Physical Chemistry Third Edition

22.4 The Rotation and Vibration of Polyatomic Molecules 939

Symmetric stretch

n~ 15 1340 cm^21

Symmetric stretch

n~ 15 1151.2 cm^21

Asymmetric stretch

n~ 35 2349 cm^21

Asymmetric stretch

n~ 35 1361 cm^21

Bend (2 of these)

n~ 25 667 cm^21

Bend

n~ 25 519 cm^21

(a)

(b)

Figure 22.6 Vibrational Normal Modes.(a) Carbon dioxide. (b) Sulfur dioxide.

Figure 22.6 shows schematically the motion corresponding to the four normal modes

of carbon dioxide (linear) and the three normal modes of sulfur dioxide (nonlinear),

and shows the frequencies divided by the speed of light, given in cm−^1. These values

are sometimes called “frequencies” and the unit is called “wave numbers.” The arrows

in the diagrams show the direction of motion of each nucleus away from its equilibrium

position during one-half cycle of the concerted motion. As each nucleus oscillates, it

first moves in the direction indicated and then reverses.

Each triatomic molecule has a normal mode called asymmetric stretchin which

both bonds shorten and lengthen simultaneously. There is also anasymmetric stretch

in which one bond lengthens while the other shortens. A linear triatomic molecule

such as carbon dioxide can bend in two perpendicular directions, so there are two

bending modes, which have the same frequency. A nonlinear triatomic molecule has

only one bending mode, because it can bend only in the plane of the molecule. A motion

perpendicular to the plane of the molecule would have no restoring force to make it

oscillate, and therefore corresponds to a rotation, explaining why there is one more

normal mode for a linear molecule than for a nonlinear molecule. There is a common

pattern of frequencies for triatomic molecules that is shown in Figure 22.6: Asymmetric

stretches usually have the highest frequency, symmetric stretches are often somewhat

lower in frequency, and bends nearly always have the lowest frequency. Some software

packages such as Spartan generate movies showing the normal mode motions.

For a molecule with more than three atoms, there are numerous normal modes,

and we do not attempt to describe all of them. Benzene, with 12 nuclei, has 30 normal

modes, including a “breathing mode” in which the ring alternately contracts and swells,

and a “pseudorotation” in which a kind of puckered wave moves around the ring.

Various techniques, including group theory, are used in studying the normal modes of

polyatomic molecules.^7

In some large molecules, some of the normal modes correspond to relatively large

oscillations of one bond length or bond angle while other bond lengths and bond angles

do not oscillate or oscillate with smaller amplitudes. The frequency of such a normal

mode is often nearly the same for the same pair of elements in different compounds. For

example, most hydrocarbons exhibit a C–H stretching frequency in the 2850 cm−^1 to

3000 cm−^1 range, and compounds with an O–H bond usually exhibit an O–H stretching

frequency in the 3600 cm−^1 to 3700 cm−^1 range. Table A.23 of Appendix A lists a few

such characteristic frequencies. Organic chemistry textbooks give longer lists.

(^7) I. N. Levine,Molecular Spectroscopy, John Wiley and Sons, New York, 1975, p. 427ff.