Question

At a given instant there are $25\%$ undecayed radioactive nuclei in a sample. After $10\text{sec}$ the number of undecayed nuclei reduces to $6.25\%$, the mean life of the nuclei is

• A. $14.43\text{sec}$
• B. $7.21\text{sec}$
• C. $5\text{sec}$
• D. $10\text{sec}$

Answer 1

The correct option is B $7.21\text{sec}$

In $10\text{sec}$, number of nuclei has been reduced to one fourth $\left(25\%\text{to}6.25\%\right)$

Using formula,

$\frac{N}{N_0}=\frac{1}{2^{t/\left(t_{1/2}\right)}}$

$\Rightarrow \frac{1}{4}=\frac{1}{2^{t/\left(t_{1/2}\right)}}$

$\Rightarrow \frac{t}{t_{1/2}}=2$

$\Rightarrow t_{1/2}=\frac{10}{2}$

$\Rightarrow t_{1/2}=5\text{sec}$.

$\therefore \text{Mean life}T=\frac{t_{1/2}}{0.693}=\frac{5}{0.693}$

$\therefore T=7.21\text{sec}$

Hence, option $\left(\text{B}\right)$ is the right choice.

Why this question? Half life $\left(T_{1/2}\right)$ : It is the time during which the number of active nuclei reduce to half of its initial value. $T_{1/2}=\frac{\ln \left(2\right)}{\lambda }=\frac{0.693}{\lambda }$