15.5
Electronic Excitation
The electronic partition function is
where g0 and g1, are, respectively, the
degeneracies of the ground state and the first
excited state.
E1 is the energy separation of the two lowest
states.
Introducing
For most gases, the higher electronic states are
not excited (θe ~ 120, 000k for hydrogen).
therefore,
  ln Z 
U  NkT 
 0
 T V
2
 U 
CV  
 0
 T V
At practical temperature, electronic excitation
makes no contribution to the internal energy
or heat capacity!
15.6
The total heat capacity
For a diatomic molecule system
Since
Discussing the relationship of T and Cv (p. 288-289)
Heat capacity for diatomic molecules
• Example I (problem 15.7) Consider a diatomic
gas near room temperature. Show that the
entropy is
3
 7
 2mk  2 T 5 2  
S  Nk   ln  2 
 ,
 2
 h  2 rot  
• Solution: For diatomic molecules
S  Strans  S vib  S rot  S excit
At room temperature
S excit  0
1
1 


Nk   

 e  T
T
2
e  1   Nk ln 

S vib 
 1  e  T
T

 0 (does not contribute!)




• For translational motion, the molecules are
treated as non-distinguishable assemblies
3
U  NkT
2
three degrees of freedom 
3
 2mkT  2
Z V

2
 h

U
S tr   Nk ln Z  ln N  1
T
3


5
V   2mkT  2 
S tr  Nk  Nk ln 

 
2
2
 N   h
 


• For rotational motion (they are distinguishable
in terms of kinetic energy)
U  NkT
S
2 rot
1

 due to homonuclea r molecule 
2

for distinguis
U
 Nk ln Z
T
 Nk  Nk ln
T
Z
hable assembly 
T
2 rot
 
5
 V
S system  Nk  Nk ln
2
N

 
  2mkT  3 2  
T




 Nk  Nk ln

  h 2   
2 rot

 

5
3/ 2



7
 V   2mk  T 2 
S system  Nk  Nk ln 


  h 2  2 rot 
2
N


 

• Example II (problem 15-8) For a kilomole of
nitrogen (N2) at standard temperature and
pressure, compute (a) the internal energy U;
(b) the Helmholtz function F; and (c) the
entropy S.
• Solution:
calculate the characteristic temperature
first!
 vib  3352k and  r  2.9k for N 2
U  U trans  U vib  U rot
1
3
1 

  NkT
U  NkT  Nk   vib   

2
 2 e T 1 
because  r T
U
1

5
1

NkT  Nk   vib   3352

2
 2 e 298  1 
1
3352 

 Nk  2.5  298   3352  11.25 
2
e
1 

3352 

 Nk  2272  11.25 
e
1 

 1.0kmol  8.314 103 J  kmol1  k 1  2272
 1.89 107 J
F   NkT ln Z  ln N  1 non  distinguis hable particles 
F   NkT ln Z Distinguis hable particles such as vib & rotation
Ft   NkT ln Z t  ln N  1
 2mkT 
Zt  V 

2
 h

3
2
 2  4.65 10 1.38110
 22.4m 
2
 68
6
.
626

10

 26
3


 22.427.3  10 
 22.4 273.86 10
19
3
2
20
3
 23
Jk  298 


1
3
2
2
3
2
 3.195 1033
 3.195 1033 
6

Ft   NkT  ln

1


NkT
ln
5
.
15

10
1
26

 6.02 10

  NkT 16.4
 
 
Fvib   NkT ln Z vib
Z vib 
e
3352
1 e
2298
3352
298
e 5.62

1  e 11.24
1 

ln Z vib  5.62  ln 1  11.24   5.62
 e

T
Frot   NkT ln Z rot Z rot 
for T  rot
2 rot
298
 102.72 ln Z rot  4.63
2 * 2.9
Frot  4.63 NkT
Z rot 
F  Frot  Fvib  Ft
 4.63 NkT  5.62 NkT  16.4 NkT
 15.41NkT
 15.41 6.02 10 26 1.38 10  23  298
 3.817 107 J
Chapter 16:
The Heat Capacity of a Solid
16.1
Introduction
1. This is another example that classical kinetic theory
cannot provide answers that agree with
experimental observations.
2. Dulong and Petit observed in 1819 that the specific
heat capacity at constant volume of all elementary
solids is approximately 2.49*104 J .kilomole-1 K-1 i.e.
3R.
3. Dulong and Petit’s result can be explained by the
principle of equipartition of energy via treating
every atom of the solid as a linear oscillator with
six degrees of freedom.
4. Extensive studies show that the specific heat
capacity of solid varies with temperature,
becomes zero as the temperature
approaches zero.
5. Specific heat capacities of certain substances
such as boron, carbon and silicon are found
to be much smaller than 3R at room
temperature.
6. The discrepancy between experimental
results and theoretical prediction leads to
the development of new theories.
16.2 Einstein’s Theory of The Heat
Capacity of a Solid
• The crystal lattice structure of a solid
comprising N atoms can be treated as an
assembly of 3N distinguishable onedimensional oscillators!
• The assumption is based on that each atom is
free to move in three dimensions!
From chapter 15:
the internal energy for N linear oscillators
is
U= Nkθ(1/2 + 1/(eθ/T -1)) with θ = hv/k
The internal energy of a solid is thus
1
U  3Nk E ( 
2
1
E
eT
)
1
Here θ is the Einstein temperature and is
denoted by θE.
The heat capacity:
3Nk E 3Nk E
(
 E
)
2
T

U


e
1
Cv  
 
T
 T v
Case 1: when T >> θE
This result is the same as Dulong & Petit’s
Case 2:
T << θE
As discussed earlier, the increase of
powered by the increase of
As a result, when
is out
If an element has a large θE , the ratio will be
large even for temperatures well above
absolute zero
When
is large,
is small
Since
hv
E 
k
A large θE value means a big
On the other hand
1 k
v
2 u
To achieve a large , we need a large k or a
small u (reduced mass), which corresponds to
a light element or elements that produce very
hard crystals.
• The essential behavior of the specific heat capacity of solid is
incorporated in the ratio of θE/T.
• For example, the heat capacity of diamond approaches 3Nk
only at extremely high temperatures as θE = 1450 k for
diamond.
• Different elements at different temperatures will poses the
same specific heat capacity if the ratio θE/T is the same.
• Careful measurements of heat capacity show that Einstein’s
model gives results which are slightly below experimental
values in the transition range of