Temperature and the Arrhenius equation

Almost every reaction goes faster when it is hotter: milk sours faster on a hot day, and food cooks faster in a pressure cooker. A rough rule: a 10 °C (10 K) rise about doubles the rate.

Why? Molecules must collide to react, and only collisions with enough energy succeed. The minimum energy needed is the activation energy, Ea. Picture an energy hill between reactants and products: the reactants must climb to the top, where they form a short-lived activated complex, before rolling down to products.

In any sample, molecules have a spread of energies (the Maxwell-Boltzmann distribution). Warming it shifts the spread towards higher energies, so the number of molecules with energy above Ea goes up sharply. That is why a small rise in temperature has a big effect.

The Arrhenius equation puts this into numbers:

k = A e^(–Ea/RT)

A is the Arrhenius factor (or frequency factor), R the gas constant, T the temperature in kelvin. The part e^(–Ea/RT) is the fraction of molecules with energy equal to or greater than Ea.

Taking logs gives a straight line: ln k = ln A – Ea/RT. A plot of ln k against 1/T has slope –Ea/R, so Ea can be read from the graph.

For two temperatures, the most useful form is:

log(k₂/k₁) = (Ea/2.303R)(1/T₁ – 1/T₂)

A catalyst speeds up a reaction without being used up. It does not push the reactants over the same hill; it offers a different path with a lower activation energy, a shortcut over a lower pass. A catalyst does not change ΔG of the reaction, so it does not change the equilibrium constant: it only helps equilibrium arrive sooner.

Collision theory adds one more condition. Molecules must collide with enough energy and facing the right way: the proper orientation. So the rate is written rate = P Z_AB e^(–Ea/RT), where Z_AB is the collision frequency (how often A and B collide) and P is the probability or steric factor, the fraction of collisions that are correctly oriented.