Rotational spectra, like those in Figures 14.11 and 14.12, show an interest-
ing intensity pattern. The intensity pattern of rotational spectra is due to the
number of molecules having a certain rotational state. This is called the popu-
lationof the state. Because the population of each rotational state is different,
the number of molecules absorbing radiation and becoming excited to the next
state is different. This population difference is responsible for the varying in-
tensities of each rotational spectral line.
The temperature determines the population of the rotational energy levels.
Rotational levels are close enough in energy that thermal energy is sufficient to
cause some of the molecules to be in excited rotational states. Therefore, there
is an increased probability of a transition from those excited rotational states
to the level next highest in J. At some value ofJ, however, the ability of the tem-
perature to thermally populate rotational levels decreases. A statistical treat-
ment of the energy levels indicates that the approximate maximum-populated
Jquantum number,Jmax,isJmax (^)
2
kT
B
1/2
(14.23)
where kis Boltzmann’s constant (1.381
10 ^23 J/K),Tis the absolute tem-
perature of the sample, and Bis the rotational constant (expressed in units of
joules in this equation). This expression is in part due to the fact that the de-
generacy of the rotational levels is 2J 1. If it were not, then the lowest rota-
tional state would always be the most populated rotational state, and such in-
tensity patterns as seen in Figures 14.11 and 14.12 would not appear. Equation
14.23 is an approximation. As a quantum number,Jis limited to whole num-
bers, so some rounding off is usually necessary when using the above equation.
Equation 14.23 allows one to estimate the temperature of a gas-phase sample
from its rotational spectrum. (The effect of temperature on the population of
energy levels will be a major topic of Chapter 17.)
The energy of each rotational state has a degeneracy of 2J 1 due to the
possible values ofMJ. However, in the presence of a strong electric field in a
particular direction, molecules rotating with different quantized values of an-
gular momentum in that direction, indicated by the MJvalues, will have slightly
different energies. Absorptions or emissions due to rotational state transitions
will appear to split into a larger number of lines, and the particular number of
lines for each transition will be determined by the MJ0,1 selection rule.
This phenomenon is an example of the Stark effect,discovered by the German
physicist Johannes Stark in 1913. In 1919, Stark was awarded a Nobel Prize for
this discovery. This effect is another case of behavior that classical mechanics
could not explain but quantum mechanics could. Examples of the splitting of
the energy levels, the additional transitions, and a Stark effect spectrum are
shown in Figures 14.13–14.15. It is important to note that the Stark effect de-
pends on the zcomponent of the angular momentum, identified with MJ, and
is not dependent on the figure-axis component of angular momentum, iden-
tified with K.
Molecules that don’t have a permanent dipole moment are rotating, of
course, but they do not follow the gross selection rule for pure rotational spec-
tra. Their rotations cannot be observed directly using microwave spectroscopy.
We now examine the rotational spectra of symmetric top molecules. The
values for the rotational energy levels are given in equations 14.16 and 14.17
for symmetric tops. The specific selection rules for the change in J,K, and MJ
quantum numbers were given above in equations 14.18–14.20. As with linear
14.6 Rotational Spectroscopy 477
Electric
field off
Electric
field on
J 4
MJ 0
MJ 4
MJ 1
MJ 2
MJ 3
J 3
MJ 3
MJ 0
MJ 1
MJ 2
Figure 14.13 An applied electric field splits
the degenerate same-Jlevels into different energy
levels, depending on the MJvalues. This is called
the Stark effect. The changes in the energy levels
are not necessarily to scale, since the change in
energy is dictated by various parameters.