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As mentioned, hydrogen bonding sites are places on molecules where a hydrogen bond could be formed. So this could mean one of two things: a N, O, F atom with an available lone pair OR a H atom attached to a N, O, F atom. In the first instance, the lone pair could allow bonding with a nearby hydrogen atom from a different molecule. In the second instance, the H atom could bond with a nearby N, O, or F atom from a different molecule.
Hydrogen bonding sites are where hydrogen bonding can occur. This includes hydrogen atoms that are bound to electronegative atoms like F, N, or O. A hydrogen bound to one of these elements would be a potential site for hydrogen bonding as well as the nitrogen, fluorine, and oxygen atoms bound to hydrogen atoms. So in a compound like HF, both the hydrogen and the fluorine would be potential hydrogen bonding sites, and in a molecule of water, oxygen and hydrogens would both be hydrogen bonding sites due to their partial charges that result from the difference in electronegativities of the atoms.
Hydrogen bonding occurs when there is a covalent bond between a H and F, O, or N, causing a large dipole moment. Then another Hydrogen can bond on the lone pairs of the F, O, or N or another F, O, N can bond to the partially positively charged H.
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