The Proton´s Potential: How Acidic Is Sulfuric Acid at All?
Authors: Radtke Valentin , Stroh Regina , Bäuerle Monika , Himmel Daniel , Krossing Ingo 
Published: December 2024

Hammett and Deyrup introduced the H
0 function to gather “acidity in terms of a basic indicator”.[1] Since then, the H
0 acidity function and those derived from it have been used to specify the acidity of various solutions or pure acids, and particularly superacids. However, over the course of time, the distinction between the H
0 acidity and the pH acidity seems to have become confused or completely obscured for a number of researchers, as if the two are to be considered the same. The activity coefficients, or more precisely, the medium effect f of an indicator base B and its protonated form BH+ are integral part of the H
0 acidity function definition:

H
0 = − lg a
H2O(H+, S) − lg (f
B / f
BH+
)
S is a non-aqueous solvent or a highly concentrated acid, the activity a
H2O refers to the aqueous reference state. In contrast, the pH acidity is introduced as unifying scale pHabs
H2O that only accounts for the chemical potential of the proton, or, since the activity of a chemical entity is defined by its chemical potential, for the activity of the proton:[2]

pHabs
H2O = − lg a
H2O(H+, S)
This scale allows the straightforward comparison of the pH acidity of all homogeneous and isotropic media and can be regarded as continuation of the aqueous pH scale defined by Sørensen into both, the acidic and basic direction.
Our investigations with potentiometric measurements on two different types of electrochemical cells, one of which is the H+ analog of the determination of Gibbs transfer energies of the Ag+- and the Cl−-ion,[3] indicate that the H
0 function, if misinterpreted as extension of the pH acidity, underestimates the proton´s chemical potential in pure sulfuric acid by at least 11 orders of magnitude. The difference of the pH and the H
0 acidity scales can be quantified by subtracting above two equations:
(pHabs
H2O − H
0) RT ln10 = Δtr
G°(B, H2O → H2SO4) − Δtr
G°(BH+, H2O → H2SO4)
High level quantum chemical computations of Gibbs solvation energies of the indicator bases B and their protonated forms BH+ that were used by several investigators[4] on the one hand, and of chemical entities expected to be present in sulfuric acid – water mixtures on the other, support our experimental findings.
We provide a rule of thumb to calculate pHabs
H2O values into H
0 values, pHabs
H2O ≈ H
0
4/3 (see Figure), it has to be noted, however, that this rule only applies to sulfuric acid.
References
[1] L. P. Hammett, A. J. Deyrup, J. Am. Chem. Soc.
1932, 54, 2721.
[2] a) D. Himmel, S. K. Goll, I. Leito, I. Krossing, Angew. Chem. Int. Ed.
2010, 49, 6885; b) A. Suu, L. Jalukse, J. Liigand, A. Kruve, D. Himmel, I. Krossing, M. Rosés, I. Leito, Anal. Chem.
2015, 87, 2623.
[3] a) V. Radtke, D. Priester, A. Heering, C. Müller, T. Koslowski, I. Leito, I. Krossing, Chem. Eur. J.
2023, 29, e202300609; b) V. Radtke, N. Gebel, D. Priester, A. Ermantraut, M. Bäuerle, D. Himmel, R. Stroh, T. Koslowski, I. Leito, I. Krossing, Chem. Eur. J.
2022, 28, e202200509; c) A. Ermantraut, V. Radtke, N. Gebel, D. Himmel, T. Koslowski, I. Leito, I. Krossing, Angew. Chem. Int. Ed.
2018, 57, 2348.
[4] a) R. J. Gillespie, T. E. Peel, E. A. Robinson, J. Am. Chem. Soc.
1971, 93, 5083; b) M. J. Jorgenson, D. R. Hartter, J. Am. Chem. Soc.
1963, 85, 878; c) M. A. Paul, F. A. Long, Chem. Rev.
1957, 57, 1.

Figure 1