Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB [email protected] 5-3375 1.
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Transcript of Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB [email protected] 5-3375 1.
![Page 1: Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1.](https://reader030.fdocuments.in/reader030/viewer/2022032605/56649e755503460f94b7630b/html5/thumbnails/1.jpg)
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Statistical ThermodynamicsCHEN 689Fall 2015
Perla B. Balbuena240 JEB
![Page 2: Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1.](https://reader030.fdocuments.in/reader030/viewer/2022032605/56649e755503460f94b7630b/html5/thumbnails/2.jpg)
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Goals of Statistical Thermodynamics
• Based on a microscopic description, predict macroscopic thermodynamic properties
• Example: what is pressure?– Molecular simulations– Probabilistic description
• Objective: Determine probability distributions and
average values of properties considering all possible states of molecules consistent with a set of constraints (an ensemble of states)
![Page 3: Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1.](https://reader030.fdocuments.in/reader030/viewer/2022032605/56649e755503460f94b7630b/html5/thumbnails/3.jpg)
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Connection between microscopic state and macroscopic state
• Example: NVE microcanonical ensemble – how to describe an “ideal gas” of
identical indistinguishable particles; – And a real condensable gas??
![Page 4: Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1.](https://reader030.fdocuments.in/reader030/viewer/2022032605/56649e755503460f94b7630b/html5/thumbnails/4.jpg)
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Quantum mechanical description of microstates
2223/2
2
8 zyxlll lllmV
hzyx
Cubic box side L = V1/3
h = Planck’s constantm = mass of the particlelx, ly, lz: quantum numbers (0,1, 2,…)
![Page 5: Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1.](https://reader030.fdocuments.in/reader030/viewer/2022032605/56649e755503460f94b7630b/html5/thumbnails/5.jpg)
Consider a macroscopic system
• N particles, volume V, and given certain forces among the particles
• Schrödinger equation:
• For an ideal gas:
• Monoatomic gas:
5
jjN jEH ˆ J = 1, 2, 3,…
N
iij VNE
1
),(
Ignore electronic states and focus on translational energies
Cubic container of side L1/3
2223/2
2
8 zyxlll lllmV
hzyx
![Page 6: Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1.](https://reader030.fdocuments.in/reader030/viewer/2022032605/56649e755503460f94b7630b/html5/thumbnails/6.jpg)
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energy states and energy levelsEnergy state lx ly lz
A 2 1 1
B 1 2 1
C 1 1 2
2223/2
2
8 zyxlll lllmV
hzyx
compute :e
each energy state gives the same energy level;this jth energy level has a degeneracy (wj) of 3
![Page 7: Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1.](https://reader030.fdocuments.in/reader030/viewer/2022032605/56649e755503460f94b7630b/html5/thumbnails/7.jpg)
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distribution of non-interacting identical
molecules (indistinguishable) in energy states
we define a set of occupation numbers: n( n1, n2, n3…) each number is associated with the ith molecular state
jj
iji nE
energy of ith microstate
(all molecular energy states) W degeneracy is the number
of microstates that havethis energy level
![Page 8: Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1.](https://reader030.fdocuments.in/reader030/viewer/2022032605/56649e755503460f94b7630b/html5/thumbnails/8.jpg)
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Statistical Mechanics Postulates
• All microstates with the same E, V, N are equally probable
• The long time average of any mechanical property in a real macroscopic system is equal to the average value of that property over all the microscopic states of the system (ergodic hypothesis)– each state weighted with its probability of
occurrence, provided that all the microstates reproduce the thermodynamic state and environment of the actual system
![Page 9: Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1.](https://reader030.fdocuments.in/reader030/viewer/2022032605/56649e755503460f94b7630b/html5/thumbnails/9.jpg)
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Boltzmann energy distribution
• how to assign probabilities to states of different energies?
