Why are all elements with atomic numbers above 82 unstable?
The stability of the atoms of different elements is dependent on their nuclear binding energy. If the nuclear binding energy of an atom is more than the repulsive forces between the nucleons, it is stable. It has been seen that binding energy starts to increase from the atoms of the lightest elements; it reaches the highest with iron and then begins to drop.
The nucleus of atoms with a higher binding energy is found to be lighter than the mass of the constituent nucleons. This decrease in mass is due to energy which is expelled. If such an atom has to be split an amount of energy equivalent to the decrease in mass has to be introduced.
In the case of atoms of elements with a mass greater than 82, the level of binding energy does not allow stability. The force of repulsion between the particles that constitute the nucleus makes the atoms split and atoms of new elements are formed as they try to reach a nuclear configuration which is stable.