Question

The decay constant of a radioactive sample is ($\lambda $). The half-life and mean-life of the sample are, respectively, given by,
$\left( A \right)$ $\dfrac{1}{\lambda }{\text{ and }}\dfrac{{\ln 2}}{\lambda }$
$\left( B \right)$ ${\text{ }}\dfrac{{\ln 2}}{\lambda }{\text{ and }}\dfrac{1}{\lambda }$
$\left( C \right)$ $\lambda \ln 2{\text{ and }}\dfrac{1}{\lambda }$
$\left( D \right)$ $\dfrac{\lambda }{{\ln 2}}{\text{ and }}\dfrac{1}{\lambda }$

Answer 1

Hint: You can start by describing half-life and mean-life first. Then you can lay out the general equations of half-life and then substitute the given values in it. Repeat the same for mean-life. Use this concept to reach the solution.

Complete step-by-step solution -
Decay Constant: A proportionality constant that gives us a relation between the number of radioactive atoms and the rate of decrease in the population.
Half-life (${t^{\dfrac{1}{2}}}$): It is the time in which the amount of a radioactive substance reduces to half the initial amount. Half-life is different for different substances, and can be seen as an indicator of how quickly or slowly a radioactive substance degrades naturally.
Mean-life or Average life ($\tau $): It is the average lifespan of all the nuclei that are present in an atomic species. This can be considered as the sum of the lifespan of all the nuclei present in an atomic species divided by the total number of nuclei.
Given a decay constant for half-life is ($\lambda $), that is also the general symbol of decay constant.
Substituting the value of decay constant in the equation for (${t^{\dfrac{1}{2}}}$) (half-life)
${t^{\dfrac{1}{2}}} = \dfrac{{\ln 2}}{\lambda }$ ......................... (1)
The general equation for mean-life is:
\[\tau = \dfrac{{{t^{\dfrac{1}{2}}}}}{{\ln 2}} = \dfrac{{\ln 2}}{{\ln 2 \times \lambda }} = \dfrac{1}{\lambda }\].......................... (2)
So the half-life and mean-life of the given sample are ($\dfrac{{\ln 2}}{\lambda }$) and ($\dfrac{1}{\lambda }$), respectively.
Thus, option B is the correct answer.

Note: The mean-life of a species is always 1.443 times longer than its half-life. For example lead-209, decays to form bismuth-209. It has a half-life of 3.25 hours and a mean-life of 4.69 hours. Radioactive elements have shorter half-life in comparison to stable elements and are considered generally more unstable.