Arrhenius equation and activation energy

Reaction rates are strikingly sensitive to temperature — a common rule-of-thumb estimate is that many reaction rates roughly double for every 10°C rise, a far larger effect than can be explained by the modest increase in how often molecules collide as they move faster at higher temperature. The Arrhenius equation, k = A·e^(−Ea/RT), explains this sensitivity by identifying its real source: not collision frequency, but the fraction of those collisions energetic enough to actually react.

Collision theory holds that a reaction between two molecules can occur only if they collide with the correct orientation and with combined kinetic energy exceeding a threshold called the activation energy, Ea — a minimum energy required to distort and partially break existing bonds on the way to forming new ones, corresponding to the energy needed to reach the transition state, the highest-energy configuration along the reaction pathway. Even at a fixed temperature, molecules in a gas or liquid have a spread of kinetic energies (described by the Maxwell-Boltzmann distribution), and only the high-energy tail of that distribution, the fraction of molecules with energy above Ea, is actually capable of reacting on collision.

This is exactly where the exponential term in the Arrhenius equation comes from: the Boltzmann factor, e^(−Ea/RT), gives the fraction of molecules with energy at least Ea at absolute temperature T, and this fraction grows exponentially, not linearly, as temperature rises — a modest temperature increase can substantially enlarge the high-energy tail of the distribution even though the average molecular speed increases only modestly. The pre-exponential factor A accounts for how often molecules collide with the correct orientation, a comparatively weak, roughly temperature-independent contribution by comparison. Because Ea sits in the exponent, reactions with a large activation energy are far more temperature-sensitive than those with a small one — a useful practical consequence is that plotting the natural logarithm of measured rate constants against 1/T gives a straight line (ln k = ln A − Ea/RT), from which both the activation energy and the pre-exponential factor can be extracted experimentally, without needing any prior theoretical knowledge of the reaction mechanism.