Physical Chemistry , 1st ed.

This is the probability function for the velocities of gas molecules in three

dimensions.

If we want to focus on the scalarpart of the velocities of the gas species, we

do not include the fact that velocity as a vector can be positive or negative. We

will get a slightly different probability distribution function. This function,

labeled G(v), also has a normalization requirement:





0

G(v) dv 1

In this case, the integration limits are 0 to , rather than to . Also, be-

cause the magnitude of any velocity vector is independent of its direction, each

value in G(v) actually represents a spherical shell of possible velocity vectors,

as demonstrated in Figure 19.3. There is thus a 4 v^2 component as part of the

infinitesimal. (This is akin to the argument used to get a physically useful de-

scription for the 1swavefunction for the hydrogen atom.) Using the linear

probability functions gx,gy, and gz,we get

G(v)dv 

2

m

kT



1/2

emvx

(^2) /2kT
 2
m
kT

1/2
emvy
(^2) /2kT
 2
m
kT

1/2
emvz
(^2) /2kT
 4 v^2 dv
This equation simplifies by collecting the exponential terms and writing them
as an exponential of the square of the overall velocity, and also by collecting
the (m/2 kT)1/2terms. We get
G(v) dv 4 
2
m
kT

3/2
v^2 emv
(^2) /2kT
dv (19.33)
This three-dimensional probability distribution function of the velocity mag-
nitudes is called the Maxwell-Boltzmann distribution.It can be plotted versus
velocity (much like the Planck distribution of light intensity can be plotted ver-
sus wavelength). This distribution depends on the mass of the gas particle and
the (absolute) temperature. Figure 19.4 shows various plots for different gases
at different temperatures. This expression is a special case of the Maxwell-
Boltzmann distribution mentioned in statistical thermodynamics, where the E
in the exponential refers to the kinetic energies of the particles moving in three
dimensions.
19.3 Definitions and Distributions of Velocities of Gas Particles 663
0.0025
0
0
Velocity, m/s
5000
Relative probability
Ar, 298 K
O 2 , 298 K
H 2 , 500 K
He, 298 K H 2 , 298 K
0.002
0.0015
0.001
0.0005
1000 2000 3000 4000
Total volume
 4 v^2 dv
Infinitesimal volume
vz  dvxdvydvz
vy
vx
dvz
dvy
v dvx
dv
Figure 19.3 In three dimensions, any infini-
tesimal change in velocity is represented by a
spherical shell about the origin. Therefore, the in-
finitesimal for integrating G(v) from 0 to must
take the spherical symmetry into account.
Figure 19.4 Distributions of speeds for various gases. Note how the curve for H 2 is shifted to
higher velocities at 500 K. These curves are collectively called Maxwell-Boltzmann distributions.