Maxwell-Boltzmann distribution

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:


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:

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 , where ,

As the probability density function is separable, its normalisation is equivalent to the separate normalisation of each of the individual functions:

Let’s evaluate just one of the integrals in eq4 by substituting  in to give



Show that .


Let and at the same time let since and are dummy variables.

In polar coordinates, . For infinitesimal changes, the change in area can be approximated as a rectangle with sides and (see diagram below), with .




Hence, eq5 becomes,

Substituting eq6 in eq3,

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

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,

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 ., which implies that the probability density function for the speed of a molecule is

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

where is the mass of one mole of the gas and is the gas constant.



Show that the most probable speed of one mole of an ideal gas is .


The most probable speed corresponds to the peak of the distribution (see above diagram). Carrying out the derivative , equating it to zero, and noting that the most probable speed is neither nor , we have .



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