Loose building blocks in the edifice of electrochemistry in a historical perspective and their impact on the teaching

Abstract

There have been several misinterpretations even in the course of the establishment of the fundamental laws and ideas of electrochemistry. Despite the contemporary and further critics, these poor ideas (loose building blocks) have survived for more than a century, and unfortunately, those still appear in books and papers. The origin and the aftermath of several important problems will be analyzed in this paper. At the same time, the present thinking and recommendations will also be summarized. Moreover, some terms, which usually cause problems in teaching, will also be discussed.

Introduction

“doppelte Vorsicht zur Pflicht, damit nicht gleich in die Grundpfeiler des neuen Gebäudes lose Bausteine eingefügt werden.” wrote Lothar Meyer (1830 − 1895) in 1889 [1] (Fig. 1) as a part of his criticism on van 't Hoff’s explanation of the origin of the osmotic pressure [2].

Fig. 1

From the introduction of the paper of Lothar Meyer (1889) [1]

According to van 't Hoff, the osmotic pressure is due to the dissolved species similar to the pressure of gases which is in connection with the gas particles [2]. Meyer rightly pointed out that the dissolved species do not exert any such pressure (in fact, there is no such thing as the pressure of a solution), and the phenomenon called osmotic pressure is a pressure that must be applied to the solution to bring it into a certain equilibrium condition. It is worth to cite Meyer’s comments: “The osmotic pressure is not the pressure of dissolved material but that of the solvent… or in general the pressure of the substance for which the wall is permeable. It was considered in this way until Herr van ‘t Hoff declared his diagonally opposite opinion.” [1].

The similarity between the equation derived by van 't Hoff for the osmotic pressure and gas law fascinated not only Jacobus H. van 't Hoff (1852 − 1911) but also Wilhelm Ostwald (1853 − 1932), Svante Arrhenius (1859 − 1927), and Walther Hermann Nernst (1864 − 1941). The critique by Meyer and others [(Johannes J. van Laar (1860 − 1938), Max Planck (1858 − 1944)], was refused and the view of van 't Hoff triumphed because of the authority of the abovementioned scientists, and the chemists liked the beauty of the simple formula. The detailed story of the osmotic pressure and its consequences including in electrochemistry can be found in Ref. [3].

The most important point that we have to emphasize is that the van 't Hoff’s equation is valid in dilute solutions only. This is true for many theories and the respective equations derived in physical chemistry (e.g., freezing point depression), and also in electrochemistry (Debye − Hückel theory, Gouy − Chapman theory on double layer, etc.) where already in the derivation—taking into account the low concentration of the solute species—simplifications had been made. The misunderstanding of the real nature of the osmotic pressure has caused problems in the field of electrolyte; however, the greatest effect and its most important and long-lasting influence are conspicuous in connection with the derivation of the Nernst equation and in its aftermath [3, 4]; see later.

It is of utmost importance to use well-established, unambiguous concepts in education. However, it causes serious problems if even in the scientific books and papers, in the textbooks of higher education, one may find misleading, outdated definitions, false calculations neglecting the limits of the usage of a given equation, etc. There are different reasons behind such practice. First, even electrochemists could not agree in the meaning of some given terms, and therefore, the International Union of Pure and Applied Chemistry (IUPAC) could not recommend a proper terminology. It is the very reason that IUPAC recommendations have been changing from time to time. Second, the teaching of electrochemistry generally occurs within physical chemistry curricula at the universities, the lecturer is not an expert in electrochemistry, the material discussed is rather limited, the curriculum is behind the modern developments with decades in this field, and old textbooks are suggested. It would be desirable to present the fundamental principles of electrochemistry both for chemistry students, and also for future teachers in chemistry, biologists, pharmacists, electrical and materials engineers, etc., as well as to teach electrochemistry in an advanced level by a professor in the subject for students specialized in electrochemistry, electroanalytical chemistry, electrochemical engineering, and corrosion and materials science. It should be noted that beside the general knowledge in electrochemical thermodynamics and kinetics, it is of importance to teach the special knowledge in a given field, e.g., new power sources like Li-ion batteries and fuel cells for future engineers, electrochemical sensors including biosensors for biologists, physicians, and electroanalytical chemists. Beside all students and teachers, and also for the general public, it is necessary to popularize the role of electrochemistry to achieve more time for its education. It can draw the attention to the importance of batteries in the transportation, pacemakers, spaceships, and their further developments for these and other purposes.

In this paper, it is intended to deal with several basic concepts including their historical background hoping that it would help the professors, students, chemistry teachers, and specialists in their teaching and research work.

Electrode: a casually defined term

The word “electrode” was coined by Michael Faraday (1791 − 1867) from Greek words ἤλεκτρον (meaning amber) and ỏdỏς a way.

“In place of term pole, I propose using that of electrode, and I mean thereby that substance, or rather surface, whether of air, water, metal, or any other body which bounds the extent of the decomposing matter in the direction of the electric current. The surfaces at which, according to common phraseology, the electric current enters and leaves a decomposing body are most important places of action, and require to be distinguished apart from the poles, with which they are mostly, and the electrodes, with which they are always, in contact…” [5] (Fig. 2).

Fig. 2

The terms that we use today was born. Faraday M (1834) [5]

Setting aside the fact that the expression poles is still used, e.g., in the car repair shops by the workmen, albeit Faraday has proven almost 200 years ago that the laws of electrostatics are not valid in the case of batteries, and therefore, he created a new word to be used; the definition of the electrode is still a problematic issue. (It is not unique that we use expressions which reflect old and outdated theory, e.g., “sun rises”).

If we ask a chemist or a chemistry student, they would be taken aback and they would say that everybody knows what the word “electrode” means. Is it indeed the case? Certainly not. We will deal with the usage of this term in electrochemistry, and neglect its appearance in technologies where the construction and function of electrodes are very different, e.g., people working with welding also use the term. The definition of the electrode in electrochemistry and electroanalytical chemistry is a tricky issue.

There are currently two usages for the term electrode [6, 7]. In the laboratory practice, it is just an electron conductor connected to an external lead, for instance, a platinum plate or disc with a platinum wire connection. In this case, we speak of a platinum electrode. We may use it for measuring the redox potential of a redox couple dissolved in an electrolyte or to produce hydrogen evolution, etc. However, because we can measure only the potential difference between two electrodes which is related to the chemical reaction occurring in an electrochemical cell (cell reaction), we need a more exact definition that allows thermodynamic calculations, i.e., to establish the relationships between the Gibbs energy of the (cell) reaction (ΔG) and the potential of the cell reaction (E_cell), to derive the dependence of E_cell on the composition.

Therefore, it is better to consider the half-cell between one electron conductor (e.g., a metal) and at least one ionic conductor. The half-cell, i.e., the electrode, may have a metal and an electrolyte solution containing its own ions. However, the situation is more complicated. Even when a pure metal is immersed into an electrolyte solution, its surface may be covered, e.g., with an oxide layer. Typical examples are magnesium or aluminum; however, even the surface of platinum is covered with oxides when it is stored in air or at higher positive potentials. Furthermore, the electronic conductor may be also an alloy (e.g., an amalgam), carbon (e.g., graphite, glassy carbon) boron-doped diamond, a semiconductor (e.g., a metal oxide, metal salt, doped silicon, germanium alloys), and metal oxides (e.g., iridium dioxide, titanium covered with ruthenium dioxide). Beside the spontaneously formed surface layers, the surface of metals or other substances is often modified on purpose to obtain electrodes for special functions. When a metal surface was covered by an electrochemically active polymer layer, we speak of polymer-modified (polymer film) electrodes which have become an important class of electrodes. A polymer film electrode can be defined as an electrochemical systems in which at least three phases are contacted successively in such a way that between a first-order conductor (usually a metal) and a second-order conductor (usually an electrolyte solution) is an electrochemically active polymer layer: this polymer is, in general, a mixed (electronic and ionic) conductor. A satisfactory definition, which includes the factors and problems mentioned above, may be as follows.

The electrode consists of two or more electrically conducting phases switched in series between which charge carriers (ions or electrons) can be exchanged, one of the terminal phases being an electron conductor and the other an electrolyte.

