Question

Define half- life of a radioactive substance and write the relation of half- life with the following:
(a) Radioactive decay constant (disintegration constant)
(b) Mean life of a radioactive substance.

Answer 1

The substance that decay or disintegrates naturally are called the radioactive substances. They decay because they possess the unstable nucleus in them and these substances release the alpha particle, beta particle and the gamma particle when they disintegrate.

Useful formula:
(1) The formula of the half-life of the radioactive substance is given as

${\tau _{1/2}} = \dfrac{{\ln 2}}{k}$

Where ${\tau _{1/2}}$ is the half-life of the radioactive substance and $k$ is the disintegration constant.

(2) The relation between the mean life and the radioactive decay constant or the disintegration constant is given as

$m = \dfrac{1}{k}$

Where $m$ is the mean life of the radioactive substance.

Complete step by step solution:
The half- life of the radioactive substance is defined as the time taken for the disintegration or the decaying of half the amount of radioactive substance.
(a) The radioactive decay constant is denoted as $k$ . Its relation with the half-life is written as

${\tau _{1/2}} = \dfrac{{\ln 2}}{k}$ -----------(1)

(b) The mean life of the radioactive substance is defined as the lifetime of the radioactive substance with the unstable nucleus or the time taken by the radioactive substance for the complete decay of itself. The relation between the mean life and the half-life of the radioactive substance is calculated as follows.
Taking the equation (1),

${\tau _{1/2}} = \dfrac{{\ln 2}}{k}$

Substituting the mean life formula in the above step, we get

${\tau _{1/2}} = m\,ln2$

The above equation denotes the relation between the half-life and the mean life.

Note: The example of the radioactive substances are uranium, potassium, thorium, radium etc. The alpha particles emitted from the disintegration of the radioactive substance are helium ions, the beta particles are electrons and the gamma particles are charge less massless photons.