The Maxwell-Boltzmann distribution )
describes the range of speeds 
among gaseous molecules of an ideal gas at a given temperature 
.
The energy of an ideal gas molecule of mass 
is characterised solely by its kinetic energy 
. For a molecule with velocity 
, its energy is 
, where 
, 
and 
are the velocity vector components of in Cartesian coordinates. Given that a molecule with the velocity magnitude 
has the same energy regardless of direction, any molecule with the ordered triple })
that lies on the surface of a sphere of radius 
in velocity space belongs to the energy state 
(see diagram below).
The fraction of molecules in the energy state 
is given by the Boltzmann distribution:
where
is the Boltzmann constant.
is the equilibrium temperature of the system.
is the number of molecules in the energy state.
is the total number of molecules in the system.
In probability theory, 
is the probability 
of molecules in state 
, with 
being the normalisation constant. In other words, 
, which we can rewrite as:
=Ke^{-\frac{m(v_x^2+v_y^2+v_z^2)}{2kT}}\;\;\;\;\;\;\;\;2)
where })
is the probability density function (probability per unit interval of ) of molecules in a particular energy state 
(we have dropped the index for convenience) and 
is the proportionality constant.
Since =\biggr\(K'e^{-\frac{mv_x^2}{2kT}}\biggr\)\biggr\(K'e^{-\frac{mv_y^2}{2kT}}\biggr\)\biggr\(K'e^{-\frac{mv_z^2}{2kT}}\biggr\)=f(v_x)f(v_y)f(v_z)})
, where 
,
=Ke^{-\frac{mv_x^2}{2kT}}e^{-\frac{mv_y^2}{2kT}}e^{-\frac{mv_z^2}{2kT}}\;\;\;\;\;\;\;\;3)
As the probability density function is separable, its normalisation is equivalent to the separate normalisation of each of the individual functions:
dv_x=1\;\;\;\;\int_{-\infty}^{\infty}f(v_y)dv_y=1\;\;\;\;\int_{-\infty}^{\infty}f(v_z)dv_z=1\;\;\;\;\;\;\;\;4)
Let’s evaluate just one of the integrals in eq4 by substituting =K^{1/3}e^{-\frac{mv_x^2}{2kT}}})
in dv_x=1})
to give
Hence, eq5 becomes,
^{\frac{3}{2}}\;\;\;\;\;\;\;\;6)
Substituting eq6 in eq3,
=\biggr\(\frac{m}{2\pi kT}\biggr\)^{\frac{3}{2}}e^{-\frac{mv_x^2}{2kT}}e^{-\frac{mv_y^2}{2kT}}e^{-\frac{mv_z^2}{2kT}}\;\;\;\;\;\;\;\;7)
Eq7 is known as the Maxwell-Boltzmann distribution for the velocity vector of a molecule. The fraction of molecules 
with velocity vectors in the infinitesimal range of to 
, to 
and to 
is
dv_xdv_ydv_z=\biggr\(\frac{m}{2\pi kT}\biggr\)^{\frac{3}{2}}e^{-\frac{m(v_x^2+v_y^2+v_z^2)}{2kT}}dv_xdv_ydv_z\;\;\;\;\;\;\;\;8)
The fraction of molecules 
with speed in the infinitesimal range of to 
is then equivalent to the fraction of molecules with velocity vectors that lies within a spherical shell of infinitesimal thickness in velocity space (see diagram below).
In other words,
^{\frac{3}{2}}e^{-\frac{mv^2}{2kT}}\sum_{shell}dv_xdv_ydv_z)
where 
, and we have assumed that the volume of the shell of infinitesimal thickness is approximately equal to the sum of infinitesimal cubes, each with an infinitesimal volume of 
.
Therefore, 
and dv=4\pi\biggr\(\frac{m}{2\pi kT}\biggr\)^{\frac{3}{2}}v^2e^{-\frac{mv^2}{2kT}}dv})
., which implies that the probability density function for the speed of a molecule is
=4\pi\biggr\(\frac{m}{2\pi kT}\biggr\)^{\frac{3}{2}}v^2e^{-\frac{mv^2}{2kT}}\;\;\;\;\;\;\;\;9)
Eq9 is known as the Maxwell-Boltzmann distribution for the speed of a molecule. It also expresses the probability of a molecule being associated with a particular energy state. For a mole of an ideal gas, eq9 becomes
=4\pi\biggr\(\frac{M}{2\pi RT}\biggr\)^{\frac{3}{2}}v^2e^{-\frac{Mv^2}{2RT}}\;\;\;\;\;\;\;\;10)
where 
is the mass of one mole of the gas and 
is the gas constant.
