Maxwell-Boltzmann Distribution
The Maxwell–Boltzmann distribution is the probability distribution of molecular speeds in a classical ideal gas in thermal equilibrium: f(v) = 4π(m/2πkBT)3/2 v² exp(−mv²/2kBT). It was derived by James Clerk Maxwell in 1860 and generalised by Ludwig Boltzmann, and it predicts the most probable, mean and root-mean-square speeds of the molecules, which scale as √(T/m).
Curious
At room temperature, nitrogen molecules have speeds ranging from nearly zero to well over 1,000 m/s, but most are near 400–500 m/s. Plot how many have each speed and you get a lopsided hump: the Maxwell–Boltzmann distribution. Heat the gas and the hump flattens and shifts to higher speeds; use lighter molecules and it shifts the same way.
The fast tail matters. Evaporation happens because the fastest molecules at a liquid's surface escape; chemical reaction rates depend on the few molecules with enough energy to react; and the lightest gases, hydrogen and helium, slowly leak from Earth's atmosphere into space because a tiny fraction move faster than escape speed (see Kinetic Theory of Gases).
Exploring
| Characteristic speed | Formula | N2 at 300 K |
|---|---|---|
| Most probable | √(2kBT/m) | 422 m/s |
| Mean | √(8kBT/πm) | 476 m/s |
| Root-mean-square | √(3kBT/m) | 517 m/s |
Deep Dive
Derivation and generalisation. Maxwell assumed that the three velocity components are independent and that the distribution is isotropic; only a Gaussian satisfies both. Boltzmann later derived it from his transport equation as the unique equilibrium solution (the H-theorem) and generalised it to the Boltzmann factor, P ∝ exp(−E/kBT), for any energy, including potential energy in a gravitational field, which gives the barometric decrease of air density with height (see Ludwig Boltzmann).
Experimental confirmation. Direct measurements of molecular speeds came with molecular beam experiments, notably Otto Stern's (1920) and the rotating-slotted-disc experiments of Miller and Kusch (1955), which confirmed the predicted distribution closely for potassium and thallium vapours.
Limits. The distribution assumes classical particles, equilibrium and weak interactions. At low temperature and high density, quantum statistics take over (see Fermi Energy and Bose-Einstein Condensation); in plasmas and astrophysical settings, non-thermal (for example, power-law) tails are common. Atmospheric escape is more complex than the instantaneous fraction above suggests, since escape happens from the exobase over time and non-thermal processes also contribute.
Concept Relationship Map
Requires: kinetic theory, temperature
Enables: reaction-rate theory, evaporation, atmospheric escape, Boltzmann factor
Commonly confused with: a Gaussian speed distribution; quantum statistics
Sources
- Maxwell, J.C. (1860). Philosophical Magazine, 19, 19–32.
- Boltzmann, L. (1872). Wiener Berichte, 66, 275–370.
- Miller, R.C., Kusch, P. (1955). “Velocity Distributions in Potassium and Thallium Atomic Beams.” Physical Review, 99, 1314–1321.