Solvent-Independent Unifying Acidity

$\mathrm{\mathbf{pH_{abs}^{H_{2}O}}}$ of proton solutions (at pH_S 0)

Below you can find $\mathrm{pH_{abs}^{H_{2}O}}$ and $\mathrm{pH_{abs}}$ values at standard conditions (i.e. $a_\mathrm{S}\left(\mathrm{H}^{+},\mathrm{S}\right)=1$) for a variety of solvents. They are based on Gibbs transfer energies of the proton $\Delta_{\mathrm{tr}}G{^\circ}\left(\mathrm{H}^{+},\mathrm{H_{2}O\rightarrow S}\right)$, which we determined using our electrochemical ILSB method. Further below you will find values for solvents that we have not determined directly, as these are derived from literature data. The Gibbs hydration energy of the proton $\Delta_{\mathrm{solv}}G{^\circ}\left(\mathrm{H}^{+},\mathrm{H_2O}\right)$ is taken from the literature.^[1] All values of $\Delta_{\mathrm{tr}}G{^\circ}\left(\mathrm{H}^{+},\mathrm{H_{2}O\rightarrow S}\right)$ and $\Delta_{\mathrm{solv}}G{^\circ}\left(\mathrm{H}^{+},\mathrm{S}\right)$ are given in kJ mol⁻¹, $\mathit{E}_\mathrm{abs}$ and $\mathrm{\mathit{E}_{abs}^{H_{2}O}}$ values are given in V. Details are published here.

$\mathrm{pH_{abs}^{H_{2}O}}$ and $\mathrm{pH_{abs}}$ values at standard conditions (i.e. $a_\mathrm{S}\left(\mathrm{H}^{+},\mathrm{S}\right)=1$) for a variety of solvents NOT determined with our ILSB method.

^a Values according to Marcus et al.^[2] 

^b Values according to Marcus et al.^[3] 

^c Values calculated with the rCCC model.^[4]

^d Values according to the COSMO-RS model.

Literature

[1] M. D. Tissandier, K. A. Cowen, W. Y. Feng, E. Gundlach, M. H. Cohen, A. D. Earhart, J. V. Coe, T. R. Tuttle, J. Phys. Chem. A 1998, 102, 7787.

[2] Y. Marcus, M. J. Kamlet, R. W. Taft, Journal of Physical Chemistry 1988, 92, 3613-3622.

[3] C. Kalidas, G. Hefter, Y. Marcus, Chem. Rev. 2000, 100, 819.

[4] D. Himmel, S. K. Goll, I. Leito, I. Krossing, Chemistry A European Journal 2011, 17, 5808.

$\mathrm{\mathbf{pH_{abs}^{H_{2}O}}}$ of H₂SO₄

Here you can find the unified acidity of water-sulfuric acid mixtures as determined inter alia with our ILSB method, along with the widely used Hammett Acidity H₀ values.^[1] The former reflects the the protochemical potential, and the latter additionally the medium effects of the indicator bases used to define and determine the H₀ values. $\mathrm{pH_{abs}^{H_{2}O}}$ values lower than 70 wt.-% H₂SO₄ are in good approximation identical to the H₀ values. $\mathrm{pH_{abs}^{H_{2}O}}$ values above 80 wt.-% H₂SO₄ were calculated with to Eq. av7. Values in between were estimated (excepting the value for 75 wt.-% H₂SO₄). Details are published here.

Further below you can find the key data of H₂SO₄ at standard conditions, i.e. $a_\mathrm{H_2SO_4}\left(\mathrm{H}^{+},\mathrm{H_2SO_4}\right)=1$.

At $\boldsymbol{a}_{\mathbf{H_{2}SO_{4}}}\left(\mathbf{H^{+},H_{2}SO_{4}}\right)\mathbf{\mathbf{=}1}$ or $\mathrm{\mathbf{\mathbf{\mathrm{\mathbf{p}}}}\mathbf{H}_{\mathbf{H_{2}SO_{4}}}}\mathbf{=0}$, respectively, following values are valid:

Literature

[1] L. P. Hammett, A. J. Deyrup, J. Am. Chem. Soc. 1932, 54, 2721.

Remarks to $\mathrm{\mathbf{pH_{abs}^{H_{2}O}}}$

The distinction between the $\mathrm{pH_{abs}}$ scales (i.e. $\mathrm{pH_{abs}}$ and $\mathrm{pH_{abs}^{H_{2}O}}$) and the $\mathrm{pH_{S}}$ scales (i.e. $\mathrm{pH_{H_{2}O}}$, $\mathrm{pH_{MeCN}}$, $\mathrm{pH_{H_{2}SO_{4}}}$, etc.) is the reference state of the proton: For the latter ones the ideal dilute proton solution in the solvent S with the relative activity $a_{\mathrm{S}}\left(\mathrm{H}^{+},\mathrm{S}\right)=1$ is used. In this (hypothetical) state, contrary to solute-solute interactions, solute-solvent and solvent-solvent interactions are still operative. Since each solvent molecule interacts differently with a solute molecule and with itself, each solvent comes along with an individual reference state, which generally differs from reference states in other solvents. As a result, this leads to incomparability of the $\mathrm{pH_{S}}$ values for different S. The reference state of the $\mathrm{pH_{abs}}$ and the $\mathrm{pH_{abs}^{H_{2}O}}$ scales is the ideal proton gas at 298.15 K and 10$^{5}$ Pa where the chemical potential $\mu_{\mathrm{abs}}\left(\mathrm{H}^{+}\right)$ of the proton arbitrarily is set to 0 kJ mol$^{-1}$. In this (hypothetical) state the “solute” proton is free of any interaction, however, it includes the standard terms of entropy and enthalpy according to Bartmess.^[1] This state is independent of any solvent or medium ensuring the comparability of all values within these scales; it is in accordance with IUPAC recommendations. For more details of our used standard states, see here.

