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Two radioactive materials A and B have decay constants $25 λ$ and $16 λ$ respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of B to that of A will be “e” after a time $\frac{1}{a λ}$ . The value of a is ______.

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answer is 9.

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Detailed Solution

Radioactive material NA have decay constant, $λ_{1} = 25 λ$

Radioactive material NB have decay constant $λ_{2} = 16 λ$

$N = N_{0} e^{- λ t} \Rightarrow \frac{N_{B}}{N_{A}} = \frac{e^{- λ 2 t}}{e^{- λ 2 t}} = e^{- λ 2 t} . e^{- λ 1 t}$

$e^{1} = e^{(λ_{1} - λ_{2}) t} \Rightarrow (λ_{1} - λ_{1}) t = 1$

$\Rightarrow t = \frac{1}{λ_{1} - λ_{2}} = \frac{1}{25 λ - 16 λ} = \frac{1}{9 λ}$

Compare with $\frac{1}{a λ}, s o a = 9$

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