The following data were obtained during the first order thermal decomposition of $N_{2} O_{5} (g)$ at constant volume
$2 N_{2} O_{5} (g) \to {2N}_{2} O_{4} (g) + O_{2} (g)$
Time (sec) Total pressure (atm)
0 0.5
100 0.6
Then the rate constant of the reaction is ___(Given that log 5 = 0.6990, log 3 = 0.4771)

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$2.55 \times 10^{- 3} s^{- 1}$

$5.1 \times 10^{- 3} s^{- 1}$

$1.02 \times 10^{- 3} s^{- 1}$

$5.1 \times 10^{- 4} s^{- 1}$

answer is A.

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Detailed Solution

$2 N_{2} O_{5} (g) \to 2 N_{2} O_{4} (g) + O_{2} (g)$

$\begin{matrix} 0.5 atm & 0 atm & 0 atm \\ 2 x & 2 x & x \\ 0.5 - 2 x & 2 x & x \end{matrix}$

$P_{t} = 0.5 - | 2 x + 2 x + x = 0.5 + x \Rightarrow x = P_{t} - 0.5$

$∴ P_{N_{2} O_{5}} = 0.5 - 2 x = 0.5 - 2 (P_{t} - 0.5) = 0.5 - 2 P_{t} + 1 = 1.5 - 2 P_{t} a t 100 \sec, P_{t} = 0.6 a t m$

$∴ P_{N_{2} O_{5}} a t 100 \sec = 1.5 - 2 (0.6) = 0.3 a t m$

$∴ k = \frac{2.303}{100} \log \frac{0.5}{0.3} \Rightarrow k = 5.1 \times 10^{- 3} S^{- 1}$

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