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 $\mathrm{X}_1$ to that of $X_2$ will be $\frac{1}{e}$ after a time
  1. $\frac{e}{\lambda}$
  2. $\lambda$
  3. $\frac{1}{2} \lambda$
  4. $\frac{1}{4 \lambda}$

Solution

$\begin{aligned} & \frac{N_{x_1}}{N_{x_2}}=\frac{e^{-5 \lambda t}}{e^{-\lambda t}}=\frac{1}{e} \\ & \Rightarrow \quad t=\frac{1}{4 \lambda} \end{aligned}$

Asked in: NEET 2008 (Mains)

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