Physical Chemistry , 1st ed.

Example 14.11

Determine the number of vibrational degrees of freedom for the following

molecules.

a.Hydrogen chloride, HCl

b.Hydrogen sulfide, H 2 S

c.Benzene, C 6 H 6

d.Acetylene, C 2 H 2

e.The sulfate ion, SO 42 

f.Hydrogen peroxide, H 2 O 2

Solution

a.By definition, HCl is linear, therefore it has 3(2)  5 1 vibrational de-

gree of freedom.

b.H 2 S is not linear, so it has 3(3)  6 3 vibrational degrees of freedom.

c.Benzene has 3(12)  6 30 degrees of freedom. Its high symmetry will

simplify matters, though.

d.C 2 H 2 is linear, so it has 3(4)  5 7 vibrational degrees of freedom.

e.The sulfate ion is not linear: 9 vibrational degrees of freedom.

f.Hydrogen peroxide is nonlinear: 6 vibrational degrees of freedom.

The solutions above hint that symmetry has a great deal to do with the num-

ber of truly unique vibrational degrees of freedom. Consider benzene, which is

planar and has D6hsymmetry. Because of its symmetry, certain vibrations of

benzene are identical to each other and have the same vibrational frequency.

This means that there will be fewer than 30 unique vibrational frequencies in

this molecule. (There are in fact only 20 unique frequencies.) Symmetry will

have similar ramifications for vibrations in other molecules, too.

14.9 The Normal Modes of Vibration

The vibrations of all molecules can be described in terms of independent mo-

tions such that for each motion the frequency of vibration for all atoms is the

same. These are the normal modes of vibration. Why are the normal modes so

important? There are several reasons, but perhaps the most important one is

this: to a good approximation, the frequencies of light that are absorbed due

to vibrational motions of atoms in molecules are those that have the same fre-

quencies as the normal modes of vibration.

Consider the vibration of the two atoms in the HCl molecule (Figure 14.23).

There is only a single vibrational mode, where the hydrogen and chlorine

atoms are alternately closer and farther away from each other (a “stretching

mode”). For simplicity, the arrows indicating an atom’s direction of motion

point in a single direction only. It is understood that in the course of a full

vibration the atoms move in the opposite direction also. Note too that the

lengths of the arrows are different: the hydrogen atom is shown “moving” far-

ther than the chlorine atom. This preserves the position of the center of mass

of the molecule, so it does not change.

A single vibration agrees with the 3N5 expression for the number of vi-

brations in a molecule: for HCl, 3N5 is 3(2) 5, which equals 1. The two

atoms in HCl can be thought of as a classical harmonic oscillator composed of

two masses (the two atoms) connected by a spring (the chemical bond). This

harmonic oscillator vibrates with a frequency of about 8.65 

1013 Hz, or

14.9 The Normal Modes of Vibration 483

H Cl

Figure 14.23 The single vibration of HCl has
the hydrogen and chlorine atoms moving alter-
nately back and forth. The hydrogen’s movement
is much larger than the chlorine’s, because of H’s
much higher mass.