Hammett acidity function

The Hammett acidity function, H₀, can replace the pH in concentrated solutions. It is defined using an equation[4][5][6] analogous to the Henderson–Hasselbalch equation:

${\displaystyle H_{0}={\mbox{p}}K_{{\ce {BH^+}}}+\log {\frac {{\ce {[B]}}}{{\ce {[BH^+]}}}}}$

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.[3]

Hammett also pointed out the equivalent form

${\displaystyle H_{0}=-\log \left(a_{{\ce {H^+}}}{\frac {\gamma _{{\ce {B}}}}{\gamma _{{\ce {BH^+}}}}}\right)}$

where a is the activity, and the γ are thermodynamic activity coefficients. In dilute aqueous solution (pH 0–14) the predominant acid species is H₃O⁺ and the activity coefficients are close to unity, so H₀ is approximately equal to the pH. However, beyond this pH range, the effective hydrogen-ion activity changes much more rapidly than the concentration.[4] 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 H₃O⁺ but rather H₃SO₄⁺, which is a much stronger acid. The value H₀ = -12 for pure sulfuric acid must not be interpreted as pH = −12 (which would imply an impossibly high H₃O⁺ concentration of 10⁺¹² mol/L in ideal solution). Instead it means that the acid species present (H₃SO₄⁺) has a protonating ability equivalent to H₃O⁺ at a fictitious (ideal) concentration of 10¹² 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.[3]

On this scale, pure H₂SO₄ (18.4 M) has a H₀ value of −12, and pyrosulfuric acid has H₀ ~ −15.[7] 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 H₂O), pH is very nearly equal to H₀. 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 H₂SO₄ in water but stronger than HCl in glacial acetic acid.[8][9])

H₀ for some concentrated acids:[10]

• Helonium: −63
• Fluoroantimonic acid (1990): −23 > H₀ > −28
• Magic acid (1974): −23
• Carborane superacids: H₀ < −18.0
• Fluorosulfuric acid (1944): −15.1
• Hydrogen fluoride: −15.1
• Trifluoromethanesulfonic acid (1940): −14.9
• Perchloric acid: −13
• Sulfurochloridic acid: −13.8; −12.78[11]
• Sulfuric acid: −12.0

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 H₀ vs mole fraction can be found in the literature for many acids.[3]