Langmuir adsorption isotherm, 145Laser,injection, 381, 3S8
Laiem heat, 281, 284enthalpy, 284
increase of entropy, 45
vaporization,2SILaw of corresponding states, 290Law of increaseof entropy, 45
Law of mass action, 268, 362, 382
Laws of rarefied gases, 413Laws of thermodynamics, 48, 49
Leff, H. S-,259
Legged, A. J., 217
Lcln, W. H-, 195
Linde cycle, 339Liouville theorem, 40SLiquid helium 11, 209
Liquid heIium-3, 217supcrliuid phases, 217
Liquid hdium-4, 207heat capacity, 113
Liquid 3He-4He mixlute, 320
mixing energy, 330
Uquiduscurve, 323 Negative temperature, 460Long tube, Row, 416 -Never,\" 53Loschmidt number, 396 N'iels-Hakkenbcrg.C.G.,114Lounasmaa, O, V., 342 Nondegenerate semiconductor, 358Low orbital free atoms, 201 Nonequiiihrium semiconductors,379
Low temperature thermometry, 448 Normal phase, 203
Lyncis, B., ill Nuclear demagnetization, 348Nuclear matter, 198
McFee, J. H., 394 Nuclear reactor,critical size, 437
Magnetic concentration, 145 Nucieatton,294Magnetic difliisivHy, 425, 437 critical radius, 295Magnetic energy, 253 Nyquist theorem, 93Mapiciic field, adsorption of Oj, 147
mobile magnetic particles. 127Occupation<W levels, 369
spm entropy 170Onsajcr relation. 406
in superconductors, 253 fVhhii 9 lV>
MaSn\302\253ic susceptibility. 81Spl,^Magneticsystem, 23
' J mMagnetic work. 252
oS^ZlZl^
Mass action, law, 268, 270, 362, 387Ovcrhauser efTeci,K4
Maxwell distribution of velocities, 392, 419Maxwell relation, 71, 272 Paramagnetic system, 69Maxwell transmission distribution, 395 . Paramagnetism, 52, 446Maxwell velocity distribution 393 Particic-antipartide equilibrium, 274Mean field method, 288 Particle diffusion, 399, 409Mean free path, 395 Partition function, 61Meanspeeds,Maxweilian distribution, 419 two systems, 85Mean value, 22 Pascal (Pa), back endpaperMdssner effect, 252 Path dependence, 240
Merzbachcr, E., 459 Pauli exclusion principle, 152
Metastable phases, 278 Peltier effect, 336
Meyer, L-, 215 Penetration, temperalure oscillation, 426
Milaer, J. H., 34S Pcnnings,N. H., 345
Minority carrier lifetime, 3S8 Peritectic systems, 330
Mixing, energy, 3S4, 330 Perpetual motion, 50entropy, 78, 578,314 pH, 269
Mixture, binary, 310 Phase,267crystalline, 319 condensed, 203
liquid 3He-4He, 320 equilibria,322phase equilibria, 322 normal, 203
Mobile magneticparticles, relations of helium, 210
chemical potential, ]27 Phasediagram, 321
Mobility, electron, 380 Phase Iransitions, 298Molecules,Earth's atmosphere, 145 Landau Iheory, 298Monkey-Hamlet,53 superconducting,306,307
Mosl probable configuration, 33, 35 Phenomcnologicailaws,398
Multiple binding of Oa, 148 Phillips,N. E-, 195,196,254
Multiplicity, 7 Phonon, 102
Multiplicity function, 15, 18 beat capacity, 113harmonic oscillator, 24
' '\342\226\240''. mode, 104 \342\226\240'\342\226\240
-
Myoglobin, 140, 142 \342\200\242\342\226\240'. . ;
\"
solids, 102 ;
Photon, Carnot engine, 258condensation, 321heal capacity, 113
thermal, HOPhoton gas, 112,114
free energy, II2
iscntropic expansion, 114one dimension. 112
Pillans. H-, 111Planck distribution function, 89, 91Planck law, 91, 95
p-n junction, 373reverse-biased,377
Poise. 403
Poisson distribution. 138.453distribution law, 455
Poision equation, 375Pollution, tlicnniil. 25S
Potjmer, 86
elasticity. 86
Population inversion. 460I'ound, R. V.,-563
Pratt, W. P.. 34gPressure. 64. 164
degenerate Fermi gas. 2!9thermal jadiation. ill
Principle of detailed balance,271Probability, 30
Proctor, W. G., 463Propagation, sound waves,430
Pulse, development. 427
Fourier analysis, 436random, 457
Pump, speed. 457Puree!!.E. M.,463
Quantum concentration, 73, 85conduction electrons, 36!holes,36!
