Arrhenius equation
k = A exp(−E_a/k_BT) — the law that ties a reaction's rate to temperature through the Boltzmann tail of molecules energetic enough to react.
Definition
The Arrhenius equation expresses the rate constant of a chemical reaction as k = A exp(−E_a/RT), with E_a the activation energy, A a pre-exponential frequency factor, and RT (or k_BT per molecule) the thermal energy. The exponential is a Boltzmann factor: it is the fraction of molecular collisions energetic enough to clear the activation barrier. A modest rise in temperature can multiply the reacting fraction many times over, which is why reaction rates are so sharply temperature-sensitive.
The form is the direct fingerprint of the Maxwell–Boltzmann tail. Only molecules in the high-energy tail of the distribution carry enough energy to react, and that tail grows exponentially with temperature. A common rule of thumb — that many reactions roughly double in rate for every 10 °C — is just the Arrhenius factor evaluated for typical activation energies.
Plotting ln k against 1/T gives a straight line whose slope is −E_a/R, the standard experimental route to measuring activation energies. The same exponential governs diffusion in solids, viscous flow, and the temperature dependence of biological processes.
History
Proposed by Svante Arrhenius in 1889, building on earlier work by van 't Hoff; later given a microscopic basis by collision theory and transition-state theory, both resting on the Boltzmann factor.