Physical Chemistry , 1st ed.

won’t go into). The symmetry number is ultimately equal to the number of

pure rotational symmetry operations in the point group of the molecule. In

our discussions, symmetry numbers will be given in all examples. In the limit

of high energies, which imply high Jvalues, the “+ 1” in the degeneracy is neg-

ligible and the summation can be replaced with an integral. We therefore have

qrot



1





0

4 J^2 exp

J(J

2

IkT

1)^2

dJ

This integral has a known solution, and in terms of the variables in the above

expression, the high-temperature limit for qrotof a spherical top becomes

qrot



1/2



2



Ik

2

T



3/2

(18.36)

If we define the rotational temperature rfor a spherical-top polyatomic mol-

ecule as

r 

2



I

2

k

 (18.37)

we have

qrot



1/2





T

r



3/2

(18.38)

Although the mathematical processes for symmetric tops and asymmetric

tops are less straightforward, the ultimate expressions for the rotational parti-

tion functions are variants of equation 18.36 (or 18.38). For a symmetric top,

two of the three moments of inertia are equal. The rotational partition func-

tion for a symmetric top is

qrot



1/2



2 Idup



lic

2

atekT



2 Iun



iq

2

uekT



1/2

(18.39)

In equation 18.39,Iduplicaterefers to the two moments of inertia that are equal,

and Iuniquerefers to the unique moment of inertia (either IAor IC, depending

on whether the molecule is a prolate or an oblate top, respectively). Equation

18.39 is applicable to both oblate and prolate tops, with the proper application.

For an asymmetric top in which all three moments of inertia are different,

we have

qrot



1/2



2 I



A

2

kT



1/2



2 I



B

2

kT



1/2



2 I



C

2

kT



1/2

(18.40)

where IA,IB, and ICrepresent the three different moments of inertia. In terms

of rotational temperatures, equations 18.39 and 18.40 can be written as

qrot



1/2





T

r, A





T

r, C



1/2

(18.41)

for a symmetric top. For an asymmetric top:

qrot



1/2





T

r, A



1/2



T

r, B



1/2



T

r, C



1/2

(18.42)





1/2



r, A

T

r,

3

Br, C



1/2

Table 18.4 lists rotational constants and symmetry numbers for some molecules.

18.6 Polyatomic Molecules: Rotations 635

Table 18.4 Rotational temperatures rfor
some polyatomic molecules
Molecule
(symmetry number) r(K)
H 2 O (2) 13.4, 20.9, 40.1
CO 2 (2) 0.561
NH 3 (3) 13.6, 13.6, 8.92
CH 4 (12) 7.54
CCl 4 (12) 0.0823
NO 2 (2) 0.590, 0.624, 11.5