Maxwell-Boltzmann Distribution Calculator

About this calculator

The Maxwell-Boltzmann distribution describes how molecular speeds are spread across a gas at thermal equilibrium: f(v) = 4π n (m/2πkT)^(3/2) v² e^(−mv²/2kT), where m is the molecule's mass, T the absolute temperature, and k the Boltzmann constant. From this single distribution three characteristic speeds fall out algebraically — the most probable speed v_p = √(2kT/m), the mean speed v_mean = √(8kT/πm), and the root-mean-square speed v_rms = √(3kT/m), each slightly larger than the last.

This calculator evaluates the probability density at a chosen speed and returns all three characteristic speeds for a given molar mass and temperature. Physics and physical chemistry students use it to check kinetic theory homework, while engineers modeling gas diffusion, effusion or reaction rates and atmospheric scientists studying how light gases like hydrogen escape a planet's atmosphere rely on the same speed distribution.