The electrode can be schematically denoted by these two terminal phases, e.g., Cu(s)│CuSO₄(aq), disregarding all other phases that may be interposed. However, in certain cases, more phases are displayed, e.g., Ag(s)│AgCl(s)│KCl (aq) or Pt(s)│Polyaniline (s)│H₂SO₄ (aq) since the consideration of those phases are essential regarding the equilibria and the thermodynamic description. The electrodes can be classified in several ways. The electrode on which reduction (transfer or electrons from the metal to the dissolved species) occurs is called cathode, and on which oxidation takes place is called anode. The positive electrode is the cathode in a galvanic cell and the anode is in an electrolytic cell. In a galvanic cell, the negative electrode is the anode, while in an electrolytic cell, it is the cathode. According to the nature of species participating in electrochemical equilibria and the realization of the equilibria, we may speak of electrodes of the first kind, electrodes of the second kind, electrodes of the third kind, redox electrodes, and membrane electrodes. When more than one electrode reactions take place simultaneously at the interface, the electrode is a mixed electrode. Another important distinction is based on whether charged species cross the interface or not. In the former case when the charge transfer is infinitely fast, the electrode is called ideally non-polarizable electrode. When no charge transfer occurs through the interface and the current (charge) that can be measured merely contributes to the establishment of the electrical double layer, the term is ideally polarizable electrode. There are also names which express the function of the electrode and refer to the whole construction including mechanical parts of the electrode, e.g., dropping mercury or hanging mercury drop electrodes, rotating disk electrode, combined glass electrode, optically transparent electrodes, and photoelectrodes. Electrodes with different properties can be constructed by using the same element or compound for the solid phase, however, in various crystal forms, morphology, surface structure, with or without additives. For instance, carbon electrodes are made of various materials, such as graphite of spectral purity, graphite powder with liquid or solid binders, glassy carbon, carbon fibers, highly oriented pyrolytic graphite, paraffin impregnate graphite (PIGE), or diamond. Platinum might be polycrystalline or in forms of different single crystals. The electrode geometry plays also an important role. We may classify electrodes according to their forms such as inlaid disk, sphere, cylinder, sheet, net, spiral wire, sponge, inlaid ring, inlaid plate, and ring-disk. The electrode size is also an important factor, and consequently macroelectrodes, microelectrodes, and ultramicroelectrodes are distinguished. The functional grouping is as follows. The electrode which is under study is called working electrode in voltammetry or indicator electrode in potentiometry. The electrode in which the potential is practically constant and used to make comparison of electrode potentials, i.e., to define the value of the potential of the electrode on the scale based on standard hydrogen electrode, is called a reference electrode. The electrode that serves to maintain the current in the circuit formed with the working electrode in voltammetric experiments in three-compartment cells is the auxiliary or counter electrode.

Electrode potential

“Depth of understanding makes the mind good.”—Lao-Tzu (Laozi), Tao Te Ching. Chapter 16, around 400 BC.

The origin and the place of the electric potential in galvanic cells have been in the center of discussion for almost 150 years. It has been settled, more or less, but most likely it is not the end of the story. The very productive covenant of electricity and chemistry has started with Alessandro Volta (1745–1827) and his pile. He constructed a device just using different metals and a liquid conductor, and which supplied electric current continuously [8]. The two different metals (e.g., silver and copper or silver and zinc) were in direct contact, and then a piece of strata of pasteboard or skin soaked with salt solution, and he used 30–60 of such elements in series. Volta gave the name of the artificial electric organ in order to distinguish it from the natural electric organ of the torpedo or electric eel, etc. In fact, Volta was lucky that he made a series from these elements because in this way, metal–electrolyte–metal cells were formed. A single cell would not work. It should also be mentioned that it was not a primary galvanic cell but presumably a Zn–air fuel cell according to the present classification or depending on the electrolyte used the dissolution of zinc and evolution of hydrogen occurred. According to Volta, the potential difference is caused by the direct contacts of the metals (contact potential). This thinking is based on the theory of electrostatics, which had been already fairly developed; e.g., the Coulomb law [9, 10], which was published by Charles Augustin de Coulomb (1736–1806) in 1785 but was discovered by Henry Cavendish (1731–1810) earlier [11], was already established.

A review of the story of the past of the static electricity from the ancient times until Volta will be too long herein, but it is worth to mention William Gilbert (1544?–1603) who studied electric and magnetic phenomena, and from the name of amber (in Greek ηλεκτρον, electron) [12] created the new Latin word electricus (like amber). Based on it, Thomas Browne (1605–1682) introduced the term “electricity.”

In the electrostatics based on the fluid theory of electricity, the nature of charge was not of interest. It was not considered that the charge always belongs to a concrete particle (and it is still not considered by physicists in electrostatics). It was Faraday who first concluded that “… there is a certain absolute quantity of the electric power associated with each atom of matter.” However, it became evident that in a galvanic cell chemical changes take place which cannot be explained within the framework of electrostatic theories. This double nature of electrochemistry is the essential issue which still causes problems. This problem became the focus of the debate when Josiah Willard Gibbs (1839–1903) [13] and Walther Nernst (1864–1941) [14] elaborated the thermodynamic theory of galvanic cells.

Wilhelm Ostwald supported the thermodynamic approach, and he declared that the place of the formation of the electromotive force is the same where the chemical processes occur, i.e., the interface between the metal and the electrolyte [15]. He expressed a fulminatory opinion about the contact potential theory of Volta (he called it “a mistake of a genius”), and its effect of on the development of electrochemistry.

We have already presented that the “mistake of a genius” has happened also in other cases (the story of van 't Hoff and the osmotic pressure), and we will also discuss the false osmotic theory of Nernst in this respect. For that matter, Nernst’s theory was strongly supported by Ostwald.

It is impossible to determine the electrode potential experimentally. One can measure the Galvani potential difference, i.e., the difference of the inner potentials in the two phases under contact (Φ^β – Φ^α) between two phases of identical composition, only. Even nowadays in several publications, this difference is used as the electrode potential. However, in this case, the chemical contribution is neglected which is a capital mistake in electrochemistry.

Already Gibbs has pointed out this problem: “Again, the consideration of the electrical potential in the electrolyte and especially 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.” [13]. Edward Armand Guggenheim (1901–1970), who further developed and popularized Gibbs’ thermodynamics, added [16]: “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 or 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.” [16, 17].

It is very useful to describe both the chemical and electrostatic work functions in one function that is called electrochemical potential, $\widetilde{{\mu }}_{i}$ which was introduced by Guggenheim [16, 17]. The electrochemical potential of ion B in a phase is the partial molar Gibbs energy of the ion. The name is somewhat misleading because the term “potential” in electrochemistry was mostly used for electric potential and expressed in Volt unit; however, the electrochemical potential of a species is a work (energy) and its unit is J mol^–1. Nevertheless, students should know the meaning of the potential functions from thermodynamics. For a charged particle B in the phase β, the electrochemical potential can be expressed by the sum of the chemical work and the electrostatic work as follows:

$$\widetilde{\mu }_{\mathrm B}^{\upbeta}\text{ = }{\mu }_{\mathrm B}^{\upbeta }+{z}_{\mathrm B}\;{\mathit{F}}\;{{\phi }}^{\upbeta }$$

(1)

where μ_B is the chemical potential of species B, z_B is the charge number of an ion or the electron, F is the Faraday constant, and Φ^β is the inner potential of phase β.

The first term is the chemical work depending on the chemical composition and the second term is the electrostatic work depending on the charge of the species and all electrostatic potential of the phase, respectively. If z_B = 0 (neutral species), only chemical work occurs. The place of the contributions is different, since all the excess charge is situated on the surface of a conducting body, and the charged species (an electron or an ion) interact electrostatically here, while chemical interactions (reactions) occur within a phase [6, 18,19,20,21,22].

The electrochemical equilibrium between two contacting phases can be expressed as follows (Fig. 3).

Fig. 3

The first page of the paper by Guggenheim EA (1930) [17]

$$\widetilde{\mu }_{\mathrm{B}}^{\alpha }\text{ = }{\widetilde{\mu }}_{\mathrm{B}}^{\upbeta }$$

(2)

When the composition of the two phases are the same (e.g., copper), we can calculate/measure the work which is needed to transfer the charge (electron in this case) from phase α to phase β since in this case no chemical interactions (reactions) occur (the difference of the chemical potentials is zero); the work is entirely electric work. It is the very reason why we need the same metal at each end of the galvanic cell to determine the electromotive force (emf) or any reliable cell voltage. This requirement practically always prevails because the cell and the voltmeter are connected with copper wires.

$${\widetilde{\mu }}_{\mathrm{B}}^{\upbeta }-\widetilde{\mu }_{\mathrm{B}}^{\alpha }=\left({\mu }_{\mathrm{B}}^{\upbeta }-\mu _{\mathrm{B}}^{\alpha }\right)+{z}_{\mathrm{B}}F\left({{\phi}}^{\upbeta }-{{\phi}}^{\alpha }\right)={z}_{\mathrm{B}}F\left({{\phi}}^{\upbeta }-{{\phi}}^{\alpha }\right)$$

(3)

In the case when α = β = Cu and B = e⁻, i.e., z_B =  − 1, therefore − F (Φ^β – Φ^α) =  − FE.

