Physical Chemistry , 1st ed.

Example 14.10

Find the energy of the J 10 →J11 transition for HCl for two cases: as-

sume a pure rigid rotor behavior, and then include the centrifugal distortion

constant. Use the value for Band DJfrom Table 14.2. Compare the two results.

Solution

Using the rigid rotor approximation:

E(21.18 cm^1 )(10 1) 233.0 cm^1

Using the expression for Ethat explicitly includes the centrifugal distortion

constant:

E(21.18 cm^1 )(10 1)

4(5.32 

10 ^4 cm^1 )(10 1)^3

233.0 0.2575 cm^1 232.7 cm^1

where the normal rules for significant figures in calculations have been ap-

plied. Notice that the change in the predicted Eisn’t much, but it does get

progressively larger as the rotational quantum number increases.

Diatomic and linear molecules aren’t the only molecules that experience

centrifugal distortions. Figure 14.17 shows diagramatically what happens to

a water molecule when it rotates at high values ofJ: the atoms are forced to

spread out somewhat due to centrifugal distortion, and this affects the val-

ues of rotational energy, as depicted in Figure 14.18. For prolate and oblate

480 CHAPTER 14 Rotational and Vibrational Spectroscopy

(b) High^ J

(a) Low^ J

No centrifugal

distortion

With

centrifugal

distortion

J  2

Energy

J  3

J  4

J  5

J  6

J  0

J  1

Figure 14.17 Centrifugal effect

on a molecule of H 2 O. Since water

has three different rotations, it will

have three different DJvalues.

Figure 14.18 Effect of centrifugal distortion

on the rotational energy levels. Because the distor-

tion depends on J^2 (J 1)^2 , higher Jvalues show a

larger deviation from ideal rotational energies.

This figure gives the general behavior of the rota-

tional energy levels and is not to scale for H 2 O.