For the reaction $A + B \rightarrow$ products, it is observed that

$(i)\,\,$on doubling the initial concentration of $A$ only, the rate of reaction is also doubled and

$(ii)$  on doubling the initial concentration of both $A$ and $B,$ there is a change by a factor of $8$ in the rate of the reaction.

The rate of this reaction is given by

  • [AIPMT 2009]
  • A

    rate $= k[A] [B]^2$

  • B

    rate $= k[A]^2 [B]^2$

  • C

    rate $= k[A] [B]$

  • D

    rate $= k[A]^2 [B]$

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$(i)$ $X + Y \rightleftharpoons M$      very rapid equilibrium

$(ii)$ $M + Z \to P$      slow

$(iii)$ $O + Y \to N$      very fast

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If $‘a’ $ is the initial concentration and $ ‘n’ $ is the order of the reaction and the half life period is $ ‘T’,$  then