Why do atoms like carbon and nitrogen not like to make ions, while sodium and chlorine do?

The chemical behavior of atoms is best understood in terms of the degree to which an atom of a particular element attracts electrons, a characteristic officially known as electronegativity. When electronegativity is either very high (as in a chlorine atom) or very low (as in a sodium atom) then you have an atom which tends to either acquire or get rid of one or more electrons, and when it does so it becomes an ion. Carbon has a moderate electronegativity and therefore it is more likely to share electrons (forming covalent bonds) rather than either giving them up or acquiring them (forming ionic bonds). Nitrogen does have a relatively high electronegativity and does form ionic bonds, but in ionic compounds it is most often found in the nitrate radical, combined with 3 oxygen atoms. Nitrogen is also found in molecules that have covalent bonds, such as proteins, but it is the moderating influence of carbon that makes this happen.

I should add that inert elements such as helium do not attract electrons but neither do they give up the ones that they have; they are in a special category, and they form no bonds, neither ionic nor covalent.

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