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, the ratio of the number of nuclei of $X_1$ to that of $X_2$ 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

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

Considering first order decay process, and same no. of initial no. of atoms:
$X_1=N_o{e}^{-10\lambda t}$
$X_2=N_o{e}^{-\lambda t}$
$\frac{X_1}{X_2}=\frac{e^{-10\lambda t}}{e^{-\lambda t}}$
$\frac{X_1}{X_2}=e^{-9\lambda t}$

Determining the time when, $\frac{X_1}{X_2}=\frac{1}{2}$
$\frac{1}{e}=e^{-9\lambda t}$
$t=\frac{1}{9\lambda }$