14.2 & 14.3 Selection Rules and
Electromagnetic Light
14.1.Why won’t a rotational spectrum be observed for the
rotation of a linear molecule about its linear axis?
14.2.Determine if the following integrals can be nonzero if
the molecular or atomic system has the given local symmetry.
Use the great orthogonality theorem if necessary.
(a)AuOˆB2uAudin D2hsymmetry
(b)A 1 OˆA 1 A 2 din C3vsymmetry
(c)g^ Oˆggdin Dhsymmetry
(d)EOˆA 2 T 1 din Tdsymmetry
14.3.What is the frequency of light having the following
wavelengths? (a)1.00 m (b)4.77
10 ^5 m (c)7894 Å
(d)1.903
103 m
14.4.The Cu(H 2 O) 62 complex has octahedral symmetry. Is
a transition from a T2gstate to an Egstate allowed if the tran-
sition moment operator has a T1usymmetry label?
14.5.What are the wavelength, speed, and energy of a pho-
ton that has a frequency of 8.041
1012 s^1?
14.6.Show that the wavenumber of any electromagnetic
light is proportional to its frequency.
14.7.Both of the units micron (wavelength) and cm^1 (wave-
number) are common in infrared spectroscopy. Use their def-
initions and relationships to derive a simple equation to con-
vert from one to the other. (Hint:the product of the two values
equals a constant.)
14.4 Rotations
14.8.The silver hydride diatomic molecule,^197 Ag^1 H, has an
internuclear bond distance of 1.617 Å. Predict the energies, in
joules, of its first four rotational levels. (Use Ir^2 .)
14.9.Classify the following molecules as linear, spherical tops,
prolate symmetric tops, oblate symmetric tops, or asymmetric
tops.
(a)Dimethylacetylene, CH 3 –CC–CH 3
(b)Sulfur hexafluoride, SF 6
(c)The phosphate ion, PO 43
(d)Glycine, CH 2 (NH 2 )(COOH)
(e)cis-1,2-Dichloroethylene
(f)trans-1,2-Dichloroethylene
(g)Hexamethylbenzene, C 6 (CH 3 ) 6
(h)Diacetylene, CHC–CCH
(i)The cyanide radical, CN
14.10.Diatomic sulfur, S 2 , was detected in the tail of Halley’s
comet when it last approached Earth in 1985–86. It has a
bond length of 1.880
10 ^10 m. Calculate the value of B, in
units of cm^1 and J, for S 2.
14.11.Calculate the values for Bof SF 6 and UF 6 , which are
both octahedral molecules. The S–F bond distance is 1.564 Å,
and the U–F bond distance is 1.996 Å. Comment on the dif-
ferences between the two Bvalues, keeping in mind that the
atomic weight of S is 32.06 and that of U is 238.0.
14.12.The moments of inertia for phosphine, PH 3 , are 5.478
10 ^47 kg m^2 , 5.478
10 ^47 kg m^2 , and 6.645 10 ^47
kg m^2. Calculate the rotational constants A, B, and Cfor phos-
phine.
14.13.Show that the degeneracy of rotational levels for sym-
metric tops is 2(2J 1) unless K0, for which the degener-
acy is 2J 1.
14.14.Calculate the values of the first five rotational energy
levels of phosphine, PH 3. Use the values of the moments of in-
ertia given in exercise 14.12, above.
14.15.Calculate the values of the first five rotational energy
levels of ethane, CH 3 CH 3 , assuming it is in its energetically
minimal staggered conformation. The moments of inertia for
ethane are 1.075
10 ^46 kg m^2 , 4.200
10 ^46 kg m^2 , and
4.200
10 ^46 kg m^2.14.5 Rotational Selection Rules
14.16.Which of the following molecules should have pure
rotational spectra?
(a)Deuterium, D 2 (D is^2 H) (b)Carbon monoxide, CO
(c)cis-1,2-Dichloroethylene (d)trans-1,2-Dichloroethylene
(e)Chloroform, CHCl 3 (f)Buckminsterfullerene, C 60
(g)Dimethyltriacetylene, CH 3 –CC–CC–CC–CH 3
(h)Cyanotetraacetylene, H–CC–CC–CC–CC–CN
(Such molecules have been detected in interstellar space.)
(i)Nitric oxide, NO (j)Nitrogen dioxide, NO 2.
14.17.The following are sets of rotational quantum numbers
(J, MJ, K). Label each indicated transition as either allowed or
forbidden. Hint:remember the rules for allowed values of the
various quantum numbers.
(a)(0, 0, 0) →(1, 1, 0) (b)(0, 0, 0) →(1, 0, 0)
(c)(3, 2, 1) →(3, 1, 1) (d)(4, 4, 1) →(2, 4, 1)
(e)(5, 4, 0) →(3, 6, 0) (f)(8, 2, 2) →(9, 2, 2)
(g)(7, 4, 2) →(7, 4, 2) (h)(4, 2, 5) →(3, 2, 5)14.6 Rotational Spectroscopy
14.18.Having used a spectrometer to measure a simple ro-
tationalspectrum, you plot it in units of wavenumbers, cm^1.
How do you expect the spectrum to look? Convert the ab-
sorption energies to units of wavelength and replot. Are the
absorptions equally spaced? Why or why not?
14.19.The rotational spectrum of^127 I^35 Cl consists of lines
equally spaced by 0.114 cm^1. Calculate the bond distance
for iodine monochloride.Exercises for Chapter 14 515EXERCISES FOR CHAPTER 14