Thermodynamics and Statistical Mechanics

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29 Οκτ 2013 (πριν από 3 χρόνια και 5 μήνες)

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Thermo & Stat Mech
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Spring 2006 Class 19

1

Thermodynamics and Statistical
Mechanics

Partition Function

Thermo & Stat Mech
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Spring 2006
Class 19

2

Free Expansion of a Gas

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Free Expansion

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Isothermal Expansion

Thermo & Stat Mech
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Spring 2006
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Isothermal Expansion

Reversible route between same states.


đ
Q =
đ
W + dU

Since
T

is constant,
dU

= 0.

Then,
đ
Q =
đ
W
.


Thermo & Stat Mech
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Spring 2006
Class 19

6

Entropy Change

The entropy of the gas increased.

For the isothermal expansion, the entropy of the

Reservoir decreased by the same amount.

So for the system plus reservoir,

S

= 0


For the free expansion, there was no reservoir.

Thermo & Stat Mech
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Spring 2006
Class 19

7

Statistical Approach

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Spring 2006
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Statistical Approach

Thermo & Stat Mech
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Partition Function

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Boltzmann Distribution

Thermo & Stat Mech
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Spring 2006
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Maxwell
-
Boltzmann Distribution

Correct classical limit of quantum
statistics is Maxwell
-
Boltzmann
distribution, not Boltzmann.


What is the difference?

Thermo & Stat Mech
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Spring 2006
Class 19

12

Maxwell
-
Boltzmann Probability

w
B

and
w
MB

yield the same distribution.

Thermo & Stat Mech
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Spring 2006
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13

Relation to Thermodynamics

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Relation to Thermodynamics

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Chemical Potential


dU = TdS


PdV +
m
dN


In this equation,
m

is the chemical energy
per molecule, and
dN

is the change in the
number of molecules.


Thermo & Stat Mech
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Spring 2006
Class 19

16

Chemical Potential


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Spring 2006
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17

Entropy

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18

Entropy

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Helmholtz Function

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Chemical Potential

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Chemical Potential

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Boltzmann Distribution

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Distributions

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Distributions

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Ideal Gas

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Ideal Gas

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Ideal Gas

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Entropy

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Math Tricks

For a system with levels that have a constant
spacing (e.g. harmonic oscillator) the partition
function can be evaluated easily. In that case,

n

=
n

, so,

Thermo & Stat Mech
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Spring 2006
Class 19

30

Heat Capacity of Solids

Each atom has 6 degrees of freedom, so based
on equipartition, each atom should have an
average energy of 3
kT.
The energy per mole
would be 3
RT
. The heat capacity at constant
volume would be the derivative of this with
respect to
T
, or 3
R
. That works at high enough
temperatures, but approaches zero at low
temperature.

Thermo & Stat Mech
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Spring 2006
Class 19

31

Heat Capacity

Einstein found a solution by treating the solid
as a collection of harmonic oscillators all of the
same frequency. The number of oscillators was
equal to three times the number of atoms, and
the frequency was chosen to fit experimental
data for each solid. Your class assignment is to
treat the problem as Einstein did.


Thermo & Stat Mech
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Spring 2006
Class 19

32

Heat Capacity