Physical Chemistry Third Edition

628 14 Classical Mechanics and the Old Quantum Theory

x 1 x 2

massm 2

spring

massm 1 centerof

mass

xc

x

Figure 14.4 A Second System Represented by a Harmonic Oscillator.

wherem 1 andm 2 are the masses of the two objects. The motion of the two-particle

oscillator is mathematically the same as the motion of a fictitious particle of massμat

the end of the same spring with the other end of the spring held fixed. For this model

the frequency of oscillation is given by

ν

1

2 π

√

k

μ

(14.2-40)

wherekis the force constant.

Exercise 14.3

The frequency of vibration of a hydrogen molecule equals 1. 3194 × 1014 s−^1. Find the value

of the force constant of chemical bond in the H 2 molecule.

PROBLEMS

Section 14.2: Classical Mechanics

14.1The vibrational frequency of a^12 C^16 O molecule is

6. 5405 × 1013 s−^1. The atomic masses are:^12 C: 12.00000

amu;^13 C: 13.00335 amu;^16 O: 15.994915 amu;

(^17) O: 16.99913 amu.
a.Find the value of the force constant.
b. Find the vibrational frequency of a^13 C^16 O mole-
cule, assuming that the force constant is
unchanged.
c.Find the vibrational frequency of a^12 C^17 O mole-
cule assuming that the force constant is
unchanged.
14.2Assume that a^12 C^16 O molecule is adsorbed on a platinum
surface in such a way that the carbon atom
is held stationary but the oxygen atom vibrates.
Find the vibrational frequency of the oxygen atom.
Assume that the force constant of the bond is the
same as that of the free CO molecule, determined in
Problem 14.1.
14.3The frequency of vibration of a^1 H^35 Cl molecule is
8. 966 × 1013 s−^1.
a.What would the frequency be if the chlorine atom were
infinitely massive?
b.What would the frequency be if the hydrogen atom
were infinitely massive?
14.4The frequency of vibration of an H 2 molecule is equal to

1. 31945 × 1014 s−^1. The atomic mass of H is 1.007825
   amu and that of^2 H is 2.014102 amu. Assume that the
   force constant of the vibration is unchanged by isotopic
   substitution.
   a.Find the frequency of vibration of an HD molecule,
   where D^2 H.
   b.Find the frequency of vibration of a D 2 molecule.
   14.5Assume that a hydrogen atom is bonded to a large crystal
   with a chemical bond that is equivalent to a spring having
   a force constant equal to 600.0 N m−^1. Assume that the
   hydrogen atom oscillates harmonically but the atom at the
   other end of the bond is stationary. Find the frequency of
   the oscillation.