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250
Heat
and
Thermodynamics
(6)
L\\
=
KA/Ax, where A'
is
the
thermal
conductivity
and A
and,
Ax
arc
the
area
and length, respectively, of the wire.
(c)
L12
=
T/R', where
A is the
electric resistance
of
the
wire.
9-25 Show
that,
in the
case of
irreversible
coupled flows
of heat
and
electricity,
(b)
(0
d
\
dr/iT
=
2/
and
d AT
rf)
=2,,.
Show that, with
AT fixed,
the equilibrium
state obtained
when
1=0 involves
a
minimum rate of
entropy
production.
(d)
Show that,
with
AS fixed, the equilibrium
state obtained when
I
s
=
involves
a
minimum rate of entropy production.
9-26
Three
identical finite
bodies of constant
heat capacity
are
at
temperatures
300, 300,
and 100°K.
If
no work
or
heat
is
supplied from
the
outside,
what is
the highest temperature
to which
any one of
the
bodies
can
be
raised by
the
operation of
heat engines
or
refrigerators?
10.
STATISTICAL
MECHANICS
10-1
Fundamental
Principles
In the
treatment of kinetic theory
given in
Chap.
6,
the molecules
of an
ideal
gas
could not be regarded as
completely
independent of one
another,
for
then they
could
not arrive at an
equilibrium
distribution of
velocities.
It was
therefore
assumed that
interaction did
take
place, but
only during
collisions
with other molecules
and
with
the
walls.
To
describe
this
limited
form of interaction we refer to
the molecules
as weakly
interacting
or
quasi-independent. The
treatment
of
strongly interacting
particles
is
beyond
the
scope
of
the
present
discussion.
The
molecules
of
an
ideal
gas
have another characteristic
besides
their
quasi-independence.
They
are
indistinguishable,
because they are
not localized
in
space. It
was
emphasized
in
Chap.
6
that the
molecules
have neither a
preferred location nor
a
preferred
velocity.
The particles occupying
regular
lattice sites in a crystal
are
distinguishable,
however,
because they are
con-
strained to
oscillate
about fixed
positions;
therefore one
particle
can be dis-
tinguished
from
its
neighbors
by its
location. The
statistical
treatment
of
an
ideal crystal as a
number
of
distinguishable,
quasi-independent
particles
will
be given
in
the
next
chapter.
In
this
chapter
we
shall
confine
our
atten-
tion to
the
indistinguishable,
quasi-independent particles
of
an ideal gas.
Suppose
that a monatomic
ideal
gas
consists of N particles,
where
N,
as
usual,
is
an
enormous
number
—say, about 10
20
. Let
the gas
be
contained in
a
cubical enclosure whose
edge has a length
L, and
let
the
energy
e
of any
particle, as
a
first step,
be entirely kinetic
energy of translation.
In the x
direction,
(mxY
m
J|
2m
=
i
2
*mx
=
where
p
x
is the x component of the
momentum.
If
the
particle is
assumed
to
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252
Ileal
and
Thermodynamics
10-1
Statistical Mechanics 253
move
freely
back and
forth
between
two
planes a distance
/,
apart,
the
sim-
plest
form of
quantum mechanics
provides that, in
a
complete cycle
(from
one
wall to the
other
and
back
again, or a total distance of
2L), the
constant
momentum
p
x
multiplied by
the total path 2L is an integer
n
x
times
Planck's
constant h.
Thus,
p
x
2L
=
n
x
h.
Substituting
this
result into the previous
equation,
we
get
,
A
2
n
x
—
j
v8me
x
.
The allowed
values
of
kinetic energy t
x
are discrete, corresponding
to
integer
values of
n
x
\
but when n
x
changes
by
unity, the corresponding change
in
e
x
is
very
small,
because
n
x
itself is exceedingly large.
To
see
that a
typical value
of n
x
is
very large,
consider a cubical box containing
gaseous helium at
300°K,
whose
edge is,
say,
10
cm.
It
was shown
in Chap.
6
that the
average
energy
associated
with each
translational
degree
of
freedom
is
\kT.
Then,
e*
=
UT<=
i
X
1.4
X
10- |3I
x
300 deg
-
deg
and
n,
=
=
2.1
X
10-'
erg,
10 cm
6.6
X
10-
erg
s
10
X
10.5
X
10~
19
6.6
X
10-*'
—
s/8
X
6.6
X
10-
g
X
2.1
X
10
H
erg
«
10
9
.
Therefore,
the
change
of
energy
when
nx
changes
by
unity
is
so
small
that,
for
most
practical purposes,
the energy may
be
assumed
to vary
continuously.
This
will
be of advantage later
on, when it
will be
found useful to replace a
sum
by
an
integral.
Taking
into account the three components
of
momentum,
we
get for
the
total
kinetic
energy
of a
particle
=
Pi
+
Pi
+
Pi
_
(10-1)
2m
a«tf
w
+ S+<»-
The
specification
of
an integer
for each n
x
,
n„,
and n
z
is
a specification
of a
quantum
state
of a particle. All states characterized
by
values
of
the ri's such
that n
x
+
»„
+
nz
=
const,
will
have
the
same
energy.
To
use
an example
given
by
Guggenheim, the
states
corresponding to the
values of
n
x ,
n
y
,
and
n
z
in
Table
10-1
all have
the energy
e
=
66A
2
,
8mL
2
. There
are
twelve
quan-
Table
10-1
ni +
n;
+
n;
=
66
1
2
3
4
5
6
7
8 9
10
11 12
n
x
8
1 1
7
1 4
7 4 1
5 5
4
y
1
8
1
4
7
1 1
7
4
5
4
5
>h
1 1
8
1
4 7
4
1 7 4
5 5
turn states
associated with the same
energy
level,
and we
therefore refer to
this
energy level
as
having
a
degeneracy
of
12.
