CHAP. 10: TRANSPORT PROCESSES [CONTENTS] 344
Note: In literature, we may sometimes encounter the term “Kinetic theory of an ideal
gas”. This, however, never relates to an ideal gas but to the hard spheres model.Hard spheres. The second simplest model is a gas whose molecules are replaced with
hard spheres, particles which do not attract each other but cannot permeate each other. This
model provides a qualitatively correct description of the behaviour of real gases. In the region
of low densities, where collisions between three and more molecules can be neglected, explicit
relations for transport quantities can be derived for this model.
10.5.3 Basic terms of kinetic theory.
Thecollision diameterσis the diameter of the range of repulsive forces of a molecule. In
the model of an ideal gas it is zero, in the hard spheres model it equals the diameter of the
sphere. In real molecules it is determined experimentally. For example, the collision diameter
of an argon atom is roughly 3.4× 10 −^10 m.
Themean free path`is the average distance a molecule travels between two collisions.
We have
`=1
√
2 π σ^2 N, (10.18)
whereN is the number of molecules in unit volume. For an argon atom at a temperature of
273 K and pressure 101 kPa, the mean free path equals approximately 10−^7 m.
Thecollision frequencyZ, the number of collisions made by a single molecule per unit
time, is given by the relation
Z=u
`=π σ^2 N, (10.19)whereuis themean velocityof the molecule. An argon atom at a temperature of 273 K and
pressure 101 kPa collides about five-thousand-million-times in one second.
Thecollision densityZis the total number of collisions per unit volume per unit time.
Z=
ZN
2
=π σ^2 uN^2
2
. (10.20)
For argon under normal conditions we obtain an unimaginably high numberN = 6× 1034
m−^3 s−^1.