pH of dilute acids in hexane

Question

Consider a strong acid $\ce{HA}$. In a $\pu{1000 mL}$ of hexane (assume it is unable to accept a proton) we "dissolve" (this can mean forming a suspension with vigorous stirring - otherwise we can use a polar solvent which cannot accept a proton) $\pu{1e-30 mol}$ of the acid along with enough water to remove all the protons.

Now we have extremely dilute solution of $\ce{H+}$ where autoprotolysis is insignificant compared to the volume of solvent, so according to $\mathrm{pH} = -\log[\ce{H+}]$, pH should be extremely high. However, we are still using an acid, so pH should not be too high. What accounts for the difference?

Answer 1

pH is defined for water solutions; using it elsewhere is either impossible or leads to contradictory results like yours. Who said pH should not be too high? Well, you might argue that high pH implies alkaline solution, does it not? No, it doesn't. Alkaline solutions are (roughly speaking) those with high concentration of $\ce{OH-}$. In water this means the same thing, because $\ce{[H+]\cdot[OH-]=10^{-14}}$, so low $\ce{[H+]}$ means high $\ce{[OH-]}$ and vice versa. In hexane, not really.

To determine pKa values which are way beyond the physically possible range of pH, we extend it with things like Hammet's $H_0$, and thus build a ladder of measures suitable for a much wider range. You may call it "effective pH" or like that, but remember: it has little to do with literal concentration of $\ce{H+}$, and your hypothetical solution of $\ce{10^{-30}M\; HCl}$ in hexane is in fact neither acidic nor basic.

Answer 2

Well, I think the pH (activity) can be theoretically applied to non-aqueous solutions (http://onlinelibrary.wiley.com/doi/10.1002/chem.201003164/abstract;jsessionid=6237E1A09F29EE5D07AA17B2658C677B.f03t03), but it isn't very convenient.