Question

The mean lives of a radioactive sample are 30 years and 60 years for α-emission and $β$-emission respectively. If the sample decays both by $α$-emission and $β$-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, $\lambda_{\left(α+β\right)} = \lambda_{α} + \lambda_{β}$
$\frac{1}{\tau} = \frac{1}{\tau_{\alpha}}+\frac{1}{\tau _{\beta}} \quad$ (As $\lambda =\frac{1}{\tau }$)
$\Rightarrow \quad \frac{1}{\tau } = \frac{1}{30}+\frac{1}{60} = \frac{1}{20}$
$\therefore \quad\tau = 20$ years.
Now, $T_{1/2} = ln\left(2\right) τ = 13.86$ years
One-fourth of sample will remain after $2$ half life $= 27.72$ years.