Two radioactive materials $X_1$ and $X_2$ have decay constants $5 \lambda$ and $\lambda$ respectively. If…

Two radioactive materials $X_1$ and $X_2$ have decay constants $5 \lambda$ and $\lambda$ respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of $X_1$ to that of $X_2$ will be $\frac{1}{e}$ after a time
  1. $\lambda$
  2. $\frac{1}{2} \lambda$
  3. $\frac{1}{4 \lambda}$
  4. $\frac{e}{\lambda}$

Solution

If $N$ is the number of radioactive nuclei present at some instant, then
$N=N_0 e^{-\lambda t}$
The constant $N_0$ represents the number of radioactive nuclei at $t=0$
Now,
$\begin{aligned}
& \frac{N_1}{N_2}=\frac{e^{-\lambda_1 t}}{e^{-\lambda_2 t}} \\
& \frac{N_1}{N_2}=\frac{e^{-5 \lambda t}}{e^{-\lambda t}}=e^{-4 \lambda t} \\
& \frac{N_1}{N_2}=\frac{1}{e}
\end{aligned}$
but $\frac{N_1}{N_2}=\frac{1}{e}$ (as provided)
Therefore, $\frac{1}{e}=\frac{1}{e^{4 \lambda t}}$
or $4 \lambda t=1$
or $t=\frac{1}{4 \lambda}$

Asked in: NEET 2008 (Screening)

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