If a unimolecular reaction, `A(g) rarr` Products, takes place according to the mechanism
I. `A+A underset(k_(-1))overset(k_(1))hArr A^(**)+A`
II. `A^(**)underset(k_(2))rarr P`
where `k_(1), k_(-1)`, and `k_(2)` are the rate constant and `P, A`, and `A^(**)` stand for Product molecule, normal molecules of reactants and activated molecules of reactants respectively.
Which of the following expresison are correct?

Options

A. `(d[A^(**)])/(dt) = k_(1)[A]^(2)-k_(1)[A^(**)][A]-k_(2)[A^(**)]`
B. `(d[A^(**)])/(dt) = 0`
C. `[A^(**)] = (k_(1)[A]^(2))/(k_(-1)[A]+k_(2))`
D. All of the above

Answer

D

Explanation

`k_(eq) = (k_(1))/(k_(-1)) ([A^(**)][A])/([A]^(2))` …(i)
`(d[A^(**)])/(dt) = k_(1)[A]^(2)-k_(-1)[A^(**)][A]-k_(2)[A^(**)]` …(ii)
Since `A^(**)` is activated molecules of reactant, applying steady state approximation for `[A^(**)]`,
i.e., `(d[A^(**)])/(dt) = 0` in Eq. (ii), we get
`k_(1)[A]^(-2)-k_(-1)[A^(**)][A]-k_(2)[A^(**)]=0`
`[A^(**)] = (k_(1)[A]^(2))/(k_(-1)[A]+k_(2))`