Physical Chemistry , 1st ed.

(Nfactorial).N, recall, is the number of molecules in the system. For indistin-

guishable particles, then, the overall partition function for the system is

Qsys

Q

N

N

!

 (18.43)

For a molecule, we can substitute for Qin several degrees of complexity:

18.8 Thermodynamic Properties of Molecules from Q 637

Qsys

Qsys
N

1

!

 (^) 

2

h

m

2

kT



3/2

V

N

(^) 

1/2


2 I



A

2

kT



1/2



2 I



B

2

kT



1/2



2 I



C

2

kT



1/2



N

(g 1 eDe/kT)N (g1,nuc)N (^) 
3 N 6
j 1

1
e


e
j

/2T
j/T
N
(18.44)
(qtransqelectqvibqrotqnuc)N

N!
(with some substitution for special molecules, like homonuclear diatomic
molecules.N in the vibrational partition function represents the number of
atoms in the molecule, to differentiate it from N, the number of molecules in
the system).
Equation 18.44 is certainly complicated. However, we note two things. First,
virtually all of the information required to evaluate equation 18.44 for a mol-
ecule is available experimentally, mostly using various spectroscopic tech-
niques. Second, because equation 18.44 is independent of the identity of the
molecule, it is a relatively simple task to write a computer program to evalu-
ate Qsysfor a given system. As we will see in the next section, once we have an
expression for Qsys, we will be able to derive expressions for various thermo-
dynamic properties. These expressions can also be evaluated by calculator or
computer program. In fact, an important use of these equations of statistical
thermodynamics is not to verify that they yield numbers that agree with ex-
periment, but to predictthe thermodynamic properties at different conditions
or for new substances whose thermodynamic properties have not been
measured.

18.8 Thermodynamic Properties of Molecules from Q

Recall that the major aim of statistical thermodynamics is to be able to calcu-

late the thermodynamic properties of systems using the mathematics of statis-

tics. It has taken us some time and effort to get to this point, because we first

had to determine the forms of the partition functions for a molecule. Having

done that now, we can turn our attention to thermodynamic properties.

First, we will state that even though the exact expression for the partition

function Qis somewhat expanded from the partition function qfor a

monatomic gas, the basic relationships between Qand various thermodynamic

functions are the same. That is,

EkT^2 



ln

T

Q

V (18.45)

pkT





ln

V

Q



T

(18.46)

HkTT





ln

T

Q



V

(^1)  (18.47)