Question

In a RA element the fraction of initiated amount remaining after its 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 $\frac{1}{e}$

$\begin{array}{c}\text{Given,}\\ \text{a Radioactive element}\\ \text{mean life time}=\frac{1}{\lambda }\end{array}$

$\begin{array}{c}\text{Let}{A}_0=\text{initial no. of nuclie}\\ \text{afiiuhter any time}=t,\text{no of nuclie left}=A_t\end{array}$

$\begin{array}{c}{A}_t=A_0{e}^{-\lambda t},\lambda =\text{decay constant}\\ A_t=A_0{e}^{-\lambda \times \frac{1}{\lambda }}\cdot \therefore t_{\text{mean}}=\frac{1}{\lambda }\\ A_t=A_0{e}^{-1}\\ A_t=\frac{A_0}{e}\end{array}$

$\left(\frac{A_t}{A_o}\right)=\frac{1}{e}=\text{fraction of ammount left}$