## 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 *v _{x}, v_{y}* and

*v*in velocity space. The volume of the spherical shell from surfaces

_{z}*v*and

*v + dv*is given by

*4pv*By analysis of the transfer of momentum during collisions between molecules, Maxwell determined that the volume element must be multiplied by the Boltzmann factor

^{2}dv.*exp(-1/2 mv*. (You will recognize

^{2}/kT)*1/2mv*as the expression for kinetic energy.) The

^{2}*(m/2pkT)*is a normalization factor required to make the integral of

^{1/2}*dN*over all velocities equal

*N*.

Below is a plot of the probability distribution of molecules as a function of velocity at three temperatures.