A hydrogenation reaction is carried out at `500 K`. If the 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)`.

Explanation

Let `E_(a)` and `E'_(a)` be the energy of activation in presence and absence of catalyst for hydrogenation reaction, then
`K=Ae^(-E_(a)//RT)`
In presence of catalyst `K_(1)=Ae^(-E_(a)//(Rxx500))`
In absence of catalyst: `K_(2)=Ae^(-E'_(a)//(Rxx400))`
Given, the two rates are same, i.e., `r_(1)=r_(2), ( :. K_(1)=K_(2))`
Therefore, `e^(-E_(a)//(Rxx500))=e^(-E'_(a)//(Rxx400))`
or `E_(a)/(Rxx500)=(E'_(a))/(Rxx400)`
or `E_(a)/500=(E_(a)-20)/(400), ( :' E_(a)-E'_(a)=20)`
`:. E_(a)=100 kJ mol^(-1)`