9.4 The Distribution of Molecular Speeds
407
Consider a spherical shell of radius v and thickness dv as shown in Figure 9.11. This
shell contains all of the points in velocity space that represent speeds between v and
v + dv. To find the probability of speeds in this shell we integrate the probability shown
in Eq. (9.4-2) over all values of θ and φ for fixed values of v and dv:
probability that
v lies in dv
f v (v) dv
π
0
2π
0
g(v)v
2 sin(θ)dφdθ
dv
g(v)v
2 dv
π
0
sin(θ)dθ
2π
0
dφ
(9.4-3)
4πv
2 g(v)dv
The speed probability distribution or probability density (probability per unit length
on the speed axis) is denoted by f v :
f v (v) 4πv 2 g(v) 4πv 2
m
2πk B T
3/2
e −mv 2 /2k B T
(9.4-4)
Exercise 9.12
a. Argue that while φ ranges from 0 to 2π, θ ranges only from 0 to π to cover all possible angles.
b. Carry out the integrals in Eq. (9.4-3) that lead to the factor 4π in Eq. (9.4-3).
v x
v z
dv
v
v y
Figure 9.11 A Spherical Shell of Thickness dv in Velocity Space.
407
Consider a spherical shell of radius v and thickness dv as shown in Figure 9.11. This
shell contains all of the points in velocity space that represent speeds between v and
v + dv. To find the probability of speeds in this shell we integrate the probability shown
in Eq. (9.4-2) over all values of θ and φ for fixed values of v and dv:
probability that
v lies in dv
f v (v) dv
π
0
2π
0
g(v)v
2 sin(θ)dφdθ
dv
g(v)v
2 dv
π
0
sin(θ)dθ
2π
0
dφ
(9.4-3)
4πv
2 g(v)dv
The speed probability distribution or probability density (probability per unit length
on the speed axis) is denoted by f v :
f v (v) 4πv 2 g(v) 4πv 2
m
2πk B T
3/2
e −mv 2 /2k B T
(9.4-4)
Exercise 9.12
a. Argue that while φ ranges from 0 to 2π, θ ranges only from 0 to π to cover all possible angles.
b. Carry out the integrals in Eq. (9.4-3) that lead to the factor 4π in Eq. (9.4-3).
v x
v z
dv
v
v y
Figure 9.11 A Spherical Shell of Thickness dv in Velocity Space.