• Assume a system at N, V, in contact with a very large heat bath at constant T
![Page 10: Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1.](https://reader030.fdocuments.in/reader030/viewer/2022032605/56649e755503460f94b7630b/html5/thumbnails/10.jpg)
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Boltzmann energy distribution
pA(En)
pB(Em)
since A and B are totally independent from each other,
pA(En) pB(Em) is the probability of finding the complete system with Aand B in the specified energy sub-states
![Page 11: Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1.](https://reader030.fdocuments.in/reader030/viewer/2022032605/56649e755503460f94b7630b/html5/thumbnails/11.jpg)
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Boltzmann energy distribution
how the energy of the composite system would change if we change En without changing Em?
as a first approximation we assume that the energy levels areclosely spaced ( continuous function)
![Page 12: Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1.](https://reader030.fdocuments.in/reader030/viewer/2022032605/56649e755503460f94b7630b/html5/thumbnails/12.jpg)
The Boltzmann distribution of energyConsider two macroscopic subsystems (no need to be identical) inside an infinite thermostatic bath (i.e., their temperatures are perfectly controlled and fixed). If the energy in A fluctuates, this has no effect on B, and vice-versa
![Page 13: Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1.](https://reader030.fdocuments.in/reader030/viewer/2022032605/56649e755503460f94b7630b/html5/thumbnails/13.jpg)
The Boltzmann distribution of energy
Probability that subsystem A is in a microstate sA with energy En
A np E
This is NOT the probability of finding subsystem A with energy En
Several microstates of subsystem A may have the same energy
These that have the same energy are equally probable (postulate 1)
![Page 14: Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1.](https://reader030.fdocuments.in/reader030/viewer/2022032605/56649e755503460f94b7630b/html5/thumbnails/14.jpg)
The Boltzmann distribution of energy
Probability that subsystem A is in an energy level of energy En
A n A nE p E
is the degeneracy of energy level En in subsystem A
A nE
![Page 15: Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1.](https://reader030.fdocuments.in/reader030/viewer/2022032605/56649e755503460f94b7630b/html5/thumbnails/15.jpg)
The Boltzmann distribution of energy
Probability that subsystem B is in a microstate sB with energy Em
B mp E
This is NOT the probability of finding subsystem B with energy Em.
Several microstates of subsystem B have the same energy.
These that have the same energy are equally probable (postulate 1)
![Page 16: Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1.](https://reader030.fdocuments.in/reader030/viewer/2022032605/56649e755503460f94b7630b/html5/thumbnails/16.jpg)
The Boltzmann distribution of energy
Probability that subsystem B is in an energy level of energy Em
B m B mE p E
is the degeneracy of energy level Em in subsystem B
B mE
![Page 17: Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1.](https://reader030.fdocuments.in/reader030/viewer/2022032605/56649e755503460f94b7630b/html5/thumbnails/17.jpg)
The Boltzmann distribution of energy
These probabilities in subsystems A and B are independent of each other. Remember our initial assumption: if the energy in A fluctuates, this has no effect on B, and vice-versa
![Page 18: Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1.](https://reader030.fdocuments.in/reader030/viewer/2022032605/56649e755503460f94b7630b/html5/thumbnails/18.jpg)
The Boltzmann distribution of energy
John enters a draw with 8% chance of winning.
Mary enters another, independent draw with 5% chance of winning.
What is the probability that John AND Mary win?
![Page 19: Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1.](https://reader030.fdocuments.in/reader030/viewer/2022032605/56649e755503460f94b7630b/html5/thumbnails/19.jpg)
The Boltzmann distribution of energy
John enters a draw with 8% chance of winning.
Mary enters another, independent draw with 5% chance of winning.
What is the probability that John AND Mary win?
& 0.08 0.05 0.004 0.4%p J M
![Page 20: Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1.](https://reader030.fdocuments.in/reader030/viewer/2022032605/56649e755503460f94b7630b/html5/thumbnails/20.jpg)
The Boltzmann distribution of energy
The probability of finding the composite system in a particular microstate sAB is only a function of the total energy of the composite system, EAB (because of postulate 1).
