A chemical reaction is carried out with and without a catalyst at 300 K . The rate constant of the catalyzed reaction is e^8 times the rate constant of the uncatalyzed reaction. If the activation energy of the catalyzed reaction is 50 kJ mol ⁻¹ , the activation energy of the uncatalyzed reaction is ________ kJ mol ⁻¹ . [Given: R = 25 3 J K ⁻¹ mol ⁻¹ , assume the pre-exponential factor A remains the same for both reac

Step-by-step solution

According to the Arrhenius equation, the rate constant is given by k = A e^ -E_a/RT . For the catalyzed and uncatalyzed reactions, we have: k_c = A e^ -E_ ac /RT k_u = A e^ -E_ au /RT Dividing the two equations gives: k_c k_u = e^ (E_ au - E_ ac )/RT Taking the natural logarithm on both sides: ( k_c k_u ) = E_ au - E_ ac RT Given that k_c k_u = e^8 , we have (e^8) = 8 . Substituting the given values ( T = 300 K , R = 25 3 J K ⁻¹ mol ⁻¹ , E_ ac = 50 kJ mol ⁻¹ = 50000 J mol ⁻¹ ): 8 = E_ au - 50000 ( 25 3 ) 300 8 = E_