Physical Chemistry , 1st ed.

Example 14.2

Water absorbs infrared radiation that has a frequency of 1595 cm^1 .Convert

this wavenumber into a wavelength in microns.

Solution

Rearranging equation 14.6 above, one finds



1

 ̃



Using the given wavenumber:



1595

1

cm^1

0.0006269 cm 6.269 microns or 6.269 

Note how the centimeter unit comes to the numerator of the fraction.

14.4 Rotations in Molecules

Atoms are not considered candidates for rotation. An atom in free space prob-

ably rotates, but how would we be able to tell? At least two atoms, bonded to-

gether and tumbling in space, are necessary in order to consider rotation on a

quantum-mechanical scale. The simplest molecule to consider is a diatomic

gas. Its molecules rotate in space. The rotations are illustrated in Figure 14.3.

Notice that there are only two ways the molecule can rotate, and the two rota-

tions are the same except that the rotational axes are 90° apart. This system is

very much like 3-D rotational motion. The energy levels for the pure rotational

motions of a diatomic molecule are quantized and, to a very good approxi-

mation, given by the expression for the energy of a 3-D rigid rotor from

Chapter 11:

Erot

J(J

2 I

1)^2

 J0,1,2,... (14.7)

where Erotis the rotational energy and Iis the moment of inertia of the

diatomic molecule, which is defined in terms of its reduced massand the

internuclear distance r:

Ir^2

The molecule also has angular momentum, which you would expect it to

have since it is rotating. The quantum number Jis used to define the total an-

gular momentum of the molecule rotating in three dimensions. The total an-

gular momentum of a molecule is given by the same eigenvalue equation from

three-dimensional rotational motion:

ˆL^2 J(J 1)^2  (14.8)

The square of the total angular momentum is the formally quantized observ-

able. In order to get the magnitude of the angular momentum, you must take

the square root of the eigenvalue from equation 14.8. There is also a zcompo-

nent of the total angular momentum for the diatomic molecule, and this

zcomponent is also quantized:

LˆzMJ MJJ (14.9)

466 CHAPTER 14 Rotational and Vibrational Spectroscopy

Figure 14.3 A diatomic (or polyatomic linear)
molecule has only two defined rotational mo-
tions, which are equivalent to each other.