Kirja

An electrochemical reaction involves always electron transfer between species. Such a reaction is called a redox reaction. It is defined:

A substance capable of oxidizing another substance is knows as an oxidant. Accordingly, a reducing substance is a reductant. If we write down a reaction

where n electrons are transferred from B to A, A is the oxidant of B and B is the reductant of A. Hereinafter in this book, the subscripts ’ox’ and ’red’ refer to the oxidized and reduced species, respectively.

The ability of a substance to oxidize or reduce another substance depends on their electron structure and, to a lesser extent, on the medium. This ability is reflected in the values of the standard reduction potentials that are tabulated for several elements and molecules. Standard potentials are discussed in Chapter 4, and a selection of the most common redox couples is given in Appendix 6. From the standard potentials it is possible to deduce which reactions would occur spontaneously or how much energy must be fed in to the system to force a reaction to take place.*

If an electron transfers from a species in the solution to another species without a mediating electrode, electron transfer is homogeneous. For example, if an aqueous solution of CeCl₄ and FeCl₂ are shaken for a while, Ce⁴⁺ oxidizes Fe²⁺ to Fe³⁺ reducing to Ce³⁺. In biology, redox reactions are mostly homogeneous. Metals can also be reduced from aqueous solutions with strong reductants without an external circuit. One of the best known processes is the deposition of nickel from a hypophosphate solution. This kind of processes are known as electroless plating. They do not, however, represent true homogeneous electron transfer because a solid surface mediates electrons and other simultaneous reactions compensate the transferred charges.

A redox reaction is electrochemical if electron transfer takes place heterogeneously, most commonly on solid electrodes.

An alternative definition would be

Electrodes traditionally used in electrochemistry are made of solid materials that have sufficient electrical conductivity, such as metals, graphite or indium-titan oxide glass (ITO). An electrode exchanges an electron with a reacting species, and receives an electron from an external circuit or donates an electron to it. The surface of an electrode can be modified with functional groups in order to enhance its selectivity towards a particular reacting species.

As a result of a redox reaction, an imbalance of charges is created in the solution, and another electrode must be inserted in the system to maintain electroneutrality in the solution. The electrodes must be connected via an external circuit. Reaction (1.1) can now be written in terms of electrode reactions as follows:

Summing reactions (1.2a) and (1.2b) gives reaction (1.1). In reaction (1.2a) the electrode donates n electrons to A_ox: it is a cathode. In reaction (1.2b) the other electrode receives the same number of electrons: it is an anode.

Reactions (1.2) and (1.2b) are the half-reactions of reaction (1.1). If we want to study, say, the rate of the half-reaction (1.2a), the cathode is called the working electrode, and the anode that only balances the number of charges in the solution the counter electrode or the reference electrode. In modern electrochemical measurement, the counter and reference electrodes are separate – this will be discussed in Chapter 4.7.

Electron transfer is a fundamental reaction which is very difficult to describe with quantum mechanics. Theories of electron transfer have traditionally been based on the famous Arrhenius equation or the absolute rate theory (aka theory of activated complexes) by Eyring. The rate of electron transfer, in other words electric current, is, however, easily measured and controlled in situ, making electrolysis a widely used chemical process.

1.2.1 Cell voltage due to an electrochemical reaction

Let’s write down a very simple and familiar reaction:

It is found from thermodynamic tables that the standard Gibbs free energy of the reaction is \( \Delta \)G⁰ = –237 kJ mol^–1. The reaction does not appear electrochemical but it is the net reaction of the hydrogen fuel cell in acidic conditions. At the anode of the fuel cell, hydrogen decomposes by means of a catalyst (oxidizes):

Protons are transported to the cathode through the electrolyte and electrons via the external circuit. At the cathode, oxygen is reduced:

Two electrons are transported per water molecule. Hence, the standard cell voltage (or potential) is

          \( \displaystyle E^0=\frac{237000\text{ J/mol}}{2 \cdot 96485 \text{ As/mol} } \)

the thermodynamic (= equilibrium) potential of the hydrogen fuel cell**.

Thus, in electrochemistry, a homogeneous reaction is split into two heterogeneous reactions and the Gibbs free energy of the reaction, \( \Delta \)G, is harnessed as electrical energy. In a reverse process, electrical energy can be used to run reactions against their natural affinity. The question of how \( \Delta \)G of the total reaction is split between the half-reactions remains still unanswered. As will be seen in Chapter 4, thermodynamics is incapable of answering this question, but it is resolved with an IUPAC resolution. This does not actually matter because measuring a single electrode potential, i.e. its energy, is not possible, only the potential difference between two electrodes can be measured.

1.2.2 Electric current due to an electrochemical reaction

where i is electric current density (A cm^–2), n the number of electrons exchanged in the reaction, F the Faraday constant (96485 As mol^–1) and r the reaction rate (mol cm^–2 s^–1). The direction of electric current is defined as the direction of positive current, i.e. opposite to the direction of electrons. In a solution, the current flows in the direction of the positively charged species. When representing current-voltage curves in x-y coordinates, it is still necessary to agree if an oxidation or reduction current is positive: according to the IUPAC resolution, an oxidation current is positive. Many text books, however, especially ones about polarography, show cathodic current (reduction current) as positive. Great care must therefore be taken when reading literature and noticing the sign convention used by authors.

It is necessary to emphasize that an electrochemical reaction is the only reaction, the rate of which can be measured in situ. With modern current amplifiers, the measurement of pA (10^–12 A) currents is rather easy. Applying the Faraday law it is simple to calculate that this corresponds to the transfer of the order of 10⁶ electrons per second.

Since electric current is a property of a closed circuit, its magnitude depends on the rate of the slower reaction. Let’s assume that the reduction reaction (1.2a) is faster than the oxidation reaction (1.2b). If the electrodes in Fig. 1.1 are short-circuited, electric current density is i = nFr_ox because the rate of the oxidation reaction limits the rate of electron flow. Superficially thinking, this observation may seem self-evident but it has a huge impact in the production of metals via electrolysis where oxygen evolution is the anode reaction. The rate of oxygen evolution (inverse of reaction (1.5)) is limited by reaction kinetics, increasing the cell voltage by several hundred millivolts which, in turn, increases the energy cost of the process significantly.

