Elsevier

Electrochimica Acta

Volume 135, 20 July 2014, Pages 604-639
Electrochimica Acta

A pH centenary

https://doi.org/10.1016/j.electacta.2014.04.006Get rights and content

Abstract

The activity coefficients (and therefore the activities) of single ionic species are concepts tentatively introduced by G. N. Lewis, which he could not define thermodynamically because of electroneutrality requirements. Guggenheim subsequently gave their formal, pseudo-thermodynamic definitions, while warning that they were imaginary constructs without physical significance. Consequently, the hydrogen ionic activity, as a purely conceptual but immeasurable quantity, cannot serve as the basis of the pH, the globally accepted experimental measure of acidity.

Various aspects of this mismatch are described, based on the original literature sources as well as on experimental data used by their proponents. An especially pernicious but apparently widespread misconception is that the hydrogen ion concentration cannot be determined by thermodynamic means, despite the extensive work of Harned, Robinson and coworkers who showed otherwise.

A pathway is indicated to facilitate a smooth return to the original, thermodynamically sound definition of Sørensen in terms of the hydrogen ion concentration

We also describe a useful formalism based on the imaginary nature of single ionic activities.

Introduction

In a recent article [1] commemorating the centenary of the concept of pH in an official IUPAC news magazine, one of the co-authors of the latest IUPAC recommendation [2] for the definition of pH wrote that pH is “most likely the most measured chemical parameter and the one most people hear or talk about” but lamented that “In fact, beyond the simple process of measuring pH, there is poor understanding of the concept, the basis for its derivation, and limitations of its applicability.” Is this a problem that physical chemists, analytical chemists, and electrochemists should address by better teaching? In this communication, which reflects a tutorial lecture I presented at the 2012 ISE meeting in Prague, I will argue that this poor public understanding is not a matter of deficient teaching, but of a poorly defined, and therefore unteachable subject and is, in fact, a problem of IUPAC's own making.

This paper will briefly review the historical development of the concept of pH since 1909, when it was introduced as –log [H+], and why it was subsequently redefined as –log aH. It will consider different aspects of this development, including direct quotes from the writings of its major players, because those who develop and advocate new concepts tend to think about them deeply, and usually explain them clearly. In order to keep the notation as simple as possible, we will assume that all dimensional parameters are made properly dimensionless where necessary.

As in any historical overview, the facts and citations must (and will) be accurate and objective, but their interpretation is necessarily subjective. In this context, the tenor of this paper is perhaps best described by a quote from G. N. Lewis [3] who, upon introducing the concept of activity, wrote about the development of the Guldberg-Waage mass action law and its consequences: “As approximations to the truth they have been of the greatest service. But now that their utility has been demonstrated, the attention of a progressive science cannot rest upon their acknowledged triumphs, but must turn to the investigation of their inaccuracies and their limitations”. Here we will apply a similar standard to pH.

In the late nineteenth century, Guldberg and Waage [4] formulated the final form of their mass action law, and Arrhenius [5] introduced the idea of the permanent presence of ions in electrolyte solutions. Ostwald [6] combined these to analyze the behavior of weak acids and bases, and his extensive work convinced many skeptics at the time to accept these novel concepts.

Friedenthal [7] introduced the idea that the negative ten-based logarithm of the hydrogen ion concentration (here, regardless of the actual speciation of solvated protons, denoted by [H+] in general, and cH or mH more specifically) would make a useful acidity scale. Moreover, he treated acids and bases in the same way rather than by using two different scales, one for [H+] and the other for [OH]. Sørensen [8] adopted Friedenthal's scale, gave it the name pH, and provided an extensive description of its experimental determination, by both spectroscopic and potentiometric means.

