A unified pH scale for all solvents: part I – intention and reasoning (IUPAC Technical Report)

2.1 The physical variable of the unified pH scale

The need for a unified pH scale arises from the impossibility to correlate the individual pH_S scales without further knowledge of the considered acid–base systems. The basic problem of knowing how acidic a proton is – that is what an acidity scale or “pH” value should reflect – is closely connected to the state of the proton in its environment, that is, on the proton’s bonding conditions, etc. One can thus deduce that the knowledge of the proton’s chemical potential μ H + (or the protochemical potential, Eq. 3) is the fundamental key to connect all pH scales. Certainly, more convenient is “one” pH scale directly anchored to the protochemical potential (see Eq. 8).

The tilde brings the dependence of an ion’s partial molar Gibbs energy on the Galvani potential φ of a phase to mind, and μ ˜ H + denotes the electrochemical potential of the proton [9], [10].^[1] n H + is the amount of H⁺ and F the Faraday constant. Equation 3 implies that the chemical potential of single ions is an unmeasurable quantity although certain combinations of individual ionic activities are measurable in a thermodynamically exact manner [11]. This instance is the source of long-lasting discussions, and widespread is the view that single-ion activities are without physical meaning. On the other hand, physical reality is attributed to the electrochemical potential, although this quantity is not measurable either. It has been pointed out that physical significance is not linked to measurability alone [12]. As long as no experiment is operable thermodynamically stringent in character the need of methods outside of exact thermodynamics is widely accepted and the determination of single-ion activities is performed.

2.2 The reference state of the unified pH scale

An appropriate reference state should have two key properties: it should be maximally simple and maximally extensive. To achieve this, we take the ideal proton gas at p = p ⊖  = 10⁵ Pa and T = 298.15 K as the reference state (Fig. 1). The adjective “ideal” suggests that no interactions between the protons (e.g., of coulombic nature) occur. Thus, we actually consider an isolated proton in the gas phase and add, according to Bartmess [13] the standard terms for entropy and enthalpy to reach 298.15 K and 10⁵ Pa.^[2] This state corresponds to the standard state of ideal neutral gases (Eq. 4) [10]. We arbitrarily set the chemical potential of the proton in this state to 0 kJ mol⁻¹.

y H + is the mole fraction of the proton in a gas mixture and, hence, unity in the pure gas.

Fig. 1:

The (hypothetical) reference state of the unified pH_abs scale and the thermodynamic relations to the Brønsted-acidity in all phases. The Gibbs energy of solvation Δ s o l v G ( H + , S ) is not specified here because different definitions exist, see text. Similarly, Δ i n t G ( H + , S ) and Δ r G ( H + , S ) are also not specified.

To achieve ideal behavior of solutes in solution, that is, the absence of interactions between the solutes, a hypothetical state of the solute is established, in which the solute behaves as if infinitely dilute, marked by the infinity symbol, but at the standard pressure p ⊖ and at the standard molality m ⊖ . This state is the standard state of solutes in solution [10]. Because solute–solvent and solvent–solvent interactions are still operative and each solvent interacts differently with a solute B and with itself, each solvent is accompanied by an individual standard state which normally differs from the standard states in (or of) other solvents. This results in relative activities, Eq. 5, and is the reason for the incomparability of conventional pH_S values (and redox potential values) in different solvents.

In accordance with the IUPAC definition, the standard chemical potential of the solvated proton in the solvent S can be expressed using Eq. 6.

Both reference states (Eqs. 4 and 6) are consistent with the IUPAC recommendations of the Commission on Thermodynamics of the Physical Chemistry Division [14]. Although the definition of the standard state of a solute in solution is comprehensible, the process of a particle’s solvation and the corresponding thermodynamic parameters may cause – although clearly defined – confusion. This is amplified by the use of different symbols and, of course, standard states and conventions. Hünenberger and Reif [15] elaborated a consistent notation expanding on the recommendations by the IUPAC [10], leaving no room for ambiguity. However, due to its complexity, the introduction of this notation goes beyond the scope of this report.

