Question

The half life of a radioactive nucleus is $50$ days. The time interval (t${}_2-t_1$) between the time t${}_2$ when $\frac{2}{3}$ ot it has decayed and the time $t_1$, when $\frac{1}{3}$ of it had decayed is

• A. 30 days
• B. 50 days
• C. 60 days
• D. 15 days

Answer 1

Answer: (B)

According to radioactive decay law
$N=N_0{e}^{-\lambda t}$
where N${}_0$ = Number of radioactive nuclei at
time t= 0
N = Number of radioactive nuclei left
undecayed at any time t
$\lambda$= decay constant
At time $t_2,\frac{2}{3}$ of the sample had decayed
$\therefore N=\frac{1}{3}{N}_0$
$\therefore \frac{1}{3}{N}_0=N_0{e}^{-\lambda t_2}\dots \left(i\right)$
At time $t_1,\frac{1}{3}$ of the sample had decayed,
$\therefore N=\frac{2}{3}{N}_0$
$\therefore \frac{2}{3}{N}_0=N_0{e}^{-\lambda t_1}\dots \left(ii\right)$
Divide (i) by (ii), we get
$\frac{1}{2}=\frac{e^{-\lambda t_2}}{e^{-\lambda t_1}}$
$\frac{1}{2}=e^{-\lambda t_2}(t_2-t_1)$
$\lambda (t_2-t_1)=In2$
$t_2-t_1=\frac{In2}{\lambda }=\frac{In2}{\left(\frac{In2}{T_{1/2}}\right)}\left(\because \lambda =\frac{In2}{T_{1/2}}\right)$
$=T_{12}=50$ days