A radioactive sample S_1 having an activity of 5 μ Ci has twice the...

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

Options

A20 yr and 5 yr, respectively
B20 yr and 10 yr, respectively
C10 yr each
D5 yr each

Correct answer

A. 20 yr and 5 yr, respectively

Step-by-step 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).

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