23.3 Rotational and Vibrational Spectra of Diatomic Molecules 971
as a function of reciprocal wavelength is produced by connecting these points. The
spectrum is obtained rapidly and it is possible to take the spectrum repeatedly and to
add the separate spectra, allowing weak spectral lines to be seen and allowing random
“noise” to be reduced by the averaging procedure. An interferogram can be taken in less
than one second using a deuterated triglycine sulfate (DTGS) detector, and as many as
20 or 30 interferograms can be taken in a second with a mercury–cadmium–telluride
detector.PROBLEMS
Section 23.3: Rotational and Vibrational Spectra of
Diatomic Molecules
23.17Using an identity from Appendix F, derive the selection
rule for a harmonic oscillator,∆v0,±1.
23.18From the spectroscopic constants of the CO molecule,
find the value of the force constant of the CO bond.
Typical force constants are near 500 N m−^1. Comment on
the size of your answer.
23.19a.Find an expression for the reciprocal wavelengths of
the microwave spectrum of a diatomic molecule
including all of the terms in Eq. (22.2-50). Assume
thatv0.
b.Find the reciprocal wavelengths and the wavelengths
for the first four lines in the microwave spectrum of
CO using your expression of part a. Assume that
v0. The values of the parameters are in
Table A.22 of Appendix A. Compare your values
with those obtained with the approximation of
Eq. (23.3-5).23.20a.Find the wavelength of the first line of theRbranch of
the IR spectrum of HBr, using the harmonic
oscillator–rigid rotor approximation.
b.Repeat the calculation of part a including the
corrections for anharmonicity, centrifugal stretching,
and rotation–vibration interaction.
23.21a.Find the wavelength of the band center of the first
overtone vibrational band of the HF molecule.
̃ve 4138 .5cm−^1 , and ̃Be 20 .956 cm−^1.
Neglect the correction terms in the energy level
expression.
b.Find the splitting in cm−^1 between the individual
lines in theRbranch of this band.
c.Find the wavelengths of the first line in thePbranch
and the first line in theRbranch of this band.23.22Find a formula for the rotational frequency (number of
revolutions per second) of a rigid diatomic molecule
assuming that classical mechanics holds, but that the
angular momentum happens to have the magnitude
h ̄√
J(J+1). Compare this formula with that for the
frequency of a photon absorbed when a quantum-
mechanical molecule makes a transition fromJtoJ+1.
Show that the two frequencies are nearly equal for large
values ofJ.
23.23Using information on the normal HF molecule from Table
A.22 of Appendix A, predict the reciprocal wavelengths
of the absorptions in the microwave spectrum of DF,
where D is deuterium,^2 H. Assuming equal oscillator
strengths, find the line of maximum intensity at 298 K.
23.24Which of the following substances will have a microwave
spectrum?
a.CO 2
b.N 2 O
c.CCl 4
d.CHCl 3
e.CH 2 Cl 2
23.25Which of the following substances will have a microwave
spectrum?
a.CH 3 Cl
b.BH 3
c.NH 3
d.C 2 H 4
e.SO 3
23.26Using 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^19 F. Find the