The spontaneous breaking of a nucleus is known as Radioactive Disintegration. According to Rutherford and Soddy made experimental study of the radioactive decay of various radioactive materials and gave the following the laws:

Radioactive decay is a random and spontaneous process. It is not influenced by external conditions such as temperature, pressure, electric field etc. each decay is an independent event occurs by a chance to take first.

In any radioactive decay, either a $$\alpha$$ particle or $$\beta$$ particle is emitted by the atom. Emission of both is impossible at a time. Moreover, an atom doesn’t emit more than one $$\alpha$$ particle or more then on $$\beta$$ particle at a time.

At any instant the rate of decay of radioactive atoms is proportional to the number of atoms present at that instant, i.e.

$$-\frac{dN}{dt}\propto N\Rightarrow \frac{dN}{dt}=-\lambda N$$,

It can be proved that, $$N={{N}_{0}}{{e}^{-\lambda t}}$$,

This equation shows that number of atoms on taken radioactive element decreases exponentially with time. Theoretically, infinite time is required for radioactivity to disappear completely and this is same for all elements. In terms of mass, $$M={{M}_{0}}{{e}^{-\lambda t}}$$

Where, N = Number of atoms remains un-decayed after time t,

$${{N}_{0}}$$ = Number of atoms present initially,

$$M=$$ Mass of radioactive nuclei at time t,

$${{M}_{0}}=$$ Mass of radioactive nuclei at time t=0,

$${{N}_{0}}-N=$$ Number of disintegrated nucleus in time t

$$\frac{dN}{dt}=$$ Rate of decay,

$$\lambda$$ = Disintegration constant (or) decay constant of the radioactive element.