Physical Chemistry , 1st ed.

Suppose we consider a sample of a pure gas. (We will consider gas mixtures
later, briefly.) How often does any one gas particle collide with other gas par-
ticles, and how far does the particle travel between collisions? We can answer
these questions by considering the hypothetical situation of one gas particle
moving while all other particles are stationary. As the moving particle P travels
through space, it will collide with any gas particle whose centergets within 2r
(twice the radius) of the center of particle P. This is illustrated two-dimensionally
in Figure 19.7. In three dimensions, the path of particle P sweeps out a cylin-
der of space, and any other particle whose center is in that space will collide
with particle P. The radius of that cylinder, which is equal to twice the radius
(2r) or the diameter (d) of the particle, is called the collision diameterof the
particle. In three dimensions, the cross section of this cylinder is a circle whose
area is d^2 ; this area is called the collision cross sectionof the gas particle.
Exact distances between colliding particles may be long or short (on the
atomic scale), but let us assume that there is some average distance a particle
travels between collisions. We call this average the mean free path(because it is
the average—or mean—distance that the particle is “free” and not colliding
with any other particle) and give it the symbol. The average volume of the
cylinder that is swept out by particle P between collisions is equal to the area
of the cylinder (the collision cross section) times its length (the mean free
path):

average volume between collisions 

d^2 (19.37)

But if we have Ntotal number of atoms in some given volume V, then the av-
erage volume per particle is V/N. Over any macroscopic time scale, these aver-
age volumes must be equal:



N

V



d^2

This lets us solve for the mean free path between collisions as



N

V

d^2

 (19.38)

This equation actually provides an estimate for the mean free path, since equa-
tion 19.37 considers a cylindrical volume swept out by travel in one dimension,

19.4 Collisions of Gas Particles 667

2 r

2 r

Figure 19.7 In two dimensions, a gas particle P sweeps out an area, colliding with any other
gas particle whose center is within 2r(2rd) of the center of particle P. In this figure, P will
collide with three other gas particles, but two additional gas particles will remain untouched by
P. In a real gas, a cross-sectional area of d^2 is swept out as particle P travels through space.