At about the same time, G. N. Lewis [3] noticed that a class of electrolytes (then already called “strong”) did not seem to follow Ostwald's dilution law [6] and, by analogy to his earlier introduction of fugacity [9], introduced molecular activity coefficients, in order to allow for the non-ideal behavior of such solutions. It was soon realized that, in dilute electrolyte solutions, long-distance electrostatic interactions were mainly responsible for such deviations, and Lewis therefore also tried to introduce activities and corresponding activity coefficients for single ionic species. However, he could not find a satisfactory thermodynamic definition for the activity of single ionic species, in contrast to those for the neutral electrolytes that contained them. As he commented in his 1923 book with Randall [10]: “In developing our equations we have made use of the activity coefficient of the separate ions, and we have shown that, for a salt like potassium chloride, the activity coefficient is the geometric mean of the activity coefficients, γ+ of potassium ion, and γ of chloride ion. It remains for us to consider whether these separate values can be experimentally considered. This is a problem of much difficulty, and indeed we are far from any complete solution at the present time.” And a few pages later in that same book, they wrote [11]: “At the present time we must conclude that the determination of the absolute activity of the ions is an interesting problem, but one which is yet unsolved”.

Unfortunately, this issue was somewhat confused at that time, because the ionic concentrations of strong electrolytes were not obtained directly from the concentrations of the electrolytes that contributed them, but rather were computed from their conductances, on the assumption (then already disproved by Kohlrausch [12] in his square-root law for the conductivity of strong electrolytes) that the ionic mobilities would remain constant. But there was also a more fundamental problem, one that Lewis recognized but could not solve.

The concentration of a solute is its mass per volume of solution and is then called molarity (with (symbol c), or its mass per mass of solvent and then called molality (symbol m). These fundamental chemical parameters are linked directly to weighing, molecular mass, and the conservation of mass. Lewis defined activity as concentration times a correction factor, the activity coefficient. The latter is a strictly thermodynamic construction, which therefore exists only within a thermodynamic framework. If it cannot be defined within that context, it simply does not exist or, if one prefers the euphemism, it is a concept without physical significance, i.e., an illusory, imaginary quantity. As we will see below, that is the case with the activity coefficient of a single ionic species, and is the root cause of the difficulty mentioned by Camões [1], because IUPAC [2] now defines the pH, an experimental measure, in terms of the hydrogen ion activity.

In their 1924 review, Sørensen & Linderstrøm-Lang [13] cited the above Lewis comments, and concluded that: “… it would seem premature at present to introduce the activity principle in place of the concentration principle for measurements of hydrogen ions in biochemical investigations generally; …” and, a little further-on: “We therefore suggest, that in accordance with the practice of Bjerrum and his collaborators, the terms cH, pH and π0 should be allowed to retain their original significance, aH, paH and aπ0 being used to denote respectively hydrogen ion activity, exponent of hydrogen ion activity (paH = –log aH) and the π0 used in calculating the activity of hydrogen ions, …” which they then repeated in their “Proposals for Standardisation” as:

  • 1)

    In electrometric measurements of hydrogen ions, a sharp distinction should be made between concentration and activity of the hydrogen ions.

  • 2)

    In statements of concentrations of hydrogen ions, the terms cH, pH and π0 should be used, retaining the same significance as hitherto.

  • 3)

    In stating the activity of hydrogen ions, the terms aH, paH and aπ0 should be used, indicating respectively activity, exponent of activity, and the π0 used in calculating the activity of hydrogen ions.

These quotes are difficult to reconcile with IUPAC's 2002 use [2] of reference [13] as its sole justification for using the hydrogen activity rather than concentration to define pH !

In the meantime, Debye & Hückel [14] solved the theoretical problem of ionic interactions in sufficiently dilute electrolyte solutions by deriving their equationlogfi=zi2AI1+BaIwhere fi is the activity coefficient of ions i with valency zi, the ionic strength I is defined [15] as ½ Σ zi2 ci, a is the distance of closest approach of the selected, ‘central’ ion i to its (predominantly counter-) ions, and A and B are known macroscopic constants reflecting the solvent temperature, dielectric constant, etc. Note that the definition of ionic strength I is in terms of ionic concentrations ci. Likewise, the ionic activity coefficient predicted by the Debye & Hückel model requires the ionic concentration to define the ionic activity ai = fi ci. Some relevant details of the derivation are given in section 1.3.

