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In redox reactions a potential difference drives the transfer electrons from an anode (negative electrode) to a cathode (positive electrode): oxidation occurs at the anode and reduction occurs at the cathode. Reactions are spontaneous in the direction of ΔG < 0, which is also the direction in which the potential (defined as E_cathode – E_anode) is positive. In a redox reaction the anode is then the half‐reaction written with electrons on the right and the cathode is the half‐reaction with electrons appearing on the left side.

The electric work done by a spontaneous redox reaction, like in a galvanic cell (E > 0), is the (measurable) electromotive force of the reacting systems and equals the Gibbs free energy change (e.g. Ottonello, 1997) via the Nernst equation:

(1.13)

with a_i the activity of the i^th component participating in the redox exchange, F as the Faraday constant (96,485 Coulomb per mole), n the number of transferred electrons, and Q the activity product. In writing redox reactions, complete electrolytes are often used because the activity coefficients are measured without extra thermodynamic assumptions, but Equation 1.13 is normally used for reactions based on individual ions. To establish a potential scale for half‐reactions, we keep using the convention that electrons are reported on the left‐hand side of the reaction, that is, in the sense of reduction. The potentials of half‐reactions can be added and subtracted, like free energies, to give an overall value for the reaction. It is also worth noting that by convention, it was decided to use a hydrogen‐electrode‐scale electric potential, by setting E⁰ = 0.0 V for reaction 1.7 with the constituents in their standard state (e.g., Casey, 2017). This arbitrary decision implies that (i) the Gibbs energy for H⁺_(aq), the electron (e^–), and H_2(g) are all 0.0 kJ/mol, and (ii) potential difference of reactions involving the hydrogen electrode (Reaction 1.7) are given by the other half‐reaction completing the redox exchange.

The electrode potential values (E⁰) hold at standard conditions: by definition, standard conditions mean that any dissolved species have concentrations of 1 m, any gaseous species have partial pressures of 1 bar, and the system is 25°C. Standard potentials represent the case where no current flows and the electrode reaction is reversible. Measuring a voltage is an indication that the system is out of equilibrium. Nernstian processes are characterized by fast electron transfer and are rate‐limited by the diffusion of the electron‐active species into the electrolyte. The system then spontaneously approaches equilibrium because negative and positive charged species can flow in opposite directions. At equilibrium, the voltage drops to zero and the current stops, like in dead batteries. The magnitude of the cell potential, E⁰ = E⁰_cathode – E⁰_anode, may be viewed as the driving force for current flow in the circuit.

The hydrogen‐electrode scale electric potential so defined, E (also indicated as E_h in aqueous solutions), is a measure of the oxidation state of a system at equilibrium relative to a hydrogen electrode. E is not a constant (for given T and P) but depends on the system composition via activities of ions entering a half redox reaction. When coupled to a compositional parameter of the system related to the activity of the ligand making up the solvent of interest, such as a_H+ for aqueous solutions, E can be used to establish a kind of phase diagram that shows which species (dissolved ion species, gases, or solids) will predominate among a chosen set in the system of interest (a solution) for a given temperature.

To easily understand all this, we can look at the reaction leading to the formation of liquid water:

(1.14)

which is given by the sum of Reaction 1.7 (H⁺/H₂ redox couple: the anode) and the following half‐reaction (the cathode):

(1.15)

which is governed by the O₂/H₂O redox couple. The presence of protons in both Reactions 1.7 and 1.15 shows that the overall Reaction 1.14 is defined for acidic conditions (pH < 7). For neutral or basic conditions (pH ≥ 7), Reaction 1.14 can be obtained from the following two half‐reactions for H₂O/H₂ and O₂/OH^– couples, respectively:

(1.16)

(1.17)

Let us now deal with Reactions 1.7 and 1.15 occurring in the acidic medium (see, for example, Ottonello, 1997). The standard potential of Reaction 1.15 is E⁰₁₆ = 1.228 V and refers to a standard state of water in equilibrium at T = 25°C and P = 1 bar with an atmosphere of pure O₂. From Equation 1.13 we obtain:

(1.18)

where a and f denote activity and fugacity, respectively, pH = –loga_H+ and it is considered that aO₂ = fO₂/fO₂⁰ with fO₂⁰ = 1 bar.

