Physical Chemistry , 1st ed.

ofEwith respect to temperature and come up with a simple expression for the
heat capacity:

CVNk



^3

2

 ^3

2

 (^) 
3 N* 6
j 1 



T

j



2



(1

e

e





j/T

j/T) 2



(18.54)

trans rot vib

The origins of each part of the heat capacity have been labeled under each
term. The derivative in the definition ofCVconverts the product function
from the vibrational partition function into a summation of terms. Notice
what equation 18.54 implies: translations and rotations contribute the same
amount to the heat capacity of a gaseous molecule, and vibrations also con-
tribute. The greater the number of atoms N* in the molecule, the more vi-
brations, and so the greater the vibrational contribution to CV. Such obser-
vations are indeed made experimentally. An expression for Cpis left as an
exercise.

18.8 Thermodynamic Properties of Molecules from Q 639

Table 18.5 Expressions for the various components of the thermodynamic state

functions of moleculesa

State function trans nuc elect

E ^32 NkT — NDe

H ^52 NkT — NDe NkT

G NkTln ^2 hm 2 kT

3/2

VNe NkT — NDe NkT

SNkln ^2 hm 2 kT

3/2

VNe

5/2

 — Nkg 1

State function vib, diatomic rot, diatomic

E NkT 2 Tv ev/Tv/T 1  NkT

H NkT 2 Tv ev/Tv/T 1 1  2 NkT

G NkT 2 Tv ln(1 ev/T) (^1)  NkT 1 ln T
r

SNkev/Tv/T 1 ln(1 ev/T) Nkln T
r
 Nk
State
function vib, polyatomic rot, polyatomic

E NkT 

3 N* 6

j 1 

 2 Tv ev/Tv/T 1 

b

^32 NkT*

H NkT 

3 N* 6

j 1 

 2 Tv ev/Tv/T 1  NkTb ^52 NkT*

G NkT 

3 N* 6

j 1 

 2 Tv ln(1 ev/T) NkTb NkTln ^ 

1/2



A

T

B

3

C

1/2

NkT*

S Nk 

3 N* 6

j 1 

ev/Tv/T 1 ln(1 ev/T)

b

Nkln ^

1/



(^2) e3/2

A
T
B
3
C
1/2
aTranslational and electronic contributions are the same for all molecules. Vibrational and rotational contributions
depend on whether the molecule is diatomic or polyatomic.Nnumber of species in the system;N number of
atoms in molecule.b
Expression is different for linear molecules. Consult the text.

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