1.3.1 Electron structure of solid materials

Electrodes are typically solid, metallic materials. In order to understand reactions taking place on them, it is useful to examine the structure of solid metals. A simple approach is to consider the structure as an ensemble of tightly packed lattices of atomic nuclei and electrons that are partly moving freely within the lattice. Let us concentrate on the cloud of these free electrons and study its emergence with an example. Lithium is the lightest alkali metal, its electron configuration is 1s²2s¹. It has one electron more than a noble gas (e.g. He). Two lithium atoms form a molecule Li₂. The molecular orbital theory tells that from the two atomic 2s orbitals of lithium, two molecular orbitals are formed in a Li₂ molecule: a binding orbital \( \sigma \)_2s that resides between the Li nuclei, and an anti-bonding orbital \( \sigma \)*_2 s on the opposite sides of the nuclei. According to the Aufbau principle, the 2s electrons of the lithium atoms that participate in the bond formation are filling the binding molecular orbital; the four 1s electrons stay on their atomic orbitals. Bringing more Li atoms into the system means more molecular orbitals are formed, including anti-bonding orbitals. Upon this process, the energy levels of the bonding orbitals get closer to each other, ultimately forming a continuous energy band when the number of atoms becomes high (molecule Li_N), see Figure 1.2. Binding electrons, one electron for each Li atom, occupy only half of this band known as the valence band.

In terms of energy, below the valence band there is a band of 1s electrons that is fully occupied (not shown in Figure 1.2). Between this band and the valence band, there is an energy gap that is so wide that no 1s electron can rise to the valence band. Above the valence band there is a conduction band that is unoccupied at ground state. (Figures 1.2 and 1.3). In the case of lithium, it is formed from the molecular orbitals that are left unoccupied. Their energy is higher than that of the occupied orbitals by such small an amount that thermal energy (k_BT) is capable of raising electrons to them, making lithium electronically conducting. Metals, in general, are good electronic conductors because their valence electrons are delocalized within the entire lattice.

Considering alkaline earth metals such as magnesium in the similar manner, the situation changes somewhat. Molecular orbitals formed from atomic 3s orbitals would be fully occupied by the 3s electrons of Mg.  So why are alkaline earth metals also good electronic conductors? Let us continue with the example of Mg. Molecular orbitals are also formed from the atomic orbitals that are unoccupied at ground state (3p orbitals). Again, when the number of atoms is high, the energy levels of these unoccupied molecular orbitals form a continuous band. In magnesium, this band partly overlaps with the valence band, making the exchange of electrons between the valence and conduction band feasible, see Figure 1.3.

Electrical conductivity can be reduced to the issue of the band gap between the valence and conduction bands, as shown in Figure 1.3. If the band gap is of the order of 5 eV or higher, the material is an insulator because thermal energy (k_BT = 0.025 eV at 300 K) is not capable of raising an electron from the valence band to the conduction band. If the band gap is of the order of 1-2 eV, some electrons can be raised to the conduction band, and the material is a semi-conductor. The conductivity of semi-conductors increases with temperature, in other words thermal energy, while the conductivity of metals decreases because the thermal motion of the metal nuclei interferes the free transfer of electrons. This is used as the definition of a metal.

1.3.2 Fermi levels and work function in solids

Electrons are fermions obeying the Pauli exclusion principle that two electrons cannot have the same quantum state in a system. As the consequence, electrons do not occupy the energy levels according to the Boltzmann but the Fermi-Dirac distribution. The probability P of finding an electron on an energy level E is

where E_f is the Fermi energy. At absolute zero (0 K) the energy levels are occupied according to the Aufbau principle, starting from the lowest energy level. The energy of the highest occupied molecular orbital, HOMO, is known as the Fermi energy that is also equal to the electrochemical potential of an electron. As Equation (1.8) manifests, at 0 K the probability of finding an electron on an energy level  E < E_f  is unity and on a level   E > E_f  zero. Since the electrochemical potential of an electron in a metal does not change significantly as a function of temperature, it has been common to call the energy where the step function occurs the Fermi level. At higher temperatures, however, Fermi-Dirac distribution is no longer a step function but changes smoothly, as shown in Figure 1.4. Yet, the distribution takes values different from unity or zero only around the energy where P(E) = 0.5, which is the definition of the Fermi level. The Fermi level is sometimes defined as the lowest energy of the conduction band of a solid which makes it possible to give numerical values to the Fermi level with respect to a reference point.

In solid state physics, metals are often sorted according to their work function, \( \Phi \), which is the work (energy) required to remove an electron from an uncharged metal, that is, from its Fermi level. The work function can be measured via, for example, the photoelectric effect.  The work function is in practice a temperature-independent quantity, but it is important to note that it does depend on the lattice structure of a material. Work functions of metals vary from 2.3 eV of potassium to 5.3 eV of gold.

1.3.3 Potential and voltage

Potential is one of the most versatile and salient quantities in electrochemistry. Voltage is often used as a synonym for potential, but voltage is a measurable quantity while only potential differences can (possibly) be measured. Voltage can thus be defined as the potential difference between two points that can be a sum of several contributions. The free energy of the net electrochemical reaction is converted to the cell voltage according to Equation (1.6) which is the sum of the contributions of the anode and cathode reactions. Free energy is the driving force of a reaction and a measure of its tendency to take place, and we know from thermodynamics that spontaneous reactions have negative Gibbs free energies. Therefore, from Equation (1.6), positive cell voltages mean spontaneous reactions that produce energy, for example in a battery.

Although thermodynamics is not capable of resolving the free energy of individual electrode reactions, but an assumption is made to fix the potential scale (Chapter 4), we can elaborate the potential of a solid electrode a little bit further. The potential of a solid electrode – be it metal or semiconductor – can be considered a measure of the Fermi level of electrons in the electrode. From electrostatics we know that an extra electric charge creates potential. If an electrode carries a net negative charge its potential is negative with respect to an uncharged electrode or to an electroneutral solution. A positive charge obviously means positive potential. With electrochemical instrumentation, we can control the electrode potential externally which means controlling the Fermi levels or, alternatively, the charge of an electrode.

Let us consider further potential related quantities encountered in electrochemistry. The inner phase of any material is electroneutral. If an object carries an electric charge it is spread over its surface. The potential measured on the surface of an object or in its immediate proximity, with respect to vacuum, is the outer potential, \( \ \psi \). This is the work required to bring an elementary charge from an infinite distance in a vacuum to the surface of a phase (see Figure 1.5). The outer potential can be measured using a Kelvin probe. It follows from the previous paragraph that a negatively charged object has a negative outer potential and a positively charged one a positive outer potential. The potential difference between two charged objects is also known as the Volta potential difference.

