Modern Engineering Thermodynamics

18.5 Molecular Velocity Distributions 735
3
2
Population growth
(α > 0)
N/N 0
1
Population decay
(α < 0)
FIGURE 18.4
Population growth and decay as predicted by Eq. (18.17).
The problem of determining the distribution of velocities among the molecules of a gas can be thought of as a
population problem, except that we are no longer interested in how the population size N varies with time, but
rather how it varies with molecular velocity. Using the general form of Eq. (18.16), we can postulate a velocity
distribution population model as follows:
dN V
= f ðVÞN (18.18)
dV
where α has been replaced by a more general function f(V), called the velocity distribution function. The problem
now is to find the mathematical form of f(V). For example, if we assume a Gaussian velocity distribution,
we have
Š
f ðVÞ = pffiffiffiffiffi
exp − V2
2π δ 2δ 2
where δ is the standard deviation.
By utilizing the assumptions stated at the beginning of this section, Maxwell was able to show that f(V) isnot
Gaussian, but instead has the following form:
f ðVÞ =
4
m 3/2V
pffiffiffi
2 exp − mV2
(18.19)
π 2kT
2kT
Substituting Eq. (18.19) into Eqs. (18.4) and (18.5), one finds that the average and root mean square velocities
have the following simple formulae:
rffiffiffiffiffiffiffiffi
V avg =
8kT
πm
(18.20)
V rms =
rffiffiffiffiffiffiffiffi
3kT
m
(18.21)
Figure 18.5 shows the shape of the distribution function f(V) for oxygen at 300 K as described by Eq. (18.19).
We call the velocity at which f(V) has a maximum the most probable velocity V mp . It is determined by setting
df ðVÞ/dV = 0 and solving for V = V mp , which, using Eq. (18.19), gives
rffiffiffiffiffiffiffiffi
2kT
V mp =
(18.22)
m
By comparing Eqs. (18.20), (18.21), and (18.22), it is clear that V mp < V avg < V rms , as shown in Figure 18.5.
Z V2
0
0 1 2 3 4 5 6 7 8 9 10
Let dN V = NðV 1 ! V 2 Þ be the number of molecules with velocities between V 1 and V 2 . Then, it follows from
V 1
Eq. (18.18) that
NðV 1 ! V 2 Þ
N
Time
Z V2
= f ðVÞ dV
V 1
(18.23)