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WBJEEA radioactive sample \(S_1\) having an activity of \(5 \mu \mathrm{Ci}\) has twice...
A radioactive sample \(S_1\) having an activity of \(5 \mu \mathrm{Ci}\) has twice the number of nuclei as another sample \(S_2\) which has an activity of \(10 \mu \mathrm{Ci}\). The half lives of \(S_1\) and \(S_2\) can be
Solution:
Activity of \(S_1=\frac{1}{2}\) (activity of \(S_2\) )
or
\(\)
\lambda_2 N_1=\frac{1}{2}\left(\lambda_2 N_2\right) \text { or } \frac{\lambda_1}{\lambda_2}=\frac{N_2}{2 N_1}
\(\)
or
\(\)
\begin{aligned}
& \frac{T_1}{T_2}=\frac{2 N_1}{N_2}\left(T=\text { half-life }=\frac{\ln 2}{\lambda}\right) \\
& N_1=2 N_2 \\
& \frac{T_1}{T_2}=4
\end{aligned}
\(\)
Given
\(\therefore\) correct option is (a).
or
\(\)
\lambda_2 N_1=\frac{1}{2}\left(\lambda_2 N_2\right) \text { or } \frac{\lambda_1}{\lambda_2}=\frac{N_2}{2 N_1}
\(\)
or
\(\)
\begin{aligned}
& \frac{T_1}{T_2}=\frac{2 N_1}{N_2}\left(T=\text { half-life }=\frac{\ln 2}{\lambda}\right) \\
& N_1=2 N_2 \\
& \frac{T_1}{T_2}=4
\end{aligned}
\(\)
Given
\(\therefore\) correct option is (a).
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