For the reaction, $2 \mathrm{~N}_2 \mathrm{O}_5 \rightarrow 4 \mathrm{NO}_2+\mathrm{O}_2$, the rate equation…
For the reaction, $2 \mathrm{~N}_2 \mathrm{O}_5 \rightarrow 4 \mathrm{NO}_2+\mathrm{O}_2$, the rate equation can be expressed in two ways $-\frac{\mathrm{d}\left[\mathrm{N}_2 \mathrm{O}_5\right]}{\mathrm{dt}}=\mathrm{k}\left[\mathrm{N}_2 \mathrm{O}_5 \quad\right] n \mathrm{nd}$ $+\frac{\mathrm{d}\left[\mathrm{NO}_2\right]}{\mathrm{dt}}=\mathrm{k}^{\prime}\left[\mathrm{N}_2 \mathrm{O}_5\right]$
$\mathrm{k}$ and $\mathrm{k}^{\prime}$ are related as:
$\mathrm{k}=\mathrm{k}^{\prime}$
$2 \mathrm{k}=\mathrm{k}^{\prime}$
$\mathrm{k}=2 \mathrm{k}^{\prime}$
$\mathrm{k}=4 \mathrm{k}^{\prime}$
Solution
Rate of disappearance of reactant $=$ Rate of appearance of products
$
\begin{aligned}
&-\frac{1}{2} \frac{\mathrm{d}\left[\mathrm{N}_2 \mathrm{O}_5\right]}{\mathrm{dt}}=\frac{1}{4} \frac{\mathrm{d}\left[\mathrm{NO}_2\right]}{\mathrm{dt}} \\
&\frac{1}{2} k\left[\mathrm{~N}_2 \mathrm{O}_5\right]=\frac{1}{4} k^{\prime}\left[\mathrm{N}_2 \mathrm{O}_5\right] \\
&\frac{k}{2}=\frac{k^{\prime}}{4} \\
&\therefore k^{\prime}=2 k
\end{aligned}
$