Physical Chemistry Third Edition

426 9 Gas Kinetic Theory: The Molecular Theory of Dilute Gases at Equilibrium

Molecular Collisions in a Hard-Sphere Gas

We now want to study the rate of molecular collisions in our model hard-sphere gas

of a single substance. In this model system a molecule moves at a constant velocity

between collisions. We first pretend that only particle number 1 is moving while the

others are stationary and distributed uniformly throughout the container. The mean

number of stationary particles per unit volume is given by

N− 1

V

≈

N

V

N (9.8-8)

where we neglect unity compared withNand whereN is the number density of

molecules in the gas, equal toN/V. The mean numberN′of particles in a volumeV′

is given by

N′NV′ (9.8-9)

As the moving particle travels along, it “sweeps out” a cylindrical volume as shown

in Figure 9.19. The radius of thiscollision cylinderis equal to twice the radius of the

molecules and is equal tod. We calldthecollision diameter. The cross-sectional area

of the collision cylinder is called thecollision cross section

(collision cross section)πd^2 (9.8-10)

If the center of any stationary molecule lies in this cylinder it will be struck by the

moving particle. The length of the cylinder that contains on the average one stationary

particle is equal to the average distance between collisions, called themean free path

1

d

Collision

cylinder

1

1

2

2

Figure 9.19 A Portion of the Collision Cylinder of Particle 1.