Question

Two radioactive materials ${\text{X}}_1$ and ${\text{X}}_2$ have decay constants $11\lambda$ and $\lambda$ respectively. If initially they have the same number of nuclei , then ratio of number of nuclei of ${\text{X}}_1$ to ${\text{X}}_2$ will be $\frac{1}{e^2}$ after a time-

• A. $\frac{1}{9\lambda }$
• B. $\frac{1}{10\lambda }$
• C. $\frac{1}{11\lambda }$
• D. $\frac{1}{5\lambda }$

Answer 1

The correct option is D $\frac{1}{5\lambda }$

At some instant time $t$, number of undecayed atoms exists is,

$N_1=N_0{e}^{-11\lambda t}$ and $N_2=N_0{e}^{-\lambda t}$

It is given that,

$\frac{N_1}{N_2}=\frac{1}{e^2}$

$\Rightarrow \frac{1}{e^2}=\frac{N_0{e}^{-11\lambda t}}{N_0{e}^{-\lambda t}}$

$\Rightarrow \frac{1}{e^2}=\frac{e^{-11\lambda t}}{e^{-\lambda t}}$

$\Rightarrow e^{-2}=e^{-(11-1)\lambda t}$

Taking log on both sides, we get,

$\Rightarrow 10\lambda t=2$

$\Rightarrow t=\frac{2}{10\lambda }=\frac{1}{5\lambda }$

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Hence, $\left(D\right)$ is the correct answer.