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A radioactive sample contains two different types of radioactive nuclei. A with half-life 5 days and B with half-life 30 days. Initially the decay rate of the A type nuclei is 64 times that of B type of nuclei. Their decay rates will be equal when time is 9n days. Find the value of n.

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answer is 4.

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Detailed Solution

Let at $t = 0, R_{B} = R_{0},$ then $R_{A} = 64 R_{0} (R = d e c a y r a t e)$

After time t, $R_{A} = (64 R_{0}) e^{- λ_{A} t}$ And $R_{B} = R_{0} e^{- λ_{B} t}$

Given, $R_{A} = R_{B}$

$∴ 64 R_{0} e^{- λ_{A} t} = R_{0} e^{- λ_{B} t}, 64 = e^{(λ_{A} - λ_{B}) t}$

$\Rightarrow {(2)}^{6} = e^{(λ_{A} - λ_{B}) t}$

$∴ 6 \ln (2) = (λ_{A} - λ_{B}) t$

$∴ t = \frac{6 (\ln 2)}{λ_{A} - λ_{B}} = \frac{6 \ln 2}{\frac{\ln 2}{5} - \frac{\ln 2}{30}} = \frac{6 \times 150}{25} = 36 days$

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