### 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

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

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

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.