Find the average rate of formation of \(\mathrm{NO}_{2(\mathrm{~g})}\), in following reaction.
\(2 \mathrm{~N}_2 \mathrm{O}_{5(\mathrm{~g})} \rightarrow 4 \mathrm{NO}_{2(\mathrm{~g})}+\mathrm{O}_{2(\mathrm{~g})}\)
\(\left[-\frac{\Delta\left[\mathrm{N}_2 \mathrm{O}_5\right]}{\Delta \mathrm{t}}\right]=x \mathrm{~mol} \mathrm{dm}^{-3} \mathrm{~s}^{-1}\)

\(\begin{aligned}
& \frac{-\Delta\left[\mathrm{N}_2 \mathrm{O}_5\right]}{\Delta \mathrm{t}}=x \mathrm{~mol} \mathrm{dm}^{-3} \mathrm{~s}^{-1} \\
& 2 \mathrm{~N}_2 \mathrm{O}_{5(\mathrm{~g})} \rightarrow 4 \mathrm{NO}_{2(\mathrm{~g})}+\mathrm{O}_{2(\mathrm{~g})}
\end{aligned}\)
Average rate of reaction
\(\begin{gathered}
=-\frac{1}{2} \frac{\Delta\left[\mathrm{N}_2 \mathrm{O}_5\right]}{\Delta \mathrm{t}}=\frac{1}{4} \frac{\Delta\left[\mathrm{NO}_2\right]}{\Delta \mathrm{t}}=\frac{\Delta\left[\mathrm{O}_2\right]}{\Delta \mathrm{t}} \\
\therefore \quad \frac{\Delta\left[\mathrm{NO}_2\right]}{\Delta \mathrm{t}}=-\frac{4}{2} \frac{\Delta\left[\mathrm{N}_2 \mathrm{O}_5\right]}{\Delta \mathrm{t}}=\frac{4}{2} \times x \\
=2 x \mathrm{~mol} \mathrm{dm}^{-3} \mathrm{~s}^{-1}
\end{gathered}\)
\(\therefore \quad\) Average rate of formation of \(\mathrm{NO}_{2(\mathrm{~g})}\) \(=2 x \mathrm{~mol} \mathrm{dm}^{-3} \mathrm{~s}^{-1}\)
[Note: In the question, \(\frac{\Delta\left[\mathrm{N}_2 \mathrm{O}_5\right]}{\Delta \mathrm{t}}\) is changed to \(\frac{-\Delta\left[\mathrm{N}_2 \mathrm{O}_5\right]}{\Delta \mathrm{t}}\) to apply appropriate textual concepts.]