In
any
actual
case, n\
+
n
y
+
n\
is
an
enormous
number,
so
that
the degeneracy
of an actual energy
level
is
extremely
large.
However
close
they
may
be,
there
is still
only
a
discrete
number
of
energy
levels
for
the
molecules
of an ideal
gas.
It
is the fundamental
problem
of
statistical
mechanics
to determine, at
equilibrium,
the
populations
of
these
energy levels—
that
is,
the
number
of
particles
Ni
having
energy
ei,
the
number
A
;
2
having energy
e
s,
and
so on. It is
a
simple
matter
to show that
(see Prob.
10-2) the
number
of quantum states
g
;
corresponding
to
an energy
level
i
(the degeneracy of
the
level)
is very
much
larger than
the
number
of
particles
occupying
that level. Thus,
Mi
»
N
u
(10-2)
It
is very
unlikely,
therefore,
that
more
than
one particle
will
occupy the
same
quantum
state at
any
one
time.
At
any one moment,
some
particles
are
moving
rapidly
and some slowly,
so that
the
particles
arc distributed
among
a
large
number ofdifferent quan-
tum states.
As
time
goes
on, the
particles collide
with
one another and
with
the
walls,
or
emit
and
absorb
photons,
so that
each
particle undergoes many
changes
from
one
quantum state
to
another.
The fundamental
assumption of
statistical
mechanics
is that
all quantum
stales have equal likelihood
of
being
occupied. The probability
that
a
particle
may
find
itself in
a
given
quantum
state
is
the same
for all
states.
t
E.
A. Guggenheim,
Boltzmann's
Distribution
Law,
Interscience
Publishers,
Inc.,
New York,
1955.
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254
I It-at
and
Thermodynamics
1
2
3
4
5
6
7
8
9 10 i 1
12 13 14 15 16
A
B
c
'
A
C
B
B
A
C
B
C
A
C
A
B
C
B
A
Fig.
10-1
There
are
six
ways in
which
three,
distinguishable
particles
A,
B, and C can
occupy
three given
quantum
slates.
Now
consider the
N
f
particles
in any of
the
g,
quantum states
associated
with the energy
«,-.
Any
one particle
would
have
$
choices
in
occupying
g<
different
quantum
states.
A
second particle
would have
the
same
gt
choices,
and
so
on. The
total
number of
ways
in which A',-
distinguishable
particles
could
be
distributed
among
g<
quantum
states would
therefore bcg
t
•'.
But the
quan-
tity
g*
1
'
is
much
too
large,
since
it
holds
for
distinguishable
particles
such
as
A, B,
and C in Fig.
10-1. This
figure
shows
six
different
ways in
which three
distinguishable
particles
can occupy
quantum
states
2, 7,
and
13.
If the
particles
had no
identity,
there would
be
only one
way to occupy
these
particular quantum states.
That
is,
one must
divide
by
6,
which
is
3
The
number
of
permutations of
A ,
distinguishable
objects is
A'; If
the
quantity
gfi
is
divided
by
this factor, the
resulting
expression will
then hold
for
indis-
tinguishable
particles. Therefore,
No.
of
ways
that
Ni
indistinguishable
}
particles
can
be
distributed
among
g.
quantum
states
Nil
(10-3)
\
It
should be pointed out
that
the
A
indistinguishable,
quasi-independent
particles were
assumed
to be
contained
within
a
cubical
box
only for
the
sake of
simplicity.
A
rectangular box
with
three
different
dimensions
could
easily
have been
chosen,
in
which case
Eq. (10-3)
would
be
unchanged.
10-2
Equilibrium Distribution
We
have seen that,
in the
case of
an ideal gas, there
are many
quantum
states
corresponding to the same energy
level
and
that
the
degeneracy
of
each
level
is
much larger than the
number of particles
which
would be
found
in
10-2
Statistical
Mechanics
255
any
one level
at any one
time. The
specification,
at any one
moment,
that
there are
A'i
particles
in energy
level
ei
with degeneracy
gi
Ni
particles
in
energy
level
et
with
degeneracy
£2
Ni
particles
in
energy
level
e,
with
degeneracy
gi
in
a
container of
volume V
when
the
gas
has
a
total
number
of
particles
N
and
an energy
V
is
a
description of
a macrostate
of the gas. The number of
ways Q
in
which
this macrostate
may
be
achieved is
given
by
a
product
of
terms of
the
type
of Eq.
(10-3),
or
=
Nx'.
N
t
l
(10-4)
The quantity
V.
is
called the thermodynamic
probability of
the particular macro-
state.
Other
names for this
quantity are the number
of
microstates
and the num-
ber
of
complexions. Whatever
its
name, the larger <i is, the greater
the proba-
bility of finding the
system
of N
particles in this state.
It is
assumed that, if
V,
N,
and
U
are
kept constant, the equilibrium state
of
the
gas
will
correspond to
that macrostate in
which Q
is
a maximum.
To
find
the
equilibrium
populations
of
the
energy levels, therefore,
we look
for the
values of
the
individual
A's that
render
52
—or
more simply, In fi—
a maximum.
Since
In
contains
factorials
of
large
numbers,
it is
convenient
to
use
Stirling's
approximation,
which
may
be derived
in the
following
way: the
natural
logarithm
of
factorial
x
is
In (x )
=
In
2
+
In 3
+
In x.
If
we
draw
a series
of
steps
on a
diagram,
as shown
in Fig.
10-2, where the
integers arc plotted
along
the x
axis and In
x
along thc^
axis, the
area
under
each step is
exactly
equal
to
the natural logarithm,
since the
width of
each
step equals unity.
The
area
under
the
steps from
x
=
1 to x
=
x
is
therefore
In
(xl).
When
x
is
large,
we may replace
the
steps by
a
smooth
curve,
shown
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256
Heat
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Thermodynamics
In
x
In
7
In 6
In 5
In 4
In
3
In
2
In
1
8
X
Fig.
