$\mathrm{r}=\mathrm{k}\left[\mathrm{A}\right]$ for a reaction, $50\%$ of $\mathrm{A}$ is decomposed in $120$ minutes. The time taken for $90\%$ decomposition of $\mathrm{A}$ is ______ minutes.

$\mathrm{r}=\mathrm{k}\left[\mathrm{A}\right]$

So, order of reaction $=1$

${\mathrm{t}}_{1/2}=120 \min$

$\mathrm{k}=\frac{0.693}{{\mathrm{t}}_{1/2}}$

For $90\%$ completion of reaction,

$$\begin{gathered} \Rightarrow \mathrm{k}=\frac{2.303}{\mathrm{t}}\log \left(\frac{\mathrm{a}}{\mathrm{a}-\mathrm{x}}\right) \\ \mathrm{a}=100,(\mathrm{a}-\mathrm{x})=10 \end{gathered}$$

By putting the values in above equation,

$\Rightarrow \frac{0.693}{{\mathrm{t}}_{1/2}}=\frac{2.303}{\mathrm{t}}\log \frac{100}{10}$

$\therefore \mathrm{t}=399 \min$