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 (i.e. velocity magnitudes) 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 the result 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
From thermodynamics, the mean kinetic energy equals 3kT/2, where k is Boltzmann’s constant and T is the absolute temperature. Thus
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
wherea, b and n are to be determined.
Let the fraction of velocities with x-components in (vx,dvx) be f(vx)dvx. Similarly for (vy,dvy) and (vz,dvz). Then the fraction with components in (vx,dvx) and (vy,dvy) and (vz,dvz) is f(vx)f(vy)f(vz)dvxdvydvz. And the number of velocities is Nf(vx)f(vy)f(vz)dvxdvydvz.
In velocity space each velocity can be represented as a point (i.e. as the tip of a "displacement" vector from the origin). Since dvxdvydvz is a volume element in velocity space, the density of the velocity vectors at point (vx,vy,vz) is Nf(vx)f(vy)f(vz). 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
In particular, for vx=v and vy= vz=0.
where the constant C equals4pf2(0). Now this equation 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
From Eq. 1,
we find that
From Eq. 2 we have
we find that
Substituting in Eq. 10,
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.