10-2
The
area under
the dashed
curve
approximates
the area
under
the
steps
{In
2
+
In
3
+
In
4
+
•
•
+
In x)
when
x
is
large.
as
a
dashed
curve
in Fig.
10-2;
therefore,
approximately,
when
x
is
large,
In (xl)
~
/ In
x dx.
Integrating by
parts,
we get
In (.v )
~
x
In
x
—
x
+
1.
Then,
if we neglect 1
compared with
x,
In
(x\)
«
x
In x
—
x.
(10-5)
This
formula
is
Stirling's approximation.
Using
Stirling's approximation in
Eq.
(10-4),
we
get
In 9.
=
A'i
In
g
x
—
Ni
In h\
4-
A'i
+
Nt
In
g
2
—
A
2
In N
2
=
V
Ni
In
gi
-
Y A ,- In
A',-
-•-
V
.V,-,
or
In Q
=
y
A',- In
#
+
A',
Zw
A;
+
#8
+
(10-6)
10-2
Statistical Mechanics
257
where
we
have
used the fact
that
2A'f
=
N.
Our problem now
is to render
In V. a
maximum
subject
to
the
conditions
that
2A
r
,-
=
A*
=
const.,
SA',«,-
=
II
=
const.
(10-7)
(10-8)
Before we proceed
to
solve this
problem
by
the
method
of
Lagrange
multi-
pliers, it is important
to
bear
in
mind that the
e's
and
g's are
constants.
The
only variables are the populations
of the
energy levels,
and their
sum A'
is
constant.
Since dN
=
0,
the
differential
of In Q
is
d In
Q
L>
=
y
d
Niln
iL\
=
N
t
I
B|^i.
(10-9)
Setting
the
differential
of
In U equal to zero
and taking the
differential of
Eqs.
(10-7) and
(10-8),
we
get
gi
g-i
gi
In
|f
dNi
+
In
U-
dN
s
+••••+
hi
#£
Mi
+
=
0,
A
i
JV
2
A,-
dNx
+
dNi
+
•
+
dNi
+
•
=
0,
and
«i
dNi
+
e
2
dN
t
+••+
u
dNi
+
=
0.
Multiplying
the
second
equation
by In A and
the
third by
—
/?,
where
In A
and
—0
arc
Lagrange
multipliers
(sec Art.
6-3), we get
In
&-
dNi
+
In
|r
«f#s
+
Ai
A'2
In
A
dNi
+
In
A
dN2
+
and —
/3<i dN%
—
fie* «W»
—
+
In
§
dN
t +
A;
+
In
A
dN +
-
0e;
dN
-
=
o,
=
o,
=
0.
If
we add these
equations,
the coefficient
of
each
dN may be
set equal
to zero.
Taking
the
ith
term,
In
4
?-
+
In
A
-
0a
=
0.
Ni
or In^i-
ln
A
=
-0
%
N
t
= Agie-e'<.
(10-10)
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Heat
and Thermodynamics
The
population
of
any
energy
level
at equilibrium
is therefore
seen to
be
proportional to
the
degeneracy
of the
level and to vary
exponentially
with
the energy
of
the level.
The
next step
is
to determine the physical
significance of
the Lagrange
multipliers A
and
/?.
1
0-3
Significance of
A
and
j3
The
population A , of the z'th
energy
level
is given
by
Ni
=
Agie-f':
Summing over all the energy
levels,
we
get
and
N
A
=
H*
r»
(10-11)
The sum in
the denominator
plays a fundamental
role
in statistical mechanics.
It
was
first
introduced by
Boltzmann,
who called
it the Zuslandsumme,
or
sum
over
states. We retain
the
first
letter
of
Zuslandsumme
as
a
mathematical
symbol,
but the
accepted
English
expression
for
this
sum
is
the
partition
junction.
Thus.
?»^nr*i
(10-12)
and
-f-
(10-13)
Substituting
this result
into
Eq.
(10-10),
we
get
Ni = A'
,gm
(10-14)
It will
be
shown
later
that
Z
is
proportional
to the
volume of the
container.
Since
the properties
of a gas
depend
on temperature
as
well as
on
volume,
one
would
expect
a relation
between
/3
and the temperature.
To introduce
the
concept
of
temperature
into
statistical
mechanics,
we
must
go back to
the
fundamental
idea
of
thermal equilibrium
between
two
systems,
just
like
the
procedure
in
Chap.
8 for
relating
the
quantity X
to
temperature.
Conse-
quently,
let
us consider
an
isolated composite
system
consisting
of
two samples
of
ideal
gas
separated
by
a
diathermic wall,
as shown in
Fig.
10-3.
For
the
10-3 Statistical
Mechanics 259
1
N
t
,N
s
Mt
SAT,
=
N
>
A
A
=
11
«J
*}i
.
. .
A
A A
N„
»„...
Nj,
. . .
A A
Wj
=
iV
//-/V/y/
W///Y/////y///z/^^^^^
^
Fig. 1 0-3
An
isolated
composite
system
of
two
samples of
ideal
gas
separated
by
a
diathermic
wall.
The
total
energy
is constant.
second sample of
gas, the
symbols
expressing
energy levels,
populations,
etc.,
are
distinguished
with
a circumflex.
The
thermodynamic
probability
of
the
composite
system
Q is
the product
of
the separate
thermodynamic
proba-
bilities,
so
that
the
logarithm is
In ii
=
Y
N
In
%
+
N
+ y^
y
In
Jf
+
N.
U Ni
L,
fy
Each sample
has
a
constant number
of molecules,
so
that
and
2A
7
,- =
A'
=
const.
2A'j
=
Jv
=
const.,
but
the
energy of
each
sample
is not
constant.
Only
the
total energy
of
the
composite
system
is constant;
thus,
=
U
=
const.
