For a first order reaction involving decomposition of $N_{2} O_{5} .$ the following information is available.
$\begin{array}{ll} 2 N_{2} O_{5} (g) \to 4 {NO}_{2} (g) + O_{2} (g) rate = k [N_{2} O_{5}] \\ N_{2} O_{5} (g) \to 2 {NO}_{2} (g) + \frac{1}{2} O_{2} (g) rate = k^{'} [N_{2} O_{5}] \end{array}$
Which of the following expressions is true?

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$k = k^{'}$

$k > k$

$k = 3 k^{'}$

$k^{'} = 2 k$

answer is D.

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Detailed Solution

For $2 N_{2} O_{5} (g) \to 4 {NO}_{2} (g) + O_{2} (g); - \frac{1}{2} \frac{d [N_{2} O_{5}]}{d t} = k [N_{2} O_{5}]$

For $N_{2} O_{5} (g) \to 2 {NO}_{2} (g) + \frac{1}{2} O_{2} (g); - \frac{d [N_{2} O_{5}]}{d t} = k^{'} [N_{2} O_{5}]$

Since $- d [N_{2} O_{5}] / dt$ does not depend upon the way the chemical reaction is formulated, we will have $k' = 2 k$ .

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