Temperature and the Arrhenius equation

Most reactions speed up with temperature. N₂O₅ in the gas phase has a half-life of about 12 min at 50 °C, 5 h at 25 °C and 10 days at 0 °C. A mixture of KMnO₄ and oxalic acid decolourises faster when warm.

For many reactions a rise of 10 °C roughly doubles the rate constant.

Arrhenius equation: k = A e^(−Ea/RT). A is the Arrhenius (frequency) factor, Ea the activation energy in J mol⁻¹ and R = 8.314 J K⁻¹ mol⁻¹.

Reactants must pass through a short-lived, high-energy activated complex before becoming products. For H₂ + I₂ → 2HI the complex has partly broken H–H and I–I bonds and partly formed H–I bonds. Ea is the energy needed to reach it from the reactants.

At a given temperature molecules have a spread of kinetic energies (the Maxwell-Boltzmann distribution). The peak is at the most probable energy. A higher temperature flattens the curve and shifts the peak to higher energy; the total area stays the same because the number of molecules is fixed.

e^(−Ea/RT) is the fraction of molecules with energy at least Ea. Raising T by 10 °C nearly doubles the area beyond Ea, which is why rate nearly doubles.

Taking logs: ln k = ln A − Ea/RT. A plot of ln k against 1/T is a straight line with slope −Ea/R and intercept ln A.

Two temperatures: log(k₂/k₁) = [Ea/(2.303R)] × [(T₂ − T₁)/(T₁T₂)]. For k = 0.02 s⁻¹ at 500 K and 0.07 s⁻¹ at 700 K, this gives Ea = 18.2 kJ mol⁻¹.

If k doubles between 298 K and 308 K, Ea = 2.303R × log 2 × (298 × 308)/10 = 52.9 kJ mol⁻¹.