JV**
+
^%
=
v
Vo
find
equilibrium
conditions,
we proceed as
before
and
get
In
A
y
dNi
=
In
A
V dNj
=
-££«4ft-lT%d$-
0.
Adding
and
setting
each
coefficient
of dN
and
dN
equal to zero, we
get two
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260
Hrat
and
Thermodynamics
sets
of
equations,
and
Nj
=
Ag,-e-^,
where
all
quantities
are
different, except
0.
When two systems
separated
by
a
diathermic
wall
come
to
equilibrium, the
temperatures
are
the same
and
the
0's
are
the
same.
The
conclusion
that
is
connected
with
the
temperature
is
inescapable.
It
was
shown in
Chap.
9
that the
entropy
of an isolated
system increases
when the
system undergoes
a
spontaneous, irreversible
process. At
the con-
clusion
of
such
a
process,
when
equilibrium
is
reached,
the
entropy has the
maximum value consistent
with
its
energy and volume.
The thermodynamic
probability also increases and
approaches
a
maximum
as equilibrium is
approached.
We therefore look for some correlation between
.9
and £2. Con-
sider two
similar
systems
A and B in
thermal contact
—one with
entropy
Sa
and
thermodynamic
probability
il
A
,
the other
with
values
Sb
and
Ob-
Since entropy
is an
extensive variable, the total entropy
of the
composite
system is
S
=
Sa
+
Ss,
The
thermodynamic probability,
however,
is the product, or
£2
=
Q
A
Q
B
.
If we
let
then.
s
=
m),
f(QjiB
)
=/(a
A
)
+f(ih<).
The
only function that satisfies this
relation is the
logarithm. Introducing
an
arbitrary
constant k', we may
write
S
=
k' In S2
(10-15)
for
the
relation
between
entropy and
thermodynamic
probability.
The
first law
of thermodynamics applied to any
infinitesimal process
of
any
hydrostatic
system is
dQ
=
dl
+
P
dV.
If
the
process takes
place
between two neighboring
equilibrium states,
it may
be
performed
reversibly,
in
which case dQ
=
T
dS,
and
dU
=
T dS
-
P
dV.
10-4
Statistical
Mechanics
261
If we now
specify that
the reversible
process take place at constant I
,
we
have
the
important
link
between
thermodynamics and
statistical
mechanics:
1
= (?£
T
\dU
(10-16)
Since
both
S
and
U
may
be
calculated
by
statistical
mechanics,
the
derivative
(dS/(ll/)v gives
the reciprocal
of
the
Kelvin
temperature. This
is
the way
in which
the
macroscopic
concept
of
temperature is injected
into
statistical mechanics.
In
employing the
Lagrange
method to find the equilibrium values
of
the
energy
level
populations
[Eqs.
(10-9) and
(10-10)],
wc found that
and
Therefore,
'
In il
=
y
In
-0-
dN,:
In
|i
=
/3<e,-
-
In
A.
'
In
Q
=
7
/3e,:
dNi
-
In A^ dN
t
where
U
is the total energy
of the
system. Therefore,
d
In
2
1
d
0-
dU
=
k>du
k
'
lnn
=
k' \dUj,
Since
(dS/dlf)v
=
V^'>
we
S
ct
tnc beautiful
result
fi
~
Yt
(10-17)
W
;
hen
the
actual
values
of
the
e's
appropriate to an ideal
gas
are
introduced,
it will
be
seen
that k'
is
none other
than
Boltzmann's
constant
k.
10-4
Partition Function
Wc have
seen that
the population
jV,- of
the ;'th energy level is
Ni
=
Agie-H'i.
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262
Heat
and
Thermodynamics
Substituting
\/k'T
for
/3
and
N/Z for
A, we
get
N
/V- =
—
(T
p->ilVT
vhere
Z
=
2^-'<'*T
(10-18)
(10-19)
The
partition
function Z contains
the heart of the statistical
information
about
the
particles
of
the
system,
so
that it is
worth
while to express other
properties
of the
system,
such as U, S,
and P, in terms ofZ. If
wc
differentiate
Z
with
respect to T,
holding V
constant,
wc get
(&-Z-
fri™ *-*
=
_- . V
f-a-p-'tili'T
=
It
z,
€(
'
v
''
ze/
/V/fcT
2
It follows that
V-«T>( jA.
(10-20)
and
E7
may
be
calculated
once
In
Z
is
known as a function
of
T and V. Also,
,9
=
k'
In
0,
where, according
to
Eq.
(10-6),
Hence,
.9
=
-£'
Y
-V,' In
—
+
k'N.
U
gi
Substituting
for
/Vj/gs
the value given in
Eq.
(10-18),
we
get
10-5
Statistical
Mechanics 263
and,
finally,
7 rr
a
=
Nk
In
N
+
r
-
Nk',
(10-21)
which
provides
us with
a
method of
calculating
.S
once In
Z
is known.
One more
equation
that is of
value is
the relation between
pressure and
the partition function.
Since
TdS
=
dU
+
PdV,
~ Apr-H)
From
the relation
between
S
and the partition function,
given in
Eq.
(10-21),
we
get
U
-
TS
=
-Nk'T In
4
-
Nk'T;
TV
therefore,
P
=
Nk'T
d
In z\
(10-22)
so
that
again
the
pressure may
be
calculated
once In
Z is known
as
a
function
of T and I '.
This
is the advantage
of statistical
mechanics. It
provides
us
with
a
simple
set of
rules
for
obtaining
the
properties
of a
system:
10-5
Use
quantum
mechanics
to
find the
e values of the quantum states.
Find
the partition function Z
in
terms of T and V.
Calculate
the
energy by differentiating In Z with respect to
7 .
Calculate
the
pressure
by
differentiating
In
Z
with
respect
to
V.
Calculate
the
entropy
from Z and
U.
Partition
Function of an Ideal Monatomic
Gas
To
apply
the rules laid down in the preceding article to
an
ideal gas, wc
must first calculate
the appropriate
partition function.
