Question

The half-life period of ${{N}^{13}}$ is 10.1 minute. Its mean lifetime is:
A. 5.05 minutes
B. 20.2 minutes
C. $\dfrac{10.1}{0.6931}$ minutes
D. Infinity

Answer 1

Hint: Half-life period of an element is the time in which the number of radioactive nuclei decay is half of its initial value. Mean life of all nuclei of radioactive elements is the mean total life of all nuclei. There is a relation between mean life and half – life i.e. $\tau =\dfrac{{{t}_{half}}}{0.693}$. Therefore we can find mean life directly from this formula. Let us see in brief how this formula came.

Complete step by step answer:
We know that from radioactive decay equation,
$N={{N}_{0}}{{e}^{-\lambda t}}$
Where:
$N$is number of nuclei at time t
${{N}_{0}}$ is the number of nuclei at t=0 or number of nuclei in the beginning
$\lambda $is the decay constant.
Decay constant of a radioactive element is defined as the reciprocal of time, the number of undecayed nuclei of that radioactive element falls to $\dfrac{1}{e}$ times of its initial value.
For half-life,
$\begin{align}
  & t={{t}_{\frac{1}{2}}} \\
 & N=\dfrac{{{N}_{0}}}{2} \\
 & \dfrac{{{N}_{0}}}{2}={{N}_{0}}{{e}^{-\lambda t}} \\
\end{align}$
Taking log on both sides,
$\begin{align}
  & {{\log }_{e}}2=\lambda {{t}_{\frac{1}{2}}} \\
 & {{t}_{\frac{1}{2}}}=\dfrac{{{\log }_{e}}2}{\lambda } \\
 & {{t}_{\frac{1}{2}}}=\dfrac{0.693}{\lambda } \\
\end{align}$ $\left( {{\log }_{e}}2=0.693 \right)$
We know that mean life $\left( \tau \right)$ is the reciprocal of decay constant,
i.e.
$\begin{align}
  & \tau =\dfrac{1}{\lambda } \\
 & \lambda =\dfrac{1}{\tau } \\
 & {{t}_{\frac{1}{2}}}=\dfrac{0.693}{\lambda } \\
 & {{t}_{\frac{1}{2}}}=0.693\tau \\
\end{align}$
This is the relation between half-life and mean life.
In this question the half-life of ${{N}^{13}}$ is 10.1 minutes, and we have to find out its mean life.
From the expression,
$\begin{align}
  & {{t}_{\frac{1}{2}}}=0.693\tau \\
 & 10.1=0.693\tau \\
 & \tau =\dfrac{10.1}{0.693} \\
\end{align}$
Hence the mean life of ${{N}^{13}}$ is $\dfrac{10.1}{0.693}$ minutes.

Therefore option C. is the correct answer.

Note: The mean life of radioactive nuclei is nearly 42% more than that of half-life.
Students must always notice the question (in this case, the unit is minute). Sometimes the unit in the question and answer can be different. Don’t try to memorize all the formulas, always try to memorize the basic formula and the way of the derivation for the further formulas.