MAXWELL-BOLTZMANN DISTRIBUTION

The distribution of molecular velocities in a gas, established first by Maxwell and later proved rigorously by Boltzmann, is given by a function F and is today known as the Maxwell-Boltzmann velocity distribution function. Since this probability function depends upon the specified velocity u, F = F(u) and is defined such that F(u) dudvdw gives the probability that a molecule selected at random will, at any instant, have a velocity u with Cartesian components in the ranges u to u + du, v to v + dv, and w to w + dw.

The Maxwell-Boltzmann velocity distribution function refers specifically to a gas which is at rest (in the sense that no macroscopic flow exists) and in a state of thermodynamic equilibrium. Subject to these assumptions, the distribution law states that

(1)

where m is the mass of one molecule, k is Boltzmann's constant, and c = |u| is the speed of the molecule. Note that F is given as the product f(u)f(v)f(w) and that the velocity components in different directions are therefore uncorrelated. In other words, the probability of the molecule possessing a specified velocity u in the x direction is not influenced by the values of v and w for that or any other molecule. The function f is thus a velocity distribution function for motion in a specified direction, and is given by

(2)

where q represents one of (u,v,w).

The distribution function Fc(c) for molecular speed, irrespective of direction, is easily found by:

(3)

The behavior of the function F_c(c) is illustrated in Figure 1 for nitrogen gas at temperatures of 300 K and 1000 K. Note that f and F_c are normalized distribution functions which, upon integration with respect to their argument on [0, ∞], yield unity. Various 'average' molecular speeds may be obtained easily from the distribution function F_c as follows:

The most probable speed (maximum Fc):	ĉ = (2kT/m)1/2
The mean speed:	c = = (8kT/πm)1/2
The root-mean-square speed:	= ( )1/2 = (3kT/m)1/2

Figure 1. Maxwell-Boltzmann distribution of molecular speeds for nitrogen gas.

These measures of average speed may be compared with the speed of sound in the perfect gas: (kTγ/m)^1/2. For nitrogen at 300 K, the speed of sound is 353 m/s; ĉ = 420 m/s, c = 478 m/s; and c^21/2 = 516 m/s: all are proportional to .