This
was defined
to be
levels
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264
Heal
and
Thermodynamics
where
the
summation was
over all the energy levels.
Exactly
the
same
result
is
obtained
if we sum the
expression
over all
quantum states.
We at
first take
into
account
only
the
kinetic
energy
of
translation
of
the particles
confined to
a rectangular box whose
x,
y,
and
z
sides
are,
respectively,
a,
l>, and c. The energy
of
any
quantum
state; is
given
by Eq.
(10-1)
as
*'
8m
\a*
+
b*-
+
?)'
where n
t
,
n,„
and
n
z
arc quantum
numbers specifying
the various
quantum
states.
The partition function is therefore
a
threefold sum: thus,
Z
=
V V V
gHh*t«mk
f
THnJt<l.*+n,
,
l&*+n
t
*{c
i
)
71x=
1
«y=
1 tli
=
1
or
2
=
y
('
C
|
.'8»'*'n( ^/ )
y
e
-(h
,
ISmk'T)(n,
,
H
t
)
V
e
-(li
,
;S»a-'r)(ri,
I
..'c
>_
Since
the
values of
n
r
,
n,„
and n
z
that give
rise to appreciable
values
of
the
energy
are
very
large and
since
a change of
n
x
or
n
y
or
n.
by unity produces
a change
of
energy that is exceedingly small, no
error is introduced by
replacing
each sum with an integral and
by
writing
z
=
r
f
x
r
tfis.*'rit,,w
d
„
i
r
/*
r
a>n«w)(«,w
rf
„
i
[ J°°
g-oto**?nwt*>
&t\
Each
integral
is
of the
type
listed in
Table
6-2:
f
g-«*
*
«
i„
Therefore,
Z
=
and
since
abc
=
V,
2\a
<?
ISirrnk'T
2
\
A
1
4
l&irm/c'T
2\
A
2
8jrm*'r
.2 \
A
Z=
F
2wmk'
T\i
h
2
(10-23)
10-5
Statistical
Mechanics 265
and In Z
=
In
f
In r
fin
(
2Trmk'\
(10-24)
Pressure
of
an
ideal monalornic
gas
P
=
Nk'T
dlnZ\
bV
)
T
-««(;)
=
_
i- T
(10-25)
Comparing
this
result with
the
expression
for
the
pressure
P
=
NkT/V
obtained
with
the kinetic
theory
of
gases, given in
Eq.
(6-10),
where
k is
Boltzmann's
constant, we
sec that the arbitrary constant
k' introduced
in
the
equation
S
=
k' In
Q,
is
none other than the
Boltzmann
constant,
or
k'
=
k
=
R
(10-26)
2
Energy
of
an ideal monatomic
gas
U
=
NkT
2
(
d
In
Z\
\
w
h
=
INkT.
(10-27)
This
is
exactly
the same
result
which was
obtained by the kinetic
theory
of
gases for
a
monatomic ideal gas and
shows
that,
when particles each
having
three translational
degrees of
freedom
come
to
statistical
equilibrium, the
energy
per
particle equals |-
k
T.
3 Entropy
of
an
ideal
monatomic
gas
^
=
Nk In
£
+
¥
+
m
In
Z
+ |
la
T
In
(2irmk
+
$M
+ Nk
=
Nk
||
In T
*5
A
. /2armk\i
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266
Heat
and Thermodynamics
If
we
take
1 mole of gas, N
=
A
T
A
and A'\k
=
R.
Therefore,
s
=
c
v
In
T + R
In v
-
R
In
^
2lrm
'
:
/
h
^
i
+
s
R
(10-28)
This
expression
is to
he
compared with
Eq.
(9-5),
namely,
s
=
cv
In
T
-\-
R\n
v
-\-
s
u
,
and
we
sec
that not
only
were we able
to
arrive at
this equation
by
the
methods of statistical mechanics
but also we
were
able
to calculate
the
con-
stant
s<,.
Equation
(10-28),
which
was
first
obtained
by
Sackur
and
Tetrode,
usually
bears
their
names.
10-6
Equipartition
of
Energy
Both
kinetic
theory and
statistical
mechanics,
when applied
to the mole-
cules of an ideal gas
(each
having three
translational
degrees
of
freedom),
yield
the
result
that
at
equilibrium the
energy
per
particle
associated
with
each degree
of translational
freedom
is
%kT.
The methods
of
kinetic
theory
could not
be applied
to
rotational
and
vibrational
degrees
of
freedom,
but
the simple
statistical
method
just
developed is
capable
of
dealing
with all
types of
molecular
energy,
not
just translational
kinetic
energy.
The
property
of
the
partition
function
which makes it
so useful
is
that,
whenever
the energy
of
a molecule is
expressed as a
sum of independent
terms
each referring
to
a
different
degree
of
freedom,
then.
6
=
t + e + t
+•;
Z
= \
e
~'
lkT
=
V
£-(«'+« +« '+•
••)/*r
i-i
Li
=
y
e
-'''
kT
y
e-< i
kT
y
r****
=
Z'Z Z'
(10-29;
If the
various
types of energy are calculated
with
classical
physics,
it is
a
simple
matter to
derive
the
classical principle
of
the
equipartition
of
energy.
We
take
Eq.
(10-20),
namely,
U
=
NkT*
d
In Z
dT
10-6
Statistical
Mechanics 267
and
rewrite
it thus:
l\
-
U
-
din Z
W
N
d(\/kT)
<«>
=
d
In Z
dp
(10-30)
Suppose e
to
consist
of terms
representing
translational
kinetic
energy
of
the
type
fymw*,
those representing
rotational kinetic energy of
the
type
i/co
2
,
those
representing vibrational
energy
gJKj
8
+ $&£*,
etc. All these
forms
of
energy
are
expressed
as squared terms of the
type
bip\.
