22.4 The Rotation and Vibration of Polyatomic Molecules 941
torsional vibration relative to each other (this is called
internal rotation). Assuming free internal rotation,
how many vibrational normal modes does ethane
have?22.36Calculate the three principal moments of inertia for the
water molecule, assuming a bond length of 96 pm and a
bond angle of 104. 5 ◦. You must first find the location of
the center of mass in the molecule. Assume its isotopes
are^16 O and^1 H. Pick a product of inertia and show that it
vanishes.
22.37a.Calculate the principal moments of inertia for the
chloroform molecule, assuming tetrahedral bond
angles, a C–H bond length of 111 pm, and a C–Cl
bond length of 178 pm. Show that the molecule is an
oblate symmetric top.
b.Find the rotational energy of theJ1,K1 level of
chloroform. The energy of an oblate symmetric top is
given by the formula^8
EJ(J+1)h ̄^2
2 IB
+K^2(
1
IC
−1
IB)
h ̄^2
222.38a.Calculate the principal moments of inertia for the
chloromethane molecule, assuming tetrahedral bond
angles, a C–H bond length of 111 pm, and a C–Cl
bond length of 178 pm. Show that the molecule is a
prolate symmetric top.
b.Find the rotational energy of theJ1,K1 level
of chloromethane. The energy of a prolate symmetric
top is given by the formula^9
EJ(J+1)h ̄^2
2 IB
+K^2(
1
IA
−1
IB)
h ̄^2
222.39Give the number of vibrational normal modes for each
molecule:
a.C 6 H 6
b.C 2 N 2
c.C 2 H 4
d.C 2 H 6
e.C 8 H 18
f.C 2 H 2
22.40Give the number of vibrational normal modes for each
molecule:
a.SF 6
b. BH 3
c.NH 3
d.C 6 H 12
e.H 2 CO
f. CH 3 COCH 3
22.41The covalent radius of an atom is an average value
derived from bond lengths in various compounds. The
covalent radius of uranium is 142 pm and that of fluorine
is 72 pm. Use these values to estimate the bond lengths in
uranium hexafluoride. Calculate the principal moments of
inertia of a UF 6 molecule. Find the energy and the
degeneracy of the first excited rotational level of a UF 6
molecule.
22.42Give the symmetry number and the number of vibrational
normal modes for each of the following molecules in its
equilibrium conformation. Assign each to a point
group.
a.N 2 O
b.C 2 N 2
c.IF 3
d.CH 2 Cl 2
22.43Using a software package such as Spartan or CAChe, the
normal modes and their predicted frequencies for carbon
dioxide, sulfur dioxide, and water, using both
semiempirical and abinitiomethods. Compare the
predicted frequencies with the experimental frequencies.
The frequencies for carbon dioxide and sulfur dioxide are
in Figure 22.6, and the frequencies (divided by the speed
of light) for water are 3657 cm−^1 , 1595 cm−^1 ,
and 3756 cm−^1.
22.44Using a software package such as CACHe or Spartan,
find the vibrational normal modes of a molecule of your
choice that contains some C–H bonds, a CO bond, an
O–H bond, and so on. Try to identify normal modes that
you could classify as a C–H stretch, a C–H bend, an O–H
stretch, and so on. Determine whether these normal
modes have calculated frequencies that lie in the regions
usually assumed for such vibrations.(^8) Davis,op. cit., p. 316 (note 1).
(^9) Davis,op. cit., p. 315 (note 1).