Catalyst A reduces the activation energy for a reaction by 10 kJ mol^$-$1 at 300 K. The ratio of rate constants, ${{{}^kT,\,Catalysed} \over {{}^kT,\,Uncatalysed}}$ is e^x. The value of x is ___________. [nearest integer]

[Assume that the pre-exponential factor is same in both the cases. Given R = 8.31 J K^$-$1 mol^$-$1]

Activation Energy : It is the minimum amount of energy required to activate the molecules/ atoms so that they can undergo the chemical reaction. <br/><br/>A catalyst increases the rate of a reaction by lowering the activation energy so that more reactant molecules collide with enough energy to surmount the smaller energy barrier. <br/><br/>In the Question, <br/><br/>Given : $\mathrm{E}_{\text {cat }}-\mathrm{E}_{\text {uncat }}=10 \mathrm{~kJ} / \mathrm{mol}$ <br/><br/>$\mathrm{T}=300 \mathrm{~K}$ <br/><br/>According to Arrhenius Equation, <br/><br/>$$ \begin{aligned} & \mathrm{K}=A \mathrm{e}^{-\mathrm{Ea} / \mathrm{RT}} \\\\ & \frac{\mathrm{E}_{\text {cat }}}{\mathrm{E}_{\text {uncat }}}=e^{\frac{E a-E_a^1}{R T}} \end{aligned} $$ <br/><br/>$$ \begin{aligned} & =e^{\frac{10 \times 10^3}{8.21 \times 300}} \\\\ & =e^4 \\\\ & =4 \end{aligned} $$