Question

Two radioactive materials $A$ and $B$ have decay constants $10\lambda \text{and}\lambda ,$ respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of $A$ to that of $B$ will be $\frac{1}{e}$ after a time:

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

Answer 1

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

Given:
Decay constant for $A,{\lambda }_A=10\lambda$
Decay constant for $B,{\lambda }_B=\lambda$
$\frac{N_A}{N_B}=\frac{1}{e}$

As, $N=N_0{e}^{-\lambda t}$

$N_A=N_0{e}^{-{\lambda }_{A}t}$

$N_B=N_0{e}^{-{\lambda }_{B}t}$

So, $\frac{N_A}{N_B}=e^{({\lambda }_B-{\lambda }_A)t}=\frac{1}{e}$

$\Rightarrow ({\lambda }_B-{\lambda }_A).t=-1$

$\Rightarrow ({\lambda }_A-{\lambda }_B).t=1$

$\Rightarrow t=\frac{1}{({\lambda }_B-{\lambda }_A)}$

$t=\frac{1}{10\lambda -\lambda }=\frac{1}{9\lambda }$

Hence option $\left(A\right)$ is correct.