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It's important to also know that the reason diatomic molecules have standard enthalpies of formation at zero is not because they are diatomic in and of itself, but it is a consequence of the fact that diatomic elements are pure substances. For example, molecules like C(s) or K(s) or O2 etc., are all composed of one element. Because standard enthalpies of formation are derived from the reaction of constituent elements making the substance at hand, it makes sense to see how a carbon atom forming itself, or more accurately remaining as a carbon atom, would not have any enthalpy change.
Diatomic molecules are made of two elements. The seven diatomic elements from the periodic table are nitrogen, oxygen, bromine, iodine, chlorine, hydrogen, and fluorine. Because they are the same elements, they have a standard enthalpy change of 0.
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