A certain endothermic reaction: `Ararr` Product, `DeltaH=+ve` proceeds in a sequence of three elementary steps with the rate constants `K_(1), K_(2)` and `K_(3)` and each one having energy of activation `E_(a), E_(2)` and `E_(3)` respectively at `25^(@)C`. The observed rate constant for the reaction is equal to `K_(3) sqrt(K_(1)/K_(2)). A_(1), A_(2)` and `A_(3)` are Arrhenius parameters respectively.
The observed energy of activation for the reaction is:

Options

A. (a) `(2E_(1)-E_(2)+2E_(3))/(2)`
B. (b) `(E_(2)-E_(1)-2E_(3))/(2)`
C. (c ) `sqrt((E_(1)E_(3))/(E_(2)))`
D. (d) `(E_(1)-E_(2))/(2)+E_(3)`

Answer

d

Explanation

`K_(1)=A_(1)e^(-E_(1)//RT), K_(2)=A_(2)e^(-E_(2)//RT),`
`K_(3)=A_(3)e^(-E_(3)//RT), K_(obs)=Ae^(-E_(obs)//RT)`,
`:' K_(obs)=sqrt(K_(1)/K_(2)).K_(3)`
`:. [A_(1)/A_(2)]^(1//2).A_(3).e^((-E_(1)+E_(2))/(2RT)-E_(3)/(RT))=Ae^(-E_(obs)//RT)`
or `(A_(1)/A_(2))^(1//2). A_(3).e^((-[E_(1)-E_(2)+2E_(3)])/(2RT))=Ae^(-E_(obs)//RT)`
`:. E_(obs)=(E_(1)-E_(2)+2E_(3))/(2)`
and `A=sqrt(A_(1)/A_(2)).A_(3)`