Thermodynamics and Statistical Mechanics

Heat CapacityOf a diatomic gas Of a solid

A diatomic molecule has degrees of freedom

3 translational

2 rotational and

1 vibrational degrees of freedom

Classical statistical mechanics — equipartition theorem:

in thermal equilibrium each quadratic term in E has an average energy .TkB21

17(3 2 2)( )

2 2kTU kT= + + =

Classical statistical mechanics — equipartition theorem

kT23

22

kT

22

kT 2 21 12 2v xE mv k x⎛ ⎞= +⎜ ⎟

⎝ ⎠

2 2 21 1 12 2 2t x y zE mv mv mv= + +

221 12 2

yxr

JJE

I I= +

3 2 6× =

Diatomic GasAccording to classical statistical mechanics a diatomic gas has

Internal energy U= 3.5NkT and Constant heat capacity Cv =3.5Nk,

derived from the translational, rotational and vibrational energy.

VV T

UC ⎟⎠⎞

⎜⎝⎛∂∂

=

Experimental results

Diatomic GasAccording to classical statistical mechanics a diatomic gas has

Internal energy U= 3.5NkT and Constant heat capacity Cv =3.5Nk,

derived from the translational, rotational and vibrational energy.

Because of the spacing (quantum effect!) of the energy levels of each type of motion, they are not all equally excited.

This shows up in the heat capacity.

VV T

UC ⎟⎠⎞

⎜⎝⎛∂∂

=

Partition Function

For one molecule, ε = εtrans + εrot + εvib

( )j trans rot vib

trans rot vib

molecule j jj j

molecule trans rot vib

molecule trans rot vib

Q g e g e

Q g e g e g eQ Q Q Q

βε β ε ε ε

βε βε βε

− − + +

− − −

= =

=

=

∑ ∑

∑ ∑ ∑

nEn

Q e β−= ∑

Partition Function( ) ( )

( )ln ln ln ln

ln

ln ln ln

N Nmolecule trans rot vib

trans rot vib

molecule

V

trans rot vib

V V V

trans rot vib

Q Q Q Q Q

Q N Q Q Q

QUN

Q Q QUNU U U U

β

β β β

= =

= + +

∂⎛ ⎞= −⎜ ⎟∂⎝ ⎠

∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − − −⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠= + +

Translational Motion

3 / 2

2

,

2

32

32

trans

trans

V trans

mkTQ Vh

U NkT

C Nk

π⎛ ⎞= ⎜ ⎟

⎝ ⎠

=

=

The Harmonic oscillator Classical and Quantum Result

2 21 12 2xE mv k x⎛ ⎞= +⎜ ⎟

⎝ ⎠classic:

quantum: 1 12 2nE n hv nε ⎛ ⎞ ⎛ ⎞= + = +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠zero-point

energy

The Vibrational Partition Function Zvib

The simple harmonic oscillator

0 0

12exp expn

vibn n

nE

QkT kT

ε∞ ∞

= =

⎛ ⎞⎛ ⎞+⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎜ ⎟= − = −⎜ ⎟ ⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠

∑ ∑

0 0exp exp exp exp

2 2

n

vibn n

nQkT kT kT kTε ε ε ε∞ ∞

= =

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − − = − −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

∑ ∑

1 1 1exp2 1 exp exp exp 2sinh

2 2 2

vibQkT

kT kT kT kT

εε ε ε ε

⎛ ⎞= − = =⎜ ⎟ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎝ ⎠ − − − −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

1 12 2nE n hv nε ⎛ ⎞ ⎛ ⎞= + = +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

The Vibrational Partition Function Zvib

2

2

, 2 2

1 11 12 21

1

1 1

vib h hkT

hhkTkT

vibV vib h h

VkT kT

U Nh Nhe

e

hh eeU kTkTC Nh NkT

e e

β ν ν

νν

ν ν

ν ν

νν

ν

⎡ ⎤⎡ ⎤ ⎢ ⎥= + = +⎢ ⎥ ⎢ ⎥−⎣ ⎦ −⎣ ⎦⎡ ⎤ ⎡ ⎤

⎛ ⎞⎢ ⎥ ⎢ ⎥⎜ ⎟⎢ ⎥ ⎢ ⎥∂⎛ ⎞ ⎝ ⎠= = =⎢ ⎥ ⎢ ⎥⎜ ⎟∂⎝ ⎠ ⎛ ⎞ ⎛ ⎞⎢ ⎥ ⎢ ⎥− −⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦

