Physical Chemistry Third Edition

964 23 Optical Spectroscopy and Photochemistry

The expression in Eq. (23.3-5) can be improved on by including the correction terms

in the energy level expression of Eq. (22.2-45).

Exercise 23.4

The equilibrium internuclear distance of HCl is 1.275× 10 −^10 m. Find the spacing between the

lines in the microwave spectrum for both^1 H^35 Cl and^1 H^37 Cl. The chlorine atomic masses are

34.96885 amu and 36.96590 amu and the^1 H atomic mass is 1.007825 amu.

The intensity of a given line in a spectrum is determined by the magnitude of

the transition dipole moment for the transition that produces the spectral line and

by the number of molecules occupying the initial state and the final state. At ther-

mal equilibrium the population of a level is given by the Boltzmann distribution of

Eq. (22.5-1):

(Population of energy levelJ)∝gje−EJ/kBT

∝(2J+1)−ehBeJ(J+1)/kBT (23.3-6)

The degeneracy increases and the Boltzmann factore−EJ/kBTdecreases asJincreases,

so the population rises to a maximum and then decreases asJincreases. If the tran-

sition dipole moments for different rotational transitions in the same molecule are

roughly equal, the level with the largest population is the one with the largest absorption

intensity.

Exercise 23.5

a.Find the rotational level with the largest population for HCl molecules at 298 K. The inter-

nuclear distance equals 1.275× 10 −^10 m.

b.Find the rotational level with the largest population for Br 2 molecules at 298 K. The inter-

nuclear distance equals 2.281× 10 −^10 m.

Vibration–Rotation Spectra of Diatomic Molecules

When transitions are observed between vibrational energy levels, infrared radiation

is emitted or absorbed. The vibrational selection rules are derived in the Born–

Oppenheimer approximation by evaluating the transition dipole moment integral

(μx)v′v′′

∫

ψ∗v′̂μ(x)ψv′′dx (23.3-7)

whereψv′andψv′′are two vibrational wave functions, expressed in terms ofxr−re,

and wheremû(x) is the operator for the molecular dipole moment. Since the vibrational

wave functions approach zero rapidly for large magnitudes ofx, taking the limits of

the integral as infinite produces no significant numerical error, but does contribute to

the fact that vibrational selection rules are only approximately correct.

We assume that the dipole moment can be represented by the truncated Taylor series

μ(x)μ(0)+

(

dμ

dx

)

0

x+ ··· (23.3-8)