Question

Two radioactive materials $X_1$, and $X_2$ have decay constants $5\lambda$ 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 $X_2$ will be $1/e$ after a time

• A. $1/4\lambda$
• B. $e/\lambda$
• C. $\lambda$
• D. $\frac{1}{2}\lambda$

Answer 1

Answer: (A)

$X_1=N_0{e}^{-\lambda 1t};X_2=N_0{e}^{-\lambda 2t}$
$\frac{X_1}{X_2}=e^{-1}=e^{\left(-{\lambda }_1+{\lambda }_2\right)t};e^{-1}=e^{\left({\lambda }_1-{\lambda }_2\right)t}$
$\therefore t=\left|\frac{1}{{\lambda }_1-{\lambda }_2}\right|=\frac{1}{(5\lambda -\lambda )}=\frac{1}{4\lambda }$