Activation energy of an uncatalysed reaction at 1000 K is 5 Kcal. In presence of catalyst the activation energy of same reaction at same temperature will be ___ cal. (Ratio of rate constants of catalyzed reaction to uncatalysed reaction is 10 : 1)

By Arrhenius Equation,

${\log }_{e}k = {\log }_{e}A - \frac{E}{RT}$

given, ratio of rate constants of the two reactions = 10:1

so, let the rate constants be 10x and x respectively.

For catalysed reaction, activation energy = E^'

${\log }_{e}10x = {\log }_{e}A - \frac{E^{\text{'}}}{RT} ...1)$

For uncatalysed reaction, activation energy = E

${\log }_{e}x = {\log }_{e}A - \frac{E^{ }}{RT} ...2)$

$1) - 2) {\log }_{e}10 = \frac{-E^{\text{'}} + E}{RT}$

$2.303\times 2\times 1000 = E - E^{\text{'}}$

$4606 = 5000 - E^{\text{'}}$

$E^{\text{'}} = 394 cal$

So, in presence of catalyst the activation energy is 394 cal