Question

The half life of a radioactive substance is 20 minutes. The approximate time interval $(t_2-t_1)$ between the time $t_2$ when $\frac{2}{3}$ of it has decayed and time $t_1$ when $\frac{1}{3}$ of it had decayed is:

• A. 7 min
• B. 14 min
• C. 20 min
• D. 28 min

Answer 1

The correct option is C 20 min

Given, $t_{\frac{1}{2}}=20min$
$\Rightarrow t_{\frac{1}{2}}=20=\frac{ln2}{\lambda }$
$\Rightarrow \lambda =\frac{ln2}{20}$ ...(i)

Now, the times for $\frac{1}{3}rd$ decayed = time when the $\frac{2}{3}rd$ Nuclei are remaining is $t_1-t_{\lambda }$
$\therefore \frac{2}{3}=e$ ...(ii)

Similarly, time when $\frac{2}{3}rd$ has decayed is $t_2$
$\frac{1}{3}=e^{-t_2\lambda }$ ...(iii)

from (ii) and (iii)
$e^{(t_2-t_1)\lambda }=2$

$\Rightarrow (t_2-t_1)\lambda =ln2$ ...(iv)

from (iv) and (i)
$t_2-t_1=20min$