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Is there a molecule that can donate a hydrogen to a hydrogen bond but not accept one? It would seem that since the donated hydrogen needs to be bonded to an electronegative atom, that electronegative atom will always have lone pairs available to accept a hydrogen bond.

For example, in ethanol, the H in the OH can participate in hydrogen bonding since the the O is electronegative. Then the oxygen with its two lone pairs can also accept hydrogen bonds.

In reverse, things like diethyl ether can accept but not donate hydrogens because the oxygen is surrounded by carbons, not hydrogen.

Are there any exceptions to this apparent rule?

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    $\begingroup$ Ammonium ions should be able to act as hydrogen donors yet lack a lone pair and are thus no hydrogen bond acceptors. $\endgroup$ – Philipp Mar 12 '16 at 22:49
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As mentioned in the comments by Philipp, ammonium ions $\ce{R3NH+}$ are a good direct answer to what you are looking for.

The nitrogen in an amide $\ce{RCONHR'}$ is similar - the N lone pair is conjugated into the C=O bond and that makes it a very poor hydrogen bond acceptor, but the N-H lone pair can be a hydrogen bond donor. For more details, see this previous question with an excellent writeup by Jan. Now an amide can of course function as a hydrogen bond acceptor via the lone pairs on oxygen, so it's not strictly what you're looking for, but it was just something related. And it is of course directly relevant to the secondary structure of proteins.

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