Quantum gas, JS2
Quantum icgime, 182Quasi-Feimilevel, 379
Quasipanicte, 212
Radiant energy fiux, 114
Radiant object, 114Radiation
black body background, 98
thermal, 111Ramsey, N. R, 463
Rarefied gases, laws of, 413 .Reaction,chemical, 266
Reaction raw, 371Read, W. T., 3S5
Reese, W., 219*Refrigeratorperformance.
Carnol coefficient. 234
coefficient, 234helium dilution. 342light bulb in, 259
Rcif. F.,215Relativistic Fermi gas, 21 %
Relativistic white dwarfs. 222Relaxation, thermal, 432
time, 433
Reservoir, 58
Resistivity. 387Reverse-biasedp-n junction, 377
Reversible isotherm^ expansion. 171Reversible process, 64
Room air conditioner. 253Rose,W. K., I(J7
Roionbliiiii. S. S., 34SRossi-Fanctti, A., 142
Rotation, diatomic molecules, 84
Sackur-Tetrodeequation.77, 165
experimental tests, 167
Schindler, H., 148Schotiky anomaly, 63
Second law, 49, 240Secondorder transiiion, 304
Segregation coefficient, 331Semiconductor. 353
degenerate,358,365
donor impurities, 273
extrinsic, 364impurity atom iontzation, 143
n~ and p-type, 363nondegenerate, 358
nonequihbrium, 379
Semi-insulaiing gallium arsenide. 372Shockley.W.. 385
Shunk, F. A., 323Simmonds. S., 142Smythe, W. R., 425
Soilj temperature variations, 437thermal diffusivity, 427
Solar constant, 110Soiidificaiionrange, 331Soljduscurve, 323
Solubility gap, 310, 311
Soliduscurve, 323
Solubility gap, 310, 311phase diagram, 32 i
Sound wave, Jieal transfer, 434propagation, 430
Specific heal, 63
Speed, pump, 417lube, 422
Spin entropy, 170
aclditivity, 53Spin excess,14Spinsystem, 10, 37. 52
Srinivasan, S., 248Stefan-Bo!tzjaann constant, 96
Stefan-Boltzmann law. 91, 94Stcycrt, W. A., 343
Stirling approximation, 19,441Stokes-Einstein relation, 404
Stmve, O., i Jl
Sudden expansion, ideal gas, 243vacuum, 175
Sun
age, I i i
interior temperature, I Ji
mass aad radius, I j r
surface temperature, 1 iO
Superconducting transition, 306
Superconductor, 252magnetic work, 252
Supercooling, 278
Supctfiuid phases, 217
Superfluidity, 212
Superheating, 278Superinsulation. 19
Susceptibility, nugnelic, 8!Swcnson, C. A., 2J0
Tacoais,K.W., 345
Taylor, B. N., 446Teelers,W. D., 259
Temperature, 41
critical, 276Earth's surface, JU
estimation of surface, 97fundamental, 4!
Kelvin, 4!
negative, 461oscillation,426
scales, 445
Sun's average, I! 1
Sun's surface, HO
variations in soil, 437
Temperature oscillation, penetration,426, 437
Thermal average, 62
Thermal conductivity, 401, 42jmetals, 421
Thermal contact, 33, 37Thermal diflusivity, 425
soil, 427
Thermal equilibrium, 36, 39values, 36
Thermal expansion, 272Thermal ionization of hydrogen, 272
Thermal photon, 110Thermal pollution, 258Thermal radiation, J J J
Thermal relaxation, 432
Thermodynamic identity, 67, 133, 177
Thermodynamic relations 71,272Tliermodynamic tbermometry, 449
Thermodynamics, superconductingtransition, 306
Thermometers, 447
Third law, 49Throughput, 415Torr,4J4Transitions, first order, 302
second order, 304Transmission distiibutton.
