Thermo & Stat Mech - Spring 2006 Class 7 1 Thermodynamics and Statistical Mechanics Tds Equations.
THERMODYNAMICS dynamics – changes thermo – heat Laws of Thermodynamics: 0 th, 1 st, 2 nd, 3 rd.
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Transcript of THERMODYNAMICS dynamics – changes thermo – heat Laws of Thermodynamics: 0 th, 1 st, 2 nd, 3 rd.
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THERMODYNAMICS
dynamics – changesthermo – heat
Laws of Thermodynamics:0th, 1st, 2nd, 3rd
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Note: Thermodynamics ……..
is not concerned about rate of changes (kinetics) but the states before and after the change
does not deal with time
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Classical Thermodynamics
Marcoscopic observables T, P, V, …
Statistical Thermodynamics
Microscopic details dipole moment, molecular size, shape
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Joule’s experiment
T mgh adiabatic wall
(adiabatic process)
U (energy change) = W (work) = mgh
w
h
Thermometer
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w
T time interval of heatingU = q (heat)
Conclusion: work and heat has the same effect to system (internal energy change)
FIRST LAW: U = q + W*
Page 59
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U: internal energy is a state function [ = Kinetic Energy (K.E.) + Potential Energy (P.E.) ] q: energy transfer by temp gradient W: force distance E-potential charge surface tension distance pressure volume
First Law: The internal energy of an isolated system is constant
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Positive: heat flows into system work done onto systemNegative: heat flows out of system work done by system
Convention:
Pressure-volume Work
P1 = P2
V2V1
MM
d work F dlext M gdhpiston
weight
AAdh
P Adhext
P dVext work P V Vext 2 1
Mg h
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If weight unknown, but only properties of systemare measured, how can we evaluate work?
PV1V2P
MM
Assume the process is slow and steady,
Pint = Pext
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Free Expansion:
Free expansion occurs when the external pressure is zero, i.e. there is no opposing force
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Reversible change: a change that can be reversed by an infinitesimal modification of a variable. Quasi equilibrium process: Pint = Pext + dP (takes a long time to complete)
infinitesimal at any time
quasi equilibrium process
ò dVPW extò dVPint
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1 2P
V
Example: P1 = 200kPa = P2
V1 = 0.04m3 V2 = 0.1m3
W PdV1 21
2
P V V2 1
200 01 0 04 3kPa m. . 12kJ
what we have consider was isobaric expansion(constant pressure) other types of reversible expansion of a gas: isothermal, adiabatic
*
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Isothermal expansion
remove sand slowlyat the same time maintain temperature by heating slowly
W PdVrev 1
2
nRT
VdV
1
2
nRTdV
V1
2
nRT
V
Vln 2
1
T
VV1 V2
P
area under curve
*
Page 65-66
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Ex. V1 = 0.04m3 P1 = 200kPa V2 = 0.1m3
W nRTV
Vrev 1
2
1
ln
PV
V
V1 12
1
ln
200 0 04
01
0 043kPa m. ln
.
.kJ 7.33
PPV
VkPa2
1 1
2
80
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Adiabatic Reversible Expansion
For this processPV = constant for ideal gas(proved later)
V1
P1 T1
V2
P2 T2
(slightly larger than 1)
C
Cp
v
W PdVPV dV
V 1 1
1
2
1
2
PV V V1 1 21
11
1
PV PV2 2 1 1
1
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Ex. V1 = 0.04m3, P1 = 200kPa, V2 = 0.1m3,
= 1.3
P kPa2
1 3
2000 04
01060 77
.
..
.
W kJ
60 77 01 200 0 04
1 136 41
. . .
..
volume change
|Wa|>|Wb|>|Wc|>|Wd|
W PdV 0
a
db
V
P
200kPaconst
isothermal
adibatic
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State Function Vs Path Function State function: depends only on position in
the x,y plane e.g.: height (elevation)
300200
100mB
AX
Y
1 2
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Path Function: depends on which path is taken to reach destination
from 1 2, difference of 300m (state function)but path A will require more effort.Internal energy is a state function, heat and work are path functions
3
2481.13K
192.45K
P
200kPa
V/m30.04 0.1
1
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5 moles of monoatomic gas
C R
U nC T R kJ
W kJ
Q U kJ kJ
W kJ kJ
Q kJ kJ kJ
V
V
3
23
2 289 5 18
12
12 30
7 33 0 7 33
7 33 18 25 33
1 2
1 2
1 3 2
1 3 2
. .
. .
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Joule’s second experiment
Energy UU(V,T) ???
