Explanation:
A Given that, half-life $\left(\mathrm{T}_{1 / 2}\right)=20 \mathrm{~min}$
Let, the initial decay substance $=\mathrm{N}_{0}$
When $20 \%$ decay, then we have $80 \%$ of the substance is left,
$\frac{80 \mathrm{~N}_{0}}{100}=\mathrm{N}_{0} \mathrm{e}^{-\lambda t_{20}}$
When $80 \%$ decay, then we have $20 \%$ of the substance is left,
$\frac{20 \mathrm{~N}_{0}}{100}=\mathrm{N}_{0} \mathrm{e}^{-\lambda \mathrm{t}_{80}}$
On dividing equation (i) by (ii), we get-
$4=\mathrm{e}^{\lambda\left(\mathrm{t}_{80}-\mathrm{t}_{20}\right)}$
$\ln 4=\lambda\left(\mathrm{t}_{80}-\mathrm{t}_{20}\right)$
$2 \ln 2=\frac{0.693}{\mathrm{~T}_{1 / 2}}\left(\mathrm{t}_{80}-\mathrm{t}_{20}\right) \quad\left[\because \lambda=\frac{0.693}{\mathrm{~T}_{1 / 2}}\right]$
$2 \ln 2=\frac{0.693}{20}\left(\mathrm{t}_{80}-\mathrm{t}_{20}\right)$
$\left(\mathrm{t}_{80}-\mathrm{t}_{20}\right)=\frac{20 \times 2 \ln 2}{0.693} \quad[\because \ln 2=0.693]$
$\left(\mathrm{t}_{80}-\mathrm{t}_{20}\right)=\frac{20 \times 2 \times 0.693}{0.693}$
$\left(\mathrm{t}_{80}-\mathrm{t}_{20}\right)=40 \text { minutes }$