Question

Two radioactive materials $X_1$ and $X_2$ have decay constant $10\lambda$ and $\lambda$ respectively. If initially they have equal 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 time________.

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

Answer 1

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

As we know,
$N_1=N_0{e}^{-10\lambda t}and{N}_2=N_0{e}^{-\lambda t}$
$\therefore \frac{N_1}{N_2}=e^{-9\lambda t}$
$\because \frac{N_1}{N_2}=\frac{1}{e}$
$\therefore e^{-1}=e^{-9\lambda t}$
$-9\lambda t=-1$
$t=\frac{1}{9\lambda }$