Physical Chemistry , 1st ed.

up to 3N6 terms. One way of writing this is to use the symbol;qvibcan

be written as

qvib

3 N 6

j 1





i 1

gieE^ j/kT (18.22)

As complicated as this expression might seem, it is simply the product of

3 N6 individual vibrational partition functions. Rather than repeat the de-

rivation of the vibrational partition function, we will simply take the result from

the case of the diatomic molecule and state that, for a polyatomic molecule,

qvib

3 N 6

j 1

(18.23)

Because we are considering all 3N6 vibrations explicitly, the degeneracies gi

are all 1 and gino longer appears in equation 18.23. For molecules, we rarely

invoke the high-temperature limit, so equation 18.23 is the preferred expres-

sion for qvib. Nonlinear polyatomic molecules have up to 3N6 different vi-

brational temperatures (there can be fewer independent values ofvif the

vibrations are doubly or triply degenerate), so in terms of the vvalues equa-

tion 18.23 is

qvib

3 N 6

j 1

(18.24)

Table 18.2 lists a few vibrational temperatures for some small molecules. Note

in equation 18.24 that the vibrational temperatures (v,j) have two labels, one

to indicate that it is a vibrational temperature, and one to indicate to which

vibration of the molecule it refers.

There are two points to consider in light of equation 18.23 or 18.24. First,

the more atoms a molecule has, the more terms will be in the product (because

as Nincreases, 3N6 increases). Second, since we should suspect that qvibwill

have some effect on the thermodynamic properties of the gas, we might also

think that as the number of atoms in the molecule increases, the thermo-

dynamic functions will deviate more from monatomic gas thermodynamic

values. This is indeed the case, as we will see in a few sections. This is one rea-

son why we confined ourselves to monatomic gases as examples in our earlier

treatments. This is also a reason why it was difficult to classically predict ther-

modynamic properties of molecules: molecules have other ways to distribute

energy. This can have a major impact on their thermodynamic properties.

Example 18.5

Determine qvibfor H 2 O at 298 K, given that its normal modes of vibration are

3720, 3590, and 1590 cm^1. Rationalize whether or not the high-temperature

expression for qvibcan be used. Assume all frequencies are singly degenerate.

Solution

The first thing to do is to calculate the three vvalues for H 2 O. We use the

definition ofvin equation 18.18, and recognize the fact that we need to

express the three vibrational frequencies in units of s^1. Using hand cap-

propriately, we find that the three vibrational frequencies are 1.115 

1014 ,

1.076 

1014 , and 4.767 

1013 s^1 , respectively. Now, using equation 18.18,

v, 15350 K

(6.626 

10 ^34 Js)(1.115 

1014 s^1 )



1.381 

10 ^23 J/K

ev,j/2T



1 ev,j/T

eh^ j/2kT



1 eh^ j/kT

18.4 Molecules: Vibrations 627

Table 18.2 Vibrational temperatures vfor
some polyatomic molecules
Molecule v(K) [degeneracy, if1]
H 2 O 2287, 5163, 5350
CO 2 954 [2], 1890, 3360
NH 3 1360, 2330 [2], 4800, 4880 [2]
CH 4 1870 [3], 2180 [2], 4170, 4320 [3]
CCl 4 310 [2], 450 [3], 660, 1120 [3]
NO 2 1900, 1980, 2330