Maxwell-Boltzmann Distribution Law
T is the absolute temperature, N is number of molecules, m is mass of a molecule, v is the velocity of a molecule, k is the Boltzmann constant 13.805 x 10_24 J deg-1.
The equation gives the fraction of gas molecules with velocities in the range v to v + dv. The velocity is described as a vector with components vx, vy and vz in velocity space. The volume of the spherical shell from surfaces v and v + dv is given by 4pv2dv. By analysis of the transfer of momentum during collisions between molecules, Maxwell determined that the volume element must be multiplied by the Boltzmann factor exp(-1/2 mv2/kT). (You will recognize 1/2mv2 as the expression for kinetic energy.) The (m/2pkT)1/2 is a normalization factor required to make the integral of dN over all velocities equal N.
Below is a plot of the probability distribution of molecules as a function of velocity at three temperatures.