# Statistical Nature of Maxwell-Boltzmann Distribution

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This Demonstration analyzes the statistical nature of the Maxwell–Boltzmann distribution. A histogram of the velocity of the molecules in an ideal gas is shown, with the velocity distribution function as the axis. You can select the order of magnitude of the number of molecules . You can change the width of the histogram bins, the temperature and the molar mass to see how the velocity distribution function changes. Also, you can show the mean, average and root mean square velocities.

Contributed by: Fis. Fernando Moncada (August 2022)
(Developed in the Physics Laboratory of Escuela Politécnica Nacional)
Open content licensed under CC BY-NC-SA

## 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).