Physical Chemistry Third Edition

934 22 Translational, Rotational, and Vibrational States of Atoms and Molecules

The other products of inertia,Iyz,Izy,Ixz, andIzx, are defined analogously. Only

three of the products of inertia have distinct values, becauseIyzIzyand so on. For

calculating the moments of inertia and the products of inertia, we neglect the masses of

the electrons and include only the nuclei in the sums since the masses of the electrons

are small compared to those of the nuclei.

There is a theorem stating that for any rigid object it is possible to choose a set

of perpendicular axes such that all products of inertia vanish. Such axes are called

principal axes, and the moments of inertia relative to them are calledprincipal moments

of inertia. For a symmetrical molecule, it is usually possible to assign a set of principal

axes by inspection. To do so, place the axes along symmetry elements as much as

possible. If there is an axis of symmetry that is at least a three-fold rotation axis

(Cnwithn≥3), a set of principal axes is obtained by choosing this rotation axis as

one of the axes and placing the other axes in any mutually perpendicular directions. If

there is a two-fold rotation axis but no higher-order rotation axis, as with a molecule

that hasC 2 vsymmetry, a set of principal axes is obtained by choosing the rotational

axis as one of the principal axes and placing the other two axes in the reflection planes.

Because the principal axes are defined relative to the molecule and rotate with it,

it is customary to call the axes by the lettersA,B, andCinstead ofx,y, andz.By

convention, the axes are ordered so that

IA≤IB≤IC (22.4-4)

If all three of its principal moments of inertia are equal to each other, an object is called

aspherical top. The name “top” is apparently chosen because of the rotating toys by

that name. A tetrahedral molecule such as methane or an octahedral molecule such as

sulfur hexafluoride is a spherical top. Any mutually perpendicular axes passing through

the center of mass of a spherical top are principal axes. If two of the principal moments

of an object are equal, the object is called asymmetric top.Aprolate symmetric tophas

a unique moment of inertia that is smaller than the other two. An American football and

a rugby ball are prolate symmetric tops if the lacing is ignored. Anoblate symmetric top

has a unique moment that is larger than the other two. A discus is an oblate symmetric

top. Any molecule with at least a three-fold rotation axis is either a symmetric top

or a spherical top. If all three principal moments of inertia are unequal, the object

is called anasymmetric top. A bent triatomic molecule such as SO 2 or H 2 Oisan

asymmetric top.

F

F

B

F

B

A

C

A-B plane

Figure 22.4 The BF 3 Molecule and
Its Principal Axes.

EXAMPLE22.8

Show that BF 3 , a trigonal planar molecule, is an oblate symmetric top.

Solution

Orient the molecule as in Figure 22.4, with the molecule in the A–B plane and with one B–F

bond on theAaxis. Let the bond length be calleda.

IA 2 mF[asin(120◦)]^2  2 mFa^2

[

1

2

√

3

] 2



3

2

mFa^2

IBmFa^2 + 2 mF[acos(120◦)]^2 

3

2

mFa^2 IA

IC 3 mFa^2 >IA