Question

A radioactive sample $S_1$ having an activity of $5\mu Ci$ has twice the number of nuclei as another sample $S_2$ which has an activity of $10\mu Ci$. The half lives of $S_1$ and $S_2$ can be

• A. 5 years each
• B. 10 years each
• C. 20 years and 10 years, respectively
• D. 20 years and 5 years, respectively

Answer 1

The correct option is D

20 years and 5 years, respectively

Activity of $S_1=\frac{1}{2}$ (activity of $S_2$)

$\Rightarrow {\lambda }_1{N}_1=\frac{1}{2}\left({\lambda }_2{N}_2\right)$

$\Rightarrow \frac{{\lambda }_1}{{\lambda }_2}=\frac{N_2}{2N_1}$

$\Rightarrow \frac{T_1}{T_2}=\frac{2N_1}{N_2}\{T=\text{half life}=\frac{lo{g}_{e}2}{\lambda }\}$

Given $N_1=2N_2$

$\Rightarrow \frac{T_1}{T_2}=4$