At sufficiently low ionic strengths, so that the term B a I ≪ 1, (1.2.1) reduces to the Debye-Hückel limiting law, without any adjustable parameters:logfizi2AI

A somewhat less restrictive approximation, again for B ai I ≪ 1, approximates (1.2.1) tologfizi2AI(1BaI)=zi2AI+zi2ABaI

Shortly after Debye & Hückel published their theory, Hückel [16] added an empirical term to represent salting-in and salting-out effects, thereby modifying the expression for log fi tologfi=zi2AI1+BaI+bI

With its additional term, this extended Debye-Hückel expression can represent many mean electrolyte activities over a rather wide range of concentrations and ionic strengths.

The Debye-Hückel model does not define the activity coefficient of a single ionic species, because the parameter a specifies a distance of closest approach to its nearest ions, not an ionic radius or diameter. Debye & Hückel were quite emphatic regarding this latter aspect, and wrote [14] “Die Gröβe a miβt dann offenbar nicht den Ionenradius, sondern steht für eine Länge, welche einen Mittelwert bildet für den Abstand bis auf welchen die umgebenden, sowohl positiven, wie negativen Ionen an das ervorgehobene Ion herankommen können”, which can be translated as “Obviously, then, the quantity a does not measure the ionic radius. Instead it represents a length equal to the average distance to which the (positive as well as negative) surrounding ions can approach the central ion.”

Each of the expressions (1.2.1) through (1.2.4) suffices to define a mean activity coefficient f±=f+f or γ±=γ+γ for the aqueous solution of a single strong 1,1-electrolyte, as was the main focus of Debye & Hückel, because in that case the distance ai of closest approach must be the same for both anion and cation, and is then indeed the only relevant distance parameter. Meanwhile it is useful to keep in mind that the Debye-Hückel expression has been (and can be) tested only in solutions of electroneutral electrolytes, i.e., for macroscopically manipulable species that can (partially or fully) dissociate into electroneutral combinations of anions and cations.

The strongest experimental evidence for the validity of the Debye-Hückel approach comes from its limiting form (1.2.2). Here is how H.S. Frank, an outspoken critic of the full Debye-Hückel equation, started a 1959 chapter on this topic coauthored with Thompson [17]: “There can be little doubt that the theory of Debye and Hückel gives a complete and correct account of activity coefficients and heats of dilution in ionic solutions which are sufficiently dilute. The finality with which it answers questions dealing with these properties is, so to speak, guaranteed by the fact that it not only gives limiting laws for log f± and L¯2 which make these quantities linear functions of c½, with slopes which, in a sufficient number of instances, are experimentally confirmed to the highest accuracy with which experiments can be carried out, but also specifies these limiting slopes as functions of temperature, dielectric properties of the solvent, and valence type of the solute, without recourse to any empirical or adjustable parameters.” (L¯ is the relative partial molar heat content of a solute in the notation of [18].)

We will here sketch how Debye-Hückel derived their result, because we will subsequently need it in section 2.5. In principle, the ionic interaction is a many-body problem involving millions of ions (because Avogadro's number, about 6 × 1023 molecules per mole, is so huge), for which a closed-form solution is well-nigh impossible [19], [20]. Fortunately, we are not interested here in the behavior of individual ions, but only in their average, statistical behavior. Debye and Hückel found an elegant simplification that allowed them to reach an approximate solution for the latter, by dividing the solution artificially into a single, arbitrarily chosen “central” ion and its resulting “ionic atmosphere”, the statistical average of the surroundings of millions of other ions. There is nothing special about the central ion, and the final result must be (and is) applicable to anions and cations alike, but the above simplification reduced the number of “particles” to be considered statistically from trillions to two, the central ion and its ionic atmosphere, thereby making it mathematically tractable. This approach was an extension of Gouy's planar diffuse double layer model [21], [22] to a spherical geometry, but Debye & Hückel [14] deleted the effect of sphericity during the derivation, see (1.3.5), making the two treatments formally equivalent.

In a homogeneous electrolyte solution, the concentration profile of a smeared-out charge density around a central ion i is given by the Boltzmann distributioncj=cjexpzjFψiRTwhere cj is the “bulk” concentration of ions j in the solution, i.e., sufficiently far apart from the central ions i, and ψ i is the distance-dependent potential around a central ion. (The original paper uses ɛ/k instead of F/R, where ɛ is the electronic charge and k is Planck's constant. Here we have multiplied both ɛ and k by Avogadro's number N in order to avoid possible confusion with the dielectric permittivity ɛ and chemical rate constants k.)

