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At any instant, two elements $X_1$ and $X_2$ have same number of radioactive atoms. If the decay constant of $X_1$ and $X_2$ are $10\lambda$ and $\lambda$ respectively, then the time when the ratio of their atoms becomes $\frac{1}{e}$ respectively will be:
Detailed Solution
$N_x=N_0e^{-\lambda_x t}$, $N_y=N_0e^{-\lambda_y t}$. $\frac{N_x}{N_y}=e^{-(\lambda_x-\lambda_y)t}=\frac{1}{e} \Rightarrow (\lambda_x-\lambda_y)t=1 \Rightarrow t=\frac{1}{10\lambda-\lambda}=\frac{1}{9\lambda}$.