Maxwell-Boltzmann Distribution
Gas molecules bounce inside a box at various speeds. The histogram of their speeds converges to the Maxwell-Boltzmann distribution — the first statistical law in physics, connecting microscopic chaos to macroscopic temperature.
About this lab
In 1860, James Clerk Maxwell derived the distribution of molecular speeds in a gas from first principles, making it the first statistical law in physics. His argument was elegant: if the velocity components in the x, y, and z directions are independent and the distribution is isotropic (no preferred direction), then the only function that satisfies both conditions is a Gaussian in each component. Transforming to speed (the magnitude of velocity) yields the Maxwell-Boltzmann distribution, with its characteristic v-squared rise at low speeds and exponential decay at high speeds. The peak — the most probable speed — depends on both temperature and molecular mass as vmp = (2kT/m)1/2.
Ludwig Boltzmann later generalized Maxwell's result through his H-theorem, showing that any initial distribution of molecular speeds will relax to the Maxwell-Boltzmann distribution through collisions. This was one of the first demonstrations that macroscopic irreversibility (the approach to thermal equilibrium) could emerge from time-reversible microscopic dynamics, a conceptual breakthrough that sparked fierce debate with Loschmidt and Zermelo. Boltzmann's statistical interpretation of entropy — S = k log W — connects the distribution directly to thermodynamics: the Maxwell-Boltzmann distribution is the one with the maximum number of microstates for a given total energy.
The high-speed tail of the Maxwell-Boltzmann distribution has profound consequences. Although the most probable speed for nitrogen molecules at room temperature is about 422 m/s, a tiny fraction travel much faster. This tail explains why Earth retains its nitrogen atmosphere but loses hydrogen: the fraction of H2 molecules exceeding escape velocity (11.2 km/s) is small but non-negligible over geological time, while for N2 it is essentially zero. This same reasoning explains why the Moon, with its low escape velocity, retains no atmosphere at all, and why the heavy noble gases (krypton, xenon) were discovered so late — they are rare in the universe but disproportionately retained by Earth because their heavy mass keeps speeds low.
The exponential factor e−E/kT that appears in the Maxwell-Boltzmann distribution is the Boltzmann factor, and it governs far more than gas kinetics. The probability of any state with energy E being occupied at temperature T follows this same law: chemical reaction rates (the Arrhenius equation), electron populations in semiconductors, the intensity of spectral lines, the populations of nuclear spin states in MRI, and even the distribution of barometric pressure with altitude. Every time you encounter a process that becomes exponentially more probable with increasing temperature, you are seeing the Boltzmann factor at work — the same statistics that Maxwell first applied to molecules bouncing in a box.