Page 67 - Modern physical chemistry

56 Gases and Collective Properties

whence
A=(;)'" [3.88J

For the mean square velocity, we now get

1 [3.89J
2a

But from formulas (3.8) and (3.24), we have

nRT kT
PV
2
vx =--=--=- [3.90J
Nm Nm m
Combining (3.89) and (3.90) leads to
1 m
a=~=--. [3.91 J
2V2 2kT
x
So distribution law (3.85) becomes
112
(
f{ vx ) = 2:T ) e-mviI2kT, [3.92J
while equation (3.83) leads to

3/2
f{ v V v) = ~ e-mv2/2kT. [3.93]
(
)
x' Y' z 27rkT
The probability f( v) dv that a molecule has a speed between v and v + dv equals the
sum of the probabilities that it is in any volume element in the spherical shell of radius
v and thickness dv. The sum of these volume elements is 47rv 2 dv. So summing (3.93) over
these volume elements yields

[3.94]

Equations (3.92), (3.93), and (3.94) are forms of the MaxweU distribution law. Since
1I2mvx2 and 1!2mv 2 are the pertinent kinetic energies and the preexponential factors are
normalized so that the 1s are probabilities, these forms agree with equation (3.82). This
equation holds regardless of the kind of energy Ej •

Example 3.10
Evaluate the integral

Carry out an integration by parts: