Question

In a radioactive element the fraction of initial amount remaining after it mean life time is

• A. $1-\frac{1}{e}$
• B. $\frac{1}{e^2}$
• C. $\frac{1}{e}$
• D. $1-\frac{1}{e^2}$

Answer 1

The correct option is C $\frac{1}{e}$

$\begin{array}{c}\text{we know that the mean life of a radioactive nuclei is -}\\ \begin{array}{c}t=\frac{1}{\lambda }\text{where}\lambda \text{is the disintegration}\\ \text{constant.}\end{array}\\ \text{By Radioactive law of disintegration -}\\ N=N_0\cdot e^{-\lambda t}\\ N_0=\text{Initial num of nuclei}\end{array}$

$\begin{array}{c}\Rightarrow \text{at}t=0\text{we have}N=N_0\text{initially.}\\ \Rightarrow \text{at}t=\frac{1}{\lambda }\text{mean life we have-}\\ N=N_0\cdot e^{-1}=\frac{N_0}{e}\\ \Rightarrow \text{Ratio of num of Nuclei left after the mean life is}\\ \frac{N}{N_0}=\frac{1}{e}\text{Hence, opt}\left(c\right)\text{is correct.}\end{array}$