Question

The half-life of a radioactive substance is $30$ min. The time (in minutes) taken between 40% decay and 85% decay of the same radioactive substance is:

• A. 15
• B. 60
• C. 45
• D. 30

Answer 1

The correct option is B 60

Ratio of remaining nuclei (N) to initial nuclei (No) having half life period of $T=30$ min at time $t$ is given by :-

$\frac{N}{N_0}={\left(\frac{1}{2}\right)}^{t/T}$

$\Rightarrow t=-T{\log }_2\left(\frac{N}{N_0}\right)$

(i) when $40\%$ decay completes in time $t_1$(i.e $\frac{N}{N_0}0.6)$ [$40\%$ decay means ,$60\%$ remaining ]

$\Rightarrow t_1=-T{\log }_2\left(0.6\right)$

(ii) when $85\%$ decay completes in time $t_2(\frac{N}{N_0}=0.15)$

$t_2=-T{\log }_2\left(0.15\right)$

So time taken ,

$\Delta t=t_2-t_1=T\left[{\log }_2\right(0.6)-{\log }_2(0.15\left)\right]$
$=T{\log }_2\left(\frac{0.6}{0.15}\right)$

$=30\times {\log }_2\left(4\right)$

$\Delta t=60min$