Figure 1.5. Left: Definitions of the inner or Galvani potential (\( \phi \)), surface potential (\( \ \chi \)) and outer potential (\( \psi \)). Right: Galvani potential difference between two phases.

\( \chi \) is the surface potential, the work required to bring an elementary charge from the surface of an object into its interior. The surface potential cannot be measured. The origin of the surface potential is the difference in the molecular (or atomic) interactions on the surface and in the interior of a material. In solids and liquids, molecules (atoms) residing on the surface experience asymmetric attraction towards the interior that gives them higher energy than that of the bulk of the material. Inserting a charged electrode into a polar solvent, the solvent molecules are oriented in a particular manner, creating surface dipoles. Now surface potential is created across the layer of these dipoles. The inner or Galvani potential, \( \phi \), is the total work required to bring an elementary charge from a vacuum to the inside of a phase. Based on what is stated above (and Figure 1.5), the Galvani potential can be expressed as the sum of the outer and surface potentials:

Galvani potential cannot be measured. On the right of Figure 1.5, the Galvani potential difference is defined. That can be measured under certain experimental arrangements (Chapter 4). Because the interiors of the both phases are electroneutral, Galvani potentials are constant within the materials and the Galvani potential difference is always formed at the interface between the phases.

1.3.4 Electron transfer on an electrode

Let us consider the reduction reaction (1.2a) in a qualitative manner. Species A has an electronic structure based on the molecular orbital theory, as depicted in Figure 1.6. In order to reduce A, an electron must be transferred to its lowest unoccupied molecular orbital (LUMO). In Figure 1.6a, the Fermi level of the electrode is not sufficiently high to inject an electron to A, but as the electrode potential is changed so that the Fermi level is set higher than that of the LUMO of A (Figure 1.6b), and electron transfer becomes feasible. It is necessary to emphasize that an electron in the solution, or merely in the solute dissolved in the solvent, has no unambiguously defined energy. It is commonly assumed that its energy demonstrates Gaussian distribution and depends on the solvent-solute interactions. The Fermi level of an electron in the solution is defined as its electrochemical potential (see Chapter 6).

The rates of electron transfer can vary in the range of several orders of magnitude. This property can be quantified as the polarizability of an electrode. Polarizability is simply a measure of the dependence of the electron transfer rate on the electrode potential. An ideally non-polarizable electrode does not change its potential when current is flowing through it, while the potential of an ideally polarizable electrode can be varied without current flowing, see Figure 1.7.

All the entities existing in a system – ions, molecules, electrons etc. – are called species, while components are those species the amount of which can be varied independently. For example, an aqueous solution of acetic acid contains the species H⁺, OH^–, H₂O, CH₃COO^– and CH₃COOH, but only two components, H₂O and CH₃COOH. A proton (or more like H₃O⁺) and a hydroxyl ion OH^– are coupled with the ionic product of water:

The dissociation constant of acetic acid couples H⁺, CH₃COO^– and CH₃COOH:

Furthermore, electrolyte solutions obey the very strong condition of (local) electroneutrality:

The general statement is that if a system contains N species, bound by m reactions and p other contraints, the number of components, M, is

In acetic acid, N = 5, m = 2 and p = 1, hence M = 2. Equation (2.4) can be generalized to the Gibbs phase rule. Assume that acetic acid is distributed between an aqueous and an organic solvent phase (o). In addition to the above mentioned aqueous species, the entire system contains (at least in principle) the species H⁺(o), OH^-(o), CH₃COO^-(o), CH₃COOH(o) and the organic solvent, making N equal to 10. The mutual solubility of water and the organic solvent is assumed to be negligible. Yet an extra variable is the Galvani potential difference created across the phase boundary. The constraints are two reactions, the electroneutrality condition and four partition equilibria. The Galvani potential difference can be eliminated with the total mass balance of acetic acid. M thus is three, the components being water, acetic acid and the organic solvent. Electroneutrality in water implicitly implies electroneutrality in the organic phase too. Writing ionic equilibria similarly to (2.1) and (2.2) in the organic phase does not result in extra constraints either. Furthermore, it should be noted that the Gibbs-Duhem equation

gives an extra constraint that in the both phases there is only one thermodynamically independent component, the solvent or acetic acid, hence two totally independent components.

We already introduced the chemical potential \( \mu_i \) of component i in the Gibbs-Duhem equation. It is defined as

where G is the Gibbs free energy of the solution. For ions, an electrochemical potential can be defined as

where F is the Faraday constant, z_i the charge number, \( \mu_i^0 \) is the standard chemical potential and a_i is the activity of ion i, respectively. Thus, for an uncharged species \( \tilde{\mu_i}=\mu_i \). Definition (2.6) is not, however, relevant for an ion because it is not possible to change the concentration of a single ion  while keeping the concentrations of the other ions constant.

The fundamental Gibbs equation expresses the change of the system’s internal energy dU as

where q is heat evolved from the system or exchanged with its environment, and dw is the work done against the system. In addition to the PV work it can contain, for example work against surface tension, \( \gamma dA \), or electrical work, \( \phi \)dq_e. (\( \gamma \) is surface tension, q_e electric charge and \( \phi \) electric potential.)

The other thermodynamic state functions, H (enthalpy), F (Helmholzin free energy) and G are easily derived from Equation (2.8):

In all these functions, the term \( \sum\limits_{i}\mu_idn_i \) can be added that takes into account the amount of material in the system. Adding the term that takes into account ions,\( \sum\limits_{i}\tilde{\mu_i}dn_i \), into Equation (2.11) gives

where the definition of the electric charge \( q_e=\sum\limits_{i}z_in_i \) is used. In an electroneutral system \( q_e=0 \), but as will be seen later on, the concept of the electrochemical potential is most useful. The Galvani potential \( \phi \) also is conceptually a bit problematic; in metals it is related to the Fermi energy of electrons (see paragraph 1.3.3). Anyway, applying the equality of the electrochemical potential between the phases at equilibrium, Galvani potential differences between phases and the electromotive force (Chapter 4) can be derived. In the Gibbs-Duhem Equation (2.5) the chemical potential can thus be replaced by the electrochemical potential.