Similarly, the redox potential related to Reaction 1.7 is then:

(1.19)

Equations 1.18 and 1.19, which can also be derived for Reactions 1.16 and 1.17, can be used to trace E‐pH diagrams (also called Pourbaix or predominance diagrams; Casey, 2017) limiting the stability field of water (Figure 1.1) and of any other systems in which E‐pH relationships can be established from the reactions of intervening species (Figure 1.2). E‐pH diagrams are most used for understanding the geochemical formation, corrosion and passivation, leaching and metal recovery, water treatment precipitation, and adsorption.

For the set of species of interest, E‐pH diagrams show boundaries that are given by:

1 lines of negative slope that limit the stability field of water (Equations 1.18 and 1.19) or related to solid–solid phase changes in which paired electron–proton exchanges occur because the ligand (water) participates in reaction, such as in the case of the hematite–magnetite boundary in Figure 1.2:Figure 1.1 E‐pH diagram reporting the stability of water at T = 25°C and P = 1 bar for different partial pressure of H2 and O2 (log‐values).(1.20) Note that boundary slope is negative because protons and electrons appear on the same side of reaction. The protons/electrons ratio determines the slope value.

2 (rare) pH‐dependent lines of positive slopes, and associated with electron–proton exchanges involving, for example, reduction of dissolved cations to the oxide with a lower oxidation number, e.g.(1.21) Boundary slope is positive because protons and electons appear on different sides of reaction.

3 horizontal lines (pure electron exchange), such as in case of half‐reaction(1.22) which participates with half‐reaction 1.7 in giving Reaction 1.3 and does not involve explicitly the water solvent.

4 vertical lines, representing no change of oxidation state but only acid–base reactions (a sole exchange of protons for aqueous solutions), such as(1.23) with the boundary plotted at the pH value for which Q23 = K23 with aFe2O3(s) = (aFe3+)2 = 1 (and also aH2O = 1).

We now see that Reactions 1.3 and 1.4 are both related to half‐reaction 1.22, but they refer to different redox conditions and then have different meanings. In Reaction 1.3 water is the oxidizing agent in acidic conditions, whereas it is the reducing agent in Reaction 1.4 (Appelo and Postma, 1996), which represents a regulating mechanism of O₂, probably the oxidative alteration of rocks containing Fe²⁺, in particular at oceanic ridges.

Figure 1.2 The E‐pH (Pourbaix) diagram for the Fe‐S‐H₂O system at 25°C at 1 bar total pressure and for total dissolved sulfur activities of 0.1 (panel a) and 10^–6 (panel b). Superimposed are the stability fields for H₂S, HS^–, HSO₄^2– and SO₄^2– dissolved species (red lines, traced from data in Biernat and Robins, 1969). Note how the field of stability of pyrite, FeS₂, shrinks and that of magnetite, Fe₃O₄, expands with decreasing total sulfur activity.

Modifed from Vaughan (2005).

Basically, E‐pH diagrams demonstrate that breaking of a redox reaction into half‐reactions is one of the most powerful ideas in redox chemistry, which allows relating the electron transfer to the charge transfer associated with the speciation state and the acid–base behavior of the solvent. Superimposing E‐pH diagrams allows a fast recognition of the existing chemical mechanism occurring in an electrolyte medium. For example, Figure 1.2 on the Fe‐H‐O‐S system can be seen as the result of the superposition of stability diagrams for H‐O‐S and Fe‐O‐H system. The resulting diagram in Figure 1.2 shows that the pyrite–magnetite boundary has a negative slope due to half‐reaction:

(1.24)

but also a positive slope well visible in Figure 1.2b due to sulfur reduction and dissolution in water as HS^–:

(1.25)

Reaction 1.25 implies of course a positive slope, because H⁺ appears on the right side and electrons on the left. We can also appreciate the reduction of sulfur from pyrite to pyrrhotite at pH > 7:

(1.26)

which has a negative slope of –0.0295pH because the number of exchanged electrons is double than protons.

These concepts can then be transferred to other solvents in which ligand–metal exchanges lead to a different speciation state and are governed by a different notion of basicity, i.e. oxobasicity, such that (see Moretti, 2020 and references therein):

(1.27)

which can be also related to redox exchanges via the normal oxygen electrode (Equation 1.6), in the same way the normal hydrogen electrode (Reaction 1.7) can be put in relation with the Bronsted‐Lowry definition of acid–base behaviour in aqueous solutions (see Moretti, 2020):

(1.28)

Figure 1.3 Limit of equilibrium potential‐pO^2– graphs in molten alkali carbonates and sulfates, at 600°C

(modified from Trémillon, 1974).