The present concept of the electrode potential

The electrode potential, E (SI unit is V), is the electric potential difference of an electrochemical cell (including the condition when current flows through the cell), and the left-hand electrode in the diagram of the galvanic cell (cell diagram) [6, 21, 23] is at virtual equilibrium, and hence acting as a reference electrode.

In electrolysis cells, the potential of the working electrode is compared to a reference electrode which is practically at equilibrium. An ideally non-polarizable electrode or an electrode which behavior is close to it may serve as a reference electrode. The choice and the construction of the reference electrode depend on the experimental or technical conditions, among others, on the current applied, the nature and composition of the electrolyte (e.g., aqueous solution, non-aqueous solution, melts), and temperature. The liquid junction potential is assumed to be eliminated. When the right-hand electrode is also at equilibrium, the measured potential is the equilibrium electrode potential.

The basic problem is that the absolute value of the electrode potential (E) cannot be measured [24]. We can measure the electric potential difference of an electrochemical cell, only. It is the very reason that the activity of a single ion cannot be determined (see pH). E is always reported relative to the potential of some reference electrode, e.g., that of a standard hydrogen electrode (SHE). However, any electrode at equilibrium can serve as a reference electrode. In this case, the exact properties of this electrode should be given.

Limitations of the use of the Nernst equation

“Nullius in verba” (from Epistles of Horace, Book I, 20 BC).

The Latin expression can be translated as “Take nobody’s word for it.” It is the motto of the Royal Society (formally The Royal Society of London for Improving Natural Knowledge) that was to signify the fellows’ determination to withstand the domination of authority and to verify all statements by an appeal to facts determined by experiments.

The Nernst equation describes the dependence of the electrode potential as a function of the composition of the electrode, usually the composition of the liquid phase.

Although the model used by Nernst was not appropriate (see below), the Nernst equation—albeit in a modified form and with a different interpretation—is still one of the fundamental equations of electrochemistry. In honor of Nernst, the equilibrium established at an electrode, i.e., between the two contacting phases of the electrode or at least at the interface (interfacial region), is called Nernst equilibrium. In certain cases (electrochemically reversible electrode processes), the Nernst equation can be applied also when current flows. If this situation prevails, we speak of reversible or Nernstian reaction (reversible or Nernstian system) [25].

Walther Nernst, while working at Wilhelm Ostwald’s laboratory in Leipzig [26], derived his famous equation [14] by using a thermodynamic equilibrium for an electrode, and van 't Hoff’s osmotic theory [2] for elucidation of the establishment of the equilibrium (Fig. 4).

Fig. 4

From the famous paper of Nernst W (1889) Die elektromotorische Wirksamkeit der Jonen. Zeitschrift für Physikalische Chemie [14]

According to Nernst, the reason for the dissolution of a metal or the deposition of metal ions is a pressure difference. Two types of pressure exist, i.e., the osmotic pressure (p) exerted by the dissolved molecules (ions) on the metal surface (which acts as a semipermeable membrane) and an opposite one, the “Lösungstension” (“dissolution tension”) of the metal (P), which drives the ions into the solution. An equilibrium will be established when p = P. The equation for the metal-electrolyte potential difference (E) in Nernst’s original work [14] was given as follows:

$$E={p}_{\text{o}}\;\mathrm{ln}\,\frac{P}{p}$$

(4)

where p_o is a constant of the Boyle-Mariotte equation (p_o = pV). In fact, dE was considered a work, i.e., dE =  − V dp, where V is the volume of the cation under the pressure p.

RT/F is a conversion factor of osmotic energy to electric energy. In 1898, Rudolf Peters (1869–1937) applied a modified form of this equation for redox systems [27].

It may be surprising that based on a false model, a correct relationship can be derived. However, it is not unique in the history of science. It is enough to recall that the nature of heat or electricity has been interpreted by the caloric theory and by fluid theories, respectively. Nevertheless, reliable experiments have been executed, and the mathematical description based on the results was correct. The importance of the achievements of Joseph Black, Sadi Carnot, Jean Fourier or William Gilbert, and Henry Cavendish cannot be denied. In our present case, the model of Nernst is a kinetic (mechanistic) one that explains how the equilibrium comes about. However, his treatment that leads to the equation is purely an equilibrium thermodynamic (energetics) one which does not rely on the time-dependent events.

In 1921, Nernst still insisted on his inappropriate model, and gave the following equation:

$$E=\frac{R\;T}{n}\mathrm{ln}\;\frac{C}{c}$$

(5)

where R is the gas constant, n is the chemical valency of the ion concerned, C is a constant specific to the electrode, and c is the ion concentration [28].

“So there arose in 1889 the osmotic theory of galvanic current generation, which has not been seriously challenged since it was put forward more than thirty years ago and has undergone no appreciable elaboration since its acceptance, surely a clear sign that it has so far satisfied scientific needs.” [28].

Albeit the original osmotic model used by Nernst is wrong, it is still the one of the mostly used equation in electrochemistry and electroanalytical chemistry.

It is surprising that Nernst entirely neglected not only the critics but especially the discovery of electron by Joseph John Thomson (1856–1940) as well as the determination of its charge/mass ratio in 1897 [29, 30].

“This constant value, when we measure e/m in the c.g.s. system of magnetic units, is equal to about 1.7 × 10⁷. If we compare this with the value of the ratio of the mass to the charge of electricity carried by any system previously known, we find that it is of quite a different order of magnitude. Before the cathode rays were investigated, the charged atom of hydrogen met with in the electrolysis of liquids was the system which had the greatest known value of e/m, and in this case the value is only 10⁴, hence for the corpuscle in the cathode rays the value of e/m is 1,700 times the value of the corresponding quantity for the charged hydrogen atom.” [30]. It is interesting to note that until the discovery of electron, H⁺ ion whose charge and mass had been determined by using electrolysis and the Faraday law was the “record holder” indicating the very important role of electrochemistry in the nineteenth century.

In 1920, a paper of György Hevesy (1885–1966) and László Zechmeister (1889–1972) appeared in that they described the results of their radiotracer experiments that proved the existence of heterogeneous and homogeneous electron transfer reactions [31, 32]. They studied isotope exchange reactions. They used ThB, i.e., ²¹²Pb for labeling. While investigating the Pb(II) acetate–Pb(IV) acetate systems, they concluded that “electrons of ions can directly transfer to isomer ions and also to the electrode and vice versa.” [31, 32]. It was a pivotal contribution to the understanding of the electron transfer processes in both homogeneous and heterogeneous systems since at that time only ionic charge transfer was considered as described in the theory of Nernst. Albeit Nernst in his Nobel lecture [28] mostly dealt with his achievement in electrochemistry, he said nothing about the possible electron transfer which opened up new vistas especially in the case of redox systems, and he did not considered it further on, either.

Unfortunately, the problem of the osmotic model and the early interpretation of the Nernst equation survived during the twentieth century and one still can find different misinterpretations even in recent publications. We show some examples for this issue.

It is not frequent but there are papers which still want to use osmotic pressure for the explanation of the electrode potential: “There is a close correspondence between osmotic pressure and electrochemical potential that might help students with better understanding both.” [33].

Nernst and his followers have published several papers where they calculated very small concentrations of ions by using the Nernst equation and claimed that those determine the actual electrode potential. It is especially so in the case of sparingly soluble salts and complex salts, electrodes of second order. It is strange that Nernst—who was an excellent physical chemist especially in thermodynamics—forgot that any thermodynamic law is valid only for systems which contain a large number of particles (n), and the error (deviation) is proportional to √n.

“Since the concentration of silvernitrate solution is 0.1 normal, log p₁ = –1, consequently: p₂ = 10^–20.7, i.e. 108 g silver exists in the form of Ag + ions in 1020.7 L cyanopotassiumsilver solution.” [34].

