Physical Chemistry Third Edition

9.8 The Hard-Sphere Gas 427

and denoted byλ. This volume is

(volume containing one particle)λπd^2 (9.8-11)

and must equalV/N, the average volume per molecule:

λπd^2 

V

N



1

N

(9.8-12)

The mean free path is therefore given in this approximation by

λ

1

πd^2 N

(approximate equation) (9.8-13)

Equation (9.8-13) is a crude approximation because all of the particles are actually

moving. When two particles collide they might initially be moving toward each other,

they might initially be moving roughly at right angles to each other, or they might be

moving in the same general direction. We assume that the “average” collision occurs

with the particles moving initially at right angles to each other as in Figure 9.20. If

prior to the collisionxis the distance of one particle from the collision site andyis the

distance of the other from this site, the separation of the two particles is given by the

theorem of Pythagoras:

r(x^2 +y^2 )^1 /^2 (9.8-14)

Therelative speedis defined as the magnitude ofdr/dt.

We assume that both particles are moving at the same speed so thatxandymust be

equal to each other in order for the collision to occur. Therefore

r

√

2 x (9.8-15)

1

x^1

r

y

2

2

Figure 9.20 An “Average” (Right-Angle) Collision of Two Hard Spheres of the Same

Type.