Maxwell, 395
Transport processes, 397Treioar, L. R. G., S6
Triple point, 284Two stale system, 62, SI
free energy,81licat capacity, 62
:erse, entropy of, 110Un
v. Karman.T., 106
Vacuum physics, 413Valence band, 355
effective density of slates, 362
van der Waals gas, critical points, 2S9energy, 305
enthalpy, 305
equation of stale, 287, 2S9
Gibbs free energy, 291helium, 350
Joule-Thomson effect, 338
Vapor prcsiurc, equilibrium,291
Vapor pressure equation, 276, 281Vaporisation, latent heat, 231
ice, 305
Velocity of sound, 432
Viiia theorem, SllViscosity, 402
kinematic, 404
Water
calculation of dT/dp, 305ionization, 269
Webb, R. A., 103Weinslock, B., 209
Weissman, M., 148
Wheatley, J.C., 217While dwarf star, 196
mass-radius relationship, 2!9relalivislic, 222
Wiebes, J., 114Wicdemaim-Franz ralio, 42!Wilks, J., 219
Woods, A. D. B., 216Work, 227, 240
chemical, 251constant pressure, 245
constant temperature, 245
definition, 227irreversible,242
isobaric, 245
isothermal, 245
magnetic, 252path dependence, 240
Zemanskv, M. W., 279,45Zerolh law, 48
Zipper problem, 85Zucca, R., 373
ZJ6'\"Y O Ga O
Unit conversions
Energy
I J s lO'erg
leal B 4.1S4JI cV = 1.60219 x IO'\"J = 1.60219x l<r'2crg = 23.061kcal mol'
1 kWh = 3.6 x 10'J
1 BTU a 1055J
Power
I W = 1 Js\021 s 1 lO'trgs\0211 hp
- 746 W = 550 ft lbs\021
Pressure
1 Pa = 1 N m~2 s O.Oi mbar s; 10\023 bar =s 10 dyn cm\022
= 7.501 x 10\"JmmHgorlorr
i bar a 10'dyn cm\021 \342\200\242=10'Nm\022 ~ 750 mm Hg
1 raraHgsa 1 iorr= 133.3Nm\023=1333dyn cm\023
1 aim b 760 mm Hg = 1.013x 10sNm\021
= 1,013 x 10s dyn cm\022 = 1.013 bar
\342\200\242Al 0'C where ihs accelerationof gravity has Ihc standard value 9.8066S in s\021.
Table of Values
Quantity
Velocity of lightProton cbarge
Planck's constant
Avogadro's number
Alomic mass unit
Electron rest mass
Proton resl massProton mass/electron mass
Reciprocal fine structureconsiani hefe*
Eieclron radius e1/mc2Electron Compton
wavelength hjmc
Bohr radius h^/me1Bohr raagnelonehjlmc
Rydberg constant meij2hl
I eieclron volt
Boitzmann consianiPermittivity of free space
Permeability of freespaceMolar gas constant NkaMolar volume idealgas, at
TQ= 273-15 K,
po = 10132SNm\"I= lali
Symbol
c
e
h
H = hj2n
N
amumM
l/\302\253
r.
K
r0
1'*
R^ or Ry
eV
eV/ft
eV/?,c
eVA-fl
k.
e0\342\226\240
fo
R
Tl
Value
2.997925
J. 602194.803256.626181.05459
6.02205 x !0IJmor
1,66057
9JO9531-67265
1836-2
137.036
2.817943.86159
5.29177
9.27408
2.17991
13.6058 eV
1.602192.41797x l0uHz8.06548
1.16045 x 10* K
1.38066
\342\200\224
8.31441
22.41383
I0l0ci__\342\226\240
[Q-1Q
io-1T
10\"iV
. 10\"\"}0~2i10\"*\"
\342\200\224
_
10\021;
10\"\0211
10\"9
10\021
10\"'
10\"'
l0*c;\342\200\224
10\"\"'
I
1
107c
103c
CGS
T1S~1'
esuergsergs
g
S
g
'cmi
cm
cm
lergG~'
'erg
*erg
sergK-'
St
108ms-'
IQ-l*Js1O-\"JS
10~27 kg
10-\"ks
lo-\"kg\342\200\224
_
10-** m10\"IJm
10\"\"m10\0234JT*110~18J
10\"J\342\200\224
iCl'm-'
\342\200\224
|0-..,ri
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Source: E. R. Cohen and B. N. Taylor, Journal of Physical and Chemical ReferenceData 2D),663 A973).
