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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)