Using the following data, calculate the value of equilibrium constant for the following reaction at `298K`
`underset("Acetylene")(3HC-=CH)hArr underset("Benzene")(C_(6)H_(6)(g))`
Assuming ideal behaviour
`Delta_(f)G^(Theta) (HC-= CH) = 2.09 xx 10^(5) J mol^(-1)`
`Delta_(f)G^(Theta) (C_(6)H_(6)) = 1.24 xx 10^(5) J mol^(-1)`,
`R = 8.314 J K^(-1) mol^(-1)`
Can the reaction be recommended for the synthesis of benzene?

Explanation

`X_(2) hArr 2X, k_(1) = 10^(5)=k_(2)`
(This is equilibrium step)
`X + Z rarr XZ, k_(3) = 10^(-4)` (Value of `k_(3)` is very less as compared to `k_(1)` and `k_(2)`).
So it slow step and is the rate-determining step
`:.` Rate `= k_(3)[X][Z]` ...(i)
`k_(eq) = (k_(2))/(k_(1)) = (10^(5))/(10^(5)) = 1`
`:. 1 = ([X]^(2))/([X_(2)])`, `[X] = [X_(2)]^(1//2)`
substittute the value of `[X]` in Eq. (i)
Rate `= k_(3)[X_(2)]^(1//2) [Z]`
`:.` Order of reaction `= (1)/(2) + 1 = (3)/(2)`