__Abstract.__
A
Velocity Space Derivation of the Maxwell/Boltzmann Molecular Speed Distribution

Following
is a derivation of g(v)dv, the Maxwell/Boltzmann distribution of molecular
speeds in a gas containing N molecules, where N is arbitrarily large and where
each molecule’s velocity is unique (so that there are N distinct velocities).
As a matter of definition, g(v)dv is the fraction of the N molecules with speeds
in the range v to v+dv. Maxwell derived g(v) in 1859.

In
the following discussion, "(v,dv)" is shorthand for "the range v
to v+dv." Let
Ng(v)dv be the number of molecular velocities with magnitudes in (v,dv). Then

.
(1)

From
thermodynamics, the mean kinetic energy equals 3kT/2, where k is Boltzmann’s
constant and T is the absolute temperature. Thus

.
(2)

In postulating a form for g(v)dv, we expect few (or no) molecules to be standing still, and none to be moving with infinite speed. And of course g(v)dv must integrate (normalize) to unity. Integrals of the form have the potential of satisfying both of these

requirements, provided n>0. Thus we postulate that

,
(3)

where a, b and
n are to be determined.

Let
the fraction of velocities with x-components in (v_{x},dv_{x})
be f(v_{x})dv_{x}. Similarly for (v_{y},dv_{y})
and (v_{z},dv_{z}). Then the fraction with components in (v_{x},dv_{x}) and (v_{y},dv_{y}) and (v_{z},dv_{z})
is f(v_{x})f(v_{y})f(v_{z})dv_{x}dv_{y}dv_{z}.
And the number of velocities is Nf(v_{x})f(v_{y})f(v_{z})dv_{x}dv_{y}dv_{z}.

In
velocity space each velocity can be represented as a point (i.e. as the tip of a
"displacement" vector from the origin). Since dv_{x}dv_{y}dv_{z} is
a volume element in velocity space, the density of the velocity vectors at point
(v_{x,}v_{y},v_{z}) is Nf(v_{x})f(v_{y})f(v_{z}).
This density can only depend on v, the "distance" from the origin.

The
number of velocities in the spherical shell, of volume 4pv^{2}dv
is Ng(v)dv, and the density in this shell is Ng(v)dv/(4pv^{2}dv).
Thus

(4)

In
particular, for v_{x }=
v and v_{y }= v_{z }=
0,

,
(5)

or

(6)

where
the constant C equals 4pf^{2}(0).
Now equation (6) can be satisfied for all v only if g(v) contains a v^{2} term.
Thus the correct choice for n in Eq. (3) is ‘2’, and we postulate that

.
(7)

From
Eq. (1),

.
(8)

Or,
since

,
(9)

we
find that

,(10)

and

.
(11)

From
Eq. 2 we have

.
(12)

Or,
since

,
(13)

we
find that

(14)

and
thus

.
(15)

Substituting
in Eq. (10),

,
(16)

and
thus

.
(17)

Or,

,
(18)

which
is the Maxwell-Boltzmann distribution.

Maxwell’s formula for g(v)dv was experimentally corroborated by Stern in 1926. Several other experiments provided further corroboration. In brief, Maxwell was right. Other derivations by Boltzmann and Gibbs arrived at the same result.