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An atom's oxidation state is a number associated with an atom that is part of a molecule through an ionic bond, i.e., a bond in which an atom gives up some electrons to another. The oxidation state is the number of electrons given up, or conversely, the oxidation state of the atom receiving the electrons is the negative of the number received (so if it is the bond of a compound of just two atoms, the sum of their oxidation states is zero). The act of losing the electrons (on formation of the bond) is termed oxidation and the act of gaining them is termed reduction. An example of an ionic bond is the bond between sodium (Na) and chlorine (Cl) in NaCl. Each forms an ion, one negative and one positive, and the resulting Coulomb force draws the two together.
Stable compounds generally consist of atoms that each have a favorable number of electrons in their outer shells, favorable numbers being complete and/or containing exactly eight electrons. Atoms within such compounds achieve this by transferring electrons to another atom (ionic bond) or alternately, by sharing electrons between two atoms (covalent bond). The type of bond that forms depends upon complex (quantum-mechanical) rules, but generally the cases where ionic bonds form involve the transfer of one or a very small number of electrons to achieve these numbers: for example sodium has 11 electrons (1 electron in its outer shell) and chlorine 17 (7 electrons in its outer shell) and sodium gives up one to chlorine, sodium becoming a negative ion and chlorine a positive ion: in sodium chloride, sodium's oxidation state is +1, and chlorine's is -1. There are calculated quantities (electronegativity) for each element that offer a simple, usually-correct means of determining the type of bond that forms, and charts to with these to allow you to do this with ease.