V1V2
thermometer
At time zero, open valve
adiabatic wall
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After time zero, V1 V1+V2
T=0 Q=0 W=0 no Pext U=0
thermometer
No temperature change
adiabatic wall
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dTCdVV
UdT
T
UdU
V
U
VTV
T
0
0 (for ideal gas)CV is constant volume heat capacity
U=U(T) Energy is only a function of temperature for ideal gas*
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*Enthalpy
Define H = U + PV state function intensive variables locating the state
Enthalpy is also a state function
H = U + PV + VP
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PPTP
CCdPP
HdT
T
HdH
PTHH
0
,
for ideal gases C C R
C
C
P V
P
V
At constant pressureH = U+PV = U - W = QH = QP constant pressure heating
H is expressed as a functional of T and P
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Thermochemistry Heat transferred at constant volume qV = U
Heat transferred at constant pessure qP = H
Exothermic H = -
Endothermic H = +
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Standard states, standard conditions do not measure energies and enthalpies absolutely but only the differences, U or H
The choice of standard state is purely a matter of convenience
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The standard states of a substance at a specified temperature is its pure form at 1 atmosphere
What is the standard state ?
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25oC, 1 atmosphere: the most stable forms of elements assign “zero enthalpy”
Ho298 = 0 used for chemical reactions
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Standard enthalpy of formation
Standard enthalpy change for the formationof the compound from its elements in their reference states.
Reference state of an element is its most stablestate at the specified temperature & 1 atmosphere
C (s) + 2H2 (g) CH4 (g) Hfo = -75 kJ
289K, 1 atm
From the definition, Hfo for elements 0
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Hess’s Law
The standard enthalpy of an overall reaction is the sum of thestandard enthalpies of the individual reactions into which areaction may be divided.
Standard reaction enthalpy is the change in enthalpy when the reactants in their standard states change to products in their standard states.
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Hess’ Law
R X Y PH H H 2 3 4
H1
H1 = H2 + H3 + H4 state function
Hess’s law is a simple application of the first law of thermodynamics
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e.g. C (s) + 2H2 (g) CH4 (g) H1o = ?
298K, 1 atm
C (s,graphite) + O2 (g) CO2 (g)
Ho = -393.7 kJ H2 (g) + ½O2(g) H2O (l)
Ho = -285.8 kJ CH4 (g) + 2O2 (g) CO2 (g) + 2H2O (l)
Ho = -890.4 kJ H1
o = -393.7 + 2(-285.8) - (-890.4)
= -75 kJ/mole
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Heat of Reaction (Enthalpy of Reaction) Enthalpy change in a reaction, which may be obtained from Hf
o of products and reactants
Reactants Products
products tsreaac
oreactionfR
oprodfp
or HnHnH
tan,,
i foi
productsreac ts
Htan
I stoichiometric coefficient, + products,
- reactantsE.g. CH4 (g) + Cl2 (g) CH3Cl (g) + HCl (g)
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Hfo/ kJ
CH3Cl -83.7
HCl -92.0
Cl2 0
CH4 -75.3
Hro = (-83.7-92.0) - (-75.3+0) = -100.4 kJ
Reactants Products
elements elements
n HR f Ro , n Hp f p
o ,
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jfo
jor HH ,
2A + B 3C + D
0 = 3C + D - 2A - B
Generally,0 = J J J
J denotes substances, J are the stoichiometric numbers
*
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Bond energy (enthalpy)
Assumption – the strength of the bond is independent of the molecular environment in which the atom pair may occur.
C (s,graphite) + 2H2 (g) CH4(g)Ho = -75.4 kJ
H2 (g) 2H (g) Ho = 435.3 kJ
C (s,graphite) C (g) Ho= 715.8 kJ C (s,graphite) + 2H2 (g) C (g) + 4H (g)
Ho = 2(435.3)+715.8 = 1586.5 kJ C (g) + 4H (g) CH4 (g) H = -75.4-1586.5
= -1661.9 kJCH Bond enthalpy = 1661.9/4 kJ = 415.5 kJ
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Temperature dependence of Hr
productsreactantsreactants
reaction,products
products, ipiRippipP
T
T
por
Tr
CnCnCC
dTCHHo
CP,R
CP,P
Hro
P
R
HrT
298 T
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What is the enthalpy change for vaporization (enthalpy of vaporization) of water at 0oC?
H2O (l) H2O (g)
Ho = -241.93 - (-286.1) = 44.01 kJmol-1 H2 = Ho + CP(T2-T1) assume CP,i
constant wrt T
H (273) = Ho(298) + CP(H2O,g) - CP (H2O,l)(273-298) = 44.10 - (33.59-75.33)(-25) = 43.0 kJ/mole
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Reversible vs IrreversibleNon-spontaneous changes vs Spontaneous
changesReversibility vs Spontaneity
First law does not predict direction of changes,cannot tell which process is spontaneous. Only U = Q + W
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Second Law of Thermodynamics
•Origin of the driving force of physical and chemical change
•The driving force: Entropy
•Application of Entropy: • Heat Engines & Refrigerators• Spontaneous Chemical Reactions
•Free Energy
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Second Law of Thermodynamics
No process is possible in which the sole result is the absorptionof heat from a reservoir and its complete conversion into work
Hot Reservoir
q
w
Engine
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Direction of Spontaneous Change
More Chaotic !!!
