Physical Chemistry Third Edition

25.4 The Calculation of Molecular Partition Functions 1073

the same formula as for atoms and diatomic molecules, Eq. (25.3-21). The electronic

partition function is obtained by explicit summation as in Eq. (25.4-2), and the one-term

approximation of Eq. (25.4-4) is usually valid if the molecule does not have unpaired

electrons.

The Rotational Partition Function

For a linear polyatomic molecule like acetylene or cyanogen, the rotational energy

levels are the same as those of diatomic molecules in Eq. (22.2-18). Equation (25.4-13)

can be used for the rotational partition function with the appropriate symmetry number

and moment of inertia. The rotational energy levels of nonlinear polyatomic molecules

are more complicated than those of diatomic molecules. The derivation of the rotational

partition function for nonlinear molecules is complicated, and we merely cite the result:^7

zrot

√

π

σ

(

8 π^2 kBT

h^2

) 3 / 2

(IAIBIC)^1 /^2

(

nonlinear polyatomic

substance

)

(25.4-22)

whereIA,IB, andICare the principal moments of inertia as defined in Eq. (22.4-2). The

symmetry numberσis equal to the number of ways of orienting a model of the molecule

such that each nuclear location is occupied by a nucleus of the same isotope of the same

element as in the other orientations, as described in Section 22.4 of Chapter 22. The

inclusion of the symmetry number corresponds to the fact that of the conceivable sets

of values of the quantum numbersJ,M, andKonly a fraction 1/σactually occur. The

rotational partition function of a polyatomic molecule is generally large enough so that

Eq. (25.4-22) is a good approximation. A more accurate expression for spherical top

molecules has been derived.^8

EXAMPLE25.11

Calculate the rotational partition function of CCl 4 at 298.15 K, assuming all chlorine

atoms to be the^35 Cl isotope. The molecule is a spherical top, with bond lengths equal to

1. 766 × 10 −^10 m, and with all three moments of inertia equal to 4. 829 × 10 −^45 kg m^2.

Solution

The symmetry number is counted as follows: With a given Cl at in a fixed position, say on

thezaxis, there are three orientations. There are four ways to place a Cl atom on thezaxis,

so the symmetry number is 3× 4 12. We can write

zrot

√

π

12

(

8 π^2 (1. 3807 × 10 −^23 JK−^1 )(298.15 K)

(6. 6261 × 10 −^34 Js)^2

) 3 / 2

(4. 829 × 10 −^45 kg m^2 )^3 /^2

 31570

This value is large enough to be a very good approximation.

(^7) N. Davidson,op. cit., p. 169ff (note 2).
(^8) R. S. McDowell,J. Quant. Spectrosc. Radiat. Transfer., 38 , 337 (1987).