Concise Physical Chemistry

c18 JWBS043-Rogers September 13, 2010 11:29 Printer Name: Yet to Come

288 EXPERIMENTAL DETERMINATION OF MOLECULAR STRUCTURE

z 0

FIGURE 18.1 A classical harmonic oscillator. The equilibrium position on the vertical axis

isz 0.

called theforce constant, andzis a function of timez(t). The sign is negative because

the force is a restoring force acting in opposition to the excursion of the mass away

fromz 0.

By Newton’s second law,f=mawhereais the acceleration,d^2 z(t)/dt^2. These

two expressions for the force can be set equal to one another:

m

d^2 z(t)

dt^2

=−kfz(t)

d^2 z(t)

dt^2

=−

kf

m

z(t)

This is a wave equation of the kind described in Section 16.2:

d^2 φ(x)

dx^2

=−

4 π^2

λ^2

φ(x)

In the analogous harmonic oscillator case, we have

d^2 z(t)

dt^2

=−

4 π^2

λ^2

z(t)

Comparing the two expressions for acceleration,

kf

m

=

4 π^2

λ^2

leads to

1

λ

=

1

2 π

√

kf

m

The speed of propagation of electromagnetic radiation isc= 2. 998 × 108 ms−^1 ,

which is the number of waves per second (frequencyν) times the distance covered

by each wave (wavelengthλ)c=νλ. When electromagnetic radiation of many fre-

quencies falls upon an idealquantumharmonic oscillator, most of it bounces off but a

selected frequency is absorbed, the one that promotes the oscillator from one quantum