“However, the concentration of mercury (I) ions in solution is equal to 10^–36 mol dm⁻³, that of potassium in the amalgam equals to 10^–45 mol dm⁻³ and the partial pressure of chlorine is 10^–57 Pa.” [35].

Even in the beginning of twentieth century, several scientists were cautious concerning the possibility of the potential determining role of ions which are present in extreme low concentrations [36, 37]. Haber proposed the reaction: Ag⁺  + 3 CN⁻ ⇄  Ag (CN)₃²⁻  + e⁻ —as we think also today—instead of Ag⁺  + e⁻ ⇄ Ag as potential-determining step.

On the basis of Nernst theory, the ideally polarizable electrodes cannot be interpreted, either. They assume that certain dissolution always should occur, and pure water does not exist since when a metal is immersed in pure water, the potential difference between the two phases would be infinite [38] (Fig. 5).

Fig. 5

Pure water in contact with a metal does not exist—Brandenburg H (1893) [38]

For a long time—even after the discovery of electrons—the interpretation of redox electrodes was a serious challenge. There is no common component of the platinum metal immersed and Fe³⁺/Fe²⁺ system which is in the contacting aqueous solution. Therefore, there are not two opposite tensions which are required by the theory of Nernst. However, an “explanation” was found [37]. In aqueous solution which contains a reducing and an oxidizing component, hydrogen and oxygen are formed at the respective electrodes (metals) which are in equilibrium with the redox couple. Consequently, we have a hydrogen electrode whose potential is determined by the pressure of H₂ and the concentration of the H⁺ ions. It is true that in some cases hydrogen evolution occurs during the dissolution of strong reducing agent, e.g., Cr(II) salts; however, in a solution Fe³⁺ and Fe²⁺, there is no such pressure of H₂ that could be potential determining. There is a problem also with the ideally polarizable electrodes.

The Nernst theory which emphasizes the equilibrium at the electrodes has hindered the development of the kinetics of electrode processes.

Bockris JO’M (1923–2013), one of the eminent electrochemists of the twentieth century, recalled the state of electrochemistry in 1949.

“The atmosphere and background of electrochemistry at this time was dominated by the dead hand of Nernst.” “Erdey-Grúz and Volmer, whom I see as the fathers of electrode kinetics and Frumkin, were seldom mentioned.” Overpotential was regarded as a kind of disease suffered by gas electrodes. There must clearly be something wrong causing “the deviation.” Perhaps it was gas bubbles, blocking the surface? Electrochemistry in 1949 was an old and breaking science. In Europe, it was dominated by industry and thoughts, e.g., of aluminum. In England and America, the textbook and reading were dominated by solution theory. Books of the time (e.g., that of Glasstone) contained mostly chapters about pH, conductivity, equilibrium, and solutions, with one chapter on “overvoltage” (“The excess pressure to form a bubble”). Who were the leading names in 1949? The most mentioned was that of Nernst. Wagner and Traud were, of course, often mentioned by Pourbaix’s presence because of his interest in corrosion. Debye and Hückel were names mentioned much in universities when one talked about electrochemistry. The most frequent exam question related to activity coefficients. “Erdey-Gruz and Volmer, whom I see as the fathers of electrode kinetics, and Frumkin, were seldom mentioned.” [39].

It is worth to stress again that according to the Nernstian perception, there is always a charge and mass transport which establishes the potential and an equilibrium process always exists. It is also true during electrolysis. Therefore, in the case of the electrolysis of water at 0.5 V, overvoltage 10¹⁷ atm hydrogen pressure exists, while at 0.1 V overpotential, the pressure of the oxygen is 10⁷ atm. We may think that this concept and the use of the Nernst equation is at non-equilibrium conditions slowly died out when the theory of electrode processes gained ground after 1930 [40]. Unfortunately, it was/is not the case. The most deterrent example is the cold fusion story in 1989 when the authors calculated astronomical pressures during electrolysis.

Fleischmann and Pons executed electrolysis of heavy water by using of a palladium (Pd) electrode [41]. 0.1 M LiOD in 99.5% D₂O + 0.5% H₂O solutions was used and the electrode potentials were measured with respect to a Pd-D reference electrode charged to the α-β-phase equilibrium. Anomalous heat (“excess heat”) was reported which was assigned to nuclear processes involving the fusion of deuterium atoms.

They considered the following reaction steps at the cathode:

• “D₂O + e^– → D_ads + OD^–

• D_ads + D₂O + e^– → D₂ + OD^–

• D_ads → D_lattice

  D_ads + D_ads → D₂

  and 4OD^– → D₂O + O₂ + 4 e^– at the anode.

They assumed that “at negative overpotentials on the outgoing interface of palladium membrane electrodes for hydrogen discharge at the ingoing interface [determined by the balance of all the steps (i) to (iv)] demonstrates that the chemical potential can be raised to high values. Our own experiments with palladium diffusion tubes indicate that values as high as 0.8 eV can be achieved readily (values as high as 2 eV may be achievable). The astronomical magnitude of this value can be appreciated readily: attempts to attain this level via the compression of D₂ [step (iv)] would require pressures in excess of 10²⁶ atm. In spite of this high compression, D₂ is not formed; i.e. the s-character of the electron density around the nuclei is very low and the electrons form part of the band structure of the overall system. A feature which is of special interest and which prompted the present investigation, is the very high H/D separation factor for absorbed hydrogen and deuterium (see Figs. 4 and 6 of ref. [2]). This can be explained only if the H⁺ and D⁺ in the lattice behave as classical oscillators (possibly as delocalised species) i.e. they must be in very shallow potential wells. In view of the very high compression and mobility of the dissolved species there must therefore be a.significant number of close collisions and one can pose the question: would nuclear fusion of D⁺ such as

• ²D + ²D → ³ T(1.01 MeV) + ¹H(3.02 MeV)

or

• ²D + ²D → ³He(0.82 MeV) + n(2.45 MeV)be feasible under these conditions?”

Such experiments have been carrying out for almost two decades at all around the world, however, less and less intensively. The production of radioactive elements, the “excess heat” etc., has been impossible to reproduce, and the explanations have been failed. Any scientific support to these experiments is finished.

Let me deal with only one aspect of the cold fusion experiment described above. It is the astronomical hydrogen pressure at an overpotential of − 0.8 V. This calculation is based on the Nernst equation as described above. As we had mentioned, the Nernst equation is only valid under equilibrium conditions. Not talking about the other problems of this and following papers on cold fusion, for an electrochemist, it is evident that this calculation is certainly wrong. Furthermore, in the case of such open cell which was used for these experiments, one cannot produce such an elevated pressure, anyway. The anode and the cathode compartments were not separated. Therefore, a reaction of 2 D₂ + O₂ → 2 D₂O with high heat evolution is possible.

I remind you “the mistake of a genius.” In this case, we would speak of “mistake of leading scientists.” Not only Fleischmann and Pons disseminated their false results but later even Bockris intensively entered the research of the cold fusion [42].

It is worth to make a  remark about the usefulness of the study of history of chemistry. Already in 1926, such experiment had been carried out [43]. Fritz Paneth (1887–1958) and Kurt Peters (1897–1978) observed helium production during the electrolysis of water which was assigned to the catalytic effect of palladium and the fusion of two hydrogen atoms (Fig. 6). Next year, they withdrew their claim because the small amount of helium entered the cell from the air outside. It that time, they could not be aware of the magnitude of nuclear energy, and thought that a good catalyst is enough for such reaction to be proceed.

Fig. 6

There is nothing new under the sun: paper by Paneth F and Peters K (1926) [43]

Electrochemical cells—the cell reaction and its thermodynamic description

Understanding the equilibria of a whole cell and in the case of emf measurements

It is worth to devote some words to the question of equilibrium. If the whole cell is at thermodynamic equilibrium, i.e., the cell reaction reaches its equilibrium Δ ${G}$ = E_cell = 0; i.e., no further work can be extracted from the cell and no current will flow if we connect the electrodes by using a conductor. This equilibrium is different from the equilibrium required for emf or E_MF (electromotive force of the cell) measurements. (The term E_MF is no longer recommended by the IUPAC. However, it is still used. One of the problems with this term is that it is an electric potential and not a force. This name originated from the nineteenth century: elektromotorische Kraft).

