Question

The relation between half-life T of a radioactive sample and its mean life is:
$
  (a){\text{ }}{{\text{t}}_{\dfrac{1}{2}}} = 0.693\tau \\
  (b){\text{ }}{{\text{t}}_{\dfrac{1}{2}}} = \tau \\
  (c){\text{ }}{{\text{t}}_{\dfrac{1}{2}}} = 2.303\tau \\
  (d){\text{ }}\dfrac{1}{{{{\text{t}}_{\dfrac{1}{2}}}}} = 0.693\tau \\
$

Answer 1

Hint: In this question use the concept that the mean life of a decay is inverse of the wavelength and the wavelength is inversely proportional to the half-life of the radio-active sample. Collaboration of these two concepts will help get the actual relation between the half-life and the mean life.

Complete Step-by-Step solution:
 As we know, the mean life ($\tau $) of a radio-active delay is equal to the inverse of the wavelength ($\lambda $).
$ \Rightarrow \tau = \dfrac{1}{\lambda }$..................... (1)
Now as we know that wavelength ($\lambda $) is inversely proportional to half-life of radio-active sample.
$ \Rightarrow \lambda \propto \dfrac{1}{{{t_{\dfrac{1}{2}}}}} = \dfrac{k}{{{t_{\dfrac{1}{2}}}}}$, where k = proportionality constant, ${t_{\dfrac{1}{2}}}$ = half-life of radioactive sample
And the value of k = ln2 = 0.693.
$ \Rightarrow \lambda = \dfrac{{0.693}}{{{\tau _{\dfrac{1}{2}}}}}$..................... (2)
Now from equation (1) and (2) we have,
$ \Rightarrow \lambda = \dfrac{1}{\tau } = \dfrac{{0.693}}{{{t_{\dfrac{1}{2}}}}}$
Now simplify this we have,
$ \Rightarrow {t_{\dfrac{1}{2}}} = 0.693\tau $
So this is the required relation between the half-life of a radioactive sample and its mean life.
Hence option (A) is the correct answer.

Note – Half-life of a radio-active substance is defined as the time over the nuclei of half of the atoms of a radioactive substance completely decays. Mean life of a radioactive substance is the average life of all the nuclei of the atoms present in that substance that undergoes radioactive decay.