2.3.1 Activity of non-electrolytes

In the expression of the chemical potential (2.6), activity a_i is introduced. It describes the ‘effective’ concentration of a solute, i.e. its deviation from ideal behavior. Depending on the concentration scale, the chemical potential can be written in different ways:

The molality and molarity scales are identical in practice, except at very high concentrations. We will mainly use the latter one; γ_i is the activity coefficient and c_i the concentration of the solute. It can be proven that the relation between the mole fraction and molarity scales is

where \( \overline{V} \) is molar volume of the solvent. Equation (2.14) is valid at relatively dilute solution where the ratio of activity coefficients f_{i :}γ_i ≈ 1. The chemical potential of the solvent is usually expressed on the mole fraction scale because when the mole fraction of the solvent x₀ → 1 also f₀ → 1, and the chemical potential approaches its standard value (Raoult’s law). For the solute f_i → 1 when x_i → 0 (Henry’s law). The standard state of the molality scale is a hypothetical solution where the molality of the solute is m* = 1.0 mol/kg and y_i = 1. The standard state of the molarity scale is also a hypothetical c* = 1.0 mol/dm³ and γ_i = 1. The chemical potentials should more likely be written

but for the sake of simplicity we leave the standard concentrations out and keep in mind that concentrations are written in mol/kg or mol/dm³ units.

2.3.2 Activity of electrolytes

Let’s consider a strong electrolyte \( \text{A}_{\upsilon+}\text{B}_{\upsilon-} \)that dissociates into ions A^z+ and B^z-.  The electroneutrality  condition now becomes \( \upsilon_+z_++\upsilon_-z_-=0 \). The chemical potential of the electrolyte is the sum of the ionic ones:

Equation (2.16) defines the mean electrolyte activity a_± and how it depends on the stoichiometric coefficients; \( \upsilon=\upsilon_++\upsilon_- \) . Since activity = concentration × activity coefficient,

for 2:1 electrolyte (e.g. CaCl₂, \( \upsilon_+=1, \upsilon_-=2, z_+=+2,z_-=-1 \)),

Biosystems always contain polyelectrolytes such as proteins, polysaccharides, polypeptides or DNA. A polyelectrolyte is a large-sized molecule that can have the molar weight as high as millions and hundreds charged groups. Polyelectrolytes have counter-ions, typically H⁺, Na⁺ or Cl^- that compensate its charge. Assume a polyelectrolyte Na₁₀₀P in the concentration c, i.e. a polyanion P^100- with 100 Na⁺ cations making it electroneutral. From Equations (2.17) and (2.18) it would be obtained that

If the mean activity a_± is measured, say, with the osmotic pressure or electrochemically, the mean activity coefficient \( \gamma_{\pm} \) can be calculated using the equation above. Without discussing this issue in greater detail, we are content to note that the activity of the counter-ion is usually much lower than its stoichiometric concentration because they do not exist in the solution as free ions due to extensive ion-binding to the polyanion. The mean activity coefficient of the polyelectrolyte can reach absurd values if Equation (2.18) is applied as such. Further discussion of this question is beyond the scope of this book.

Melting points of common salts are very high because the ions in the lattice are bound together with very strong electrostatic forces. They can dissolve, however, in polar solvents at rather high concentrations. The reason for this is the ability of the solvent molecules to solvate, i.e. to gather around ions, mainly using electrostatic forces. In the case of water, we talk about hydration and hydration number that is the number of water molecules bound by salt. For LiCl, the hydration number is 7, NaCl 3.5 and MgCl₂ 14. The hydration number is often assigned to the cation.

Free energies of hydration are negative and rather large, of the order of -300 to -4000 kJ/mol, which reflects the strength of the interactions between ions and water. The solubility of salts decreases with the relative permittivity, i.e. the polarity of the solvent, and in non-polar solvents common salts practically do not dissolve at all.

The interaction between a solvent and an ion is traditionally estimated with the Born model that, despite being very simple, gives a good picture of the electrostatics of solvation. In addition to the electrostatic interaction, there are also other forces between ions and solvents that are addressed in, for example, the scaled particle theory [1].

In the Born model, ion solvatation is thought to take place in three steps:

1. Uncharging the ion in vacuum.

2. Bringing the neutral species into the solution.

3. Charging the species in the solution.

The work to charge a species in vacuum is (a = ion radius)

The electrostatic energy of ion solvation (IS) is therefore

The electrostatic energy is, however, so dominating and negative that solvation is a spontaneous process. Assume that a = 0.1 nm, z = 1 ε_r = 78.4, p = 1 atm ja γ = 0.0728 J/m². The electrostatic energy is now -7.1 eV = -684 kJ/mol, while Step 2 has energy of only 0.057 eV = 5.5 kJ/mol. Although there are other forces in Step 2, we are no longer concerned with them.

If an ion is partitioned between an aqueous and an organic solution, Equation (2.27) can be applied via a thought process where an ion is brought from a vacuum into the both phases. The difference between the solvation energies becomes

Since \( \Delta G=\Delta H-T\Delta S \),

Table 2.1. Standard Gibbs free energy, enthalpy and entropy of ion hydration and the molar volume in water for selected ions. From: Y. Marcus in, A.G. Volkov and D.W. Deamer (Eds.), Liquid/liquid interfaces. Theory and methods, CRC Press, Boca Raton, 1996. With the Publisher’s (CRC Press) permission.

2.5.1 Mean activity coefficient

Activity coefficients of non-electrolytes are close to unity even at moderately concentrated solutions while long-range electrostatic forces between ions mean the activity coefficients of ions must be taken into account even at rather dilute concentrations (c > 1 mM). Ion-ion interaction is traditionally estimated using the Debye-Hückel theory whereby the solvent is taken as a dielectric continuum with a single static relative permittivity \( \varepsilon \)_r.

Around an arbitrarily chosen central ion there are more ions with an opposite charge than ions with a like charge.

Ions are arranged in the electric field of the central ion according to the Boltzmann distribution:

The solution of Equation (2.33) in spherical coordinate is (φ(r) = 0 when r → ∞)

The constant C is solved from the condition that the ion cloud around the central ions compensates its charge z_ce:

Here, a is the distance

of closest approach to the central ion. Using straightforward algebra, the constant C and the potential distribution is obtained as

Figure 2.3. Scaled potential e^−κr as the function of the distance from the central ion at varying concentrations (1-100 mM) of a 1:1 electrolyte; central ion assumed a point charge, a = 0.

