Question

The half life of a radioactive nucleus is $50$ days. The time interval $\left(t_2-t_1\right)$ between the time $t_2$ when $\frac{2}{3}$ of 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 $N=N_0{e}^{-\lambda t}$ Number of radioactive nuclei at time $=0$
$N=$ Number of radioactive nuclei left undecayed at any time $\lambda =$ decay constant At time $t_2,\frac{2}{3}$ of the sample had decayed
$\therefore N=\frac{1}{3}{N}_0$
$\therefore \frac{2}{3}{N}_0=N_0{e}^{-{\lambda }_1}$
Divide (i) by (ii), we get $\frac{1}{2}=\frac{e^{-\lambda t_2}}{e^{-\lambda l_1}}$
$\frac{1}{2}=e^{-\lambda \left(t_2-4\right)}$
$\lambda \left(t_2-t_1\right)=ln2$
$t_2-t_1=\frac{ln2}{\lambda }=\frac{ln2}{\left(\frac{ln2}{T_{1/2}}\right)}\left(\because \lambda =\frac{ln2}{T_{1/2}}\right)$
$=T_{1/2}=50$ days