Question

The mean lives of a radioactive sample are 30 years and 60 years for α-emission and $\beta$-emission respectively. If the sample decays both by $\alpha$-emission and $\beta$-emission simultaneously, the time after which, only one-fourth of the sample remains is

• A. 14 years
• B. 20 years
• C. 28 years
• D. 45 years

Answer 1

Answer: (C)

Here, ${\lambda }_{\left(\alpha +\beta \right)}={\lambda }_{\alpha }+{\lambda }_{\beta }$
$\frac{1}{\tau }=\frac{1}{{\tau }_{\alpha }}+\frac{1}{{\tau }_{\beta }}$ (As $\lambda =\frac{1}{\tau }$)
$\Rightarrow \frac{1}{\tau }=\frac{1}{30}+\frac{1}{60}=\frac{1}{20}$
$\therefore \tau =20$ years.
Now, $T_{1/2}=ln\left(2\right)\tau =13.86$ years
One-fourth of sample will remain after $2$ half life $=27.72$ years.