9.8 The Hard-Sphere Gas
431
1
1
x
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:
r t c
v 1
2
+ v 2
2
1/2
(9.8-25)
We denote the mean relative speed by v 12 :
v rel v 12
v 1 2 + +v 2 2
8k B T
πm 1
+
8k B T
πm 2
v 12
8k B T
πµ 12
(9.8-26)
where m 1 and m 2 are the two molecular masses and where µ 12 is called the reduced
mass of 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, µ is equal to m/2, so that Eq. (9.8-26) is valid
for that case as well as for two different substances.