It is then possible to define pO^2– = ‐logaO^2– and introduce E‐pO^2– diagrams, in which acid species will be located at high pO^2– values. These diagrams were first introduced by Littlewood (1962) to present the electrochemical behaviour of molten salt systems and provide an understanding of the stability fields of the different forms taken by metals in these systems. Reference potential for molten salt is chosen either from anion or from cation, but anion, making up the ligand, is normally selected because there may be several different cations in the system.

For molten solvent diagrams, such as carbonate and sulfate melts, the stability area of the bath depends on the salt itself and can be seen by using as examples oxyanion solvents (Figure 1.3). Limitations on the pO^2– scale of oxoacidity (Reaction 1.27) are given by the values of the Gibbs free energy of the formation reactions of alkali carbonates or sulfates at the liquid state, which depends on temperature as well as on pressure. On the basic side (low pO^2– side) the limit is imposed by the solubility threshold of the generic M^ν+O_ν/2 oxide in the electrolyte medium, i.e., pO^2–min ≈ M^ν+O_ν/2 solubility, whereas on the acidic side the limit is imposed by P_CO2 or P_SO3 = 1 bar. For example, it is 11 units in the case of the ternary eutectic Li₂CO₃+Na₂CO₃+K₂CO₃ at 600°C and 19.7 units in the case of the ternary eutectic Li₂SO₄ + Na₂SO₄ + K₂SO₄ at the same temperature (Trémillon 1974; Figure 1.3).

The upper stability limit is related to the O^–II/O_2(g) redox system (Reaction 1.6), i.e., to the oxidation of CO₃^2– and SO₄^2– anions:

(1.29)

(1.30)

which results from acid–base exchanges of the type:

(1.31)

(1.32)

coupled to half‐reaction 1.6.

Both Reactions 1.29 and 1.30 yield the E‐pO^2– relationship:

(1.33)

The lower stability limit of the solvent can be given by either the reduction of alkaline cation in the corresponding metal:

(1.34)

whose potential is independent of pO^2–, or the reduction of CO₃^2– and SO₄^2– anions given by:

(1.35)

Figure 1.4 Log fO₂ ‐ Ph diagrams for 290°C (left panel) and 145°C (right panel) at saturated vapor pressure, showing predominance fields for aqueous sulfur species (dashed lines), stability fields for Fe–O–S minerals and bornite–chalcopyrite (solid grey lines). The solubility contours (left panel: 1, 10, 100 ppm; right panel: 0.1, 1 ppm) are for gold in the form Au(HS)₂^–.

Modified from Raymond et al. (2005).

(1.36)

and to which the following E‐pO^2– relationships correspond (CO₃^2– and SO₄^2– anions having unitary activity):

(1.37)

(1.38)

Figure 1.3 shows the results on carbonate and sulfate melts (modified from Trémillon, 1974, and references therein). The utilizable regions appear as quadrilaterals on the E‐pO² graph. If in sulfates, the region is a parallelogram similar to the E‐pH region in aqueous solution, the theoretical range of potential in molten carbonates appears more restricted in an oxoacidic medium than in an oxobasic medium, because the lower limit varies versus pO^2‐ with a slope greater than that of the upper limit (Trémillon, 1974).

Silicate melts have been so far an underestimated electrolytic medium acting as a solvent for oxides. This is mainly because E‐pO^2– diagrams cannot be based on predictive thermodynamics and physical chemistry assessments such as the dilute electrolyte concept and its developments in the case of previous solvents, aqueous solutions particularly (Allanore, 2015). In silicate melts, and more generally molten oxides, oxygen tout‐court cannot be identified as the solvent, despite its abundance. Silicate melts are in fact a high‐temperature highly interconnected (polymerized) matrix in which solvation units cannot be easily defined and both ionic and covalent bonds rule the reactive entities that make up the melt network. Because of this, some approaches have been formalized in terms of the Lewis acid–base definition (network formers and their oxides, such as SiO₂ and Al₂O₃ are acids; network modifiers and their oxides such as MgO, CaO, Na₂O are bases) by using electronegativity and optical basicity that allow distinguishing and calculating three types of oxygen (bridging, non‐bridging, and so‐called free oxygen) whose mixing determines the polymerization and the thermodynamic properties of the melt mixture as a function of composition (Toop and Samis, 1962a,b; Allanore, 2013, 2015 and references therein; Moretti, 2020 and references therein). Nevertheless, silicate melts still lack a fully developed acid–base framework formalizing the thermodynamic properties of reactive species formed during the solvolysis, as the solvent itself changes its polymerization properties upon introduction of other oxide components, which are highly soluble contrary to what observed for salts in aqueous solutions. The most general thermodynamic approaches postulate mineral‐like molecular structures to interpolate existing data.