The definition of the $\mathrm{pH_{abs}}$ is:

At constant pressure and temperature, the absolute chemical potential of the proton equals the Gibbs energy of solvation, and the $\mathrm{pH_{abs}}$ is calculable via:

We use the value for the Standard Gibbs energy of hydration of the proton obtained by Tissandier et al., $\Delta_{\mathrm{solv}}G{^\circ}\left(\mathrm{H}^{+},\mathrm{H_{2}O}\right)=-1104.8\:\mathrm{kJ\,mol^{-1}}$.^[2] Thus, an aqueous solution with the $\mathrm{pH_{H_{2}O}}=0$ has the $\mathrm{pH_{abs}}=193.5$ at 198.15 K. To align the zero point of the $\mathrm{pH_{abs}}$ scale with that of the $\mathrm{pH_{H_{2}O}}$ scale, thereby obtaining the $\mathrm{pH_{abs}^{H_{2}O}}$ scale, this value has to be subtracted from the $\mathrm{pH_{abs}}$ value:

In addition to the “familiarity” of the values (e.g. 0 vs. 193.5), this has the advantage that if a more reliable value for the Gibbs energy of the proton hydration is available, in a majority of cases it is not required to recalculate the $\mathrm{pH_{abs}^{H_{2}O}}$ values. This is the case when the values were determined using Gibbs transfer energy values $\Delta_{\mathrm{tr}}G{^\circ}\left(\mathrm{H}^{+},\mathrm{H_{2}O\rightarrow S}\right)$ for the transfer of the proton from water to the solvent S:

The $\mathrm{pH_{abs}^{H_{2}O}}$ is connected to the unified reducity of the hydrogen electrode reaction, $\mathrm{\mathit{E}_{abs}^{H_{2}O}\left(H^{+}/H_{2},S\right)}$.

The $\mathrm{pH_{abs}^{H_{2}O}}$ also can be related to other acidity scales as we have shown exemplarily with the Hammett Acidity Function $H_{0}$ and its values for the entire mixing range of water and sulfuric acid, see Eq. av6.

With $_{m}f$ is the medium effect of the indicator base B and its protonated form BH$^{+}$ that is used for the determination of $H_{0}$ of an $\mathrm{H_{2}O-H_{2}SO_{4}}$ mixture or pure sulfuric acid S. In the range below about 70 wt.-% $\mathrm{H_{2}SO_{4}}$ we state that $\mathrm{pH_{abs}^{H_{2}O}}\thickapprox H_{0}$, at concentrations higher than 80 wt.-% $\mathrm{H_{2}SO_{4}}$ we find the empirical relation Eq. av7, and in between we cannot provide a sharp relation.

Final note: Since protons are electrically charged they are affected by electric fields. Thus, in principle, one is free to choose a quantity including the energy contribution of the electric field; it is in this way that the IUPAC defines the absolute electrode potential of the standard hydrogen electrode in water $\mathrm{SHE_{H_{2}O}}$. Click here for implications if the pH was defined with such a quantity instead of $\mathit{\mu}$.

Literature

[1] J. E. Bartmess, J. Phys. Chem. 1994, 98, 6420.
[2] M. D. Tissandier, K. A. Cowen, W. Y. Feng, E. Gundlach, M. H. Cohen, A. D. Earhart, J. V. Coe, T. R. Tuttle, J. Phys. Chem. A 1998, 102, 7787.

Calculation of $\mathrm{\mathbf{pH_{abs}^{H_{2}O}}}$

You can find here a calculator that approximates the $\mathrm{pH_{abs}^{H_{2}O}}$ values according to the below given formulas. It accesses databases that contain not only data by our group but also data obtained with the TATB assumption, depending on the solvent.

To get other $\mathrm{pH_{abs}^{H_{2}O}}$ values in the solvent S than at standard conditions (i.e. $\mathrm{pH_{S}}$ 0), use Eq. ca1.

You can find the $\mathrm{pH_{abs}^{H_{2}O}}\,°$ values here. The $\mathrm{pH_S}$ value can be obtained in the usual way:

Usually, this theoretical approach is of little help in daily lab work, since the activity $a_{\mathrm{s}}\left(\mathrm{H^{+},S}\right)$ is unknown. For ideal solutions and as approximation you can use the concentration instead of the activity.

The (approximated) proton concentration $c_{\mathrm{s}}\left(\mathrm{H^{+},S}\right)$ can be obtained from the acid concentration (initial concentration) $c_0\left(\mathrm{HA}\right)$. Depending on the $\mathrm{p}K_{\mathrm{a,S}}$ of the acid HA in S and on the autoprotolysis constant $\mathrm{p}K_{\mathrm{solv,S}}$ of S, following cases have to be considered, Eqs. ca4 – ca7.

With $\mathrm{p}K =-\mathrm{lg} K$. Some $\mathrm{p}K_{\mathrm{solv,S}}$ values can be found here. $\mathrm{p}K_{\mathrm{a,S}}$ values of some hundred compounds in common organic solvents were determined and collected by the group of Prof. Ivo Leito and can be found at the Department of Analytical Chemistry at the University of Tartu.