The charge density ρ in the ionic atmosphere is thenρ=jcjzjF=cjFjzjexpzjFψiRTand electroneutrality of the central ion plus its surrounding ionic atmosphere requires thata4πr2ρdr=ziF/Nwhere a denotes the (statistically averaged) distance of closest approach of ions i and j (otherwise considered as point charges), and F/N is the electronic charge. Assuming a constant dielectric permittivity ɛ, the Poisson-Boltzmann equation for spherical symmetry now yieldsdivgradψ=1r2ddrr2dψdr=ρε=FεjcjzjexpzjFψiRT

Debye & Hückel expanded the exponential and then truncated the resulting series after its second term, thereby reducing it to a planar problem,1r2ddrr2dψdr=Fεjcjzj1zjFψiRT+12zjFψiRT2...Fεjcjzj1zjFψiRT=jcjzj2F2ψiεRT=F2ψiεRTjcjzj2=κ2ψi

when zji/RT << 1, where the first term of the series expansion, jcjzj, is zero because of macroscopic electroneutrality, and where we have used the abbreviationsκ2=F2jzj2cεRT=2F2IεRTandI=1/2jcjzj2

With the further abbreviation u = i we can rewrite (1.3.5) as du2/dr2 = κ2u, which finally yields the solutionψi=ziFexp[κaκr]4πεrN(1+κa)where the necessary integration is from a to infinity, a being the distance of closest approach of the centers of the ions j in the ionic atmosphere to the center of the central ion i. This result should be compared with the potential around an isolated charge zi F/N in an infinitely large dielectric medium of permittivity ɛ without other ions,ψi"=zjF4πεrN

The difference between (1.3.7) and (1.3.8),ψi'=ψiψi"=ziF4πεrNexp[κaκr](1+κa)1ziF4πεrNκ(1+κa)forramust therefore be the resulting lowering of the electrical energy of the central ion i,ziFψi'2Nzi2F28πεN2κ(1+κa)due to its interaction with its surrounding ionic atmosphere in which, on average, a counterion will be its closest neighbor. Identification with the ionic activity term (RT/N) ln fj then yieldslnfi=zi2F28πεNRTκ(1+κa)=zi2AI(1+BaI)where all constants have been incorporated in A, B. Equation (1.3.11) is the Debye-Hückel result (1.2.1). Numerical values for A, B for aqueous solutions are listed by, e.g., Bates [23] for concentrations expressed as either molarity or molality.

The early electrochemists had introduced a salt bridge to separate the two electrodes of an electrochemical cell in the hope that, by so doing, any change in the solution composition around one (‘indicator’) electrode would not be sensed by the second (‘reference’) electrode, and that any resulting liquid junction potential could either be made constant or negligibly small. While both Lewis and Sørensen had accepted that approach, it soon became clear that there was a problem with the liquid junction, a term we use here to denote an ion-permeable connection that is not fully selective to just one of the various ionic species present. As Harned [24] wrote: “We are thus confronted with the interesting complexity that it is not possible to compute liquid junction potentials without a knowledge of individual ionic activities, and it is not possible to determine individual ion activities without an exact knowledge of liquid junction potentials. For the solution of this difficult problem, it is necessary to go outside the domain of exact thermodynamics.”

Taylor [25] examined the thermodynamic basis of the liquid junction more closely, prompted by a then recent paper by Harned [26], and possibly also by the earlier statement of Lewis & Randall [27] that “It is to be hoped that in the future we may be spared the uncomfortable necessity of guessing at the values of liquid potentials, since it seems to be possible in nearly all if not all cases to obtain the data that are of thermodynamic value, solely by means of cells which contain no liquid junctions.”

Taylor's paper [25] started as follows: “A recent paper by Harned on the thermodynamic behavior of individual ions is representative of the persistent attempts which have been made to establish a basis for the determination of the free energies of ions by means of cells with transference, i.e., a cell containing a junction of two (different) electrolytes. The present analytical study leads to the conclusion that the EMF of the cell with transference is a function of free energies which are molecular only, that it can not possibly be manipulated to yield ionic free energies, and that the ionic free energy has not been thermodynamically defined. It is to be thought of rather as a purely mathematical device, which may indeed be employed safely with considerable freedom.”