In molten silicates the electric charge is primarily transported by cations, whose contribution increases with concentration of network modifiers, hence basicity. The major element, oxygen in its three forms and particularly O^2– and O‐based anionic complexes do not contribute substantially to the charge transport (Dickson and Dismukes, 1962; Dancy and Derge, 1966; Cook and Cooper, 1990, 2000; Cooper et al., 1996a,b; Magnien et al., 2006, 2008; Cochain et al., 2012, 2013; Le Losq et al. 2020). This is a striking difference when comparing silicate melts to previously described electrolytes, particularly with the aqueous electrolytes, in which the concentration of hydroxide ions or water is always large enough to sustain high current densities (Allanore, 2013). From a practical standpoint the anode reaction producing oxygen (the inverse of Reaction 1.6) is strongly impacted by the low transport of free oxide ions, that is free oxygen, which is at very low concentration. Besides, the anode design and technical performance would greatly benefit of the precise knowledge of oxygen physical chemistry in silicate melts, which for the moment is still too limited to narrow compositional ranges that have been investigated spectroscopically (Allanore, 2015).

Nevertheless, because of their nature, silicate melts can dissolve important amounts of metals. Besides, they exhibit a large range of thermal stability, with high temperature conditions that favor fast kinetics of redox exchanges. Upper temperature limits for electrochemical applications are given by the formation of gaseous silicon monoxide or by alkali oxide thermal decomposition in alkaline systems or by high vapor pressure for Mn‐bearing systems (Allanore, 2015). In terms of transport properties, silicate melts are a solvent with high viscosity, a fact that is however compensated by the high diffusivity of the metal cation (Allanore, 2013), i.e., the cathode reactant.

Many measurements have however been carried out on melts of geological interest, also fostered by the interest in silicate electrolysis to produce on site metals and particularly molecular oxygen for terraformation of extraterrestrial planets (e.g., Haskin et al., 1992). Electrochemical series were then established in binary SiO₂‐MO systems (e.g., Schreiber, 1987) but also in ternary joins such as the diopside one (Semkow and Haskin 1985; Colson et al., 1990) for redox exchanges of the type:

(1.39)

in which half‐reactions of the type

(1.40)

combine with Half‐reaction 1.7. Nevertheless, such series do not consider the effect of the solvent, the melt and its structure, in determining the speciation state (e.g., anionic or cationic) following the definition at Reaction 1.27. The effect of the solvent also includes the amphoteric behaviour of some dissolved oxides such as Fe₂O₃ or Eu₂O₃, which can behave either as acids, yielding FeO₂^– (i.e., FeO₄^5– tetrahedral units) and EuO₂^– (i.e., EuO₄^5–) or bases, yielding Fe³⁺ and Eu³⁺ cations (Fraser, 1975; Ottonello et al., 2001; Moretti, 2005; Le Losq et al., 2020). The multiple speciation behaviours determined by pO^2– can be summarized by the following reaction mechanism (e.g., Moretti, 2005; Pinet et al., 2006):

(1.41)

Predominance and stability diagrams (e.g., E‐pH, E‐pO^2–, E‐logfO₂, or logfO₂ vs. the log‐fugacity of pH or other gaseous species in the system such as SO₂, CO₂) depend on the availability of good thermodynamic data and especially a well‐established testament of acid‐base properties of the investigated system and its solvent(s). For silicate melts and glasses, such a testament is represented by the oxobasicity scale from the Lux definition (Reaction 1.27). Electrochemical experiments should then be envisioned to complete and validate the database in order to ensure predictions about forming species and measure their activities.