Δ s o l v G ∗ describes the transfer of a particle from a fixed point in the ideal gaseous phase ( G ∗ , at c ⊖  = 1 mol L⁻¹) to a fixed point in the ideal solvation state ( G ⊖ c , at c ⊖  = 1 mol L⁻¹) at the same p and T in both phases (conveniently p ⊖ and 298.15 K) and is considered as reflecting the solvation process physically most appropriate [16]. The standard Gibbs energy of solvation Δ G ⊖ s o l v c or Δ G ⊖ s o l v m is then defined, specifying the transition of the proton from the gaseous standard state (at p ⊖  = 10⁵ Pa) to the solution standard state in the solvent S, either at the standard amount concentration c ⊖  = 1 mol L⁻¹ ( G ⊖ c ), Eq. 7a, or at the standard molality m ⊖  = 1 mol kg⁻¹ ( G ⊖ m ), Eq. 7b. The conversion from concentration- to molality-based standard states and vice versa is straightforward, Eq. 7c; therefore, in Fig. 1, Δ s o l v G ( H + , S ) is not specified.

Further discussion on the distinction between molality and amount concentration scale is not required as long as it is consistently used.

In this way, the standard states of each solvent are traced to the more extensive gaseous standard state and the direct comparability between different solvents is ensured. This also holds for all homogeneous and isotropic solid or liquid media. It follows from Eq. 3 that under standard conditions, Δ G ⊖ s o l v ( H + , S ) = μ a b s , H + , S ⊖ holds true, keeping in mind the chosen reference state and using the subscript “abs” to indicate that all protochemical potentials obtained in this way can be compared to each other.

2.3 The definition of the unified pH scale

With Eqs. 4 and 5, all absolute protochochemical potentials μ abs , H + can be related to each other. On this basis, the pH_abs scale was defined unifying all phases (Eq. 8) [6].

The absolute state is referenced to the gas phase connecting all media; thus, no indication in terms of medium is necessary and all pH_abs values can be compared to each other regardless of the medium. Because the conventional pH_S (Eq. 1) is defined by the relative activity, it can be formulated in an atypical way as p H S = − μ H + , S − μ H + , S ⊖ R T   l n 10 . Conversely, using the absolute activity [4] λ H + = a a b s , H + = exp ( μ a b s , H + R T ) , one can derive a more familiar form of pH_abs, Eq. 9a.

pH_abs can even be translated in terms of H⁺ pressure, Eq. 9b.

Because the aqueous pH scale (i.e., p H H 2 O ) is the most prominent, it is favorable to define the p H a b s H 2 O value to align the zero values (not the reference states!) of the pH_abs and the conventional p H H 2 O scale (Eq. 10) [7].

The magnitude of one pH_S, pH_abs, and p H a b s H 2 O unit is identical, that is, unity in terms of pH measure, according to RT·ln10 in terms of caloric measure or (RT/F)⋅ln10 in terms of electric measure. Thus, the p H a b s H 2 O value serves as a thermodynamically well-defined link between the acidity in water and the acidity in any other medium with respect to the aqueous system (Fig. 2).

Fig. 2:

The absolute protochemical potential μ abs , H + and the relation to the p H a b s , p H H 2 O , and p H a b s H 2 O scales based on Δ s o l v G ⊖ ( H + , H 2 O )  = −1104.5 kJ mol⁻¹. The colored areas represent the protochemical windows of some selected media, determined by the respective autoprotolysis constant pK _AP, with the vertical bars being their respective neutral points (defined with pK _AP/2); data were taken from Ref. [17].

2.4 Assembling the unified pH scale

For solutions, Eq. 8 can be expressed in terms of the Gibbs energy of solvation (Eq. 11).

pH_S is the conventional pH value in the solvent S according to Eq. 1. Thus, the knowledge of the Gibbs energy of solvation and of the relative activity of the proton in the solvent S suffices to determine the corresponding pH_abs value. However, the determination of Gibbs energies of solvation for single ions is anything but straightforward, and data are scarce.^[3] The difference of Gibbs energies of solvation of a species B in two solvents S₁ and S₂ is called the Gibbs energy of transfer between these solvents and is related to the medium effect (Eq. 12).

With Eqs. 10, 11, and 12 (if S₁ is water), one can formulate the p H a b s H 2 O as Eq. 13.^[4]

The determination of Δ t r G ⊖ ( i , S 1 → S 2 ) of single ions i is an intricating task best described as a vicious circle and has been tackled by quite a number of distinguished scientists during the past century. So far, extra-thermodynamic assumptions have been used to get around the problem [19]. The values of Gibbs energies of transfer obtained with extra-thermodynamic assumptions, however, are afflicted with uncertainties, which cannot be determined exactly but only can be estimated. These estimations depend on the method. Optimistic estimations lie in the range of 1 pH unit, cautious estimations in the range of roughly 2 to 12 pH units, and rise up to 20 pH units applied on model systems [20]. Here, we suggest approaching the problem with a direct method based on potentiometric measurements with a determined consistency of about 0.1 pH unit (see below).

