Question

The activation barrier for an uncatalyzed reaction is estimated to be 15.3 kcal/mol. The activation barrier for the catalyzed reaction is estimated to be 8.7 kcal/mol. How many times faster is the catalyzed rate versus the uncatalyzed rate? In other words, by what factor/coefficient do you have to multiply the uncatalyzed rate to equal the catalyzed rate? Assume the temperature is 298 K, and enter your answer to the nearest ones.

Answer 1

Reaction rate is proportional to the exp(-Ea/RT), where Ea- activation energy in Joules, R-gas constant and T -temperature in K.
So, the ratio of rate constants of catalyzed and uncatalyzed reactions will be equal to exp(-Ea₁/RT) / exp(-Ea₂/RT) = exp((Ea₂-Ea₁)/RT), where Ea₂ - activation energy of catalyzed reaction - 15.3 * 4184 J = 64015.2 J; and Ea₁ - activation energy of uncatalyzed reaction - 8.7 * 4184 J = 36400.8 J
Ratio = exp ( ( 64015.2-36400.8 ) / 8.314 * 298 ) = 69269
Catalyzed reaction will be 69269 times faster than uncatalyzed.