A radioactive sample having an activity has twice the number of nuclei as another sample which has an activity of . The half-lives of and can be [2008]

Question

A radioactive sample $S_{1}$ having an activity $5 μ Ci$ has twice the number of nuclei as another sample $S_{2}$ which has an activity of $10 μ Ci$ . The half-lives of $S_{1}$ and $S_{2}$ can be [2008]

Smart Achievers · Accepted Answer

(1)

Let $λ_{1}$ and $λ_{2}$ be the decay constants and $N_{1}$ and $N_{2}$ be the number of active nuclei present for the two samples $S_{1}$ and $S_{2}$ respectively.

i.e., $λ_{1} N_{1} = 5 μ Ci$ and $λ_{2} N_{2} = 10 μ Ci$ $\Rightarrow λ_{2} N_{2} = 2 λ_{1} N_{1}$

Also, $N_{1} = 2 N_{2}$

$∴$ $λ_{2} N_{2} = 2 λ_{1} (2 N_{2})$ or $λ_{2} = 4 λ_{1}$ $\Rightarrow \frac{λ_{2}}{λ_{1}} = 4$

Also, $T_{1 / 2} = \frac{0.693}{λ}$

$∴$ ${(T_{1 / 2})}_{1} = 4 {(T_{1 / 2})}_{2}$

Solution

Q.
A radioactive sample $S_{1}$ having an activity $5 μ Ci$ has twice the number of nuclei as another sample $S_{2}$ which has an activity of $10 μ Ci$ . The half-lives of $S_{1}$ and $S_{2}$ can be [2008]

1 20 years and 5 years, respectively

2 20 years and 10 years, respectively

3 10 years each

4 5 years each

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