### Theory:

The law states that ‘in any radioactive substance, the number of nuclei disintegrating per second is directly proportional to the number of nuclei present’.

$\begin{array}{l}\frac{\mathrm{\Delta}\phantom{\rule{0.147em}{0ex}}N}{\mathrm{\Delta}\phantom{\rule{0.147em}{0ex}}t}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha}\phantom{\rule{0.147em}{0ex}}N\\ \\ \frac{\mathrm{\Delta}\phantom{\rule{0.147em}{0ex}}N}{\mathrm{\Delta}\phantom{\rule{0.147em}{0ex}}t}\phantom{\rule{0.147em}{0ex}}=-\mathrm{\lambda}\phantom{\rule{0.147em}{0ex}}N\end{array}$

Where \(λ\) is the decay constant and \(N\) is the number of nuclei in an atom. The negative sign denotes that the number of nuclei decreases with time.

Half-life period:

The half-life period (\(T\)) of a radioactive substance is defined as the time during which half the amount of the substance disintegrates.

A statement 'four atoms with a half-life period of \(2\ seconds\)' means that exactly two atoms will disintegrate approximately after \(2\ seconds\).

The half-life period (\(T\)) varies for different substances. The disintegration of a particular atom in a radioactive element is unpredictable. Usually, a smaller value of \(T\) indicates a faster decay.

The decay constant is a characteristic property of a radioactive element. The formula is given as

$\mathit{Decay}\mathit{constant}(\mathrm{\lambda})=\frac{0.693}{T}$

The higher value of \(λ\) denotes a faster decay.

Let the initial number of radioactive nuclei be \(N_0\). Consider the number of nuclei present as \(N_0\)\(\times \frac{1}{2}\) after a half-life period.

After two half-lives, the number of nuclei would be

$\frac{1}{2}\times \left({N}_{0}\times \frac{1}{2}\right)\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}{N}_{0}{\phantom{\rule{0.147em}{0ex}}\left(\frac{1}{2}\right)}^{2}$

After '\(n\)' half-lives, the number of nuclei would be

$N\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}{N}_{0}{\phantom{\rule{0.147em}{0ex}}\left(\frac{1}{2}\right)}^{n}$

If '\(t\)' is the given time period and '\(T\)' is the half-life period of the radioactive substance, then the value of half-life is

$n=\frac{t}{T}$

If '\(m_o\)' is the initial mass of the substance and '\(m\)' the mass of un-decayed nuclei, then

\(\)

\(\)

$\frac{N\phantom{\rule{0.147em}{0ex}}}{{N}_{0}}=\phantom{\rule{0.147em}{0ex}}\frac{m}{{m}_{0}}$

The above equation can be re-written as

$m=\phantom{\rule{0.147em}{0ex}}{m}_{0}{\phantom{\rule{0.147em}{0ex}}\left(\frac{1}{2}\right)}^{n}$

This equation helps to find the mass of un-decayed particles.