2.5 Connection to electrochemistry

The aforementioned connection from the gaseous state to the solution state also leads in a straightforward manner to the standard state used in electrochemistry. However, one must note that the standard state, that is, the standard hydrogen electrode (SHE),^[5] in electrochemistry is not as unambiguously defined as the pH value is in acid–base chemistry (see Eq. 1). In Ref. [10], the SHE definition includes “a solution of H ⁺ at unit activity” without specifying the concentration scale. Although sometimes it is clearly defined in the molality scale, tables of standard redox potentials list values under atmospheric pressure in preference to 10⁵ Pa [22].

The Gibbs energy of a reaction, divisible in a reduction (electronation) and oxidation (de-electronation) partial reaction, is connected to the electric potential difference E _cell of an appropriate electrochemical cell by Eq. 14.

n is the number of electrons in the cell reaction and F is the Faraday constant.

3.1 Primary pH measurement

The primary method of measuring the pH of dilute aqueous solutions is laid out in the 2002 IUPAC Recommendations [2]. This method, commonly referred to as the Harned cell method, relies on the potential difference measurement of cell I.

Cell I is a cell without transference and is, therefore, assumed to exclusively reflect the Gibbs energy of the associated cell reaction.^[6]

Similarly, for some organic solvents and their mixtures with water, primary standards were defined and measured with cell I [3], [24]. However, measurements concerning intersolvental pH scales demand cells with concomitantly different solvents inside (solvation cells). Consequently, the primary method cannot be applied.

3.2 Secondary pH measurement with one solvent

Due to the restrictions imposed by the primary method, pH measurements more commonly include the measurement of potential difference of two electrodes immersed in two separate aqueous solutions, known as secondary pH measurements. In effect, the used electrochemical cell contains one or more liquid–liquid junctions, each accompanied by a potential drop, called liquid junction potential (LJP).

Secondary pH measurements can be performed with, e.g., cell II, which enables the measurement of the pH difference between the solutions on the left and the right side of cell II [25].

d ( ≠ 0 ) is a factor indicating that the (relative) proton activity in both solutions may or may not be equal. Ind is an indicator electrode selective exclusively (as far as practically possible) to protons and SB is a salt bridge. Note that both solutions contain H⁺ (solv) within the identical solvent S₁. Each of the single vertical bars represent a phase boundary, and the double-dashed vertical bars pertain to liquid–liquid junctions, in which the LJP has been assumed to be eliminated. In the case of aqueous solutions and using a SB filled with an aqueous solution of KCl (c ≥ 3.5 mol L⁻¹), this assumption works quite satisfactorily and the double-dashed vertical bars are virtually reasonable. Secondary pH measurements are generally accepted and in widespread use. To some extent, this is also true for solvents other than water [3], [8].

3.3 Secondary pH measurement with two different solvents

As discussed above, the measurement of pH difference between different solvents requires solvation cells, that is, cells with different solvents in each half cell. Although possible in principle, solvation cells can be implemented, e.g., as cell II for secondary pH measurements (note that in this case, the solutions on the left and the right side of the cell contain H⁺ (solv) within different solvents S₁ and S₂). To the authors’ knowledge, such secondary pH measurements have been performed by only one team so far, and only for aqueous organic solvents [7], [25]. UnipHied is working on filling the gap.

The measurement of pH values and the measurement of transfer energies of the proton can be considered equivalent. Consider the transfer reaction of a proton from one solvent S₁ to another S₂, Eq. 15.^[7]

The assembly of cell II with suitable indicator electrodes permits the use of Eq. 15, and in combination with Eq. 14, one obtains Eq. 16; E _II is the potential difference in cell II according to the Stockholm conventions [26].

Using different solvents, the double-dashed vertical bars in cell II are not reasonable anymore. Thus, cell III better reflects reality.

The single-dashed vertical bars represent liquid–liquid junctions including the potential drop occurring at these boundaries – the liquid junction potentials (LJPs). Thus, Eq. 16 has to be modified (Eq. 17).

E _j1 and E _j2 are LJP1 and LJP2. For further discussion, it is helpful to first take a glimpse at the topic of LJPs.

3.4 The liquid junction potential LJP

The LJP results from a non-equilibrated state at the phase boundary, in which flows of particles (ions and solvent molecules) and forces (chemical and electric potential differences) are mutually dependent. Its description demands the implementation of non-equilibrium thermodynamic principles. Applying the Onsager reciprocal relations results in the generally accepted Eq. 18 [27].

t i r is the reduced transference numbers of all involved particles in the considered system [28]. Many attempts have been undertaken to split the sum of Eq. 18 into different parts to reach an equation that is simpler to interpret and calculate. Probably, the most elaborated model is given by Izutsu, who proposed the three-component breakdown (Eq. 19; here, i refers to ionic species), specifically in the context of differing solvents on either side of the liquid junction [29].

