Physical Chemistry , 1st ed.

study of the rotational-vibrational spectra of nonlinear polyatomic molecules

is as rewarding as it is complex.

Example 14.21

The peaks representing the Rbranch get closer and closer together as Jincreases.

a.Use equation 14.41 to estimate at what rotational state the rovibrational

lines will cease to be separated from each other, and will start moving to

lower energy. (This point is called the band headof the series of absorptions.)

To simplify the problem, neglect the centrifugal distortion term. Use B 0 

10.44 cm^1 and B 1 10.14 cm^1.

b.What is the maximum populated rotational level of HCl at 298K? (Use

equation 14.23.)

c.On the basis of the previous answer, would one expect to observe the band

head in a spectrum measured at this temperature?

Solution

a.Using the expression from equation 14.41, the separation between two ad-

jacent absorptions, labeled (E), is independent of the harmonic vibra-

tional frequency and the anharmonicity and is equal to

(E) (B 1  B 0 ) (B 1 B 0 )(2J 3)

where (E) is the symbol for the difference between the Evalues of the

spectrum. Evaluating the sum and difference of the Bvalues in the above ex-

pression:

(E) 20.58 cm^1 0.30(2J 3) cm^1

This separation is usually positive, since the first term normally overwhelms

the second, negative term. However, at a high enough Jvalue, the second term

will cancel out the first term so that the lines in the spectrum will first over-

lap and then start progressing toward higher energies. We want the value of

Jwhere (E) is about zero:

0 20.58 cm^1 0.30(2J 3) cm^1

Solving for Jalgebraically yields

J32.8  33

for the approximate position of the band head.

b.Using

Jmax (^) 
2
kT
B

1/2
we will first have to convert one of the Bvalues to appropriate units. We get
B10.44 cm^1 

2.9979

s

1010 cm

6.626 

10 ^34 J s 2.074 

10 ^22 J

where we have arbitrarily used the ground-state B 0. Substituting:

Jmax (^)

1/2
All of the units cancel, so that we are taking the square root of a unitless num-
ber. We get
Jmax 4.5

(1.381

10 ^23 J/K)(298K)



2(2.074 

10 ^22 J)

510 CHAPTER 14 Rotational and Vibrational Spectroscopy