In the decomposition of `H_(2)O_(2)` at 300 K , the energy of activation was found to be 18 kcal/mole , while it decreased to 6 kcal/mole when the decomposition was carried out in the presence of a catalyst at 300 K . How many times is the catalysed reaction faster than the uncatalysed one ?

Explanation

Suppose `E_(1) and E_(2)` are the energies of activation when the reactionis carried out in the absence and presence of a catalyst respectively .
Thus , `k_(1) = Ae^(-E_(1)//RT)`
`k_(2)=Ae^(-E_(2)//RT)`
Taking log , `In k_(1) = 1n A - (E_(1))/(RT)`
In `k_(2)` = 1n A - `(E_(2))/(RT)`
`:. " "In k_(2) -In k_(1) = - (E_(2))/(RT) +(E_(1))/(RT)`
or `(k_(2))/(k_(1))=(E_(1))/(RT) =(E_(2))/(RT) +(E_(1))/(RT)`
or In `(k_(2))/(k_(1)) =(E_(1))/(RT) -(E_(2))/(RT) =1/(RT) (E_(1)-E_(2))`
or `2.303 log. (k_(2))/(k_(1))= 1/(0.002 xx300) (18-6) = 12/(0.002 xx300)`
` 2.303 log. (k_(2))/(k_(1)) = 20 `
`log . (k_(2))/(k_(1)) = 20/(2.303)`
`= 8.6843`
Taking antilog , `(k_(2))/(k_(1)) = 4.8 xx10^(8)`
The ratio of rate constants shows that the reaction will be faster by `4.8 xx10^(8)` times due to the addition of the catalyst .