The first term on the right-hand side, called part A, accounts for differences in relative activities and mobilities of the involved ions. The second term, part B, originates from different Gibbs solvation energies of the ions. The third term, part C, is assigned to solvent S₁–solvent S₂ interactions. Without going into more detail, the rigorous calculation of E _j is almost never possible because for part B, Gibbs energies of transfer of the ions are needed, and part C is estimated rather than calculated.^[8]

3.5 The ionic liquid salt bridge ILSB

Because LJP calculations are difficult and inaccurate, it is reasonable to minimize or eliminate LJP contributions experimentally. Besides the above-mentioned KCl-SB, Kakiuchi presented a salt bridge containing an ionic liquid without any solvent and showed its applicability for all-aqueous cells, in which both LJPs cancel within the 95 % confidence interval of 0.5 mV [30]. Those ILSBs are robust and led to IL-filled glass electrodes [31], which are capable of pH measurement of, e.g., rainwater samples [32].

Recently, the use of an ideal ILSB in solvation cells was proposed [20]. The ideal salt bridge “would always generate the same diffusion potential, or, better still, no difference of potential, across the liquid junction, Bridge ┊ Soln. X no matter what the composition or pH of solution X might be” [5]. From the specified requisites for salt bridges [33],^[9] one can derive the specifications for an ideal ILSB: the IL is a strong (i.e., the degree of dissociation approaches unity) binary electrolyte – and this is of utmost importance – of which cations and anions exhibit equal transference numbers through the whole cell. Because no solvent is present, the ionic concentration is much higher than in the half cells, and as a pure substance, the activity coefficients of the IL-constituting ions are per definition unity. The absence of solvent is crucial in the context of solvation cells because no solvent–solvent interface is present in cells with the ILSB.

The potentials of cells IV with the redox system Ag⁺(solv)/Ag(s) turned out to be astonishingly stable against changes of the ionic strength of the half cells and of influx of the solvent into the salt bridge. In such cells, parts A of each LJPs cancel [34].

One ILSB, which essentially meets the above-mentioned requisites, is the salt bridge filled with the IL [N₂₂₂₅][NTf₂] (amyltriethylammonium bis(trifluoromethanesulfonyl)imide). The use of cell IV leads to Gibbs energies of transfer of the silver ion from one solvent S _k to another solvent S _i in quite good agreement with literature data.

3.6 Implementation of the pH_abs scale: secondary pH_abs measurement

The combination of cells III and IV leads to cell V and is intended to perform secondary pH_abs measurements between different solvent solutions.

This is possible because, with Eqs. 13 and 16 in mind, the potential of cell V E _V directly gives the p H a b s H 2 O value via Eq. 20: if S₁ is water, a H + , S 1 = 1 and d = 1 , thus referring to an aqueous solution containing solvated protons under standard conditions (Eq. 20 is written assuming Nernstian slopes for both Ind electrodes).

If additionally a H + , S 2  = 1, and thus the solvent S₂ contains solvated protons under standard conditions, Δ t r G ⊖ ( H + , H 2 O → S 2 ) can be obtained. The condition, of course, is that LJP1 and LJP2 of cell V cancel or are ineffective for some other reason.

However, even if this is not the case, cell V can support valuable services. The evidence of stability and reproducibility of the difference of E _j1 – E _j2 would lead to a consistent set of data. Using cell V, the intersolvental pH measurement is in principle traceable to the primary pH method with a quite good (if the LJPs are ineffective) or acceptable (if the LJPs allow consistent data) accuracy. Nevertheless, very recent results show that parts C of the ILSB LJPs are absent and indicate that parts B are small and the difference of the LJPs in cell IV is negligible [35].^[10] There are no indications why this should be different in cell V.

3.7 UnipHied tasks

The main task of UnipHied is the assembly of cell V such that a reliable and reproducible pH_abs measurement procedure can be defined. This includes a number of different issues:

• – the choice of suitable indicator electrodes and their characterization

• – the choice of a suitable ionic liquid

• – the choice of suitable buffer solutions

• – the assessment of LJPs contributions

• – the preparation of an uncertainty budget

• – the definition of calibration standards

The chosen solvents for these tasks are acetonitrile, ethanol, methanol, and some mixtures of them with water. The execution of these tasks involves interlaboratory comparison.