The value of emf can be measured if no current flows through the cell, and all local charge transfer and chemical equilibria are being established. However, it is valid for a functioning cell, and practically no current flows because either the potential difference of the cell under study is compensated by another galvanic cell or the resistance of the external conductor (the input resistance of the voltmeter) is several orders of magnitude higher than the internal resistance of the galvanic cell. The equilibria are related to the processes occurring at the interfaces, i.e., to the electron or ion transfers, within the electron conducting part of the electrodes and the electrolyte, ionically conducting phase. It is worth to mention to the students that this equilibrium is dynamic in nature and the rate of exchange is measurable, and we may introduce the exchange current density herein.

Another point we have to discuss that in textbooks, electrode potential is usually derived from the potential of cell reaction (${E}_{\text{cell}}$), and consequently, it is referred to and identified as the equilibrium electrode potential. It may be a good approximation in the case of a galvanic cell when no current flows. On the other hand, it is certainly a false approach if current flows in the case of galvanic cell as the consequence of the Gibbs energy of the given cell reaction (ΔG) and electrolysis cells when current generated from an external source to execute chemical changes. For instance in the case of cyclic voltammetry, the potential represented on the x-axis is also electrode potential.

Thermodynamic derivation of the potential of the cell reaction

The whole derivation can be found in [6, 23]; however, we discuss the most important steps.

$$\Delta G=\sum\limits_{\alpha }\sum\limits_{i}{\nu }_{i}\;{\mu }_{i}^{\alpha }=-{nFE}_{\text{cell}}$$

(6)

where ν_i is the respective stoichiometric number which is negative for reactants and positive for the products and ${\mu }_{1}^{\alpha }$ is the chemical potential of i-th species in phase α, n is the charge number of the cell reaction, and F is the Faraday constant. The minus (−) sign in Eq. (6) is a convention, i.e., for a spontaneous reaction, where $\Delta G$ is negative, ${E}_{\text{cell}}$ will be positive.

The electric potential difference (E or ΔV) is equal in sign and magnitude to the electric potential of metallic conducting lead on the right minus that of the similar lead on the left (The right and left refer to the cell diagram. The use of similar terminal leads is a thermodynamic requirement because the inner potential difference can be measured between phases of the same composition; otherwise, the difference of the chemical potentials would also be involved).

The cell diagram [6, 21] should be constructed in such a way that the cell reaction should proceed in direction when positive electricity flows through the cell from left to right.

It is in accordance with the so-called Stockholm convention of 1953 that was accepted by the IUPAC and the electrochemical community [44, 45].

Why is it of importance to deal with this definition? When we use a voltmeter at any electrochemical measurements, we may measure either positive or negative potential differences. It may lead confusion especially when we may determine the sign of the potential of a given electrode. For a long time, it has been the case, indeed. We can find opposite signs for the standard electrode potential in different books which is rather disturbing. For instance, in the excellent book of Latimer, the standard potential ($E\;^{\circ \kern -4.6pt -}$) for the zinc electrode (Zn²⁺/Zn) =  + 0.76 V, and it is in most of the American books, while in European books, the same value is − 0.76 V [46,47,48,49].

In is useful to define a scale, i.e., to select a reference electrode, and fix its potential. For this purpose, the standard hydrogen electrode (SHE) was chosen; i.e., the reference system is the oxidation of molecular hydrogen to solvated (hydrated) protons [1,2,3, 9, 11, 18,19,20,21,22,23,24,25]. In aqueous solutions:

$${^{1}/_{2}}\;{\mathrm{H}}_{2}\; ({\mathrm{g}}) \to {\mathrm{H}}^{+} ({\mathrm{aq}}) + {\mathrm{e}}^{-}$$

(7)

The notation H⁺ (aq) represents the hydrated proton in aqueous solution without specifying the hydration sphere.

Therefore,

$$\begin{aligned}{E}_{cell}=-\frac{1}{nF}\Sigma \,{\nu }_{i}\,{\mu }\;^{{\circ \kern -4.6pt -}}_{i}-\frac{RT}{nF}\Sigma \,{\nu }_{i}\,\mathrm{ln}\,{a}_{i}=E\;^{{\circ \kern -4.6pt -}}_{cell}-\frac{RT}{nF}\Sigma \,{\nu }_{i}\,\mathrm{ln}\,{a}_{i}\end{aligned}$$

(8)

that can be derived from Eq. (6) [6, 23, 49] we may write that

$$E\;^{\circ \kern -4.5pt -}_{\mathrm{cell}}=-\frac{1}{F}(\mu\;^{\circ \kern -4.5pt -}_{{\mathrm{H}}^{+}}-0.5\,\mu\;^{\circ \kern -4.5pt -}_{{\mathrm{H}}_{2}})-\frac{1}{nF}\Sigma \,{\nu }_{i}\,\mu \;^{\circ \kern -4.5pt -}_{i}$$

(9)

The standard electrode potential of the hydrogen electrode is chosen as 0 V. Thermodynamically, it means that not only the standard free energy of formation of hydrogen ($\mu \;^{\circ \kern -4.5pt -}_{{H}_{2}}$) is zero—which is a rule in thermodynamics since the formation standard free energy of elements is taken as zero—but that of the solvated hydrogen ion, i.e., $\mu \;^{\circ \kern -4.5pt -}_{{\mathrm H}^{+}}=0$ at all temperatures which is an extra-thermodynamic assumption. In contrast to the common thermodynamic definition of the standard state, the temperature is ignored. The zero temperature coefficient of the SHE corresponds to the conventional assumption of the zero standard entropy of H⁺ ions. This extra-thermodynamic assumption induces the impossibility of comparing the values referred to the hydrogen electrodes, in different solvents. However, there are recommendations for reference electrodes, reference redox systems, and pseudo-reference electrodes that can be used in non-aqueous and mixed solvents [50,51,52].

Prior to 1982, the old standard values of $E\;^{\circ \kern -4.5pt -}$ were calculated by using $p\;^{\circ \kern -4.5pt -}$ = 1 atm = 101325 Pa. The new ones are related to 10⁵ Pa (1 bar). It causes a difference in potential of the standard hydrogen electrode of + 0.169 mV; i.e., this value has to be subtracted from the $E\;^{\circ \kern -4.5pt -}$ values given previously in different tables. Since the large majority of the $E\;^{\circ \kern -4.5pt -}$ values have an uncertainty of at least 1 mV, this correction generally can be neglected.

When all components are in their standard states (a_i = 1 and $p\;^{\circ \kern -4.5pt -}$ = 1 bar), E_cell = $E\;^{\circ \kern -4.5pt -}_{cell}=E^{\circ \kern -4.5pt -}$. However, a_i is not accessible by any electrochemical measurements; only the mean activity (a_±) can be determined as we will show below. Nevertheless, in practice, ${a}_{\mathrm{H}+}$ is as taken 1 in 1 mol dm^–3 HClO₄, H₂SO₄, and HCl aqueous solutions. Of course, it is a rather rough approximation due to the variation of the unmeasurable ${a}_{\mathrm{H}+}$ and a_± in different acids. However, there are other uncertainties. For instance, when we use hydrogen gas bubbling, the vapor pressure of water (3.2 kPa at 25 °C) decreases the partial pressure of hydrogen to ca. 98 kPa at atmospheric pressure. It may cause 5 mV deviation which is about the same effect than using H⁺ concentration instead of relative activity. These two effects being opposite more or less compensate each other. Note that in 1 mol dm⁻³ H₂SO₄, H⁺ and HSO₄^– ions exist; the second dissociation step does not take place since pK_a = 2. Several books can be recommended [46,47,48,49, 53,54,55,56,57] where reliable standard potentials can be found; however, their values have been changing not only because of the reason mentioned above but also because of the accuracy of the novel measurements.

We will show shortly the changes of the standard potential of the Li⁺/Li electrode during a century. There are electrodes whose standard electrode potential cannot be determined by electrochemical measurements owing to kinetic reason; i.e., the reaction is too slow or too violent. In this case, ${E}\;^{\circ \kern -4.5pt -}_{cell}$ can be calculated from the thermodynamic data of caloric measurements [58] or if the equilibrium constant of the reaction is known. ${E}\;^{\circ \kern -4.5pt -}_{cell}$ can be calculated from the standard molar Gibbs energy change ($\Delta G\;^{\circ \kern -4.5pt -}$) for the same reaction with a simple relationship:

$$E\;^{\circ \kern -4.5pt -}_{\text{cell}}=\Delta G\;^{\circ \kern -4.5pt -}\;/\;n\,F=(RT\,/\,nF)\,\text{ln}\;K$$

(10)

where K is the equilibrium constant of the reaction.

