Physical Chemistry , 1st ed.

over all of the Jvalues, so we can substitute ^12 times the summation over all J’s:

qrot,nuc(2I 1)^2 ^12  

all J’s

(2J 1)eJ(J+1)r/T

Now we can replace the summation with an integral, and perform the same

simplification of the integral as we did for a heteronuclear diatomic molecule.

We g e t

qrot,nuc(2I 1)^2 ^1

2





T

r

(2I^

2 

1

r

)^2 T (18.32)

as the high-temperature limit. This can be separated, approximately, as

qnuc(2I 1)^2 (18.33)

qrot

2

T

r

 (18.34)

for homonuclear diatomic molecules. Notice what the additional symmetry

(that is, a plane of symmetry bisecting the molecule) of the homonuclear di-

atomic molecules does to qrot: it introduces a factor of 2 in the denominator

of the partition function. The factor 2 is called a symmetry number.Symmetry

numbers also appear in rotational partition functions of polyatomic mole-

cules. Table 18.3 lists some rotational temperatures for homonuclear diatomic

molecules.

18.6 Polyatomic Molecules: Rotations

In most cases, for polyatomic molecules the nuclear partition function is again

neglected, since it usually has a very small effect on the overall thermodynamic

properties of polyatomic molecules. (Indeed, the only reason why we had to

consider it for diatomic molecules is because it imposes an obvious, measur-

able effect on various observations, like spectra and thermodynamic proper-

ties to be considered in section 18.8.) In the high-temperature limit, a linear

polyatomic molecule has the same rotational partition function as a homonu-

clear diatomic molecule:

qrot



T

r



where is the symmetry number, which is 1 for nonsymmetric linear mole-

cules (like OCS) and 2 for symmetric linear molecules (like C 2 H 2 ). This ex-

pression is essentially the same as equation 18.34 (except for the presence of

), and ultimately comes from the fact that a linear molecule has only one de-

fined rotational moment of inertia.

A nonlinear polyatomic molecule can have up to three different moments

of inertia, labeled IA,IB, and IC. By convention,IAis less than IB, which is less

than IC. Polyatomic molecules that have some symmetry may have some of

their moments of inertia equal. If all three are equal, then the molecule is called

a spherical top(see Chapter 14) and the rotational partition function can be

written as

qrot



1





J 1

(2J 1)^2 exp

J(J

2 Ik

1

T

)^2

 (18.35)

where the rotational degeneracy for a spherical top is (2J 1)^2 (for reasons we

634 CHAPTER 18 More Statistical Thermodynamics