Maxwell distribution is an equilibrium velocity distribution of molecules (particles) of a classical gas. It determines the probable number of molecules with velocities in the range v_x from v_x+dv_x, v_y to v_y+dv_y, v_z to v_z+dv_z in the unit volume

The statistical sense of Maxwell distribution can be demonstrated with the aid of Galton board which consists of the wood board with many nails as shown in animation. Above the board the funnel is situated in which the particles of the sand or corns can be poured. If we drop one particle into this funnel, then it will fall colliding many nails and will deviate from the center of the board by chaotic way. If we pour the particles continuously, then the most of them will agglomerate in the center of the board and some amount will appear apart the center. After some period of time the certain statistical distribution of the number of particles on the width of the board will appear. This distribution is called normal Gauss distribution (1777-1855) and described by the following expression:

y = j(x) = Aexp (-ax²)

where A and a are the constants which depends on the parameters of the system. The molecular-kinetic theory proves that the velocities of the molecules are distributed according to the same normal (Gauss) law. Constants A and a can be found from the normalizing conditions and other additional assumptions. Finally, we can come to well-known Maxwell distribution determined by Maxwell in 1859:

f(v) = n (m/2pkT)^3/2exp( -mv²/2kT)

where v is the vector velocity of a particle, m is the mass of the molecules, n is the number of the molecules in the unit volume. The other form of Maxwell distribution describes the probable density of the molecules with absolute velocities in the range from v to v+dv:

dn=F(v)dv=4pn(m/2p kT)^3/2exp(-mv²/2kT)v²dv

This expression achieves the maximum at velocity v_p = (2kT/m)^1/2 called the most probable velocity. With the aid of Maxwell distribution we can calculate the average value of any function dependant on velocity of molecules. So, for example, the average velocity of the molecules is <v> = (8kT/pm)^1/2 and <v²>^1/2 = (3kT/m)^1/2.

Thermodynamics (theory and practice)