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Why are hydrogen bonds relatively weak bonds?

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Hydrogen bonds are relatively weak due to their electrical characteristics. In hydrogen bonds involving oxygen and nitrogen (the most common ones observed biologically), there is a small, partial negative charge on the oxygen/nitrogen atom and a small, partial positive charge on the hydrogen atom. By "partial" I mean that the electrons are only slightly displaced from their neutral position and that there is no net charge on either atom. Each atom behaves as though it has only a small electrical difference from neutral. Since--by Coulomb's Law--the strength of the electric force between two charges depends on the size of those charges, there is a relatively small force between the two atoms and thus hydrogen bonds are relatively weak. See an explanation of Coulomb's law here:
http://www.glenbrook.k12.il.us/gbssci/phys/Class/estatics/u8l3b.html

This is not always the case--the hydrogen bond in bifluoride (HF2-) is 155 kJ/mol, which is comparable to weak covalent bonds. Most hydrogen bonds have an energy in the 5-30 kJ/mol range, which is definitely weaker than most covalent bonds. They are notably stronger than many other intermolecular forces, however; all van der Waals forces and many dipole-dipole interactions are weaker than hydrogen bonds. Also, while a single hydrogen bond may be weak, the ability of a molecule to form multiple hydrogen bonds makes this force a significant factor in molecular stability. A single water molecule is able to form at least four hydrogen bonds, for instance.

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