Question

Two radioactive materials $X_1$ and $X_2$ have decay constant $10\lambda$ and $\lambda$, 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. $\frac{1}{\left(10\lambda \right)}$
• B. $\frac{1}{\left(11\lambda \right)}$
• C. $\frac{11}{\left(10\lambda \right)}$
• D. $\frac{1}{\left(9\lambda \right)}$

Answer 1

The correct option is D

$\frac{1}{\left(9\lambda \right)}$

$\frac{N_1}{N_0}=e^{-{\lambda }_{1}t}$

$\frac{N_2}{N_0}=e^{-{\lambda }_{2}t}$

$\Rightarrow \frac{N_1}{N_2}=e^{-({\lambda }_1-{\lambda }_2)t}$

Putting $\frac{N_1}{N_2}=\frac{1}{e}$ we obtain $({\lambda }_1-{\lambda }_2)t=1$

$\Rightarrow t=\frac{1}{({\lambda }_1-{\lambda }_2)}=\frac{1}{(10\lambda -\lambda )}=\frac{1}{9}\lambda$