Physical Chemistry Third Edition

972 23 Optical Spectroscopy and Photochemistry

reciprocal wavelength of the center of the

(high-temperature) band corresponding to the transition

fromv1tov2.

23.27Using the ̃veand ̃vexevalues from Table A.22 of
Appendix A, find the reciprocal wavelength of the band
center of the fundamental band, the first overtone band,
and the second overtone band of^1 H^81 Br. Find the
reciprocal wavelength of the center of the “hot band”
(high-temperature band) corresponding to the transition
fromv1tov2.

23.28Using values of parameters in Table A.22 of Appendix A,
find the reciprocal wavelength of the radiation absorbed
in the transition from thev0,J6 state to thev1,
J7 state of^1 H^81 Br. To which branch does this line
belong, and how many lines lie between it and the band
center?

23.29Assuming that the transition dipole moments are roughly
equal, estimate the temperature at which the spectrum of
Figure 23.9 was taken.

23.30a. Using the full expression for the energy levels in
Eq. (22.2-50) obtain the expression for the recip-
rocal wavelengths of the lines in thePandRbranches
of the fundamental band of a diatomic molecule.
b. Use the expression derived in part a to find the
reciprocal wavelength of the first line of thePbranch
and the first line of theRbranch of the fundamental
band of CO.

c.Obtain the expression for the reciprocal wavelength of

the band center of the (n+1)th harmonic (nth

overtone) and find the reciprocal wavelength of the

band center of the third and fourth overtones for the

CO molecule.

23.31If the energy level expression of Eq. (22.2-50) is used, the

reciprocal wavelengths of the lines in theRbranch of the

first overtone band are given by

̃vR

1

λR

 2 ̃ve− 6 ̃vexe+ 2 ̃Be(J+1)− 4 D ̃(J+1)^3

− ̃α(2J^2 + 7 J+4)

a.Verify this equation.

b.Obtain the analogue of this equation for thePbranch

of the first overtone band.

c.Find the reciprocal wavelength of each of the first

three lines in theRbranch of the first overtone of the

CO molecule.

23.32a.Find the Fourier transformc(ω) of the function

I(t)Ae−(t−t^0 )

(^2) /D
whereA,t 0 , andDare constants.
b.Sketch graphs ofI(t) andc(ω).
c.Explain in physical terms whatc(ω) represents ifI(t)
represents the intensity of a pulse of radiation as a
function of time. Explain whyc(ω) depends ont 0 as it
does and describe what happens ift 0 0.

23.4 Electronic Spectra of Diatomic Molecules

The electronic spectra for most diatomic molecules are found in the ultraviolet and

visible regions. The electronic transitions are usually accompanied by rotational and

vibrational transitions. The following selection rules are obtained for electronic tran-

sitions in diatomic molecules:^13

∆Λ0,± 1 (23.4-1a)

∆S 0 (23.4-1b)

Parity of electronic state changes: (u→gorg→u) (23.4-1c)

(^13) I. N. Levine,op. cit., p. 298ff (note 8).