Li⁺/Li electrode is a typical one because the reaction between the Li metal and water is extremely violent. However, applying some tricks, e.g., by using a series of dilute amalgams of the metal, therefore decreasing the rate of the reaction with the solvent, the emf measurement can be carried out. Of course, several further steps should be executed, e.g., different calculations by using data from other sources and extrapolation.

The first electrochemical measurement of Li⁺/Li electrode in aqueous solution has been executed by Lewis and Keyes in 1913 [59] (Fig. 7). They studied two cells:

• Li (s) | Li⁺, I⁻, propylamine | Li (Hg)

• Li (Hg) | Li⁺, OH⁻, H₂O ¦ LiCl (0.1 M), KCl (0.1 M) ¦ KCl (1 M) ¦ Hg₂Cl₂ | Hg

Fig. 7

The first attempt to determine the standard potential of Li electrode electrochemically: Lewis GN, Keyes FG (1913) Journal of the American Chemical Society 35: 340–344 [59]

After corrections of junction potential (taken 25.9 mV which might contain several mV error) and conversion of the potential of the normal calomel electrode to the potential of normal hydrogen electrode (normal means 1 mol dm⁻³ concentration, and this term has been used for a long time instead of standard), a standard potential of lithium electrode − 3.0243 V was calculated. Later, Lewis and co-workers reported − 3.0247 V and − 2.957 V. In 1926, Latimer reported − 3.045 V for the value of this standard potential that was calculated from thermodynamic data. Fifty years later, Huston and Butler tried to determine again the standard potential of lithium electrode. They used a cell without liquid junction as follows, Pt (s) H₂ (g) | Li⁺, OH⁻, H₂O | Li (Hg), and reported $E\;^{\circ \kern -4.5pt -}$ =  − 3.0401 V [60] (Fig. 8).

Fig. 8

Houston R, Butler JN (1968) The standard potential of the lithium electrode in aqueous solutions. Journal of Physical Chemistry 72: 4263–4264 [60]

Recently, $E\;^{\circ \kern -4.5pt -}$ (25 °C, 1 atm) Li⁺ (aq) + e⁻  → Li (s) =  − 3.04 ± 0.005 V [49] is the mostly accepted value, e.g., ${E}\;^{\circ \kern -4.5pt -}$ =  − 3.0401 V [57]. In some books, slightly different values, e.g., − 3.045 V [56], are given; however, it is not always evident that which value was used as the standard pressure.

The formal potential (${E}\;^{\circ \kern -4.5pt -\prime}_{c}$)

Beside ${E}\;^{\circ \kern -4.5pt -}_{\mathrm{cell}}$ and $E\;^{\circ \kern -4.5pt -}$, the so-called formal potentials (${E}\;^{\circ \kern -4.5pt -\prime}_{\mathrm{cell,c}}$ and $E\;^{\circ \kern -4.5pt -\prime}_{\mathrm{c}}$) are frequently used [5, 7]. The purpose of defining formal potentials is to have “conditional constant” that take into account activity coefficients and side reaction coefficients (chemical equilibria of the redox species), since in many cases it is impossible to calculate the resulting deviations because neither the thermodynamic equilibrium constants are known, nor it is possible to calculate the activity coefficients. This approach involves the problem of the initial and final states; the knowledge of those is prerequisite for any thermodynamic calculation. It is a problem even in the pro forma simple case of metal/metal ion equilibrium. Metal ions in aqueous solutions exist in the form of aqua complexes. However, depending on the nature of the electrolyte, a series of mixed aqua-anion complexes are present whose composition is the function of the anion concentrations. There are also complications in the solid phase. For instance, the electrochemical preparation of PbO₂ yield a two-phase mixture of tetragonal and orthorhombic PbO₂ (s, cr), and their ratio depends on the conditions of preparation. This obviously affects the measured E_MF value, and as follows $E\;^{\circ \kern -4.5pt -}$) determined for PbO₂ (s, cr) and PbSO₄ (s, cr) reversible electrode. The novel techniques have been allowing to study small clusters containing less than 20 metal atoms. It was calculated from results of the measurements that $E\;^{\circ \kern -4.5pt -}$) was shifted by almost 2 V. It was concluded that the change of $E\;^{\circ \kern -4.5pt -}$) is due to the high surface energy of small clusters [20, 61]. While the electrode potential derived from the measurements can be shifted due to several reasons (e.g., phase formation at the same formal composition), in this case, the laws of thermodynamics cannot be applied because those are only valid for large number of species.

Therefore, the potential of the cell reaction and the potential of the electrode reaction are expressed in terms of concentrations:

$$E_{\mathrm{cell}}=E\;^{\circ \kern -4.5pt -\prime}_{\mathrm{cell, c}}-\frac{RT}{nF}\Sigma\,\nu_{i}\;\text{ln}\;\frac{c_{i}}{c^{\circ \kern -4.5pt -}}$$

(11)

$$E=E\;^{\circ \kern -4.5pt -\prime}_{\mathrm{c}}-\frac{RT}{nF}\,\Sigma\,\nu_{i}\;\text{ln}\frac{c_{i}}{c\;^{\circ \kern -4.5pt -}}$$

(12)

where

$$E\;^{\circ \kern -4.5pt -\prime}_{\mathrm{cell, c}}=E\;^{\circ \kern -4.5pt -}_{\mathrm{cell}}-\frac{RT}{nF}\;\Sigma\,\nu_{i}\;\text{ln}\;\gamma_{i}$$

(13)

and

$$E\;^{\circ \kern -4.5pt -\prime}_{\mathrm{c}}=E\;^{\circ \kern -4.5pt -}-\frac{RT}{nF}\,\Sigma\,\nu_{i}\;\text{ln}\;\gamma_{i}$$

(14)

when SHE is the reference electrode ($a_{\mathrm{H}^{+}}=\frac{p_{{\mathrm{H}}_{2}}}{p\;^{\circ \kern -3.2pt -}}=1$). Equation (12) is the well-known Nernst-equation:

$$E=E\;^{\circ \kern -4.5pt -\prime}_{\mathrm{c}}+\frac{RT}{nF}\;\text{ln}\;\frac{\pi\,c\,^{{\nu_{\mathrm{ox}}}}_{\mathrm{ox}}}{\pi\,c^{{\nu_{\mathrm{red}}}}_{\mathrm{red}}}$$

(15)

where π is for the multiplication of the concentrations of the oxidized (ox) and reduced (red) forms, respectively. The formal potential is sometimes called as conditional potential indicating that it relates to specific conditions (e.g., solution composition) which usually deviate from the standard conditions. In this way, the complex or acid–base equilibria are also considered since the total concentrations of oxidized and reduced species considered can be determined, e.g., by potentiometric titration, however, without a knowledge of the actual compositions of the complexes. In the case of potentiometric titration, the effect of the change of activity coefficients of the electrochemically active components can be diminished by applying inert electrolyte in high concentration (almost constant ionic strength). If the solution equilibria are known from other sources, it is relatively easy to include their parameters into the respective equations related to $E\;^{\circ \kern -4.5pt -\prime}_{\mathrm{c}}$. The most common equilibria are the acid–base and the complex equilibria. In acid media, a general equation for the proton transfer accompanying the electron transfer is

$${\mathrm{Ox} + \mathrm{n} \;\mathrm{e}^{-} + \mathrm{m} \;\mathrm{H}^{+} \rightleftarrows \mathrm{H}_\mathrm{m} \;\mathrm{Red}^{(\mathrm{m}\,-\,\mathrm{n})+}}\quad E\;^{\circ \kern -4.5pt -\prime}_{c}$$

(16)

$${\mathrm{H}_{m}\; \mathrm{Red}^{(\mathrm{m}\,-\,\mathrm{n})+}\rightleftarrows \mathrm{H}_{\mathrm{m}\,-\,1}\; \mathrm{Red}^{(\mathrm{m}\,-\,\mathrm{n}\,-\,1)+} + \mathrm{H}^{+}}\quad K_{a1}$$

(17)

$${\mathrm{H}_{\mathrm{m}\,-\,1} \;\mathrm{Red}^{(\mathrm{m}\,-\,\mathrm{n}\,-\,1)+} \rightleftarrows \mathrm{H}_{\mathrm{m}\,-\,2}\; \mathrm{Red}^{(\mathrm{m}\,-\,\mathrm{n}\,-\,2)+} + \mathrm{H}^{+}}\quad K_{a2}$$

(18)

etc. For m = n = 2

$$E=E\;^{\circ \kern -4.5pt -\prime}_{c}+\frac{RT}{nF}\;\text{ln}\left(\frac{c_{\mathrm {ox}}}{c_{\mathrm {red}}}\;\frac{1\;+\;K_{a1}a_{\mathrm H^{+}}+a^{2}_{\mathrm H^{+}}}{K_{a1}K_{a2}}\right)$$

(19)

The complex equilibria can be treated in a similar manner; however, one should not forget that each stability constant (K_i) of a metal complex depends on the pH and ionic strength [23].

