Question

Two radioactive materials $X_1$ and $X_2$ have decay constant $10\lambda$ and $\lambda$ respectively. If initially they have same number of nuclei, then ratio of number of nuclei $X_1$ and $X_2$ is equal to $\frac{1}{e}$ after time interval of

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

Answer 1

The correct option is A $\frac{1}{9\lambda }$

Given

$\frac{N_1}{N_2}=\frac{1}{e}$
$N_1$ number of nuclei of $X_1$ after time $t$
$N_2$ number of nuclei of $X_2$ after time $t$

$\frac{N_0{e}^{-10\lambda t}}{N_0{e}^{-t\lambda }}=\frac{1}{e}$ .

We get $e=\frac{e^{-\lambda t}}{e^{-10\lambda t}}$ i.e.,

$e=e^{9\lambda t}$ i.e., $9\lambda t=1$

$t=\frac{1}{9\lambda }$