
A function describing the probability of finding particles at different speeds in a gas.The velocity distribution describes how the speeds of particles in a gas are spread out. In a classical ideal gas at thermal equilibrium, this is given by the Maxwell–Boltzmann distribution, which shows that most particles have speeds near a characteristic value, with a long tail extending to high speeds. The distribution depends on temperature and particle mass: heavier particles move slower on average, and higher temperatures broaden the distribution. This concept is central to statistical mechanics and explains phenomena like evaporation, chemical reaction rates, and the broadening of spectral lines.
James Clerk Maxwell first derived the distribution in 1860, and Ludwig Boltzmann generalized it in the 1870s to include external forces and non-uniform conditions.
In a gas, particles constantly collide and exchange energy. Over time, the system reaches a steady state where the probability of finding a particle with a given speed follows a specific functional form. The Maxwell–Boltzmann distribution arises from the assumptions of molecular chaos and equal a priori probabilities of energy states. The distribution function f(v) gives the probability density for speed v, and it is derived from the Boltzmann factor exp(-E/kT) combined with the density of states in velocity space. The most probable speed, the mean speed, and the root-mean-square speed are all related to the temperature and mass.
The velocity distribution is used to calculate effusion rates in vacuum systems, to model gas dynamics in astrophysics, and to design molecular beam experiments.
The velocity distribution explains why some molecules in Earth's atmosphere move fast enough to escape gravity, which is why light gases like hydrogen and helium are rare in our atmosphere.










