Question

Two radioactive materials $X_1$ and $X_2$ have decay constants $10\lambda$ 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 $1/e$ after a time.

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

Answer 1

Answer: (D)

$\frac{1}{9\lambda }$

Here, $\frac{N_{X_1}\left(t\right)}{N_{X_2}\left(t\right)}=\frac{1}{e}$
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 $e=\frac{e^{-\lambda t}}{e^{-10\lambda t}}=e^{9\lambda t}$
or $9\lambda t=1$
or $t=\frac{1}{9\lambda }$.