Two radioactive materials ' M 1 ′ and ' M 2 ′ have decay constants ' 9 λ ' and ' λ ' respectively. Initially they have the same number of nuclei. The ratio of the num

Question

Two radioactive materials ' M 1 ​ ′ and ' M 2 ​ ′ have decay constants ' 9 λ ' and ' λ ' respectively. Initially they have the same number of nuclei. The ratio of the number of nuclei of M 1 ​ to that of M 2 ​ will be ' ( e 1 ​ ) ′ after a time interval

TestHub · Accepted Answer

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

For material $M_{1}$ , the number of nuclei at time $t$ is $N_{1} = N_{0} e^{- 9 λ t}$ .

For material $M_{2}$ , the number of nuclei at time $t$ is $N_{2} = N_{0} e^{- λ t}$ .

The ratio of the number of nuclei of $M_{1}$ to that of $M_{2}$ is given as $\frac{1}{e}$ .

$\frac{N _{1}}{N _{2}} = \frac{N _{0} e ^{- 9 λ t}}{N _{0} e ^{- λ t}} = \frac{1}{e}$

$e^{- 8 λ t} = e^{- 1}$

Equating the powers of $e$ , we get:

$8 λ t = 1$

$t = \frac{1}{8 λ}$

Answer: $\frac{1}{8 λ}$

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