ln vibvib

V

N QU

β∂⎛ ⎞

= −⎜ ⎟∂⎝ ⎠

12

1

h

vib h

eQe

β ν

β ν

−

−=−

The simple harmonic oscillator

High and Low Temperature Limits

kTh

vibV

vibV

kTh

kTh

vibV

ekThNkChkT

Nk

kTh

kTh

kTh

NkChkT

e

ekTh

NkC

ν

ν

ν

νν

ν

νν

ν

ν

−⎟⎠⎞

⎜⎝⎛=<<

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛ −+

⎟⎠⎞

⎜⎝⎛ +⎟

⎠⎞

⎜⎝⎛

=>>

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎟⎠⎞

⎜⎝⎛

=

2

,

2

2

,

2

2

,

11

1

1

Rotational Motion2

2 2

2 ( 1)2

2

2 22 20

0

( 1) (2 1)2

For , let ( 1) 2 12

2 2

ln 1

l lIkT

l rotl

x xI I

rot

rotrot rot

V

l l Z l eI

kT l l x l dxI

I IQ e dx e

N QU N NkT U NkT

β β

ε

β β

β β

− +

∞− −∞

= + = +

>> + ⇒ + ⇒

⎡ ⎤= = − =⎢ ⎥

⎢ ⎥⎣ ⎦∂⎛ ⎞

= − = = =⎜ ⎟∂⎝ ⎠

∑

∫

/ 2h π=

Rotational Motion2

,

For 2 rot

V rot

kT U NkTI

C Nk

>> =

=

2( 1)

2(2 1)l l

IkTrot

lZ l e

− += +∑

2

,

For 02

0

rot

V rot

kT UI

C

<< ⇒

⇒

2

For 2

kTI

≈ Full partition function Zrot must be used

Diatomic Gas Overall

NkChkT

NkChkTI

NkCI

kT

Ih

V

V

V

27

25

2

23

2

2

2

2

2

=>>

=<<<<

=<<

>>

ν

ν

ν

Experimental results

The Einstein Model for the Solid

http://www.physics.ohio-state.edu/~heinz/H133/lectures/LectureT4.pdf#search='einstein%20solid'

Heat Capacity

m

kykx

kz

The solid of N atoms is assumed to be

a collection of 3N independent harmonic oscillators!

Solid

Classical Heat Capacity

For a solid composed of 3N classical atomic oscillators:

Giving a total energy per mole of sample:

1 3 BU NU Nk T= =

33 3B

A BNk TU N k T RT

n n= = =

So the heat capacity at constant volume per mole is:

3 24.94 JV mol K

V

d UC RdT n

⎛ ⎞= = ≈⎜ ⎟⎝ ⎠

2 21 12 2xE mv k x⎛ ⎞= +⎜ ⎟

⎝ ⎠classical:

Classical statistical mechanics — equipartition theorem: in thermal equilibrium each quadratic term in E has an average energy .

122 BU k T=

Before Einstein, Dulong and Petit formulated a law which described the high-temperature prediction for heat capacity.

Law of Dulong and Petit

1.94.243 −≈= moleJkNC AV

Experimental resultsClassical result in agreement with law of Dulong and Petit (1819) is approximately obeyed by most solids at high T ( > 300 K). But by the middle of the 19th century it was clear that CV → 0 as T → 0 for solids.

So…what was happening?

Einstein uses Planck’s Work

Planck (1900): vibrating oscillators (atoms) in a solid have quantized energies 0, 1, 2, ...nE hvn n nε= = =

Einstein (1907): model a solid as a collection of 3N independent 1-D quantum oscillators, all with constant ν, and use Planck’s equation for energy levels.

We will show the Einstein work but with the correct QM result

(What is the result when you use the Planck result?)

