Two radioactive materials A and B having decay constant '7 λ ' and ' λ ' respectively, initially have same number of nuclei. The time taken to have the ratio of number of nuclei of

Question

Two radioactive materials A and B having decay constant '7 λ ' and ' λ ' respectively, initially have same number of nuclei. The time taken to have the ratio of number of nuclei of material B to that of A as ' e ' is

TestHub · Accepted Answer

Let the initial number of nuclei of both materials be $N_{0}$ .

The number of nuclei of material A at time $t$ is given by $N_{A} = N_{0} e^{- 7 λ t}$ .

The number of nuclei of material B at time $t$ is given by $N_{B} = N_{0} e^{- λ t}$ .

Given that the ratio of the number of nuclei of material B to that of A is $e$ , we have:

$\frac{N _{B}}{N _{A}} = e$

$\frac{N _{0} e ^{- λ t}}{N _{0} e ^{- 7 λ t}} = e$

$e^{6 λ t} = e^{1}$

Equating the powers of $e$ , we get:

$6 λ t = 1$

$t = \frac{1}{6 λ}$

Answer: $\frac{1}{6 λ}$

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