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On my test, there was a caffeine molecule and students needed to identify hydrogen bonding sites. There were 6, 4 due to lone pairs 4 N atoms and 2 due to lone pairs on 2 O molecules. I have been hearing that DNA was used as an example for other tests, so you may want to double check with someone if your test had this question, and not the question with caffeine.
I don't remember the exact number, but this question confused me too. It's supposed to be the total number of potential H-bonding sites on the molecule, and this can be for H-bond sites with ANY molecule, not only more of the same molecule. So, it will be the number of Hydrogens bonded to N,O, and F, as well as the number of partially charged N,O and F with lone pairs.
I'm a bit confused by the mixed answers too, I put 8 and got it right on my test. There were 4 N with 1 lone pair each, so that's 4 hydrogen bonding sites; there were 2 O with 2 lone pairs each, so that's another 4 hydrogen bonding sites, giving me 8 in total. In theory, if there were any H attached to N, O, or F, that would be a hydrogen bonding site too, but all the H in the caffeine molecule were bonded to C.
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