Let there
be
/
such
terms, or
e
=
hp\
+ b
2p\
+ +
b,p).
Then,
since
the
partition
function
is
the product
of
the
separate partition
functions,
Z
=
J
*-**«»>
dp
x
J
e-1*** dpi
J
rw dp,.
yt
=
fflpi
and d
yi
=
& dpf,
J*
f+rt
dp,
=
p
g-»ji.
f
ie
d
yi
=
(S-i
J
r*an
dyi
where
Ki
does
not
contain
p.
The partition function now becomes
Let
then
Z
=
0-iKi
/3-»/T,
• •
0-lK
f
,
=
p-z^JCff,
• •
Kf,
where none
of
the
K's
contains
0.
Since
(e)
=
—
3(ln Z)/dp,
to -
-
35
(-
2
ln
+
ln
Kl
+
ln
K
*
+
'
'
'
2/3'
and
since
/?
=
1/kT,
<<>
=
{hT.
In K
(10-31)
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268 Heat and
Thermodynamics
It
has therefore
been
proved
thai,
when
a
large
number
of
nondistinguishable,
quasi-
independent
particles whose
energy
is
expressed
as the
sum
of
f
squared
terms
come
to
equilibrium,
the
average energy per particle
is
f
times
^k
T.
This
is
the
famous
principle
of
the equipartition
of energy
that was
men-
tioned
but not proved in Art. 6-5.
It was
stated
that the
principle
broke
down
badly
when applied to polyatomic
molecules
which
have
many
vibrational
degrees
of
freedom.
It
is
a
simple
matter
to introduce
quantum
mechanical
expressions
for
the
energy
of rotation
and
vibration
in the
partition
function
and to calculate
the resulting
thermodynamic
properties.
For discussion
of
this
the
student is referred
to any of
the
books
on
statistical
mechanics
listed
in
the appendix.
10-7
Statistical
Interpretation
of
Work and
Heat
We have been considering
the
statistical
equilibrium
of a large
number
N
of nondistinguishable,
quasi-independent
particles
in
a
cubical
container
of
volume
V. The
energy
levels
e,- of
individual
particles
undergoing
trans-
lational
motion
only were given
by
A
2
8mL
I
(»i
+K+
nl).
Since
L
3
=
V,
then
l.
1
=
V',
and letting
B
t
be
the
sum
of the
squares
of the
quantum
numbers
appropriate
to
the
z'th energy
level,
we get
h
2
U-£-
BiV-i.
8m
Given
the
set
of
quantum
numbers
that
determines
B
f,
we
may
say that
the
corresponding
energy
e
t
depends
on volume
only. Taking
the logarithm
of
e,-,
It
2
In
a
=
In
—
+
In
B
t
-
£
In
V.
8m
3
The
effect
of a small
change
of
V
on e is given
by taking
the
differential
of
this
equation;
whence
dti
=
2dV
ti
3
V'
Therefore,
and
Kde^-^dV,
lN
idu=-\pV.
(10-32)
10-8
Statistical
Mechanics 269
Now, it
has
been
shown both
by
kinetic
theory and
by statistical mechanics
that
the pressure of
an
ideal
gas
is given by
P
=
NkT
Since
the
energy
per particle
is
translational kinetic
energy only, with
three
degrees
of
freedom,
U
=
$MsT.
It
follows that
p m
2
U
Substituting this result into Eq.
(10-32),
we get
S»<fe
=
-PdV.
(10-33)
(10-34)
A
change
of volume,
therefore,
causes
changes in
the
energy values
of
the
energy
levels,
without producing changes in the
populations of
the
levels.
When
the
JV;
change
and
the
u
remain
constant,
we
have from page
261,
d
In it
=
S«
;
dNi.
Since
kd In
=
dS,
k 2e,- dNi
=
dS,
and
setting
1:0
equal
to
l/T, we
get
finally,
2
£i
dNi
=
T
dS.
We
see
that a
reversible
heat transfer
produces changes in
the
populations
of the
energy levels without
changes in
the energy values
of
the
levels them-
selves. Thus, the equation
dU
=
2e,- dN
{
-\-
2JV<
de,- expresses
the first
and
sec-
ond
laws
of
thermodynamics,
with
2e,
dNi
= T
dS and
2A';
de{
=
—PdV.
10-8
Disorder,
Entropy, and Information
Whenever work
or kinetic energy
is
dissipated
within
a
system because
of
friction,
viscosity,
inelasticity,
electric
resistance,
or
magnetic
hysteresis,
the
disorderly
motions
of
molecules
arc
increased.
Whenever
different substances
arc
mixed
or
dissolved
or
diffused with one
another,
the
spatial
positions
of
the
molecules
constitute
a
more disorderly arrangement.
Rocks
crumble,
iron
rusts,
some
metals
corrode,
wood rots,
leather
disintegrates,
paint
peels, and
people
age. All
these
processes
involve
the
transition
from
some sort
of
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270
Heat
and
Thermodynamics
orderliness to
a
greater disorder. This
transition is
expressed
in the
language
of
classical
thermodynamics
by
the statement
that
the entropy
of
the
universe
increases. Molecular
disorder
and
entropy
go
together, and
if
we
measure disorder by the number of
ways a particular
macrostate
may
be
achieved, the
thermodynamic
probability S2
is a measure
of disorder.
Then
the
equation
$
=
k
In
U is the simple relation
between
entropy and
disorder.
The number
of
ways
in
which
a
particular
macrostate
may
be
achieved
can be
given another
interpretation.
Suppose
that
you
are called upon
to
guess
a
person's
first
name.
The number
of
choices of
names
of
men and
women is staggeringly large.
With
no hint
or
clue,
the number
of
ways
in
which
one can arrive at
a
name
is very
large,
and
the
information
at one's
disposal
is
small.
