Question

Two radioactive materials $X_1$ and $X_2$ have decay constants $6\lambda$ and $3\lambda$ 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.

• A. $\frac{1}{6\lambda }$
• B. $\frac{1}{3\lambda }$
• C. $\frac{3}{6\lambda }$
• D. $\frac{6}{9\lambda }$

Answer 1

Answer: (B)

$\frac{1}{3\lambda }$

The ratio of number of nuclei of $X_1$ to that of $X_2$ after time $t,\frac{N_1}{N_2}=\frac{1}{e}$
$\therefore \frac{N_0{e}^{-6\lambda t}}{N_0{e}^{-3\lambda t}}=\frac{1}{e}(\because N=N_0{e}^{-\lambda t})$
or $\frac{1}{e^{3\lambda t}}=\frac{1}{e}$ or $3\lambda t=1\therefore t=\frac{1}{3\lambda }$.