Here, we have to look at two different stabilities of atoms. One is the radioactive stability that involves forces between the particles that make up the nucleus and the other is the chemical stability that involves the electronic configuration of atoms.
The nucleus of atoms consists of protons and neutrons. There is a strong force of repulsion between the protons as they have the same electric charge. These forces are balanced by a strong nuclear force that ensures the nucleus of atoms does not break apart.
For atoms which have more than 83 protons in their nucleus the strong nuclear force is less than the force of repulsion between the protons; this makes them unstable and they undergo radioactive decay processes and convert to forms with a stable nucleus.
Chemical stability is reached when atoms have the electronic configuration of the closest noble gas. This can be achieved by either sharing electrons as in covalent bonds or giving up or accepting electrons as in ionic bonds.
Chemical stability is reached by the rearrangement of electrons and nuclear stability is reached by a change in the constituents of the nucleus of atoms.