Deduce the rate law for the converison of `H_(2)` and `I_(2)` to `HI` at `440^(@)C` corresponding to the following proposed mechanism:
step I: `I_(2) hArr 2I ("fast") (k_(1))`
Step II: `I+H_(2) hArr IH_(2) ("fast") (k_(2))`
Step III: `IH_(2) + I rarr 2HI` (slow) `(k)`

Explanation

1. `I_(2)overset(k_(1))rarr2I` (fast) `(k_(1))`
2. `I + H_(2) overset(k_(2))rarr IH_(2)` (fast) `(k_(2))`
3. `IH_(2) + I overset(k)rarr 2HI` (slow) `(k)`
Rate `= k[HI_(2)] [I]` …(i)
Step (I) and (II) are fast and reverisble.
Form step (1), `k_(1) = ([I]^(2))/([I_(2)])`
`:. [I]^(2) = k_(1)[I_(2)]`
`[I] = k_(1)^(1//2)[I_(2)]^(1//2)` ...(ii)
ismilarly, form step (II),
`k_(2) = ([IH_(2)])/([I][H_(2)])`
`[IH_(2)] = k_(2)[I][H_(2)]` ...(iii)
Substitute Eqs. (ii) and (iii) in (i),
Rate `= k_(2)k[I][H_(2)] k_(1)^((1)/(2)) [I_(2)]^((1)/(2))`
`= kk_(2) k_(1)^((1)/(2))k_(1)^((1)/(2))[I_(2)]^((1)/(2))[H_(2)][I_(2)]^((1)/(2))`
`= kk_(2)k_(1)[I_(2)]^((1)/(2))[H_(2)][I_(2)]^((1)/(2))`
`= kk_(2)k_(1)[I_(2)][H_(2)]`
`= k'[I_(2)][H_(2)] [k' = k k_(2) k_(1)]`