Page 65 - Modern physical chemistry

54 Gases and Collective Properties

number of ith particles per unit area in thickness dz then gives the net pressure due to
the change in potential. w:;:eC,N A ( _ a:,: ) dz ~ --c,N A d.,. [3.781

Combining equations (3.77) and (3.78) leads to

[3.79]
or
dCi = _ d1>i [3.80]
Ci kT
where k is the Boltzmann constantRIN A • Integrating equation (3.80) from potential energy
o to potential energy ei yields
[3.81 ]
Since c i and ciQ are proportional to the number of molecules of the ith species at levels
ei and eiQ, Ni and NiO respectively, equation (3.81) implies that

N · - N· e-CilkT . [3.82]
t -
to
From the way it has been derived, we see that equation (3.82) gives us the equilibrium
distribution of molecules in a potential field. In the next section, we will derive an equiv-
alent form for the distribution over kinetic energy.
The form encompassing both potential and kinetic energies is known as the Boltz-
mann distribution law.

Example 3.9
A gold sol containing particles averaging 4.5 x 10- 7 cm in radius reached equilibrium
at 25° C. With an ultramicroscope, 89 particles were observed in 32 successive counts at
a certain level. How many particles should be observed in 32 successive counts in a layer
0.200 cm higher?
For the volume of a typical particle, we have

3
v = ~ nr = ~ n( 4.5 x 10- cm t = 3.82 x 10- 19 cm •
7
3
Since the water buoys up the particle, its effective density equals its density in air minus the
density of water:
3
3
p'= Pgold - Pwater = 19.3 -1.0 g cm- = 18.3 g cm- •
So the gravitational force acting on the particle is

3
F = 18.3 g cm- (3.82 x 10- 19 cm 3 X9.807 N kg-I) = 6.85 x 10- 20 N.
1000 g kg- 1
On moving 0.200 cm up, its potential energy increases by

2
1> = (6.85 x 10- 20 N X 0.200 x 10- m) = 1.37 x 10- 22 J.