# Gas Chromatography - Tandem Techniques - The Characteristics of Infrared Absorption > Vibrational Rotational Spectra > Page 25

(6)

where (I) is the moment of inertia, of the molecule about its axis of

rotation,

** ** and (J) is the rotational quantum number

In the transition of a molecule from an upper level (J') to a lower level (J") the energy transmitted will be,

which assumes the moment of inertia (I) will be he same at the two levels. Generally, rotational energies are restricted to those in which DJ =1 so that,

J'-J" =1.

Thus,

According to the quantum theory,

Thus,

where (J') has been replaced by (m) representing any integer.

## Vibrational Rotational Spectra

According to the equations of wave mechanics, the energy (ep) of a harmonic oscillator (one in which the restoring force is proportional to the displacement) is given by,

(7)

As a general rule, the movement is not strictly harmonic but, for simplicity, it will assumed to be in this argument. It may also be assumed that the rotational and vibrational energies can be added and so from equations (6) and (7),

For the simultaneous vibrational and rotational transition from p' to p" and from J' to J" it follows that,

For the fundamental band p'-p"=1 and the change in the rotational quantum number, as in pure rotational spectra, is restricted to unity.

Then J'- J" = 1, and,

Thus, as e'-e" = hn

The first term on the right hand side of the equation gives the origin or center of the fundamental band and the second term gives the rotational fine structure. Examples of the vibrational modes of a non-linear molecule are shown in figure 15.

**Figure 15. Vibrational Modes of Non-Linear Molecules**