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# In a radioactive element, the fraction of initiated amount remaining after its mean life time will be given as,\begin{align} & A.1-\dfrac{1}{e} \\ & B.\dfrac{1}{{{e}^{2}}} \\ & C.\dfrac{1}{e} \\ & D.1-\dfrac{1}{{{e}^{2}}} \\ \end{align}

Last updated date: 04th Aug 2024
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Hint: the amount of radioactive nuclei left can be found by taking the product of the initial amount of radioactive nuclei present, and the exponential of the negative of the product of the decay constant and the time taken. The mean lifetime can be found by taking the reciprocal of the exponential decay constant. This can be substituted in the first equation. This will help you in answering this question.

Complete step by step answer:
let us assume that the initial amount of the radioactive nuclei is ${{A}_{0}}$. The amount of radioactive nuclei which is left even after the time $t$ be $A$. The amount left can be found by the equation given as,
$A={{A}_{0}}{{e}^{-\lambda t}}$
As we all know that the mean lifetime can be found by taking the reciprocal of the exponential decay constant. Therefore this can be written as an equation mentioned as,
${{t}_{mean}}=\dfrac{1}{\lambda }$
Substituting this in the above obtained equation can be written as,
$A={{A}_{0}}{{e}^{-\lambda \times \dfrac{1}{\lambda }}}$
This can be simplified further and can be written as,
$A={{A}_{0}}{{e}^{-1}}$
Let us take the fraction of the initiated amount remaining after its mean lifetime. That is we can write that,
$\dfrac{A}{{{A}_{0}}}=\dfrac{1}{e}$
Therefore the fraction of the initiated amount remaining after its mean lifetime has been calculated.

So, the correct answer is “Option C”.

Note: If the decaying quantity whether it is radioactive will be the number of discrete elements in a particular set, we can compute the average length of the time that an element will remain in the set. This is referred to as the mean lifetime or simply the lifetime of the particle. Half-life time is the time taken by the particle in order to become half of its initial value.