Question

Mean life of radioactive samples is 100s. Then its half-life (in minute) is
a) 0.693
b) 1
c) ${{10}^{-4}}$
d) 1.155

Answer 1

Hint: In the question we are given the mean life of a radioactive sample and we are asked to determine its half life. First we need to know what does the mean life and the half life of radioactive samples mean and their respective expressions. Once we know the expression of mean life and the half life of a radioactive sample, then we can accordingly relate the two and obtain the required relation between them. Further using this relation we will determine the half life of the particular radioactive sample.
Formula used:
${{T}_{1/2}}=\dfrac{0.693}{\lambda }$
$\tau =\dfrac{1}{\lambda }$

Complete answer:
To begin with let us understand what half life and the mean life of a radioactive sample mean.
Half life of a radioactive sample is defined as the time interval in which one-half of the radioactive nuclei originally present in the radioactive sample disintegrate. Let us say the radioactive sample has a decaying constant $\lambda $, than the half life (${{T}_{1/2}}$)of the sample is given by,
${{T}_{1/2}}=\dfrac{0.693}{\lambda }...(1)$
The mean life is defined as the average time interval for which the radioactive sample exists. It is also called the average life of the radioactive sample. The mean life ($\tau $) of the same sample having decay constant $\lambda $is given by,
$\tau =\dfrac{1}{\lambda }...(2)$
Comparing equation 1 and 2 we get,
$\begin{align}
  & {{T}_{1/2}}=0.693\tau \text{, }\because \tau \text{=100s} \\
 & \Rightarrow {{T}_{1/2}}=0.693\times 100 \\
 & \Rightarrow {{T}_{1/2}}=69.3\sec \text{, }\because 1\min \text{=60s} \\
 & \Rightarrow {{T}_{1/2}}=\dfrac{69.3}{60}\min =1.155\min \\
\end{align}$

So, the correct answer is “Option D”.

Note:
The decay constant is defined as the reciprocal of the time interval during which the number of active nuclei in a given radioactive sample reduces to 36.8% of its initial value. We can also say that it reduces 1/e times as well where e is 2.718. Different radioactive samples can be distinguished using this constant.