where f is the force constant of the bond and μ the reduced mass, given by (m 1 m 2 )/(m 1 + m 2 ).
It is important to note that the stiffer the bond or the lighter the atoms, the higher is the wavenumber of
the fundamental and vice versa. Qualitative conclusions can therefore be reached from the relative
positions of absorption bands in the infrared region (vide infra).
Polyatomic Molecules
For polyatomic molecules, whose spectra are usually recorded as liquids, solids or in solution, the
complexity of the spectrum increases rapidly with the number of atoms N in the molecule. The number
of fundamentals, or normal modes of vibration is given by 3N – 6 and 3N – 5 for bent and linear
molecules respectively. A normal mode of vibration is defined as the movement of all the atoms of the
molecule in phase. In the case of polyatomics, the number of fundamentals includes both stretching and
bending vibrations, the latter involving changes of bond angle. There are associated overtones with each
fundamental and, in addition, other absorption bands may arise due to interactions between certain of
the fundamentals and overtones. These are known as combination and Fermi resonance bands. The
overall appearance of the spectrum is that of a large number of resolved and partially resolved bands of
varying intensity (Figures 9.17 to 9.24). The most intense bands are fundamentals in which there is a
large change of dipole moment during the vibration. Conversely, a band may be weak or absent if the
dipole change is small or zero. Many bands can be assigned to the vibration of particular chemical
groups within a molecule, the wavenumber being largely independent of the rest of the structure. Such
characteristic vibration frequencies are of paramount importance in obtaining structural information
and in the identification of unknown compounds. For the most part it is in the analysis of organic and
organometallic compounds that infrared spectrometry has had its greatest impact.
Characteristic Vibration Frequencies
The occurrence of characteristic vibration or group frequencies can be explained in terms of relative
masses and of force constants using the classical analogy of weights and springs as depicted in Figure
9.14.
(a)—
Mass Effect
If the masses A, B, C, D... are all similar and the force constants f 1 , f 2 , f 3... are of the same magnitude,
the vibrations of individual atoms are strongly coupled with the result that no band can be assigned
solely to any particular group of atoms. If, however, the mass of atom A is considerably smaller than
those of the other atoms, one mode of vibration will involve the stretching of the bond between A and
the rest of the molecule. The system