Variation of constants and its consequences

“Who knows the constant, becomes wise”—Lao-Tzu (Laozi), Tao Te Ching. Chapter 16, around 400 BC.

Constants used in physical chemistry have been changed several times during the last century. Therefore, one should be careful when choosing the source of the constant to be used.

The International System of Units (SI system) has been established in 1960, however several times has been updated since then. Earlier, even different metric systems were used, and different frames of references were applied. Recently, the value of several constants has been fixed at an exact value. The results of more sensitive and accurate measurements also affected the values of the constants; however, the changes of the SI units cause minor effects in electrochemistry.

Conférence générale des poids et mesures (CGPM) (General Conference on Weights and Measures), which was established by the Metre Convention of 1875, brought together many international organizations to establish the definitions and standards of a new system and to standardize the rules for writing and presenting measurements. The SI system was published in 1960 as a result of an initiative that began in 1948. It was based on the meter–kilogram–second system of units (MKS) used previously rather than any variant of the centimeter–gram–second (CGS) based on the ideas of Carl Friedrich Gauss, James Clerk Maxwell, and William Thomson, which was also a popular alternative system especially in science. Nernst also used the CGS system in the calculation for comparison of the equation derived and experimental results in his famous paper; however, he used a constant called “elektrolytische Gaskonstate” containing both R and F [14]. It comes from that in his osmotic theory Nernst used Mariotte-Boyle law for the derivation of emf.

Faraday constant

The Faraday constant, denoted by the symbol F is the electric charge per mole of elementary charges. Since the 2019 redefinition of SI base units [62, 63], the Faraday constant has the exactly defined value given by the product of the elementary charge e and Avogadro constant N_A which were also fixed:

$$\begin{aligned}F=&\;e\times N_{A}=1.602176634\times 10^{-19}\;\text{C}\\&\times 6.02214076\times 10^{23}\;\text {mol}^{-1}\\&=9.64853321233100184\times 10^{4}\;\text{C}\cdot\text{mol}^{-1}.\end{aligned}$$

The value of this fundamental constant has been changed several times. The atomic mass of the elements has been related to that of ¹H, ¹⁶O, and ¹²C, respectively (Figs. 9 and 10). The largest variation occurred when the base of comparison of the molar (atomic) masses changed from ¹⁶O to ¹²C in the 1960s: after 1963, F = 9.6483 × 10⁴ C⋅mol⁻¹ and before 1963, two old values have been used: (i) related to a ¹⁶O taking into account the molar mass of the oxygen including the natural distribution of its isotopes (chemical atomic weight), F = 9.6487 × 10⁴ C⋅mol⁻¹, and (ii) related to ¹⁶O (physical atomic weight): F = 9.6515 × 10⁴ C⋅mol⁻¹.

Fig. 9

Official atomic weights in 1903 related to ¹⁶O and ¹H, respectively

Fig. 10

Atomic weights used by Dmitri Ivanovich Mendeleyev (1834–1907) (I) as well as determined by Jean Servais Stas (1813–1891) in different years (the false atomic weight of iodine (1865) is a misprint in the original publication). The IUPAC data in 1961 (III)

Electrochemical methods based on the Faraday’s laws have played an important role in the determination of atomic mass in the nineteenth century (Fig. 11).

Fig. 11

Electrochemical equivalent of ions determined by Faraday M (1834) VI. Experimental Researches in Electricity—Seventh series. Philosophical Transactions 124: 77–122 [5]

The IUPAC also publishes the actual values of the constants. Before the redefinition of the SI units, F = 9.648530929 × 10⁴ C⋅mol⁻¹, e = 1.6021773349 × 10⁻¹⁹ C exact fixed, and N_A = 6.022136736 × 10²³ mol⁻¹ exact fixed [64].

In different books and papers, rather diverse values of the constants can be found. In practice, it does not cause higher deviation than 1–3%; however, when teaching electrochemistry or in general physical chemistry, we have to draw the attention of the students to the actual values of constants and to explain the consequences of the usage of different units because it may surprise the students reading old books.

Concerning the Faraday constant, it is worth to mention that in the industry and in several schoolbooks, a rather approximate value is given. It is 96500 C mol⁻¹ and where e = 1.59 × 10⁻¹⁹ C and N_A = 6.07 × 10²³ mol⁻¹ [65].

If we want the students to study the original papers and books of Nernst and van 't Hoff [2], it would cause a problem because they used R = 2 instead of R = 8.314510 J K⁻¹ mol⁻¹ because in that time they calculated with R = 1.9865 cal fok⁻¹ mol⁻¹. For instance, we can find the equation as follows: d ln K / dT =  − Δ_rH$^{\circ \kern -4.5pt -}$ / 2T² [2] where the factor 2 was not explained.

We have to take into account the change of N_A and e, and because we frequently use the RT/F factor also the actual value of the gas constant R.

The acidity of a solution: pH

It is certainly confusing if a scientific term has two or more definitions. It is even more problematic if the definition of a concept called notional definition by the IUPAC [21, 66] involves a quantity (single or individual ion activity) that cannot be measured. The so-called notional definition of the pH is as follows:

$${\mathrm{pH}=-{\mathrm{lg}}\;\mathit{a}_{\mathrm{H}_{3}\mathrm{O}^{+}}}$$

(20)

or

$${\mathrm{pH} = - \mathrm{lg} \;\mathit{a}_\mathrm{H} = - \mathrm{lg} }(m_{\mathrm{H}} \;\gamma_{\mathrm{H}}\,/\,m\;^{\circ \kern -4.5pt -}),$$

(21)

where a_H is the relative activity on molality basis, γ_H is the molal activity coefficient of the hydrogen ion H⁺ at the molality m_H, and m is the standard molality (1 mol kg⁻¹).

Earlier IUPAC preferred another definition which does not contain an unmeasurable quantity pH ≈ − lg [c (H⁺) / mol dm⁻³] or pH =  − lg(c_H/c) where c_H is the hydrogen ion concentration in mol dm⁻³, and c = 1 mol dm⁻³ is the standard amount concentration [67].

In fact, the latter definition is practically the same as the original definition of Søren Peter Lauritz Sørensen (1868–1939) in 1909 [68, 69]. However, later Sørensen also accepted the definition of pH in terms of the relative activity of hydrogen ions [70].

Sørensen suggested two methods for the measurements of pH: the electrometric and the colorimetric method. The electrometric method was described as follows. “If a platinum plate that’s covered with platinum black is dipped into an aqueous – acidic, neutral, or alkaline – solution and if the solution is saturated with hydrogen, then one finds, between the platinum plate and the solution, a voltage difference whose magnitude depends on the hydrogen ion concentration of the solution according to a law.” “The colorimetric method. The sudden change of the indicator during a typical titration means, as is known, that the concentration of hydrogen ions in the solution at hand has reached or exceeded – from one direction or the other – a certain magnitude.” Nowadays, mostly the electrochemical method is used; however, colorimetric (spectrophotometric technique) is also applied (see later Hammett acidity function).

The main problem is that correct measurements and calculation can be executed only in the case of dilute acidic or alkaline solutions, which does not contain other electrolytes in high concentration. For instance, in a solution containing 0.1 mol dm⁻³ HCl and 10 mol dm⁻³ LiCl, the pH is certainly not 1 because the activity coefficient regarding the hydrogen ions or even the mean ionic activity coefficient (y_±) which can be measured is much higher than one. Therefore, the protonation activity is also higher; i.e., the concentration of the protonated form of an organic compound will be higher than expected in the case of this dilute acidic solution. Even the symbol pH is an exception to the general rules for the symbols of physical quantities [67] in that it is a two-letter symbol and it is always printed in Roman (upright) type instead of italic as suggested for all other symbols for physical quantities.

