Physical Chemistry Third Edition

9.4 The Distribution of Molecular Speeds 407

Consider a spherical shell of radiusvand thicknessdvas shown in Figure 9.11. This

shell contains all of the points in velocity space that represent speeds betweenvand

v+dv. To find the probability of speeds in this shell we integrate the probability shown

in Eq. (9.4-2) over all values ofθandφfor fixed values ofvanddv:

(

probability that

vlies indv

)

fv(v)dv

(∫

π

0

∫ 2 π

0

g(v)v^2 sin(θ)dφdθ

)

dv

g(v)v^2 dv

∫π

0

sin(θ)dθ

∫ 2 π

0

dφ (9.4-3)

 4 πv^2 g(v)dv

Thespeed probability distributionor probability density (probability per unit length

on the speed axis) is denoted byfv:

fv(v) 4 πv^2 g(v) 4 πv^2

(

m

2 πkBT

) 3 / 2

e−mv

(^2) / 2 kBT
(9.4-4)
Exercise 9.12
a.Argue that whileφranges from 0 to 2π,θranges only from 0 toπto cover all possible angles.
b.Carry out the integrals in Eq. (9.4-3) that lead to the factor 4πin Eq. (9.4-3).
vx
vz
dv
v
vy
Figure 9.11 A Spherical Shell of Thicknessdvin Velocity Space.