# A sample of radioactive material A, that has an activity of 10 mCi (1 Ci = 3.7 x 1010 decays/s), has twice the number of nuclei as another sample of a different radioactive material B which has an activity of 20 mCi. The correct choices for half-lives of A and B would then by respectively :Option 1)5 days and 10 daysOption 2)10 days and 40 daysOption 3)20 days and 5 daysOption 4)20 days and 10 days

$-\frac{dN}{dt}= \lambda N$

- wherein

Ratio of disintegration is propotional to number of nuclei

$\lambda$= disintegration constant

Activity A = $\lambda N$

For A , A = 2 No $\lambda_{A}$ = 10

For B , A =  No $\lambda_{B}$ = 20

$\frac{10}{20} = \frac{2 \lambda_{A}}{\lambda_{B}}$

$\lambda_{B}$ = 4 $\lambda_{A}$

$t_{\frac{1}{2} }= \frac{ln 2 }{\lambda}$

$\frac{t_{\frac{1}{2}A }}{t_{\frac{1}{2}B }}= \frac{\lambda_{B}}{\lambda_{A}} = \frac{\lambda_{B}}{\frac{\lambda_{B}}{4}} = 4$

$t_{\frac{1}{2}A } = 4 t_{\frac{1}{2}B }$

So option 20 days and 5 days satisfies this condition .

Option 1)

5 days and 10 days

Option 2)

10 days and 40 days

Option 3)

20 days and 5 days

Option 4)

20 days and 10 days

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