Physical Chemistry , 1st ed.

where now the quantum number for the molecule’s z-component rotational

motion is represented by MJ.MJis bounded by J, just as mwas bound by .

The total rotational energy of the molecule, equation 14.7, is determined only

by Jand not by MJ, the zcomponent. Each rotational energy level is thus de-

generate by a factor of 2J 1. Equations 14.8 and 14.9 are applicable not only

to diatomic molecules but to all linear molecules. (For polyatomic linear mol-

ecules, however, the reduced mass is more difficult to calculate. We will not

deal with such molecules here.)

Nonlinear molecules can rotate in threeindependent, mutually perpendic-

ular directions, as illustrated in Figure 14.4. However, there is absolutely no

guarantee that the rotation in one dimension is equivalent to rotations in the

other two dimensions. For most molecules, all three rotations are different spa-

tial motions. Also, the moment of inertia Ifor each dimension of each rota-

tion is usually different. This makes the rotations of nonlinear molecules

somewhat complicated. In general, there will be three different independent

rotations, but even consideration of those can get substantially complicated.

Generally, such asymmetric nonlinear molecules are treated using various

levels of approximation to more symmetric systems, a sort of perturbation-

theory kind of approach. We will not concern ourselves with such systems here.

Nonlinear molecules that have certain symmetry elements may, however,

qualify for simpler treatment (leading, ultimately, to a better understanding of

their properties). The key is the value of the moment of inertia of the mole-

cule in the three perpendicular directions. One can always define a set of axes

so that the totalmoment of inertia of the molecule can be described using

three perpendicular components (one of which is exactly zero for linear mol-

ecules). These axes are called the principal inertial axes,or simply principal

axes,of the molecule. One of the principal axes always lies coincident with the

highest-order symmetry axis, if one exists. If all three moments of inertia are

the same, then the molecule is called a spherical topand the rotational energy

of the molecule is quantized and given by

Erot, spherical

J(J

2 I

1)^2

 (14.10)

This is the same expression used to define the energy levels of diatomic (or

other linear) molecules. It can be shown that any molecule that has two or

more noncoincident threefold or higher axes is a spherical top. Examples in-

clude methane (CH 4 ), sulfur hexafluoride (SF 6 ), and cubane (C 8 H 8 ).

If the molecule has three different moments of inertia, it is called an asym-

metric top.The rotations of such molecules are complicated. As mentioned

above, specific treatment is beyond the scope of this text, although this defin-

ition includes most molecules.

If a nonlinear molecule has a single threefold or higher axis, then it will have

two of its three moments of inertia equal. Such molecules are called symmet-

ric tops.There are two types of symmetric tops. If the two equal moments of

inertia are lower than the unique moment of inertia, then the molecule is

called an oblate top.If the two equal moments of inertia are higher than the

unique moment of inertia, then the molecule is called a prolate top.Generally,

oblate tops are flat and round, like a disk, and prolate tops are long and nar-

row, like a cigar. (See Figure 14.5.) Benzene (C 6 H 6 ) is an oblate top, whereas

ethane (CH 3 CH 3 ) is a prolate top. The symmetry classification of many mol-

ecules is easy to do by inspection. However, in many cases, explicit calculation

of the moments of inertia are necessary. NH 3 is one example since it is difficult

14.4 Rotations in Molecules 467

Figure 14.4 A nonlinear molecule has three
different rotational motions. The symmetry of
the molecule determines whether any or all of
them are equivalent to each other.

H

H

H

H C CCC

(a)

(b)
Figure 14.5 (a) A prolate symmetric top is
long and cylindrical, like methyldiacetylene.
(b) An oblate symmetric top is disk-shaped, like
benzene.