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

Arrhenius eqaution may be given as,
`k=Ae^(-E_(a)//RT)`
Let `k_(500) and k_(400)` be the rate constants at temperatures 500 K and 400 K (in presence of catalyst) respectively. `E_(500) and E_(400)` be the activation energies at temperatures 500 K and 400 K respectively.
`k_(500)=Ae^(E_(500)//Rxx500)" "....(i)`
`k_(400)=Ae^(E_(400)//Rxx400)" "....(ii)`
Given `k_(500)=k_(400)` (same rates in presence and absence of a catalst) ltbr On comparing eq. (i) with eq. (ii)
`(E_(500))/(Rxx500)=(E_(400))/(Rxx400)`
or `(E_(500))/(5)=(E_(400))/(4)`
or `E_(500)=(E_(400))/(4)xx5`
Given, `E_(500)=E_(400)+20`
Substituting in eq. (iii),
`E_(400)+20=E_(400)xx1.25`
or `E_(400)=(20)/(0.25)=80 kJ "mol"^(-1)`
So, `E_(500)=80+20=100 kJ "mol"^(-1)`