There's no reason for them to split apart at all, they should remain in their current state. Electrons, due to the nature of Quantum Mechanics, should remain in their respective orbits, and the nucleus should stay bound, unless exposed to extreme conditions, such as the core of a star, where they fuse together.
Some radioactive isotopes decay into different kinds of atoms, too.
Atoms cannot "die" in the normal sense. However I expect you already knew this but didn't know how to word the question. So, what CAN happen to the atom? The most common thing is that it bonds with another atom, or several, to form a molecule by sharing, borrowing, or giving up electrons. Since electrons are considered part of the atom, it isn't the same as it used to be. We also have nuclear decay, where particles in the nucleus are transformed into other particles, ejected completely out, or the nucleus splits into large pieces, with each piece forming a new atom of a different element. Finally there is nuclear fusion where the nuclei of two atoms fuses together to form another one of a different element.