Suppose,
now,
that
we are
told
the
person
is
a
man. Imme-
diately the number
of choices of
names
is reduced,
whereas
the information is
increased. Information
is increased
further if
we
arc
told that
the man's name
starts
with
H,
for then
the number
of choices
(or ways
of
picking
a
man's
name) is
reduced
very greatly.
It is
clear
that
the
fewer
the number of
ways
a particular situation or a
particular state
of
a
system
may
be
achieved,
the
greater is the
information.
A
convenient
measure
of
the
information
conveyed
when
the
number
of
choices is reduced from
Q
n
to il y i s given
by
r ,
,
^o
/
=
tln
n7
The
bigger
the
reduction, the bigger
the information.
Since
k
In
9. is the
entropy
S,
then
I
-
&
-
S
u
or
Si
=
S -I,
which
can be
interpreted
to mean that the
entropy
of a
system is reduced
by
the
amount
of
information
about the
state
of a
system.
In
the words
of
Biillouin,
Entropy
measures
the
lack
of
information
about
the
exact
state
of
a
system.
As
an
example of
the
connection
between
entropy and
information,
con-
sider
the isothermal
compression
of
an ideal
gas
(N molecules)
from a volume
''ci
to
a
volume
Vj,
We
know
that
the
reduction
of entropy is
equal to
&
-
Si
m
Nk In
&
But,
when
we decrease
the
volume
of the
gas,
we
decrease the number
of
10-8
Statistical
Mechanics 271
ways
of
achieving this state,
because
there arc
fewer microstatcs
with
position
coordinates
in the
smaller volume.
Before the
compression, each molecule
is
known to
be in
the volume
V
u
. The number
of locations each molecule
could occupy is
Vo/AV,
where AV is some
arbitrary small
volume. After
the
compression,
each
molecule is to be
found
in volume
V\,
with a
smaller
number of possible
locations
Fj/AF.
It follows that
,
0,
V
/AV
,
V
I
=
kln-
r
kln
w
^
=
kln
Yi
,
and
for
the
entire gas of N
molecules,
I
=
Nk
In
\y
,
v
\
in
agreement
with the
result of classical
thermodynamics.
The increase of
information
as
a
result of the compression is
seen
to be identical with
the
corresponding
entropy
reduction.
The connection
between
entropy
and information can be applied
to
the
problem
of
MaxwelPs
demon.
Maxwell
imagined a
small creature
stationed
near
a
trap door separating two
compartments of
a vessel containing
a
gas.
Suppose
that the demon
opened the trap
door only
when
fast
molecules
approached,
thereby
allowing the fast
molecules
to
collect
in
one
compart-
ment and slow ones in the
other. This
would obviously result
in
a
transition
from disorder
to
order
—
thus
violating the second
law. According
to Biillouin,
the
demon could not
tell the difference
between
one kind
of
molecule and
another because he and the
molecules are
in an enclosure at a uniform
temperature
and
all
are bathed
in isotropic blackbody radiation. The demon
could not sec
the
individual
atoms. However,
suppose
that we allow
the
demon, according to the analysis of Rodd, to use
a
flashlight
whose radiation
is
not in
equilibrium with the
enclosure.
Then
the
demon can
get
information
about the
molecules
and
thereby
decrease the entropy of the system.
But
other
phenomena
come
into
the
discussion:
(1)
the
filament
of the lamp in
the flashlight undergoes an increase
of entropy;
(2)
a photon scattered by
a
molecule
is
absorbed
by the demon and serves to increase his entropy;
(3)
the action
of the
demon in opening the
trap
door reduces
the
number of
microstatcs available
to
the molecules. (The
entropy
change of the battery
ofthe
flashlight
can
be
ignored.)
If all
these processes
are
taken into
account
and the corresponding entropy changes
are
calculated
from
the
standpoint
of the
increase
or
decrease
of
information,
then
Rodd
was
able to
demon-
strate
that
the total entropy change
is
positive. The second
law is not
violated.
Much
has
been
written
about reversibility and
irreversibility, order
and
8/16/2019 Zemansky Statistical Thermodynamics
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272
Heat
and Thermodynamics
Statistical
Mechanics
273
disorder,
and supposed violations
of the
second
law.
It would
be hard
to
find
anything
in any language
to
compare
with
the
exposition
given in
Feynman's Lectures
on
Physics
(Chap.
46). This
chapter
is
recommended
wholeheartedly
to all students
whether
naive
or
sophisticated
in their
level
of
attainment
—
for
its brilliance,
its depth,
and its human
warmth.
PROBLEMS
(Values
of constants:
k
=
1.38
X
lO
16
erg/deg
and h
=
6.63
X
10'
erg
•
s.)
10-1
A mercury
atom
moves in
a
cubical
box
whose
edge
is
1
m long.
Its kinetic energy
is equal
to the average
kinetic
energy
of a
molecule
of
an
ideal
gas
at
1000°K. If
the
quantum
numbers
n
x ,
n
y
,
and n.
arc
all equal
to n, calculate
n.
10-2
The
quantum
states available
for
gas
molecules
of energy
e
in
a cubical
box
of
length
L correspond
to
integer
values
for
each
n
x,
«„,
and
n
z
according
to
Eq.
(10-1).
In
a
three-dimensional
Euclidean
space with
coordinates
n
x
,
n
v
,
and
n
z,
each
unit
volume will
contain
one
quantum
state.
The
total
number
of
quantum
states
g'
with
energy
less than
«'
is
equal to
the volume
of the
positive octant
of a sphere
of
radius
r
=
Li^mt^ih.
(a)
Show
that
S
3A»
In
a
volume
of
1
cm'
of helium
gas at
300°K and 1
atm pressure,
e' is about
10~
12
erg.
Calculate:
(b)
g'
and
(c) the number
A'
of helium
atoms,
(d)
Show
that
g'
»
N,
10-3
Consider
a function
/
defined
by
the relation
/(0.,fi
;( ) =f(ii
A
) +/(Q ll
).