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Entropy (S) is a measurement of the randomness of the system, and is a state function!
S Q S 1 / T
Spontaneous change is usually accompanied by a dispersal of energy into a disorder form, and its consequence is equivalent to heating
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T
dqdS rev
Entropy S
For a reversible process, the change of entropy is defined as
Another expression of the Second Law:
The entropy of an isolated systems increases in the course of aspontaneous change:
Stot > 0
where Stot is the total entropy of the isolated system
(thermodynamic definition of the entropy)
*
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Entropy S
The entropy of an isolated systems increases in the course of areversible change:
Stot = 0
where Stot is the total entropy of the isolated system
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Carnot’s Theoretical Heat Engine
Heat flows from a high temperature reservoirto a low temperature body. The heat can be utilized to generate work.
e.g. steam engine.
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The efficiencies of heat engines
Hot Reservoir
q
w
Engine
S = - |q|/Th < 0 not possible! contrary to the second law
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The Nernst Theorem
The entropy change accompanying any physical or chemical transformation approaches zero as the temperature approaches zero.
S 0 as T0
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Third Law of Thermodynamics
If the entropy of each element in its most state is taken as zero at the absolute zero of temperature, every substance has a positive entropy. But at 0K, the entropy of substance may equals to 0, and does become zero in perfect crystalline solids.
Crystalline form: complete ordered, minimum entropy
Implication: all perfect materials have the same entropy (S=0) at absolute zero temperature
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Statistical Interpretation of S
S = 0 at 0K for perfect crystals S = k ln
Boltzmann number of arrangements postulate of entropy Boltzmann constant
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Entropy Change of Mixing
one distinguish arrangement
S k S kA B ln!
!, ln
!
!
4
40
4
40
A B
A B
A B
A B
N N
N N
S k
!
! !
ln
8 7 6 5
4 3 2 170
70
number of arrangement increased
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In general mixing NA, NB
!!
!ln
BA
BAmix NN
NNkS
Entropy change of mixingStirling’s approximation: ln N! N ln N + 0(N)
for large N
0
BBAABA
BBAABABA
BBAABABAmix
XXXXNNk
NXNXNNNNk
NNNNNNNNkS
lnln
lnlnln
lnlnln
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Extensions of 2nd Law
TdS dq Clausius Inequality
For adiabatic process,
TdS or dS 0 0
Entropy will always attain maximum in adiabatic processes.
A similar function for other processes?
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Define Helmholtz free energyA = U - TS Thermodynamic
State Function dA = dU - TdS - SdTSubstitute into Clausius Inequality
0
0
0
dq TdS
dq dA dU SdT
PdV dA SdTfor isothermal, isochoric (constant volume)process,
0dA
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change in Helmholtz free energy = maximum isothermal work
Example of isothermal, isochoric process: combustion in a bomb calorimeter
O2 +fuel
Temp. bath
O2, CO2,H2O
Higher P heat givenout
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Gibbs Free Energy
Define Gibbs free energy G = H - TS Thermodynamic
state function = U + PV -TS dG = dU +PdV + VdP -TdS -SdT
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constant pressure, constant temperature
dG0
G will tend to a minimum value equilibrium spontaneous change
dG 0dG 0
More applications since most processes areisothermal, isobaric
chemical reactions at constant T, PReactants Productsendothermic H is positiveexothermic H is negative
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Change of Gibbs free energy with temperature(constant pressure)
ST
G
SdTGG
VdPSdTdG
P
12
0
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dP
dT
H
T V
H
TV
P H
RT
dP
Pd P
H
RTdT
PH
RTconst
P
P
H T T
RT T
vap
vap
vap
vap
vap
vap
sat vap
sat
vap
2
2
2
1
2 1
1 2
ln
ln .
ln1/T
In P
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Example: What is the change in the boiling point of water at 100oC per torr change in atmospheric pressure?
Hvap = 9725 cal mol-1
Vliq = 0.019 l mol-1
Vvap = 30.199 l mol-1
dP
dT
H
T V V
calmol l atmcal
K lmol
atmK
torr K
dT
dPK torr
sat
sat
vap
v l
9725 0 04129
37315 30180
0 03566
2710
0 0369
1 1
1
1
1
1
.
. .
.
.
.