Write a rate expresison for each speices. Assuming `[B]` is constant, write a rate expresison for `[A]` and `[C]` so that they do not contain `[B]` term. Under what conditions will the reaction be pseudo first order to second conditions will the reaction be pseudo first order to second order ?
Mechanism of reaction is
`A underset(k_(2))overset(k_(1))hArr B`
`{:(B+Aoverset(k_(3))rarrC:}/{:(2ArarrC)/`

Explanation

The rate expresison are:
`(-d[A])/(dt) = k_(1)[A]-k_(2)[B]+k_(3)[B][A]`
`(d[B])/(dt) = k_(1)[A]-k_(2)[B] - k_(3)[A][B]`
`(d[C])/(dt) = k_(3)[A][B]`
Uisng the steady state approximation for `[B]`, i.e.,
`(d[B])/(dt) = 0` and solving for `[B]`, gives
`(d[B])/(dt) = k_(1)[A] -k_(2)[B] - k_(3)[A][B] = 0`
`[B] = (k_(1)[A])/(k_(2)+k_(3)[A])`
Substituting this expresison for `[B]` into the expresison for
`(-d[A])/(dt)` and `(d[C])/(dt)` gives
`(-d[A])/(dt) = k_(1)[A] - k_(2)(k_(1)[A])/(k_(2) + k_(3)[A]) + k_(3)(k_(1)[A])/(k_(2) + k_(3)[A])[A]`
`= (2k_(1)k_(3)[A]^(2))/(k_(2)+k_(3)[A])`
`(d[C])/(dt) = k_(3)[A] = (k_(1)k_(3)[A]^(2))/(k_(2)+k_(3)[A])`
Note that `(-d[A])/(dt) = 2[(d[C])/(dt)]`. If `k_(3)[A] gt gt k_(2)`, then condition for pseudo first order
`(d[C])/(dt) = (k_(1)k_(3)[A]^(2))/(k_(3)[A]) = k_(1)[A]`
and if `k_(3)[A] lt lt k_(2)`, then condtion for pseudo secnd order
`(d[C])/(dt) = (k_(1)k_(3)[A]^(2))/(k_(2))`