I wanted to know whether hydrogen bonding is stronger in phenol or methanol. I saw on Wikipedia about the enthalpies of hydrogen bonding in various cases. but could not find the answer to this question of mine.
I also want to know whether the hydrogen bond strength depends upon the carbon atom to which oxygen is attached.
Let me answer the second question first, because the answer is so much easier: Of course the hydrogen bond strength depends among others upon the carbon oxygen is attached to. Consider phenol and methanol or methanol and formic acid.
The guess as to which hydrogen bond is stronger is — I have to admit it — my speculation. But I think that methanol should be able to undergo stronger hydrogen bonding.
Phenol, of course, is aromatic and capable of delocalising negative (or positive) charges across its entire ring system — the reason why its $\mathrm{p}K_{\mathrm{a}} = 10$, a good five units lower than methanol’s. That means that at the same pH value of the solution, there should be less strong a negative charge on phenol’s oxygen than on methanol’s.
But hydrogen bonds are, to a certain extent, electrostatic interaction. So the system which has a stronger negative charge on the oxygen (methanol) should be able to build up stronger hydrogen bonds by attracting hydrogen more strongly.
One could also picture it differently: Phenol gets deprotonated at a lower pH, so the $\ce{O-H}$-bond must be a lot weaker than methanol’s. But if the $\ce{O-H}$-bond is weak, the oxygen has a larger tendency to simply loose the hydrogen to another electronegative partner (deprotonation/protonation) rather than share it (hydrogen bond).
Phenol is a stronger hydrogen bond donor [1] but methanol is a better acceptor [2].
[1] http://pubs.acs.org/doi/abs/10.1021/jp410027h
[2] http://pubs.acs.org/doi/abs/10.1021/jm801331y
Probably, these effects compensate, and the hydrogen bond in methanol-methanol is of similar strength to phenol-phenol.
I'd guess phenol (though I'm not 100% sure about that). My reasoning is that in phenol, there are several resonance structures that put extra negative charge on the oxygen, thus allowing for a stronger interaction.
There is a strong hydrogen bonding of methanol in water than phenol in water because the partial negativity in oxygen in phenol is decreased as a result of pi electron delocalization towards the ring. Hydrogen bonding is a result of interaction between a partial negative oxygen and partial positive hydrogen. Thus, to decrease the partial negativity of the oxygen would weaken the interaction between the hydrogen of water to the oxygen in phenol.
Phenol has week acidic effect due to the presence of benzene ring produces a partial positive charge on O atom that facilitates the separation of the H+ ion.
