A certain endothermic reaction: `Ato` Product, `DeltaH=+ve` proceeds ina sequence of three elementary steps with the rate constant `K_(1),K_(2)` and `K_(3)` and each one having energy of activation `E_(1),E_(2)` and `E_(3)` respectively at `25^(@)C`. The observed rate constannt for the reaction is equal to `K_(3)sqrt((K_(1))/(K_(2)))`.
`A_(1),A_(2)` and `A_(3)` are Arrhenius parameters respectively.
For a reversible `Aunderset(K_(2))overset(K_(1))(hArr)B,DeltaH=q` if pre exponential factors are same. The correct relation is

Options

A. `K_(eq)=e(-q//RT)`
B. Rate of reaction `=(-d[A])/(dt)=K_(1)[A]-K_(2)[B]`
C. At equilibrium `K_(1)[A]=K_(2)[B]`
D. Either of these

Answer

D

Explanation

`r_(1)=K_(1).[A]^(1),K_(1)=A.e^(-epsilon_(1)//RT),r_(2)=K_(2).[B]^(1),K_(2)=A.e^(-epsilon_(2)//RT)`
net rate `r=r_(1)-r_(2)=K_(1)[A]^(1)-K_(2)[B]^(1)` at eq state `r_(1)=r_(2)`
`K_(1)[A]^(1)=K_(2)[B]^(1),K_(eq)=([B]^(1))/([A]^(1))=(K_(1))/(K_(2))=(A.e^(-epsilon_(1)//RT))/(A.e^(-E_(2)//RT)=A^(0).e^(((-epsilon_(1)+epsilon_(2))/(RT))))=A^(0).e^(((-epsilon_(1)-epsilon_(2))/(RT)))`
`K_(eq)=A^(0).e^(-epsilon_(a)//RT)`.So `epsilon_(a)=(epsilon_(1)-epsilon_(2))`