Top Banner
Thermo & Stat Mech - Spring 2006 Cla 1 Thermodynamics and Statistical Mechanics Partition Function
32

Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Dec 21, 2015

Download

Documents

Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Page 1: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - Spring 2006 Class 19

1

Thermodynamics and Statistical Mechanics

Partition Function

Page 2: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - Spring 2006 Class 19

2

Free Expansion of a Gas

Page 3: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - Spring 2006 Class 19

3

Free Expansion

Page 4: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - Spring 2006 Class 19

4

Isothermal Expansion

Page 5: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - Spring 2006 Class 19

5

Isothermal Expansion

Reversible route between same states.

f

iTQd

SđQ = đW + dU Since T is constant, dU = 0. Then, đQ = đW.

dVV

nRTPdVWd

VdV

nRVdV

TnRT

TWd

TQd

dS

Page 6: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - Spring 2006 Class 19

6

Entropy Change

2ln2

ln2

nRVV

nRVdV

nRSV

V

The entropy of the gas increased.For the isothermal expansion, the entropy of theReservoir decreased by the same amount. So for the system plus reservoir, S = 0

For the free expansion, there was no reservoir.

Page 7: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - Spring 2006 Class 19

7

Statistical Approach

2)!2/(

!

)!2/()!2/(

!

1!0!

!

lnlnln

N

N

NN

Nw

N

Nw

w

wkwkwkSSS

f

i

i

fifif

Page 8: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - Spring 2006 Class 19

8

Statistical Approach

2ln2ln

2/ln)]2/ln(ln[

}]2/)2/ln()2/{(2ln[

])!2/ln(2![ln

)!2/(

!lnln

2

nRNkS

N

NNkNNNNkS

NNNNNNkS

NNkS

N

Nk

w

wkS

i

f

Page 9: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - Spring 2006 Class 19

9

Partition Function

Z

egNN

Zeg

eg

egNN

j

j

j

j

jj

n

jj

n

jj

jj

FunctionPartition 1

1

Page 10: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - Spring 2006 Class 19

10

Boltzmann Distribution

Z

Z

Z

N

U

eg

eg

NUN

eg

egNN

n

jj

n

jjjn

jjj

n

jj

jj

j

j

j

j

ln

1

1

1

1

Page 11: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - Spring 2006 Class 19

11

Maxwell-Boltzmann Distribution

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

What is the difference?

Page 12: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - Spring 2006 Class 19

12

Maxwell-Boltzmann Probability

n

j j

Nj

MB

n

j j

Nj

B

n

j jjj

jFD

n

j jj

jjBE

N

gw

N

gNw

NgN

gw

gN

gNw

jj

11

11

!

!!

)!(!

!

)!1(!

)!1(

wB and wMB yield the same distribution.

Page 13: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - Spring 2006 Class 19

13

Relation to Thermodynamics

YdX

dN

dXdX

dNdNdU

dXdX

ddX

dNdNdU

NU

j

jj

j

jj

jjj

jjjj

jjj

jjj

jjj

Call

so ,)(

Page 14: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - Spring 2006 Class 19

14

Relation to Thermodynamics

YdXdNTdSdNdU

dX

PdVTdSdUYdXTdSdU

YdXdNdU

YdX

dNdX

dX

dNdNdU

jjj

jjjX

jjj

j

jj

j

jj

jjj

and )(

0 If

)(or

like is This

and

Page 15: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - Spring 2006 Class 19

15

Chemical Potential

dU = TdS – PdV + dN

In this equation, is the chemical energy per molecule, and dN is the change in the number of molecules.

Page 16: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - Spring 2006 Class 19

16

Chemical Potential

VTN

F

dNPdVSdTdF

SdTTdSdNPdVTdSdF

TSUF

dNPdVTdSdU

,

Page 17: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - Spring 2006 Class 19

17

Entropy

j j

jj

jj

jjj

jjj

jj

jjj

n

j j

Nj

MBMB

g

NNNkS

NNNgNkS

NgNkS

N

gwwkS

j

ln

lnln

!lnln

! whereln

1

Page 18: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - Spring 2006 Class 19

18

Entropy

)1ln(ln

lnln

ln

NZNkT

US

kTNZNNNNkS

eZ

N

g

N

g

NNNkS

j

jj

jj

jj

kT

j

j

j j

jj

j

Page 19: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - Spring 2006 Class 19

19

Helmholtz Function

)1ln(ln

)1ln(ln

)1ln(ln

NZNkTF

NZNkTT

UTUTSUF

NZNkT

US

Page 20: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - Spring 2006 Class 19

20

Chemical Potential

Z

NkTZNkT

NNkTNZkT

N

F

NZNkTF

VT

ln)ln(ln

1)1ln(ln

)1ln(ln

,

Page 21: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - Spring 2006 Class 19

21

Chemical Potential

kT

Z

Ne

Z

N

kTZ

NkT

kT

So,

Then

ln so ln

Page 22: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - Spring 2006 Class 19

22

Boltzmann Distribution

n

jj

jj

n

jj

n

jj

n

jj

jj

j

j

j

j

j

eg

egNN

eg

Ne

egeNN

egeN

1

1

11

Page 23: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - Spring 2006 Class 19

23

Distributions

Dirac-Fermi 1

1

Einstein-Bose 1

1

Boltzmann 1

jj

j

jj

j

jj

j

feg

N

feg

N

feg

N

j

j

j

Page 24: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - Spring 2006 Class 19

24

Distributions

Dirac-Fermi

1

1

Einstein-Bose

1

1

Boltzmann 1

j

kTj

j

j

kTj

j

j

kTj

j

f

eg

N

f

eg

N

f

eg

N

j

j

j

Page 25: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - Spring 2006 Class 19

25

Ideal Gas

1ln2

ln2

3ln

)1ln(ln

2

2

2/3

2

Nh

mkTVNkTF

NZNkTF

h

mkTVZ

Page 26: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - Spring 2006 Class 19

26

Ideal Gas

nRTNkTPV

VNkT

V

FP

Nh

mkTVNkTF

NT

1

1ln2

ln2

3ln

,

2

Page 27: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - Spring 2006 Class 19

27

Ideal Gas

kTZ

N

U

h

mVZ

h

mV

h

mkTVZ

2

31

2

3ln

ln2

32lnln

22

2/3

2

2/3

2

2/3

2

Page 28: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - Spring 2006 Class 19

28

Entropy

2/3

2

2/3

2

2/3

2

2ln

2

5

12

ln2

3

2lnln

)1ln(ln

h

mkT

N

VNkS

h

mkT

N

VNk

T

NkTS

h

mkTVZ

NZNkT

US

Page 29: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - Spring 2006 Class 19

29

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,

.1for 1

1

1

1

0

000

εβ

n

n

n

n

n

n

n

eex

x

eeeZ n

Page 30: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - 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 3kT. The energy per mole would be 3RT. The heat capacity at constant volume would be the derivative of this with respect to T, or 3R. That works at high enough temperatures, but approaches zero at low temperature.

Page 31: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - 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.

Page 32: Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Thermo & Stat Mech - Spring 2006 Class 19

32

Heat Capacity

Heat Capacity

-5

0

5

10

15

20

25

30

0 100 200 300 400

T (K)

Cv

(J/K

)