In Figure 2.3 above, the potential profiles have been shown as the function of the distance from the central ion. The reference value is the potential created by a single ion

When a solution is diluted from concentration c₁ to concentration c₂, the work of dilution is

Assuming that electrostatic interactions is the root of an activity coefficient, the latter term in the above equation is equal to w_ion-ion. Since γ₂ = 1 (assuming infinite dilution)

It is customary to write Equation (2.41) down with the Briggs logarithm:

\( \displaystyle A= \frac{1.8246 \cdot10^6 }{(\varepsilon_rT)^{3/2}} \) mol^-1/2dm^3/2K^3/2 ;   \( \displaystyle B=\frac{50.29 \cdot10^8}{(\varepsilon_rT)^{1/2}} \) cm^-1dm^3/2K^1/2  

Applying the definition of the mean activity coefficient (2.16) the following result is reached:

The parameter a is known as the distance of closest approach, the minimum distance between a cation and an anion. In aqueous solutions, the Debye-Hückel model works satisfactorily up to a concentration of 0.1 M. Often only the numerator of Equation (2.43) is used (Debye-Hückel limiting law) if c < 0.01 M. Various more or less empirical equations have been suggested for more concentrated solutions; perhaps the best known is the Davies formula that takes the following form for aqueous solutions at 298 K:

2.5.2 Ion pairing

In solvents with low relative permittivity, ions tend to form ion pairs. Although an electrolyte might dissolve completely in the solvent, it does not dissociate completely into free ions. Dissociation constants can be determined experimentally using, for example, conductivity measurements. The dissociation constant of an electrolyte \( \text{A}_{\upsilon+}B_{\upsilon-} \) , K_d is

where \( \alpha \) is the degree of dissociation and \( \upsilon=\upsilon_++\upsilon_- \) The activity coefficient of ion pairs is usually taken as unity because they carry no charge. Since α often is rather low, making the ionic strength also low, the mean activity coefficient \( \gamma_{\pm} \) is close to unity. In that case, for a 1:1 electrolyte applies K_d = α²c/(1 - α).

Ion pairing (or ion association) is traditionally analyzed using the theories of Bjerrum or Fuoss. The Bjerrum theory assumes that close to an ion the potential profile is essentially V(r), Equation (2.37). The number of ions i residing in the sphere element of the thickness of dr is therefore

Function (2.46) has a minimum at the point

Bjerrum postulated that all ions with the sign opposite to that of the central ion within a distance smaller than q form an ion pair. At the distance q the electrostatic energy of an ion is

The fraction of paired (associated) ions, 1 - α, is therefore

With the change of variables x = 2q/r the above equation can be written in the form of

The integral (Bjerrum integral) in the above equation must be evaluated numerically. The ion-pairing (association) constant K_a (= 1/K_d) is relatively easy to calculate. Assume an ion-pairing reaction A⁺ + B^- ↔ AB. Now, from Equation (2.45)

Because K_a is the same at each concentration, in a very diluted solution where α ≈ 1  and  \( \gamma_{\pm} \) ≈ 1, K_a ≈ (1 - α)/c. Inserting Equation (2.50) into this and changing the species number density N_i^b (m^-3) to the molar concentration gives

where N_A is the Avogadro number, 6.023×10²³ mol^-1. The weakest point in the Bjerrum theory is its rather arbitrary choice q = 2k_BT. For a 1:1 electrolyte in water at room temperature (298 K) the value of K_a is ca. 1.27 dm³/mol (a = 0.2 nm), proving that common electrolytes do not make ion pairs in water to a significant extent.

The Fuoss theory presumes contact between ions in order to form an ion pair. Let’s consider again a 1:1 electrolyte. Let the number of free ions in the solution be N₊ = N_- = N₁ and N₂ the number of ion pairs. After adding δN anions and cations in the solution a fraction of them remains free and a fraction makes ion pairs. The probability that an anion remains free is proportional to the amount of δN added and to the volume that is not occupied by cations:

where V is the total solution volume and ν₊ the volume of a cation. Accordingly, the probability that an anion makes an ion pair is, in addition to space occupied by cations, N₁v₊, and to the added amount, δN also proportional to the Boltzmann factor e^-E(a)/kBT:*

The electrostatic energy of an anion in contact with a cation, E(a) is obtained from Equation (2.37). Let the number of anions that make an ion pair be δN₂ and the number of anions remaining free δN₁. A similar analysis can of course be done for cations. Due to electroneutrality, the number of cations that make ion pairs must also be δN₂, making the total number ion pairs 2δN₂. Probabilities are proportional to the numbers. Therefore,

Assuming that the solution is so dilute that the volume occupied by cations, N₁v₊, is insignificant with respect to the solution volume V, Equation (2.55) can be integrated with respect to N₁. The result is of course

The concentration of free ions, N₁/V, is in terms of molarity

Accordingly, the concentration of ion pairs is

Combining Equations (2.51), (2.57) and (2.58) the following formula is reached:

The mean activity coefficient is again assumed to be unity due to the dilute concentration.

2.5.3 Super acids and Hammett acid function[1]

In very strong acids, such as in pure sulphuric acid or in its mixtures with organic solvents, ion association and activity coefficients become rather ambiguous quantities. Yet it has been experimentally demonstrated that these kinds of solutions are very acidic, having a negative pH: they are called super acids. Analogously, mixtures of strong bases with organic solvents are vary basic, having a pH beyond the scale of ordinary pH sensors because they can make the sensor unstable or even destroy it.

For the analysis of super acids and bases, Hammett introduced an acid function H₀ that can be considered to be an extension of the common pH scale. In order to determine the value of H₀, a weak indicator base B is added to the super acid solution, and the concentration of B is determined with, for example, UV spectrophotometer. The base protonates to BH⁺, and its dissociation equilibrium is

Taking the negative of the Briggs logarithm as usual

Equation (2.62) defines the Hammett acid function H₀ that can be experimentally defined because the concentration ratio [B]/[BH⁺] can be measured with spectroscopy. For example, for water-free suphuric acid H₀ = -12.0. Nitroanilines are typical indicator bases. The values of their equilibrium constants (K_d) in solvents other than water pose a problem of its own that must be solved with various means of comparison and extrapolation⁶.

