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The half life of a radioactive substance is 20 minutes. The approximate time interval $\left(t_2-t_1\right)$ between the time $t_2$ when $\frac{2}{3}$ of it has decayed and time $t_1$ and $\frac{1}{3}$ of it had decayed is :
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$20 \mathrm{~min}$
$20 \mathrm{~min}$
$\mathrm{t}_{\frac{1}{2}}=20$ minutes
$\mathrm{N}=\mathrm{N}_0 \mathrm{e}^{-\lambda \mathrm{t}_2} \quad \lambda \mathrm{t}_1=\ln 3$
$\frac{2}{3} \mathrm{~N}_0=\mathrm{N}_0 \mathrm{e}^{-\lambda \mathrm{t}_2} \mathrm{t}_1=\frac{1}{\lambda} \ln 3$
$\frac{2}{3} \mathrm{~N}_0=\mathrm{N}_0 \mathrm{e}^{-\lambda t_2}$
$\mathrm{t}_2=\frac{1}{\lambda} \ln \frac{3}{2}$
$\mathrm{t}_2-\mathrm{t}_1=\frac{1}{\lambda}\left[\ln \frac{3}{2}-\ln 3\right]$
$=\frac{1}{\lambda} \ln \left[\frac{1}{2}\right]=\frac{0.693}{\lambda}$
$=20 \min$
$\mathrm{N}=\mathrm{N}_0 \mathrm{e}^{-\lambda \mathrm{t}_2} \quad \lambda \mathrm{t}_1=\ln 3$
$\frac{2}{3} \mathrm{~N}_0=\mathrm{N}_0 \mathrm{e}^{-\lambda \mathrm{t}_2} \mathrm{t}_1=\frac{1}{\lambda} \ln 3$
$\frac{2}{3} \mathrm{~N}_0=\mathrm{N}_0 \mathrm{e}^{-\lambda t_2}$
$\mathrm{t}_2=\frac{1}{\lambda} \ln \frac{3}{2}$
$\mathrm{t}_2-\mathrm{t}_1=\frac{1}{\lambda}\left[\ln \frac{3}{2}-\ln 3\right]$
$=\frac{1}{\lambda} \ln \left[\frac{1}{2}\right]=\frac{0.693}{\lambda}$
$=20 \min$
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