Question

Two radioactive materials A and B have decay constants $5\lambda and\lambda$ respectively. Initially both A and B have the same number of nuclei. The ratio of the number of nuclei of A to that of B will be $\frac{1}{e}$ after a time

• A. $\frac{1}{5\lambda }$
• B. $\frac{1}{4\lambda }$
• C. $\frac{5}{4\lambda }$
• D. $\frac{4}{5\lambda }$

Answer 1

The correct option is B $\frac{1}{4\lambda }$

Using the law of radioactivity, we can write:

$\frac{N\left(5\lambda \right)}{N\left(\lambda \right)}=\frac{N_0{e}^{-5\lambda t}}{e^{-\lambda t}}$

$\frac{1}{e}=\frac{e^{-5\lambda t}}{e^{-\lambda t}}$

$e^{-1}=e^{-4\lambda t}$

$-1=-4\lambda t\Rightarrow t=\frac{1}{4\lambda }$