Taylor expressed the liquid junction potential “entirely in terms of transference numbers and EMF's of cells without transference,” so that it can be described completely in terms of dynamic (mass transport) and static (equilibrium) equations for all species involved, “the solution of which together with the arbitrary boundary conditions in time and space completely determine the state of the system”. However, the practical problem of finding the pH of an unknown solution remains, and is well-nigh insoluble. As Taylor wrote: “In particular the determination of pH numbers by such a cell is not the simple thing it is sometimes assumed, for the cell EMF depends not only on the acid activity but also on the activity of every molecular species in the cell and mobility of every ion. If these are sufficiently well known to be allowed for, the acid activity is likely to be sufficiently well known not to need measurement.”

The most common approach is to make the liquid junction potential as small as possible through the use of dominating concentrations of near-equitransferent salts, such as KCl or NH4NO3 or, better yet, RbCl [28] or CsCl [29], so that any effects due to unknown sample constituents are effectively ‘swamped’. While this is admittedly a crude approach, it seems to be the best currently available method for potentiometric measurements. The present communication will not address this problem.

Taylor's paper was quite influential, because it led directly to two important developments: (1) Guggenheim's work [30], [31] defining the ionic free energy in terms of a pseudo-thermodynamic formalism (which led to the development of “irreversible” thermodynamics), and (2) studies by Harned and coworkers [32], [33] on the experimental determination of the hydrogen ion concentrations from emf measurements on cells without liquid junctions.

It is not necessary here to go into the details of Taylor's paper, which to my knowledge has never been contested; instead, it has recently been expanded by Malatesta [34]. It will suffice to quote Taylor [25] once more, because he is quite explicit: “The EMF of the cell with transference is thus a function of molecular free energies solely and is not a function of ionic free energies. It therefore can yield no information whatsoever concerning ionic free energies. In fact no thermodynamic information can be gained from a cell with transference which could not better be gained from a cell without transference. Conversely, within our present purview a knowledge of the ionic energies is never necessary for an account of the thermodynamics of electrolytes. Indeed, with the possible exceptions of single electrode potentials and rates of reaction there appears to be no occasion for the use of ionic free energies as experimental quantities but only as a mathematical device.” The italics in the above quote are those of Taylor. Note that, at the time Taylor wrote this, the existence of “possible exceptions” of single electrode potentials had already been disavowed by Gibbs [35], [36], and for reaction rates by the work of Brønsted [37] and Christiansen [38], to which we will return in sections 2.3 through 2.5.

The hydrogen ionic activity cannot be defined thermodynamically because of macroscopic electroneutrality. The thermodynamic definition of the chemical potential μi of species i at constant temperature T and pressure P, i.e., its partial molar Gibbs free energy G, isμi=GniT,P,njiwhere ni is the number of moles of species i. However, we cannot add, say, either Na+ or Clions in substantial (e.g., macroscopically weighable) quantities without charge-compensating counterions, because of the coulombic, strongly repulsive forces resulting from their net charge density. This definition is therefore not applicable to individual ionic species, even though it works fine for neutral electrolytes such as acids, bases, and salts that contain (or may even be composed entirely of) such ions. And because activity is a purely thermodynamic construction, when a supposedly thermodynamic quantity cannot be defined thermodynamically, it doesn’t exist, and hence cannot be measured.

Lewis was well aware of this problem, and wrote [39]: “An interesting type of solution is furnished by electrolytes dissolved in water or other dissociating solvent. In this case it is customary to assume the existence of molecular species, namely the ions, which cannot be added independently to the solution; for example, we have no practical means of adding a mol of sodium ions or a mol of chloride ions alone to a solution of sodium chloride in water. We have therefore no means of determining the partial molar volumes, or other partial molar quantities for such substances as sodium ion and chloride ion.”

