Physical Chemistry , 1st ed.

Instead of using V^12 kx^2 as the vibrational potential energy function of real

diatomic molecules, it is common to use the following expression:

VDe(1 ea(rre))^2 (14.36)

This potential is called the Morse potentialand is plotted in Figure 14.29, along

with the potential curve for the ideal harmonic oscillator.Deis the molecular

dissociation energy as measured to the bottomof the potential energy curve, as

shown in Figure 14.29. The constant ais related to the force constant kof the

molecule by the expression

a

2 D

k

e



1/2

(14.37)

The constant ahas units of (length)^1.

Example 14.15

Predict the value of the Morse potential constant afor HCl if its Deis

445.0 kJ/mol and the force constant is 5.16 mdyn/Å.

Solution

Although a straightforward substitution into equation 14.37 is warranted, the

units for the given values are inconsistent. Consider Defirst. We need to find

the amount of energy to dissociate a single HCl molecule, not a mole of mol-

ecules. The following steps provide the conversion:

445.0 kJ/mol

1

1

00

k

0

J

J

 7.39
10 ^19 J

for one molecule. The force constant, 5.16 mdyn/Å, also needs to be con-

verted (10^5 dynes 1 newton):

5.16 mdyn/Å

100

1

0

d

m

yn

dyn



10

1

5

N

dyn



1

1

01

m

(^0) Å
516 N/m
Substituting these numbers into equation 14.37, and recalling that a joule
equals a newton meter:
a
2 7.

5

3

1

9

6

N/

1

m

0 ^19 J



1/2



1/2



3.49

m

2

1020



1/2

a1.87 

1010 m^1

In units of Å, this value is 1.87 Å^1. Its magnitude is understandable consid-

ering that in the course of a molecular vibration, the change in bond distance

is on the order of 0.1Å.

Deis the dissociation energy with respect to the bottom of the potential en-

ergy curve. However, this is not what is measured experimentally, since mole-

cules have a zero-point vibrational energy even at absolute zero. The energy

that it actually takes to dissociate a diatomic molecule is determined from the

v0 vibrational level, which has an energy of^12 hhigher than the potential

energy minimum. This amount of dissociation energy is labeled D 0 (the zero

516 N/m



2 7.39 

10 ^19 N m

1 mol



6.02 

1023 molecules

492 CHAPTER 14 Rotational and Vibrational Spectroscopy

Internuclear distance

re

Do De

Energy

V ^12 kx^2

Figure 14.29 The Morse potential is a better
fit to the potential energy curve of a real molecule
than is the harmonic oscillator potential energy
surface, superimposed.