It is a question whether we need the notional definition at all or it would be enough to just use the operational definition alone for expressing the acidity of a solution. IUPAC practically replaced the notional definition by an operational definition which is, in fact, an instruction for the correct measurement of pH. It is a unique practice in this area. However, it is understandable taking into account the original goal of the introduction of this term. It was nothing else than the simple handling of exponents like 10⁻⁵, 10⁻⁷, or 10⁻¹². We have to think that assistants and workers, i.e., many people, work on the measurements and the determination of pH in pharmaceutical industry (very accurate pH requirements especially for infusion), processing agricultural goods (orange juice, vines, pickles, etc.), textile industry (dyeing), etc. It is the very reason that a replacement of the definition of pH with another one is a rather a problematic issue that would require the changes of hundreds of patents, standard protocols (SOPs), etc. It is worth to mention that the first pH meter (called acidimeter) was invented by Arnold Orville Beckman (1900 − 2004) in 1935 [71]. In fact, it was the beginning of the electrochemical instrumentation, and it made possible the easy mass measurements of pH. Earlier both the industrial chemists and researchers used the traditional color tests for acidity. It was an accurate method especially after the invention of artificial buffer solutions by Paul von Szily (1878–1945) in 1903 [72, 73]. In principle, the hydrogen electrode is the plausible choice for pH sensitive electrode; however, glass electrodes are the most widely used pH electrodes both in laboratories and industry for pH measurements [74, 75].

Operational definition

The method suggested [21, 66, 67] requires two emf (electromotive force) measurements, one is with the cell containing a buffer solution with a defined pH (standard) (E_stand) and another one where the buffer is replaced by a solution whose pH to be measured (E_x) in a cell as follows.

$${\mathrm{Cu}\; |\; \mathrm{Pt} \;(\mathrm{s}) \;| \;\mathrm{H}_{2}\; (\mathrm{g}), \;\mathit{p} = \mathit{p}\;^{\circ \kern -4.5pt -}\;| \;\mathrm{H}^{+}, \;{\mathit{a}_\mathrm{H}}^{+}, \mathrm{X}^{-}, \mathit{a}_{\mathrm{X}^{-}} \;(\mathrm{aq})\; {\begin{array}{l}\vspace{-1.7mm} \shortmid \shortmid \\ \shortmid\shortmid\end{array}}\;\mathrm{KCl}\,(\mathrm{aq})},\;m > 3,5\;\mathrm{mol\; kg}^{-1}\;| \;\text{reference electrode}\; | \; \text{Cu}$$

A single vertical bar (│) represents a phase boundary, the dashed vertical ( ¦ ) and the double, dashed vertical bars ( ¦¦ ) represent junctions between miscible liquids, in the case of ( ¦¦ ), the liquid junction potential is practically eliminated. The reference electrode is usually a silver/silver chloride or a calomel electrode.

The unknown pH (x) of the solution HX (aq) is given

$${\mathrm{pH}\; (\mathrm{x}) = \mathrm{pH} \;(\mathrm{standard}) + (\mathit{E}_\mathrm{stand}- \mathit{E}_\mathrm{x})} \;F\, /\, \,R \;T \;\text{ln}\; 10$$

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The primary method for the measurement of pH involves the use of a cell without transference, known as the Harned cell [21]:

$${\mathrm{Cu}\; |\; \mathrm{Pt}(\mathrm{s}) \;|\; \mathrm{H}_{2}(\mathrm{g})\; |\; \mathrm{Buffer}\;\mathrm{S},\; \mathrm{Cl}^{-}(\mathrm{aq})\; | \;\mathrm{AgCl}(\mathrm{s}) \;| \;\mathrm{Ag(s)} \;| \;\mathrm{Cu}}$$

Any buffer of known pH can be used; however, there is a reference value of the pH standard is an aqueous solution of potassium hydrogen phthalate at a molality exactly 0.05 mol kg⁻¹ at 25 °C which has a pH of 4.005.

In dilute solutions (< 0.1 mol dm⁻³) (2 < pH < 12), pH =  − lg [γ_± c (H⁺) / mol dm⁻³] ± 0.02. More accurate measurement of pH cannot be done. However, by using the so-called Harned cell for measurements, and Debye-Hückel theory for calculation of the activity coefficient and the Bates–Guggenheim convention, it is claimed that ± 0.003 accuracy can be achieved [21].

Because the liquid junction potential cannot be fully eliminated and in practice glass electrode is used instead of the Pt/H₂ electrode, one should calculate with an uncertainty of ± 0.02.

$${\mathrm{Cu(s)}\;\vert\;\mathrm{Hg(l)}\;\vert\;\mathrm{H}_2\mathrm{C{l}}_2\mathrm{(s)}\;\vert\;\mathrm{KCl(aq)}}\;{\begin{array}{l}\vspace{-1.7mm} \shortmid \shortmid \\ \shortmid\shortmid\end{array}}\;{\mathrm{solution}\;\vert\;\mathrm{glass}\;\vert\;\mathrm{HCl(aq)}\;\mathrm{or}\;\mathrm{buffer}\;\vert\;\mathrm{AgCl(s)}\;\vert\;\mathrm{Ag(s)}\;\vert\;\mathrm{Cu(s)}}$$

$$m > 3.5 \;\text{mol} \;kg^{-1}\quad a_{\pm} = \;? \;pH =\; ?$$

Hammett acidity function

The acidity function (H_o) [76] was introduced by Louis Planck Hammett (1894–1987) and co-workers [77] for the characterization of the acidity especially of strong acid solutions of high concentrations. The fundamental idea is the utilization of the protonation of weak bases. The primary systems used by Hammett were based on anilines [e.g., 2,3,6-trinitro-difenilamine (pK_a = 10.38) or 2-trinitro-aniline (pK_a = 17.88)]; however, several indicators have been studied, and the tabulated H_o values obtained for different strong acids can be found in several papers and books [78,79,80]. The extent of protonation of the indicator molecules depends on the activity of the hydrogen ions in the solution; therefore, a determination of the concentration ratio of the protonated and unprotonated form of the indicator molecules gives an information on the hydrogen ion activity of the solution under study. If the light absorption (color) of the two forms is different, the ratio can easily be determined by UV–visible spectrometry. Other techniques, e.g., NMR measurements, have also been applied [76].

H_o can be defined as follows:

$$\mathrm{Ho} = \mathrm{p}{K}_{\mathrm{a}} - \log \mathit{I} = - \log \mathit{a}_{\mathrm{H}} + - \log {\gamma}_{\mathrm{B}}/{\gamma}_{BH} +$$

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where pK_a is the negative logarithm of the dissociation constant of the protonated weak base (BH⁺), I is ratio of ${c}_{\mathrm{BH}+}/{c}_{\mathrm{B}}$, a_H + is the relative activity of hydrogen ions, and $\gamma$_B and $\gamma$_BH + are the respective activity coefficients.

At low acid concentrations, H_o = pH.

Since the reference system is the pure water, at high acid concentrations—when, in fact, a mixed solvent is present—pK_a may be varied. The H_o values depend on the nature of the indicator (weak base). The accuracy of the determination I, i.e., the ${c_{{\mathrm{BH}+}}}/{c_{{\mathrm{B}}}}$ ratio, is in connection with the indicator used. Therefore, the indicator has to be varied when a wide range of acid concentration is studied. For instance, in the case of H₂SO₄ solution, when the concentration is less than 20%, o-nitroaniline or 4-chlor-2-nitro-aniline can be used, while at ${c}_{\mathrm{cell},\mathrm{ r}}$ > 35%, 2,4-dichlor-6-nitro-aniline is the suitable indicator due to its higher pK_a value.

Conclusions

Selected fundamental concepts and terminology in the field of electrochemistry are discussed in this paper in order to support the teachers and the students alike to acquire a solid knowledge regarding these basic topics. The subjects are treated in historical perspective, i.e., in the light of their earlier phases and subsequent evolution. While I intended to convey the present knowledge accurately that is suitable in the education in the third decade of the twenty-first century, the method used may raise the obvious idea that our knowledge will be deeper and our thinking on these issues will change also in the future, even during the life of the students attending the universities all around the world in these days.