In alkaline solutions the corresponding function is H₋ that corresponds to the equilibrium

where n is the hydration number of the hydroxide ion. In pure water its value is approximately 5.5. In aqueous solutions the function H₋ can be estimated with the equation

In Equation (2.64), K^w is the ionic product of water and [H₂O] is the concentration of free water in the solution. Their values naturally depend on the solution composition.

When a semi-permeable membrane separates two electrolyte solutions of different concentrations, osmotic pressure is created across the membrane. It should be emphasized that an ideal osmotic membrane is not permeable to any species other than water. Osmotic pressure is formed across an ion-selective membrane, but since no real membrane is ideal, the tiny amount of co-ions* in the membrane allows for slow diffusion of salt across the membrane, ultimately dissipating osmotic pressure. In biology, osmosis is an important issue but a cell membrane is not, strictly speaking, an osmotic membrane. Ionic concentrations are maintained by active membrane transporters.

Despite of the above critical comments, let us consider the emergence of osmotic pressure across an ideal osmotic membrane. Osmotic pressure is formed due to the imbalance of the chemical potential of water. Because only water can cross the membrane, the equilibrium condition of water at a constant temperature (dT = 0) between two aqueous phases α and β is written as

The chemical potential of water is therefore a function of the concentrations of all species and the pressures of the phases. (Subscript ’0’ is reserved for the solvent and the others for solutes.) The pressure dependence of the chemical potential is obtained using Equations (2.6) and (2.11) (note that the order of differentiation is free):

where \( \overline{V_i} \) is the partial molar volume of species i. On the mole fraction scale

Equation (2.67) is formally reached by an integration between pure water and a solution with the mole fraction x₀ and activity coefficient f₀ of water. Equilibrium between phases \( \alpha \) and \( \beta \) is therefore

Water is incompressible, and therefore its partial molar volume is constant unless the concentrations in the two solutions are very different. Osmotic pressure is:

The ratio of the actitivy coefficients ≈ 1. Assume that phase β is pure water. Then

Equation (2.71) is known as the Van’t Hoff equation. Inserting, for example, c = 1.0 M = 1000 mol/m³, we see that \( \pi \) ≈ 24.5 atm (T = 298 K) corresponding to approximately 250 m height of a water column. This has led to the idea of a power plant that would harness osmotic pressure. An osmotic membrane is, however, very dense, making the flux of water very low. The power density hence remains very low, which makes such a power plant commercially unviable.

Equation (2.70) can be improved with an osmotic coefficient

that takes the unideality of a solution into account. Its values are tabulated for common electrolytes.

The correlation between the mean activity coefficient of an electrolyte and the osmotic coefficient is given without derivation in a binary system with the concentration c:

If γ_± is known in the concentration range [0,c], Φ can be calculated from Equation (2.73). Accordingly, if the osmotic coefficient is measured in the same range, γ_± can be calculated from (2.74). Measuring the mean activity coefficient is relatively easy using electrochemical methods (electromotive force of a Galvanic cell), and the osmotic coefficient can be evaluated from an experiment with an osmometer; the principle of such an experiment is depicted in Figure 2.6.

The Van’t Hoff equation can also be used to determine the molar mass of a macromolecule. Inserting a sample with mass m into the measurement chamber with the volume V, and measuring osmotic pressure, the molar mass M is obtained from the formula

In biological experiments it is important to keep solutions isotonic, i.e. their osmocity equal to that of a living cell. Osmocity S is defined as the concentration of NaCl solution that has the same freezing point \( \Delta \)T_f  as the solution being investigated. Osmolality O = ΔT_f /K_f  (Os/kg water) where K_f is the cryoscopic constant. ΔT_f = -K_f m where m is the molality of a solute (≈ c). For water K_f  = 1.86. The relation between the freezing point depression and osmotic pressure is (T = 273 K)

Equation (2.76) can be improved to also include the second term of the series expansion of Equation (2.70). In that case,

Table 2.2 shows a few constants of freezing point depression and boiling point elevation for selected molecules; ΔT_b = K_b m.

Table 2.2. Constants of freezing point depression and boiling point elevation for selected molecules. From: Handbook of Chemistry and Physics, 83. p. CRC Press, Boca Raton, 2002. With Publisher’s permission (CRC Press).

Quantities introduced in this paragraph are all colligative properties, i.e. the nature of the dissolved species does not, in principle, matter, only their concentration. A careful reader will note that osmotic coefficients are different for different electrolytes, which conflicts with the definition of the colligative properties. The theory presented above therefore only applies at moderately low concentrations.

In the Introduction of this book, we explained that the analysis of transport processes in electrolyte solutions constitutes the salient part of electrochemistry. Because electric current can occur only in a closed circuit, it also flows across the electrolyte solution that contributes to the properties of an electrochemical cell. More importantly, transport of electroactive species provides the boundary condition for the solution of electrochemical problems, as will be seen later.

From the point of view of transport, an electrolyte solution can be roughly divided into two domains: the interior or the bulk solution and the polarization layer close to the electrode surface. The bulk solution is chemically homogeneous, merely determining the total resistance of the cell via its conductivity. The thickness of the polarization layer is of the order of a few microns where concentration changes take place. These changes are due to electrode reactions. Let us consider a simple one-electron transfer reaction

where Cu²⁺ reduces to Cu⁺. As the reaction occurs on the electrode surface, the concentration of Cu²⁺ on the surface (x = 0) decreases and that of Cu⁺ simultaneously increases. Transport, in other words diffusion (and migration) tries to compensate for the emerging deficit of Cu²⁺ and excess of Cu⁺. The reaction rate, r, is a scalar while the flux of a species, J, is a vector although they both have the unit mol cm² s^-1. These two quantities with different tensorial dimensions are coupled via the mass balance and the Faraday law:

Equation (3.2) shows that an electrode reaction, i.e. electric current density, is the boundary condition of a transport problem. This is the specific feature – and the mathematical difficulty – of electrochemical analysis. In an electrochemical experiment, electric current, the boundary condition of a transport problem, is usually measured, not the concentration that is the quantity appearing in the transport equations. This means there is a need for the evaluation of concentration (and potential) gradients at the surface. Often numerical methods must be resorted to in order to solve a transport problem, and the calculation of numerical derivatives is prone to errors in accuracy.

