Activation Energy, Collision Theory and Arrhenius Equation — Notes

Reactants must cross an energy barrier to become products. Threshold energy: the minimum energy colliding molecules must have to react. Activation energy $E_\text{a}$ = threshold energy − average energy of the reactants. Activated complex: the unstable, high-energy arrangement at the top of the barrier (bonds partly broken and partly formed); it may give products or revert to reactants.

Lower $E_\text{a}$ → faster reaction at a given temperature.

Exothermic: products below reactants, $\Delta H < 0$. Endothermic: products above reactants, $\Delta H > 0$. In all cases $\Delta H = E_\text{a(forward)} - E_\text{a(reverse)}$, so for an endothermic reaction $E_\text{a(f)} > \Delta H$.

Catalyst: provides an alternative path of lower $E_\text{a}$ for both forward and reverse reactions; does not change ΔH, ΔG, K or the position of equilibrium; is regenerated; equilibrium is reached sooner. Enzymes are biological catalysts.

Reactant molecules (treated as hard spheres) must collide. A collision is effective only if (i) the energy of the colliding molecules is at least the threshold energy and (ii) the molecules have the proper orientation. Rate $= P\,Z_\text{AB}\,e^{-E_\text{a}/RT}$: $Z_\text{AB}$ = collision frequency, $e^{-E_\text{a}/RT}$ = fraction of collisions with enough energy, $P$ = probability (steric) factor.

Raising temperature increases the collision frequency slightly but the fraction of energetic molecules greatly (tail of the energy distribution), so the rate rises sharply. (The mathematical treatment of collision theory is not required.)

$k = A\,e^{-E_\text{a}/RT}$: $A$ = frequency (pre-exponential) factor (same unit as $k$); $e^{-E_\text{a}/RT}$ = fraction of molecules with energy ≥ $E_\text{a}$.

Log form: $\log k = \log A - \dfrac{E_\text{a}}{2.303RT}$. Plot of $\log k$ against $1/T$: straight line, slope $= -E_\text{a}/2.303R$, intercept $= \log A$.

Two temperatures: $\log\dfrac{k_2}{k_1} = \dfrac{E_\text{a}}{2.303R}\left(\dfrac{T_2 - T_1}{T_1T_2}\right)$.

Temperature coefficient $= k_{T+10}/k_T$, usually 2–3; larger for larger $E_\text{a}$. A catalyst lowering $E_\text{a}$ by $\Delta E_\text{a}$ (same $A$) multiplies $k$ by $10^{\Delta E_\text{a}/2.303RT}$.