The rate constant for a first order reaction is $20 \, \text{min}^{-1}$. The time required for the initial…

For a first order reaction, we know:
${\mathrm{t}}_{1/2}=\frac{0.693}{\mathrm{k}}=\frac{0.693}{20}=0.03465 \min$
From the relation of half lives(where 'n' is the no. of half-lives):

$\Rightarrow {\left(\frac{1}{2}\right)}^{\mathrm{n}}=\left(\frac{1}{32}\right)$

$={\left(\frac{1}{2}\right)}^5$

Thus, No. of Half life $=5$

$\therefore$Time required $=0.03465\times 5=0.17325 \min =17.325\times {10}^{-2} \min$