Details

The Maxwell–Boltzmann velocity distribution, which describes the speeds of the molecules of an ideal gas, is given by

,

where is the molar mass of the ideal gas, its temperature, the velocity of a given molecule, the probability for a given molecule to have a velocity within the range , and is the ideal gas constant. This distribution function is normalized so that

.

The most probable velocity is found by setting the first derivative with respect to the velocity equal to zero:

.

This is the velocity of the peak of the distribution.

The mean value of the velocities of this distribution is given by

.

The root mean square velocity is

.

The Maxwell–Boltzmann distribution can be expressed in terms of the most probable velocity by

.

In a simulation with molecules, the expected number of molecules with velocities in the interval is related to the distribution function by

,

where represents the number of molecules in a given range of velocities (bin of the histogram).