Question

During mean the life of a radioactive element, the fraction that disintegrates is
A. \[1-\dfrac{1}{e}\]
B. \[\dfrac{e-1}{e}\]
C. \[\dfrac{1}{e}\]
D. \[\dfrac{e}{e-1}\]

Answer 1

Hint:There are two lives when one talks about radioactivity one is a half life and the other is mean life. Mean life of a radioactive substance is defined as the inverse of the decay constant. We can also define it in terms of the half-life. The mean life is equal to the half-life of that atom divided by the natural log of two.

Complete step by step answer:
The half life of a radioactive substance is defined as the time required for the radioactive nuclei to become one half of its initial value. The half life is denoted by \[{{T}_{1/2}}\] and the mean life is denoted by \[\tau \]. Now we want to find out during the mean life of a radioactive element, the fraction that disintegrates?
We know \[N={{N}_{0}}{{e}^{-\lambda t}}\], where N is the number of nuclei left after time t and \[{{N}_{0}}\]is the number of nuclei at the starting at the time, t=0. \[\lambda \]is the decay constant.
We know average lifetime and decay constant are related as \[t=\dfrac{1}{\lambda }\]
So modifying above mentioned equation,
\[\begin{align}
& N={{N}_{0}}{{e}^{-\lambda \times \dfrac{1}{\lambda }}} \\
&\Rightarrow N={{N}_{0}}{{e}^{-1}} \\
& \Rightarrow\dfrac{N}{{{N}_{0}}}=\dfrac{1}{e} \\
\end{align}\]
\[\begin{align}
&\Rightarrow 1-\dfrac{N}{{{N}_{0}}}=1-\dfrac{1}{e} \\
&\therefore 1-\dfrac{N}{{{N}_{0}}} =\dfrac{e-1}{e} \\
\end{align}\]
This matches with the option (B), hence (B) is the correct answer.

Note: Mean Life is always longer than half life for any radioactive sample. mean life equals the half life divided by the natural logarithm of 2. The mean life gives us an intuitive feel for radioactive decay. Both the half-life and mean life are measured in units of time and it varies from a few seconds for some samples to months and years for others.