Nitric oxide (NO) reacts with oxygen to produce nitrogen dioxide
$2 N O (g) + O_{2} (g) \to 2 N O_{2} (g)$ ,
If the mechanism of reaction is
$N O + O_{2} \overset{K}{⇌} N O_{3} (f a s t)$
$N O_{3} + N O \overset{K_{1}}{⇌} N O_{2} + N O_{2} (s l o w)$ then rate law is

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$R a t e = K' {[N O]}^{3} [O_{2}]$

$R a t e = K' [N O] {[O_{2}]}^{2}$

$R a t e = K' [N O] [O_{2}]$

$R a t e = K' {[N O]}^{2} [O_{2}]$

answer is C.

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Detailed Solution

$N O + O_{2} \overset{K}{\underset{}{⇌}} N O_{3} (f a s t s t e p - 1)$

$N O_{3} + N O \overset{K_{1}}{\to} N O_{2} + N O_{2} (s l o w s t e p - 2)$

$∵ R D S$ is slowest step

Rate $= K_{1} [N O_{3}] [N O]$

$\begin{array}{l} K = \frac{[N O_{3}]}{[N O] [O_{2}]} \\ [N O_{3}] = K [N O] [O_{2}] \end{array}$

$\begin{array}{l} R a t e = K_{1} K [N O] [O_{2}] [N O] \\ = K^{1} {[N O]}^{2} [O_{2}] (∵ k^{1} = k_{1} k) \end{array}$

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