First
differentiate
partially
with
respect
to
Q,
;
, and
then
with
respect
to il
A
.
Integrate
twice
to show
that
/(12)
=
const.
In
Q
+
const.
10-4
Take
the
expression
for
the
kinetic
energy
of
a
particle in
a
cubical
box and
imagine
a
space
defined
by
the
cartesian
coordinates
n
x
,
n
u
,
and
n
z
.
Note
that
a
single
quantum
state
occupies
unit
volume in
this
space.
(a)
Setting
n
2
= n
x
+
n\
+
n\,
show
that the number
of
quantum
states
in
the small
interval
dn
is
(4tt«
2
dn).
(b)
Prove that
the
number
of
quantum states
dg, in
the
energy interval
dek(2ir//i
i
)V(2m)h
i
d6.
(c)
Show that
the
number
of
ideal-gas
molecules
dN
(
occupying these
quantum states is
given
by
d
^
=
2
Sjm
iie
~'
lkTdi
-
(d) Derive the
Maxwellian
law
of the
distribution
of
speeds,
that
is,
Eq. (6-25).
10-5
In
the
case
of
N
distinguishable
particles,
the
number
of
ways
in
which
a
macrostate
defined by
A i
particles in
gi
quantum states
with
energy
«jj
A
2
particles in
gi
quantum states
with
energy
62;
and so
on, may-
be achieved
is
given
by
Q
=
AM
Si
Sl_>
A
r
i A'
2
• •
when g,- » A
7
,-.
(«)
Using
the
Stirling
approximation, calculate In 9..
(b) Render
In
il
a maximum subject to
the equations
of constraint
2A
r
j
=
N
=
const,
and
2A
r
,e;
=
U
=
const.,
and
explain
why
U
and
P
should
be
the
same
as
for
indistinguishable
particles but
S
should
be
different
10-6
Given
A' indistinguishable, quasi-independent
particles capable
of
existing
in
energy
levels
e%, (2,
• •
,
with
degeneracies
gi,
gz,
. .
.
,
respectively.
In
any given macrostate in which there
are
A'i
particles
in
energy level
ei;
A'2
particles with
energy
etl
and so on,
assume the
thermo-
dynamic
probability
to be given by
the Bose-Einstein expression,
o
=
(gi+Ni)\(g2+N
2
)\
liE
giWiig*m\
Using
Stirling's
approximation
and
the
Lagrange
method,
render
In
f2
U
E
a
maximum subject
to
2A
r
,-
=
A'
=
const, and 2A
r
,e,-
=
U
=
const.,
and
show
that
Si
Nt
=
Ae-?»
-
1
10-7
Given
the same
system
as in Prob. 10-6,
except
that the thermo-
dynamic
probability is.
given by the
Fermi-Dirac expression
Bra
=
gi\g*i
Nil(gt-
N{)W
t
l(g
t
-
N
2
)\
8/16/2019 Zemansky Statistical Thermodynamics
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274
Heat and
Thermodynamics
Using
Stirling's approximation and
the Lagrange
method,
render
In
Qfd
a
maximum
subject to 2iV,-
=
A
r
=
const, and 2A ,e;
=
U
=
const., and
show
that
Ni
=
gi
AeS«
+
1
10-8
Given
a
gaseous
system
of
jV
a
indistinguishable,
weakly
inter-
acting
diatomic
molecules:
(a)
Each molecule may
vibrate
with the same
frequency
v
but
with
an
energy e,-,
given
by
U
=
(k
+
t}hv
(i
=
0,l,2, . .
.).
Show that the vibrational partition
function Z„
is
e
-lh,lkT
z
t
=
1
-
«-*'/«
(b)
Each molecule may
rotate, and
the
rotational partition
function
Z
T
has
the
same
form
as that
for
translation,
except
that
the
volume
V is
replaced by the total solid
angle Air, the
mass is replaced
by
the moment
of
inertia
/,
and the
exponent
§
(referring
to
three
translational
degrees of
freedom) is replaced
by
-§-,
since there
are
only
two rotational degrees
of
freedom. Write
the
rotational partition
function.
(V) Taking
into account translation,
vibration,
and
rotation,
calculate
the Helmholtz
function.
(d) Calculate
the
pressure.
(e)
Calculate the
energy.
(/)
Calculate the
molar
heat capacity
at constant volume.
11.
PURE
SUBSTANCES
11-1
Enthalpy
The
laws
of
thermodynamics were
stated and their consequences
were
developed in a
sufficiently general
manner
to
apply
to systems
of
any
number
of coordinates. When there
are
three
or
more independent coordinates, one
speaks of
isothermal
surfaces and isentropic (adiabatic reversible)
surfaces.
If, as
is
often
the case,
there
are
only two
independent coordinates,
these sur-
faces reduce to
simple
plane curves. The most important
system
of two
inde-
pendent coordinates is a hydrostatic one, consisting of a single
pure substance
of constant
mass.
Once
the
thermodynamic equations arc developed
for
this
system,
we shall see how simple
it
is
to
write down the analogous equations
for any
other
two-coordinate
system.
In
discussing
some
of
the
properties of
gases
in Chap.
4,
the sum
of
U
and
PV
appeared several
times
(see
Probs.
4-8, 4-9,
and
4-11). It
has
been
found
very useful
to define
a
new function
H,
called
the
enthalpy,
1
by
the relation
H
=
U +
PV.
(11-1)
In
order to study the properties of this function, consider
the change
in
enthalpy
that
takes
place when a
system
undergoes an
infinitesimal
process
from
an
initial
equilibrium
state
to a final
equilibrium
state. We have
dH
=
dU+
P dV
+
V dP;
but dQ
=
dli
+
P dV.
Therefore, dH
=
dQ +
V
dP.
t
Pronounced en-thal'-pi.
(11-2)
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