Physical Chemistry Third Edition

9.8 The Hard-Sphere Gas 431

1

x^1

r

y

2

2

Figure 9.21 An “Average” (Right-Angle) Collision of Two Hard Spheres of Different

Types.

separation is given by the theorem of Pythagoras:

rtc

[

〈v 1 〉^2 +〈v 2 〉^2

] 1 / 2

(9.8-25)

We denote themean relative speedby〈v 12 〉:

〈vrel〉〈v 12 〉

√

〈v 1 〉^2 +〈v 2 〉^2 

√

8 kBT

πm 1

+

8 kBT

πm 2

〈v 12 〉

√

8 kBT

πμ 12

(9.8-26)

wherem 1 andm 2 are the two molecular masses and whereμ 12 is called thereduced

massof particles 1 and 2:

1

μ 12



1

m 1

+

1

m 2

(9.8-27)

μ 12 

m 1 m 2

m 1 +m 2

(9.8-28)

Our derivation is crude, but Eq. (9.8-26) is the correct expression for the mean relative

speed. For a pair of identical particles,μis equal tom/2, so that Eq. (9.8-26) is valid

for that case as well as for two different substances.