# Hammett acidity function

The Hammett acidity function (H$_0$) is a measure of acidity that is used for very concentrated solutions of strong acids, including superacids. It was proposed by the physical organic chemist Louis Plack Hammett and is the best-known acidity function used to extend the measure of Brønsted–Lowry acidity beyond the dilute aqueous solutions for which the pH scale is useful.
In highly concentrated solutions, simple approximations such as the Henderson–Hasselbalch equation are no longer valid due to the variations of the activity coefficients. The Hammett acidity function is used in fields such as physical organic chemistry for the study of acid-catalyzed reactions, because some of these reactions use acids in very high concentrations, or even neat (pure).

## Definition

The Hammett acidity function, H$_0$, can replace the pH in concentrated solutions. It is defined using an equation analogous to the Henderson–Hasselbalch equation:
    $H_{0}={\mbox{p}}K_{{\ce {BH^+}}}+\log {\frac {{\ce {[B]}}}{{\ce {[BH^+]}}}}$
where log(x) is the common logarithm of x, and pK$_{BH+}$ is −log(K) for the dissociation of BH$^+$, which is the conjugate acid of a very weak base B, with a very negative pK$_{BH+}$. In this way, it is rather as if the pH scale has been extended to very negative values. Hammett originally used a series of anilines with electron-withdrawing groups for the bases.
Hammett also pointed out the equivalent form
    $H_{0}=-\log \left(a_{{\ce {H^+}}}{\frac {\gamma _{{\ce {B}}}}{\gamma _{{\ce {BH^+}}}}}\right)$
where a is the activity, and the γ are thermodynamic activity coefficients. In dilute aqueous solution (pH 0–14) the predominant acid species is H$_3$O$^+$ and the activity coefficients are close to unity, so H$_0$ is approximately equal to the pH. However, beyond this pH range, the effective hydrogen-ion activity changes much more rapidly than the concentration. This is often due to changes in the nature of the acid species; for example in concentrated sulfuric acid, the predominant acid species ("H$^+$") is not H$_3$O$^+$ but rather H$_3$SO$_4$$^+$, which is a much stronger acid. The value H$_0$ = −12 for pure sulfuric acid must not be interpreted as pH = −12 (which would imply an impossibly high H$_3$O$^+$ concentration of 10$^{+12}$ mol/L in ideal solution). Instead it means that the acid species present (H$_3$SO$_4$$^+$) has a protonating ability equivalent to H$_3$O$^+$ at a fictitious (ideal) concentration of 10$^{12}$ mol/L, as measured by its ability to protonate weak bases.
Although the Hammett acidity function is the best known acidity function, other acidity functions have been developed by authors such as Arnett, Cox, Katrizky, Yates, and Stevens.

## Typical values

On this scale, pure H$_2$SO$_4$ (18.4 M) has a H$_0$ value of −12, and pyrosulfuric acid has H$_0$ ~ −15. Take note that the Hammett acidity function clearly avoids water in its equation. It is a generalization of the pH scale—in a dilute aqueous solution (where B is H$_2$O), pH is very nearly equal to H$_0$. By using a solvent-independent quantitative measure of acidity, the implications of the leveling effect are eliminated, and it becomes possible to directly compare the acidities of different substances (e.g. using pK$_a$, HF is weaker than HCl or H$_2$SO$_4$ in water but stronger than HCl in glacial acetic acid.)
List of acids by Hammett acidity :
| Name | H$_0$ |
| Trifluoroacetic acid | −2.7 |
| Phosphoric acid | −4.66 |
| Nitric acid | −6.3 |
| Methanesulfonic acid | −7.86 |
| Sulfuric acid | −11.93 |
| Perchloric acid | −13 |
| Triflic acid | −13.7 |
| Chlorosulfuric acid | −13.79 |
| Oleum (50 mole% SO$_3$) | −14.44 |
| Fluorosulfuric acid | −15.07 |
| Hydrogen fluoride | −15.1 |
| Magic acid (1:1 FSO$_3$H-SbF$_5$) | −23.0 |
| Fluoroantimonic acid (1:1 HF-SbF$_5$) | −28 |
For mixtures (e.g., partly diluted acids in water), the acidity function depends on the composition of the mixture and has to be determined empirically. Graphs of H$_0$ vs mole fraction can be found in the literature for many acids.