Selection Rule: Position of Rotational Lines

1. Selection Rule: Position of Rotational Lines
According to the quantum mechanical theory, transitions between all rotational states are not allowed. Only transitions between states whose rotational quantum number differ by unity are allowed. Thus the selection rule for rotational transition is:
?J=±1 ….(9)
For absorption ?J=1 and hence, in equation (7) J =J+1. Thus equation (7) may be written as:
…..(10)

For transition from the lowest state (J=0), the rotational is observed at 2B. The successive rotational lines are observed at frequencies 4B, 6B, 8B … for J=1,2,3…, respectively. Evidently, the energy (frequency) spacing between two successive rotational lines is 2B.

2. Intensity of Rotational Lines: Maxwell Boltzmann Distribution
The intensity of a rotational line depends on the number of molecules present in the initial state J. According to Maxwell Boltzmann Distribution law, the number of molecules (NJ) at a rotational level J is given by:
…(11)

Note that the factor 2J+1 arises from the degeneracy of the rotational levels. For the most populated level:


…..(12)
Using this condition it may be easily shown that the quantum number (Jmax) for the most populated level (i.e. NJ highest) is:

…..(13)


3. Non-Rigid Rotator
Though rotational lines are expected to be equispaced, in many cases the energy gap between successive rotational lines decreases slightly with increase in J. This discrepancy is attributed to the non-rigidness of a molecule. It is suggested that with increase in J as rotational energy increases the moment of inertia and hence, bond length increases. When this is taken into account the rotational energy is given by:

….(14)
The constant D is approximately given by:

….(15)
where ? is the frequency of vibration of the molecule. Since B is of the order of 10 cm-1 and ? is ~1000 cm-1, D is ~10-3 cm-1 i.e. very small. Using equation (14) the expression of frequency of rotational line is given by:

….(16)

4. Isotope Effect
On isotopic exchange the reduced mass of a molecule changes. This causes a change in the moment of inertia and hence, the rotational energy. Suppose, for two isotopically substituted molecules of moment of inertia I1 and I2, the frequencies of the rotational lines are ?1r and ?2r. From equation (10) it is easy to see that:

….. (17)