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Simply put, electrostatic attraction. For example, table salt NaCl. The Na ion is missing one electron, making it positive. The Cl ion has an extra electron, making ti negative one. The two join together to form a neutral compound. More complex explanation would refer to the orbitals. There are four levels of orbitals (s,p,d,f). S can hold 1 pair of electrons (electrons normally come in pairs because one spins clockwise and the other spins counterclockwise). P hold 3 pair. D holds 5 pair. F holds 7 pair. The outermost orbital or "shell" has the "valance shell electrons." These electrons are responsible for bonding, since they are most likely not completely filled. So the atom will fill it's empty spaces by sharing or transfering electrons from another atom. For example, water (H2O, two hydrogens and one oxygen). The oxygen atom has 6 electrons. The electron configuration of oxygen fills the S-orbital entirely. Four electrons remain. Those four electrons partially fill the P-orbital. Remember, the P holds 3 pair of electrons and we just gave it two. So where would those last two electrons come from? 2 Hydrogens! Each having 1 electron to share, filling the P-orbital. And that's why atoms bond.

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