A hydrogenation reaction is carried out at 500 K. If same reaction is carried out in the presence of a catalyst at the same rate, the temperature required is 400 K. Calculate the activation energy of the reaction if the catalyst lowers the activation barrier by 20 kJ mol^–1

According to Arrhenius equation
$\mathrm{k}={\mathrm{Ae}}^{-{\mathrm{E}}_{\mathrm{a}}/\mathrm{RT}}$
Let E_a of the reaction in presence of catalyst
$=\mathrm{x}-20{\mathrm{kJmol}}^{-1}$
The arhenius equations in the two conditions can thus be written as
$\begin{array}{l}\mathrm{k}={\mathrm{Ae}}^{-\frac{\mathrm{x}}{\mathrm{R}\times 500}}\dots \dots \dots \left(\mathrm{i}\right)\\ \mathrm{k}={\mathrm{Aee}}^{-\frac{\mathrm{x}-20}{\mathrm{R}\times 400}}\dots \dots \dots .\left(\mathrm{ii}\right)\end{array}$
Dividing equation(i) by (ii),we get
${\mathrm{e}}^{\frac{\mathrm{x}}{500\mathrm{R}}}={\mathrm{e}}^{\frac{\mathrm{x}-20}{400\mathrm{R}}}\Rightarrow \frac{\mathrm{x}}{500}=\frac{\mathrm{x}-20}{400}\text{ or }\mathrm{x}=100\mathrm{kJ}{\mathrm{mol}}^{-1}$