Q. Two radioactive materials $X_{1}\ and\ X_{2}$ have decay constants 10$\lambda\ and\ \lambda$ repectively. 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 $1/e$ after a time
A) $\frac{1}{10\lambda}$
B) $\frac{1}{11\lambda}$
C) $\frac{11}{10\lambda}$
D) $\frac{1}{9\lambda}$
Explanation
(d) Here, $\frac{N_{x_{1}}(t)}{N_{x_{2}}(t)} = \frac{1}{e}$
$or\ \frac{N_{0}e^{- 10\lambda t}}{N_{0}e^{- \lambda t}} = \frac{1}{e}$
(Because initially, both have the same number of nuclei, $N_{0}$).
or $e = \frac{e^{- \lambda t}}{e^{- 10\lambda t}} = e^{9\lambda t}$
$9\lambda t = 1$
$t = \frac{1}{9\lambda}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.