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Two radioactive material $A$ and $B$ have decay constants $5 \lambda$ and $\lambda$ respectively. Initially both $A$ and $B$ have the same number of nuclei. The ratio of the number of nuclei of $A$ to that of $B$ will be $1 / e$ after a time
Option: 1
$\frac{1}{5 \lambda}$

Option: 2
$\frac{1}{4 \lambda}$

Option: 3
$\frac{5}{4 \lambda}$

Option: 4
$\frac{4}{5 \lambda}$

Answers (1)

best_answer

Let initially both have No number of nuclei

$\lambda_A=5 \lambda$ $\quad \lambda_B=\lambda.$

At $t=t\, \, ,$ $\frac{N_A}{N_B}=\frac{1}{e} \text {. }$

$\Rightarrow \frac{N_0 \cdot e^{-\lambda_A t}}{N_0 e^{-\lambda_B \cdot t}}=\frac{1}{e}$ $\Rightarrow e^{\left(\lambda_B-\lambda_A\right) t}=e^{-1}$

$\Rightarrow \quad t=\frac{1}{\lambda_A-\lambda_B}=\frac{1}{5 \lambda-\lambda}=\frac{1}{4 \lambda} .$

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Riya

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