The $\mathrm{p}K_\mathrm{a}$ value of o-fluorophenol is $8.7$, while that of the p-fluorophenol is $9.9$. It's obvious that the inductive effect is more dominant at ortho-position, which results in a more acidic nature.

However, won't the $\ce{H}$ attached to $\ce{O}$ form a hydrogen bond with the $\ce{F}$ at ortho-position resulting in a less acidic nature than p-fluorophenol?

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    $\begingroup$ What you've described is a classic case of two competing factors. Having two competing factors doesn't mean that it is simultaneously more and less acidic - all it means is that one factor is more important than the other. From the pKa values it should be clear which one is more important, and you seem to have realised that too. $\endgroup$ – orthocresol Aug 27 '17 at 17:12
  • $\begingroup$ chemistry.stackexchange.com/questions/7683/… $\endgroup$ – Mithoron Aug 27 '17 at 17:35

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