Physical Chemistry Third Edition

10.3 The Gas Kinetic Theory of Transport Processes in Hard-Sphere Gases 461

situation can be approximated in the laboratory by using two substances that differ only

by isotopic substitution. Diffusion of such actual substances is calledtracer diffusion.

We now define a gaseous model system that contains two kinds of hard spherical

molecules with the same size and mass. The system is at a uniform temperature

and pressure and is contained in a rectangular box with four vertical sides. The

concentrations of the two substances depend on the vertical coordinatezbut not on

the horizontal coordinatesxandy. The sum of the two concentrations is independent

of position so that the pressure is uniform. Figure 10.7 depicts this model system.

In the interior of the system we place three imaginary horizontal planes. The center

plane is located atzz′, the upper plane is located atz′+λ, and the lower plane

is located atz′−λ, whereλis the mean free path between collisions of a molecule

with any kind of molecule.

We assume that all molecules passing upward through the center plane last suffered

collisions at the lower plane and were equilibrated at that location. Molecules passing

downward through the center plane are assumed to have been equilibrated at the upper

plane. This assumption is not correct for every molecule since thezcomponent of

every free path is not equal to the mean free path and since equilibration might not be

complete at each collision. It should be roughly valid on the average.

The number of molecules of substance 1 passing upward through the center plane

per unit area per unit time is given by Eq. (9.6-6):

ν 1 (up)

1

4

N 1 (z′−λ)〈v〉 (10.3-1)

Location of equilibration

of molecule coming

from above at z 5 z' 1 λ

Location of equilibration

of molecule coming

from below at z 5 z' 2 λ

z 5 z' 1 λ

z 5 z'

z 5 z' 2 λ

Imaginary

planes

x

z

y

Figure 10.7 The Model System Showing Three Imaginary Planes for Analysis of

Self-Diffusion in a Hard-Sphere Gas.