As Harned & Owen [40] later explained: “In solutions of an electrolyte, electro-neutrality imposes the condition that the number of mols of the individual ionic species cannot be varied independently. We must be careful, therefore, to refer to ionic species as constituents of the solution rather than as components, so that the latter term may retain the precise meaning assigned to it by Gibbs. A component is an independently variable constituent of a solution. Thus, in the system NaCl and H2O there are two components whose chemical potential can be measured by the application of thermodynamics alone. They are, of course, NaCl and H2O. Although the ionic constituents Na+ and Cl are of fundamental importance in determining the behavior and properties of the system, their concentrations are not independent variables. Thermodynamics does not permit the evaluation of the chemical potentials, free energies, activities, etc., of the individual ionic species. In spite of this limitation it is advantageous to express a number of thermodynamic developments in terms of “hypothetical” ionic activities, with the strict understanding that only certain ionic activity products, or ratios, have any real physical significance.”

Guggenheim [30], [31] developed a pseudo-thermodynamic formalism for the single ionic activity and its activity coefficient, but explicitly warned that it was “a conception which has no physical significance”, echoing Taylor's warning [25] that the ionic free energy is “a purely mathematical device”. As Guggenheim [30] explained, “… it is clear that the interionic energy is stored in the whole assembly and any partition of it amongst the separate types of ions would be arbitrary. In the theory of Debye and Hückel, … which treats the ions as rigid spheres, this shows itself by the fact that the specific quantities, which distinguish solutions of the same electric type, are not the diameters of the individual ions, but the distances of closest approach of the various pairs of ions.” In section 1.2 we already quoted Debye & Hückel [14] stressing that same point.

Not all potentials (or potential differences, which it often means) are measurable. Gibbs, in an 1899 letter to Bancroft [35], [36] already wrote about single electrode potentials that “…the consideration of the difference of potential in electrolyte and electrode, involve the consideration of quantities of which we have no apparent means of physical measurement, while the difference of potential in pieces of metal of the same kind attached to the electrodes is exactly one of the things which we can and do measure.”

Or, as Guggenheim [30] put it: “The general principle referred to may be expressed as follows. ‘The electric potential difference between two points in different media can never be measured and has not yet been defined in terms of physical realities; it is therefore a conception which has no physical significance.’ The electrostatic potential difference between two points is admittedly defined in electrostatics, but this is the mathematical theory of an imaginary fluid ‘electricity,’ whose equilibrium and motion is determined entirely by the electric field. ‘Electricity’ of this kind does not exist, only electrons and ions have physical existence, and these differ fundamentally from the hypothetical fluid electricity in that the particles are at all times in movement relative to one another; their equilibrium is thermodynamic, not static.” And [31]: “… we therefore have no knowledge of the value of the electric potential between any pair of phases, nor therefore of the chemical potential, the activity or the activity coefficient of any individual ion.”

Guggenheim [30] defined the ionic activity ai as a function of an electrochemical (rather than chemical) potential, μ˜i, an entity that cannot be thermodynamically defined either, by introducing the purely formal expressionsμ˜i=μ˜i+RTlnai+ziFψ=μ˜i+RTlnci+RTlnfi+ziFψ=μ˜i+RTlnmi+RTlnγi+ziFψwhere the tilde ∼ identifies the electrochemical (rather than chemical) potential and its standard state, the latter denoted by the superscript o, and where c and m are the concentrations on a solution volume (mol L−1) or solvent weight (mol kg−1) scale respectively. However, not only the electrochemical potentials μ˜i and μ˜i, but also the ionic activity ai, the ionic activity coefficient γi or fi, and the solution potential ψ are in general immeasurable quantities. Unfortunately, Guggenheim did not label the latter as such, but we will do so here by also placing tildes on ai, fi, γi, and ψ in order to identify them as immeasurable, as Guggenheim had clearly stated they are. Here we will therefore writeμ˜i=μ˜i+RTlna˜i+ziFψ˜=μ˜i+RTlnci+RTlnf˜i+ziFψ˜=μ˜i+RTlnmi+RTlnγ˜i+ziFψ˜in order to emphasize their pseudo-thermodynamic status. Note that the chemical potential μ has no term in (nor need for) ψ because all terms in ziFψ˜ cancel for neutral electrolytes.

As Waser wrote [41]: “Thermodynamics is a phenomenological theory, concerning macroscopic quantities such as pressure, temperature, and volume. It is both its strength and weakness that the relationships based upon it are completely independent of any microscopic explanation of physical phenomena, …” or, in the words of Smith [42], thermodynamics “… is essentially a practical subject that interrelates quantities that can be measured in the laboratory …”.