2 21 12 2xE mv k x⎛ ⎞= +⎜ ⎟

⎝ ⎠Classical:

Quantum theory: 1 12 2nE n hv nε ⎛ ⎞ ⎛ ⎞= + = +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠zero-point

energy

The Vibrational Partition Function Qvib

The simple 1-D harmonic oscillator

0 0

12exp expn

vibn n

nE

QkT kT

ε∞ ∞

= =

⎛ ⎞⎛ ⎞+⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎜ ⎟= − = −⎜ ⎟ ⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠

∑ ∑

0 0exp exp exp exp

2 2

n

vibn n

nQkT kT kT kTε ε ε ε∞ ∞

= =

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − − = − −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

∑ ∑

1 1 1exp2 1 exp exp exp 2sinh

2 2 2

vibQkT

kT kT kT kT

εε ε ε ε

⎛ ⎞= − = =⎜ ⎟ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎝ ⎠ − − − −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

1 12 2nE n hv nε ⎛ ⎞ ⎛ ⎞= + = +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

The Vibrational Partition Function Zvib

2

2

, 2 2

1 11 13 32 21

1

3 3

1 1

vibkT

kTkT

vibV vib

VkT kT

U N Ne

e

eeU kTkTC N NkT

e e

β ε ε

εε

ε ε

ε ε

εε

ε

⎡ ⎤⎡ ⎤ ⎢ ⎥= + = +⎢ ⎥ ⎢ ⎥−⎣ ⎦ −⎣ ⎦⎡ ⎤ ⎡ ⎤

⎛ ⎞⎢ ⎥ ⎢ ⎥⎜ ⎟⎢ ⎥ ⎢ ⎥∂⎛ ⎞ ⎝ ⎠= = =⎢ ⎥ ⎢ ⎥⎜ ⎟∂⎝ ⎠ ⎛ ⎞ ⎛ ⎞⎢ ⎥ ⎢ ⎥− −⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦

,3 ln vib SHOvib

V

N QU

β∂⎛ ⎞

= −⎜ ⎟∂⎝ ⎠

12

. 1vib SHOeQ

e

β ε

βε

−

−=−

The simple harmonic oscillator

High and Low Temperature Limits

2

, 2

2

, 2

2

,

3

1

13 3

1 1

3

kT

V vib

kT

V vib

kTV vib

ekTC Nk

e

kT kTkT C Nk Nk

kT

kT C Nk ekT

ε

ε

ε

ε

ε ε

εε

εε−

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠= ⎢ ⎥⎛ ⎞⎢ ⎥−⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞ ⎛ ⎞+⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥>> = =⎢ ⎥⎛ ⎞+ −⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

⎛ ⎞<< = ⎜ ⎟⎝ ⎠

(Law of Dulong and Petit)

We can define the “Einstein temperature”:

to

Ehv

k kεθ ≡ =

( )( )2/

/2

13)(

−=

T

TT

VE

EE

eeRTC

θ

θθ

2

, 23

1

kT

V vib

kT

ekTC Nk

e

ε

ε

ε⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠= ⎢ ⎥⎛ ⎞⎢ ⎥−⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

and re-write

Limiting Behavior of CV (T)

Low T limit:

These predictions are qualitatively correct: CV → 3R for large T and CV → 0 as T → 0:

High T limit: 1<<TEθ ( ) ( )

( ) RRTCT

TTV

E

EE

311

13)( 2

2

≈−+

+≈

θ

θθ

1>>TEθ ( )

( ) ( ) TTT

TT

VEE

E

EE

eRe

eRTC /2

2/

/2

33)( θθ

θ

θθ−≈≈

3RC

V

T/θE

A Closer Look:

High T behavior: Reasonable agreement with experiment

Low T behavior: CV → 0 too quickly as T → 0 !

The Debye model (1912)Despite its success in reproducing the approach of CV → 0 as T → 0, the Einstein model is quantitatively not correct at very low T. What might be wrong with the assumptions it makes?

3N independent oscillators, all with frequency ν

Discrete allowed energies: 1( ) 0, 1, 2, ...2nE n nε= + =

The 3N harmonic oscillators are not independent and as a collection can have different frequencies

Debye improved and refined this model by considering the quantum harmonic oscillators as collective modes, called phonons.

His model accurately described specific heats for low-temperature solids.3412

5vD

TC Rπ ⎛ ⎞= ⎜ ⎟Θ⎝ ⎠

Low T:

Debye versus Einstein

Debye Model: Theory vs. Experiment

Better agreement than Einstein model at low T

Universal behavior for all solids!

Debye Model at low T: Theory vs. Expt.

Impressive agreement with predicted CV ∝

T3 dependence for Ar! (noble gas solid)