The rate constant k, for the reaction $N_{2} O_{5} (g) \to 2 {NO}_{2} (g) + \frac{1}{2} O_{2} (g) is 2 .3 \times 10^{- 2} s^{- 1}$ . Which equation given below describes the change of $[N_{2} O_{5}]$ with time? ${[N_{2} O_{5}]}_{0}$ and ${[N_{2} O_{5}]}_{t}$ correspond to concentration of $N_{2} O_{5}$ initially and at time, t?

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${[N_{2} O_{5}]}_{t} = {[N_{2} O_{5}]}_{0} + kt$

$\log_{10} {[N_{2} O_{5}]}_{t} = \log_{10} {[N_{2} O_{5}]}_{0} - kt$

$\ln \frac{{[N_{2} O_{5}]}_{0}}{{[N_{2} O_{5}]}_{t}} = kt$

${[N_{2} O_{5}]}_{0} = {[N_{2} O_{5}]}_{t} e^{kt}$

answer is D.

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Detailed Solution

Rate constant $= 2 .3 \times 10^{- 2} \sec^{- 1}$

It means it is a first order reaction (because unit of rate constant is $s e c^{- 1}$ )

For first order reaction $k = \frac{1}{t} \ln \frac{a}{a - x}$

$kt = \ln \frac{a}{a - x} = \ln \frac{{[N_{2} O_{5}]}_{0}}{{[N_{2} O_{5}]}_{t}}$

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