Unfortunately, in terms of measurability, electroneutrality prevents us from determining ionic activity coefficients independently (i.e., individually) without making additional, arbitrary model assumptions. Only those combinations that do not violate electroneutrality can be measured. In sections 2.6 through 2.8 we will introduce a formalism that shows clearly which combinations of these immeasurable ionic activities and ionic activity coefficients can be expressed in terms of measurable, thermodynamic quantities. The electroneutrality (or charge balance) condition links the anionic and cationic activity coefficients, as in γ˜+γ˜=γ±2 for a single 1,1-electrolyte, where γ± is an example of such a directly measurable combination.

Lewis & Randall [43] had hoped that single ionic activity coefficients might be obtainable from electrochemical cells with liquid junctions, if only one could compute the liquid junction potential, and thereby separate the anionic and cathodic responses of the two electrodes. But because cells with or without liquid junctions do not require single ionic activities for their description, such quantities cannot be extracted from them either, which is why Taylor [25] considered the single ionic activity a “purely mathematical device”.

Incidentally, Malatesta [44], [45], [46] showed what hidden assumptions or outright mistakes underlie some recently published claims of having measured unbiased single ionic activity coefficients, and Zarubin [47] derived the uselessness of a common test purportedly validating such claims. Moreover, the literature is full of earlier, since abandoned attempts to specify how single ionic activities could be measured. But hope springs eternal.

Section snippets

Why measure paH?

If the hydrogen ionic activity is indeed a concept without physical significance, we must ask ourselves whether the hydrogen ionic activity is a useful parameter to define acidity in practical, experimental terms. (Debye & Hückel firmly established its usefulness for theoretical calculations.) To answer this question, we will first pose the related question: What would we learn if we could somehow measure the ionic activity of hydrogen? In other words: Why measure pH?

The Nernst equation defines

Measuring pH

Defining pH is a matter of finding the optimal quantity representing solution acidity. It boils down to three questions: which parameter do we need to know, which one can we compute, and which one can we measure. There are no conflicts between the parameter we need to know and that we can compute, since any quantity firmly based in chemical theory will serve both purposes. Whether we can measure it, and by what means, is a different matter. Here we will first take a brief look at the opinion of

Measuring mH

There is no question that mH is known for strong acids and bases, alone or in combination with other strong electrolytes. However, there seems to be a myth that there is no thermodynamic way to determine the concentration of hydrogen ions in solutions of weak acids, bases, or buffer mixtures from potentiometric measurements, and that some arbitrary, extra-thermodynamic quantity (adopted through some “convention”) is therefore needed to quantify acidity. This is undoubtedly the case for

What's wrong with a purely metrological definition of pH?

The development of modern science has often used somewhat arbitrarily chosen metrological definitions to facilitate the exchange of ideas, concepts, and measurement results. There is nothing “‘fundamental” about 1/40,000,000th of the diameter of the earth, idealized as a perfect sphere, as the unit measure of distance, or 1/24th of 1/60th of 1/60th of a solar day as the standard unit of time.

In China, 26 centuries ago, a common unit of length, the tehid, was apparently a decimal average of

Summary & conclusions

In this communication we have considered different aspects of the definition of pH as –log[H+], as originally proposed by Sørensen, and its more recent redefinition in terms of the hydrogen ion activity. These are summarized as follows, with the corresponding section numbers (shown in bold) where details can be found.

  • (1)

    The activity (or, more precisely, the corresponding activity coefficient) of a single ionic species is a supposedly thermodynamic concept, introduced by G. N. Lewis. However, it

Acknowledgments

This manuscript was started in the centenary year of Sørensen's definition of pH, and has since gone through many revisions in order to condense it to its essentials. It is a pleasure to acknowledge helpful suggestions by Jim Butler, Ron Christensen, Ron Fawcett, Lydia Hines, the late Normand Laurendeau, Jeff Nagle, Panos Nikitas, Keith Oldham, Roger Parsons, Carl Salter, and an anonymous reviewer. I am, of course, fully responsibility for its final form, and for the opinions expressed therein.

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