The rate of a reaction increases by a factor of e^8 when a catalyst is added at 300 K . If the activation energy of the uncatalysed reaction is 100 kJ mol ⁻¹ , the activation energy of the catalysed reaction is _____ kJ mol ⁻¹ (nearest integer). [Assume that the pre-exponential factor is the same in both cases. Given R = 8.314 J K ⁻¹ mol ⁻¹ ]

Step-by-step solution

According to the Arrhenius equation, the rate constant is given by k = A e^ - E_a RT . For the uncatalysed reaction: k_ uncat = A e^ - E_ a,uncat RT For the catalysed reaction: k_ cat = A e^ - E_ a,cat RT Taking the ratio of the two rate constants: k_ cat k_ uncat = e^ E_ a,uncat - E_ a,cat RT Given that k_ cat k_ uncat = e^8 , we can equate the exponents: E_ a,uncat - E_ a,cat RT = 8 E_ a,uncat - E_ a,cat = 8RT Substituting the given values ( R = 8.314 J K ⁻¹ mol ⁻¹ , T = 300 K ): E_ a,uncat - E_ a,cat = 8 8.314 3