Question

Two radioactive substances $A$ and $B$ have decay constants 5 $\lambda$, and $\lambda$ respectively. At $t\, =\, 0$, they have the same number of nuclei. The ratio of number of nuclei of $A$ to those of $B$ will be $(1/e^2)$ after a time

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

Answer 1

The Correct Option is C

At $t = 0, N = N_0$ for both the substances $A$ and $B$ $\therefore N_{A}=N_{0}e^{-\lambda A^t}$ and $N_{B}=N_{0}e^{-\lambda B^t}$ $\frac{N_{A}}{N_{B}}=\frac{e^{-\lambda}A^{t}}{e^{-\lambda}B^{t}}=e^{\left(\lambda_{B}-\lambda_{A}\right)t}=e^{\left(\lambda-5\lambda\right)t}$ $e^{-4\lambda t}=\left(\frac{1}{e}\right)^{4\lambda t}$ As $\frac{N_{A}}{N_{B}}=\left(\frac{1}{e}\div\right)^{2}$ [According to question] $\therefore 4\lambda t=2$ or $t=\frac{2}{4\lambda}=\frac{1}{2\lambda}$