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 $\frac{1}{2}=\frac{e^{- \lambda t_{2}}}{e^{- \lambda l_{1}}}$
$\frac{1}{2}=e^{- \lambda \left(t_{2} - 4\right)}$
$\lambda \left(t_{2} - t_{1}\right)=ln2$
$t_{2}-t_{1}=\frac{ln 2}{\lambda }=\frac{ln 2}{\left(\frac{ln 2}{T_{1 / 2}}\right)}\left(\because \lambda = \frac{ln 2}{T_{1 / 2}}\right)$
$=T_{1 / 2}=50$ days