Suppose:
AB n mE E E
AB n m A n B mp E E p E p E
![Page 21: Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1.](https://reader030.fdocuments.in/reader030/viewer/2022032605/56649e755503460f94b7630b/html5/thumbnails/21.jpg)
The Boltzmann distribution of energy
Other states with same energy of the composite system, EAB, will have the same probability (again because of postulate 1)
For example:
AB n m A n B m
A n B m
p E E p E p E
p E p E
![Page 22: Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1.](https://reader030.fdocuments.in/reader030/viewer/2022032605/56649e755503460f94b7630b/html5/thumbnails/22.jpg)
The Boltzmann distribution of energy
Let us now examine another issue: what is the effect of changing the value of En without changing Em?
m m
AB n m AB n m n m
n n m nE E
AB n m
n m
p E E dp E E E E
E d E E E
dp E E
d E E
![Page 23: Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1.](https://reader030.fdocuments.in/reader030/viewer/2022032605/56649e755503460f94b7630b/html5/thumbnails/23.jpg)
The Boltzmann distribution of energy
Using:
m
m
AB n m A nB E
n nE
p E E dp Ep
E dE
AB n m A n B mp E E p E p E
We also have that:
![Page 24: Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1.](https://reader030.fdocuments.in/reader030/viewer/2022032605/56649e755503460f94b7630b/html5/thumbnails/24.jpg)
The Boltzmann distribution of energy
Combining the results of the two previous slides:
m
AB n m A nB E
n m n
dp E E dp Ep
d E E dE
An analogous development for the effect of Em gives:
n
AB n m B mA E
n m m
dp E E dp Ep
d E E dE
![Page 25: Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1.](https://reader030.fdocuments.in/reader030/viewer/2022032605/56649e755503460f94b7630b/html5/thumbnails/25.jpg)
The Boltzmann distribution of energy
Then:
m n
A n B mB E A E
n m
dp E dp Ep p
dE dE
1 1
n m
A n B m
n mA E B E
dp E dp E
p dE p dE
ln lnA n B m
n m
d p E d p E
dE dE
![Page 26: Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1.](https://reader030.fdocuments.in/reader030/viewer/2022032605/56649e755503460f94b7630b/html5/thumbnails/26.jpg)
The Boltzmann distribution of energy
This equation deserves careful examination:
ln lnA n B m
n m
d p E d p E
dE dE
Its left hand side is independent of subsystem BIts right hand side is independent of subsystem A
Then, each side of the equation should be independent of both subsystems and depend on a characteristic that they share
![Page 27: Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1.](https://reader030.fdocuments.in/reader030/viewer/2022032605/56649e755503460f94b7630b/html5/thumbnails/27.jpg)
The Boltzmann distribution of energy
The two subsystems share their contact with the thermal reservoir that keeps their temperature constant and equal
ln lnA n B m
n m
d p E d p E
dE dE
: a function of temperature
The minus sign is introduced for convenience
![Page 28: Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1.](https://reader030.fdocuments.in/reader030/viewer/2022032605/56649e755503460f94b7630b/html5/thumbnails/28.jpg)
The Boltzmann distribution of energy
Integrating these two differential equations:
nEA n Ap E C e
, : integration constantsA BC C
mEB m Bp E C e
![Page 29: Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1.](https://reader030.fdocuments.in/reader030/viewer/2022032605/56649e755503460f94b7630b/html5/thumbnails/29.jpg)
The Boltzmann distribution of energy
But the result of adding the probabilities of each subsystem should be equal to 1
states n of states n of states n ofsystem A system A system A
1n nE EA n A Ap E C e C e
states n ofsystem A
1
n
AE
Ce
Canonical partition function
![Page 30: Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1.](https://reader030.fdocuments.in/reader030/viewer/2022032605/56649e755503460f94b7630b/html5/thumbnails/30.jpg)
The Boltzmann distribution of energy
But the result of adding the probabilities of each subsystem should be equal to 1
states m of states m of states m ofsystem B system B system B
1m mE EB m B Bp E C e C e
states m ofsystem B
1
m
BE
Ce
Canonical partition function
![Page 31: Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1.](https://reader030.fdocuments.in/reader030/viewer/2022032605/56649e755503460f94b7630b/html5/thumbnails/31.jpg)
The Boltzmann distribution of energy
The canonical partition function of a system is:
,
states i
, , iE V NQ V N e The probability of a particular microstate i with energy Ea is:
, ,
,
states i
, ,i
E V N E V N
iE V N
e ep E
Q V Ne