Page 185 - Academic Press Encyclopedia of Physical Science and Technology 3rd Analytical Chemistry

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Encyclopedia of Physical Science and Technology EN007C-340 July 10, 2001 14:45

796 Infrared Spectroscopy

FIGURE 6 Infrared spectrum of benzonitrile in a 0.01-mm NaCl
cell.

units are used, the wave number ¯ν of the radiation that has FIGURE 7 Normal modes of vibration of carbon dioxide.
the same frequency as the molecular vibration is given by
1/2
1 1
internal degrees of freedom change the size or shape of
¯ ν = 1303 k + . (5) the molecule without rotating it or translating its center of
m 1 m 2
mass.
Some examples of diatomic molecules that absorb in the
It can be shown by the methods of classical mechanics
IR are listed in Table I. Equation (5) relates the wave num-
that the 3n − 6 (or 3n − 5) internal degrees of freedom of
ber of the absorption band given to the force constant and motion correspond to 3n − 6 (or 3n − 5) different normal
atomic masses of the molecule. modes of vibration. In a normal mode of vibration the
Cartesian displacement coordinates of every atom change
periodically, each oscillating with the same frequency and
D. Polyatomic Vibrations
passing through the equilibrium configuration at the same
Polyatomic molecules have more than one vibrational fre- time. The molecule does not translate its center of mass
quency. The number can be calculated from the follow- or rotate.
ing. One atom in the molecule can move independently in The vibrational form can be described by specifying
three directions, the x, y, and z directions in a Cartesian the relative amplitudes of the Cartesian displacements of
coordinate system. Therefore, in a molecule with n atoms, each mass (Fig. 7). The vibration can also be described in
the n atoms have 3n independent ways they can move. terms of the relative changes in the internal coordinates
The center of mass of the molecule can move in three in- of the molecule, namely, changes in the bond lengths and
dependent directions, x, y, and z. A nonlinear molecule bond angles. For example, in Fig. 7 the CO 2 molecule
can rotate in three independent ways about the x, y, and has two bonds; in one vibration, both bonds stretch at the
z axes, which pass through the center of mass. A linear same time (inphase stretch), whereas in another vibration
molecule has one less degree of rotational freedom since one bond stretches while the other bond contracts (out-
rotation about its own axis does not displace any atoms. of-phase stretch). There are two mutually perpendicular
These translations of the center of mass and rotations can bending vibrations that have the same frequency.
be performed with a rigid molecule and do not change its For each normal mode of vibration, a single coordinate
shape or size. Substracting these motions, there remain can be defined called a normal coordinate. When one nor-
3n − 6 degrees of freedom of internal motion for non- mal mode of vibration is activated, one normal coordinate
linear molecules and 3n − 5 for linear molecules. These periodically changes in value. At the same time that the
normal coordinate changes, each Cartesian displacement
coordinate changes in a specified proportion (positive or
TABLE I Diatomic Molecules negative) to the change in the normal coordinate, so that
Wave number Force constant the resulting motion is a normal mode of vibration. Nor-
Molecule (cm −1 ) (mdyne/ ˚ A) mal coordinates are very useful for theoretical studies of
molecular vibrations.
CO 2143 18.6
NO 1876 15.5
HF 3962 8.9 E. Infrared Absorption
HCl 2886 4.8
In a spectrometer, a source of IR radiation sends all IR
HBr 2559 3.8
wavelengths of interest through a sample. The IR radiation