Physical Chemistry , 1st ed.

b.Calculate the rotational constants A,B, and C.

c.What is the value of the lowest nonzero rotational energy?

d.What is the value of the next lowest nonzero rotational energy?

Solution

a.Ammonia is an oblate symmetric top, having IaIbIc. Therefore, the

higher of the two defined inertial moments is the unique one:Ic4.413

10 ^47 kg m^2. The lower of the inertial moments is both Iaand Ib:IaIb

2.806 

10 ^47 kg m^2.

b.From equations 14.11–14.13 above (and recognizing that since IaIb,

then AB):

AB

2



I

2

a

1.982

10 ^22 J

C

2



I

2

c

1.260

10 ^22 J

c.The lowest nonzero value of the rotational energy has J1 and K1 for

an oblate top. Therefore, from equation 14.17:

ErotBJ(J 1) (CB)K^2

1.982 

10 ^22 J 1(1 1) (1.260 

10 ^22 J 1.982 

10 ^22 J)1^2

Elowest3.242 

10 ^22 J

d.The next lowest rotational energy level has the same value for J,but K0.

Substituting these quantum numbers into a similar expression, one gets

Enext lowest

1.982 

10 ^22 J 1(1 1) (1.260 

10 ^22 J 1.982 

10 ^22 J)0^2

The second term is zero, making the total energy dependent on only the first

term of the expression:

Enext lowest3.964 

10 ^22 J

It is perhaps counterintuitive that a higher quantum number leads to a lower

quantized energy.

14.5 Selection Rules for Rotational Spectroscopy

Given an understanding of the energy levels of molecules due to various types
of motions, it is a simple step to consider the spectroscopy involving those
motions, because spectroscopy involves a transition from one energy level to
another. Spectroscopy uses the Bohr frequency condition, originally proposed
by Neils Bohr in his theory of the hydrogen atom. This was discussed in sec-
tion 14.2. Since the energy levels of rotations for simple molecules are known
from the treatment above, determining the change in the energy levels is
straightforward.
Classically, light is an oscillating electromagnetic field that will interact
with other oscillating electromagnetic fields, much the same as two magnets
or electrical charges will interact with each other. Consider a molecule whose
only motion is rotation so that we can consider a “pure” transition involving
only rotational quantum levels. Such a rotating molecule does not provide an
oscillating electromagnetic field unless the molecule has a permanent dipole

(6.626 

10 ^34 J s)^2



(2)^22 4.413 

10 ^47 kg m^2

(6.626 

10 ^34 J s)^2



(2)^22 2.806 

10 ^47 kg m^2

14.5 Selection Rules for Rotational Spectroscopy 471