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The rate constant $k$, for the reaction
$$
\mathrm{N}_2 \mathrm{O}_{5(\mathrm{~g})} \rightarrow 2 \mathrm{NO}_{2(\mathrm{~g})}+1 / 2 \mathrm{O}_{2(\mathrm{~g})}
$$
is $2.3 \times 10^{-2} \mathrm{~s}^{-1}$. Which equation given below describes the change of $\left[\mathrm{N}_2 \mathrm{O}_5\right]$ with time? $\left[\mathrm{N}_2 \mathrm{O}_5\right]_0$ and $\left[\mathrm{N}_2 \mathrm{O}_5\right]_t$ correspond to concentration of $\mathrm{N}_2 \mathrm{O}_5$ initially and at time $t$.
Options:
$$
\mathrm{N}_2 \mathrm{O}_{5(\mathrm{~g})} \rightarrow 2 \mathrm{NO}_{2(\mathrm{~g})}+1 / 2 \mathrm{O}_{2(\mathrm{~g})}
$$
is $2.3 \times 10^{-2} \mathrm{~s}^{-1}$. Which equation given below describes the change of $\left[\mathrm{N}_2 \mathrm{O}_5\right]$ with time? $\left[\mathrm{N}_2 \mathrm{O}_5\right]_0$ and $\left[\mathrm{N}_2 \mathrm{O}_5\right]_t$ correspond to concentration of $\mathrm{N}_2 \mathrm{O}_5$ initially and at time $t$.
Solution:
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Verified Answer
The correct answer is:
$\ln \frac{\left[\mathrm{N}_2 \mathrm{O}_5\right]_0}{\left[\mathrm{~N}_2 \mathrm{O}_5\right]_t}=k t$
$$
\begin{aligned}
& \text { } k=\frac{1}{t} \ln \frac{a(\text { initial })}{a-x(\text { after time } t)} \\
& k t=\ln \frac{\left[\mathrm{N}_2 \mathrm{O}_5\right]_0}{\left[\mathrm{~N}_2 \mathrm{O}_5\right]_t}
\end{aligned}
$$
\begin{aligned}
& \text { } k=\frac{1}{t} \ln \frac{a(\text { initial })}{a-x(\text { after time } t)} \\
& k t=\ln \frac{\left[\mathrm{N}_2 \mathrm{O}_5\right]_0}{\left[\mathrm{~N}_2 \mathrm{O}_5\right]_t}
\end{aligned}
$$
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