The surface concentrations \( c_i^s \) adjust themselves so that the reaction rate equals to the rate of transport:

Let’s consider the situation in Figure 3.1: a particle (ion) with the charge q = ze feels a coulombic force \( F_c \) in an electric field E:

The coulombic force makes the particle move with the velocity v (no vector notation):

In Equation (3.6) F_f  is the frictional force and f  the friction coefficient for which Einstein derived an expression

At equilibrium, the friction force and the coulombic force cancel each other out, and we get:

Comparing Equation (3.5) with (3.8) it is immediately seen that

where \( \kappa \) is the conductivity of the solution. It can be written as the sum of the product of molar conductivities, \( \lambda \)_k, and concentrations, c_k, of all ions:

The above equation means that the mechanical and electrical mobility are assumed to be equal. Stokes' law defines the friction coefficient of a spherical object as

where η is the viscosity of the solution and a is the particle radius. Quite surprisingly, Stokes' law applies also to ions although they are of the same order of magnitude as solvent molecules, and although the law was derived via hydrodynamic considerations for macroscopic objects moving in a homogeneous medium. Assuming a as the ion radius, it follows from Equation (3.7) that

Multiplying this equation by η, only solvent independent constants are left on the right hand side: we have derived Walden's rule:

Walden's rule can be used to estimate the values of diffusion coefficients in solvents where there is no measured data. The rule does not, however, apply particularly well to ions in aqueous solutions due to their strong hydration (see Figure 3.2).

Figure 3.2. The product \( \eta\ \lambda_i \) for K⁺ (●) and Cs⁺ (■) cations in selected solvent: DMSO = dimethyl sulphoxide, NMF = n-methyl formamide, DMF = dimethyl formamide, THF = tetrahydrofurane, DMOE = dimethoxy ethane, ACN = acetonitrile. Note water! A.K.Kontturi et al., Ber. Bunsenges. Phys. Chem. 99 (1995) 1131. (P = poise = 0.1 Ns/m²).

The conductivity of a solution depends on the ionic concentrations, radii and viscosity of the solvent. Measuring the conductivity of the solution thus is – in principle – a simple means to determine ionic concentrations if molar conductivities are known, but there are a couple of problems. First, the above theory only concerns interactions between ions and solvent, i.e. it is valid in dilute solutions only. Second, an ionic radius in a solution is not an unambiguous quantity due to ion solvation. For example, the mobility of Li⁺ in water is lower than that of Na⁺, although its crystallographic radius is smaller. This is because Li⁺ is so strongly hydrated that it drags a mantle of water with it during transport. Polyatomic ions such as sulphate are not completely spherical either.

Nevertheless, measuring conductivity is one of the oldest electrochemical measurements, and data is available for common electrolytes in a wide range of concentrations and temperatures in aqueous solutions. The division of electrolytes into strong and weak ones, and classical theories of electrolytes are based on this experimental data.

Conductivity is measured with AC current in order to prevent concentration polarization. The electrodes are usually porous platinum plates to minimize the resistance that may occur as a result of electrode reactions. The measured resistance of a conductivity cell (Figure 3.3), R_s is

3.2.1 Strong electrolytes

Ion-ion interaction is very strong compared with other intermolecular forces. It is inversely proportional to distance while, van de Waals interaction, for example, is inversely proportional to the sixth power of distance. Typical electrostatic energy between two ions can be as high as 250 kJ mol^-1, which is of the order of the energy of a covalent bond, while the energy of a hydrogen bond is only approximately 10-40 kJ mol^-1. Coulombic forces may cause some extra trouble in molecular dynamics simulations of electrolytes because it is not sufficient to compute the interactions with the neighboring molecules only but computation must be extended over several layers of molecules. Periodical boundary conditions are often needed because electrostatic forces can stretch over the limits of the simulation domain.

Due to coulombic interactions the activity coefficients of electrolytes deviate from unity even in rather dilute solutions. For the same reason, ionic molar conductivity also very much depends on the concentration. The concentration dependence of a diffusion coefficient is obtained as (see paragraph 3.3)

For a 1:1 electrolyte (I = c = c_i):

Molar conductivity of an electrolyte, Λ, is the sum of ionic conductivities,

Hence, combining Equations (3.16), (3.25) and (3.26) Kohlrausch's law is achieved,

3.2.2 Weak electrolytes

The conductivity of weak electrolytes that only partially dissociate to free ions follows a different kind of dependence on the concentration (see paragraph 2.5.2 for a discussion of dissociation). Since the degree of dissociation, \( \alpha \), is often quite low, the concentrations of free ions also remain low, and their molar conductivities are practically equal to the values at infinite dilution. For a 1:1 electrolyte the conductivity \( \kappa \) is

Arranging the expression of the dissociation constant K_d = α²c/(1 – α) in the form of

a plot of Λ^-1 vs. Λc = κ is linear with the intercept \( \Lambda_ \infty^{-1} \) on the y axis; K_d can be calculated from the slope. Equation (3.30) is known as Ostwald's dilution law; it applies relatively well in aqueous solutions.

In organic solvents, electrolyte dissociation is nearly always incomplete. Constants A and B in the Debye-Hückel theory include the relative permittivity of the solvent ε_r (previously called the dielectric constant). Calculating their values it transpires that even at low ionic strengths (c < 10^-3 M) the mean activity coefficient deviates from unity. Ostwald's dilution law therefore does not apply as such but the activity coefficients need to be retained in the analysis.

Molar conductivity can be written in the form of (1:1 electrolyte)

Transport processes can be approached in various theoretical ways. The most traditional theory is thermodynamics of irreversible processes. In this book we use the most practical means, the Nernst-Planck equation. We bypass the simplifying approximation with which the Nernst-Planck equation can be derived from the theory of irreversible thermodynamics. The Nernst-Planck equation expresses an ionic flux, J_k (mol cm² s^-1), as the sum of three contributions:

The first term of Equation (3.33) relates to diffusion, the second one to migration and the third one to convection. The first term alone is equal to Fick’s first law stating that concentration differences tend to level out due to Brownian movement. The second term describes the movement of a charged species in an electric field, in other words electrophoresis. The velocity \( \vec{v} \) found in the convection term is practically equal to the solvent velocity. Convection comes from stirring the solution (forced convection) or density gradients formed in the solution (natural convection).

