Maxwell’s Derivation of the Maxwell/Boltzmann Speed Distribution

James Clerk Maxwell

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_xdv_ydv_z. And the number of velocities is Nf(v_x)f(v_y)f(v_z)dv_xdv_ydv_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_xdv_ydv_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 4pv²dv is Ng(v)dv, and the density in this shell is Ng(v)dv/(4pv²dv). Thus

(4)

In particular, for v_x = v and v_y = v_z = 0,

, (5)

or

(6)

where the constant C equals 4pf²(0). Now equation (6) can be satisfied for all v only if g(v) contains a v² 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. In brief, Maxwell was right!