A gaseous reaction is carried out both with and without a catalyst at 500K. The uncatalysed pathway requires 220kJ mol−1, whereas the catalysed pathway lowers the activation energy to 150kJ mol−1. The Maxwell–Boltzmann distribution of molecular kinetic energies for the reactant gas at 500K shows that 1.6% of the molecules have energies ≥220kJ mol−1.
Assume ideal‐gas behaviour and that the catalyst does not alter the energy distribution of the molecules.
(a) Using the given information, estimate the fraction of reactant molecules at 500K whose energies are at least 150kJ mol−1. Clearly state any assumptions or simplifying approximations you employ.
(b) Derive an expression for the ratio of the catalysed reaction rate to the uncatalysed reaction rate at 500K, given that the rate is proportional to the fraction of molecules with energy ≥Ea.
(c) The temperature is then raised to 520K while all other conditions remain the same. Describe qualitatively how the high‐energy tails relevant to both activation energies change, and predict whether the rate enhancement due to the catalyst (the ratio you derived in part (b)) will increase, decrease, or remain roughly the same. Justify your prediction.
(d) Discuss one limitation of using kinetic‐energy distributions alone to predict catalytic rate enhancements in real systems.