In the bulk solution no concentration gradients are usually found and, therefore, transport takes place only as a result of migration and convection or, in the absence of stirring, plain migration. In the polarization layer, diffusion plays the main role because, depending on the viscosity of the solvent, an unstirred stagnant hydrodynamic layer is formed on the surface of the electrode. Inert electrolytes that do not react on the electrode are usually added in an electrochemical cell. Such an electrolyte is commonly called as a supporting electrolyte because it carries most of electric current. As a consequence, the flux of an electroactive species is in practice solely diffusion; more detailed evidence is provided later on. In the absence of a supporting electrolyte, the mathematical analysis of transport becomes rather complex and the ohmic losses in the cell may grow in an uncontrollable way.

Equation (3.34) can be written in a compact form with the gradient of electrochemical potential:

Comparing the above equation with the Nernst-Planck equation it is seen that [1]

If the dependence of the activity coefficient on concentration is omitted, the first two terms of the Nernst-Planck equation are thus obtained. The convection term comes from the notion that flux can always be written in the form \( \vec{J}_k=c_k \vec{v}_k \) and interpreting \( \vec{v} \) as the solvent velocity.

The Nernst-Planck equation must be written out for each species in the solution. Constraints of the group of equations are the electroneutrality condition and electric current density:

The solution of the group is possible in closed form,² but let us highlight a couple of its important features with the help of a binary electrolyte \( A_{\upsilon_+}B_{\upsilon_-} \). Division by diffusion coefficients gives

Ionic concentrations are c₊ = \( \upsilon \)₊c and c_- = \( \upsilon \)_-c, and the condition of electroneutrality can be written as  z₊\( \upsilon \)₊ + z_-\( \upsilon \)_- = 0. Summing above equations, \( \nabla\phi \) is eliminated:

With equation (3.14) and electroneutrality, the left had side of Equation (3.39) can be written as

and equation (3.38) using straightforward algebra:

The coefficient of the concentration gradient is the electrolyte diffusion coefficient D_±:

Equation (3.42) is known as the Nernst-Hartley equation. It is important to notice that, in principle, only components, viz electrolyte diffusion coefficients can be measured, although an electrochemical measurement with a supporting electrolyte gives a very accurate approximation of an ionic diffusion coefficient – this case known as a trace ion case is discussed later. It is easy to use Equation (3.41) to check that z₊J₊ + z_-J_- = i/F. The coefficient in front of \( (i/F) \) is \( (t_i/z_i) \) where t_i is the transport number of the ion. It is a measurable quantity (see paragraph 3.4.1) that tells us how large a fraction of electric current the ion carries in the solution. Adding up the convective terms of either Equation (3.33) or (3.41) gives \( v\sum_iz_ic_i=0 \), which tells us something elser important: electric current does not depend on convection. [1]

The ratio of the electrolyte diffusion coefficient and the cation transport number gives:

Combining Equations (3.38) and (3.44), a very important result is achieved: [2]

Equation (3.46) states that the potential gradient in the solution is formed from two contributions, diffusion potential and ohmic loss. Diffusion potential arises from the differences in the ionic mobility. Although, say, a cation has a higher mobility than an anion (see Equation (3.5)), they must move at the same velocity in order to retain electroneutrality. This means that the anion is slowing down the movement of the cation and, accordingly, the cation is speeding up the movement of the anion. This is seen as diffusion potential. The magnitude of diffusion potential is, however, rather small because the ionic mobilities are usually of the same order of magnitude (H⁺ and OH^- are exceptions due to their different transport mechanism). Ohmic loss, in contrast, is always present in the measurements; it is irreversible in nature because it leaves a system as a heat loss (Joule heat). Check with a 2:1 electrolyte that Equation (3.46) truly gives the term \( 1/ \kappa \).

It is easy to show that Equation (3.46) can be written in the general form of

3.4.1 Measurement of transport number

The measurement of the transport number is one of the oldest electrochemical measurements. There are basically three methods: the Hittorf method, the method of a moving boundary and the electromotive force of a cell with transference. In the Hittorf method, cathode and anode compartments are connected via a thin capillary, with no concentration gradient allowed to be formed during the experiment. When a charge of \( Q=I\Delta t \) is run through the cell, Q/(z₊F) mol of cations has been removed from the cathode compartment but the amount of cations that have arrived is only t₊Q/(z₊F) mol. Their difference thus is \( \frac{(t_+-1)Q}{z_+F} \). A similar analysis can be done for the anode compartment. The transport number can be calculated by analyzing the composition of the cathode and anode compartments after the experiment.

The disadvantage of the Hittorf method is that the duration of an experiment cannot be very long in order to prevent a concentration gradient from forming in the capillary connecting the compartments. During a brief experiment only rather small changes in concentrations occur, meaning the chemical analysis may be inaccurate. Current cannot be increased to an unlimited extent either because it would dissipate in the cell as heat.

In the method of a moving boundary, a vertical capillary is equipped with a cathode on the top and an anode on the bottom. Two solutions with a common ion (M⁺X^– and N⁺X^–) are poured into the capillary, with the denser solution on the bottom. In order to keep the boundary between the solutions sharp, the mobility of M⁺ must be higher than that of N⁺. The boundary resides initially at position A (see Figure 3.7). The system is electrolyzed with the charge \( Q=I\Delta t \), causing M⁺ to move upwards towards the cathode. The amount of M⁺ transferred is (t₊Q)/(z₊F) mol; the boundary is now at a new position B. The amount of M⁺ removed from the interval A-B is \( c_{MX}A\Delta x \) (A is the area of the cross section of the capillary and \( \Delta x \) the distance A-B). Equating these two expressions we find that

Perhaps the most elegant method of determining a transport number is based on the electromotive force of a cell with transference, that is, its open circuit potential. For example, if two NaCl solutions at different concentrations are separated by a porous membrane or a frit that allows ions to diffuse across, Equation (3.46) gives the following for the diffusion potential (i = 0):

The transport equations presented above apply only in steady state where the flux AJ_i (mol s^-1) is not a function of the position but constant throughout the entire system; the flux density J_i (mol cm² s^-1) can vary depending on the cell geometry which is known as primary current distribution. In a non-stationary or transient state, solution concentrations vary as a function of location, and the flux is at its highest on the surface of the electrode. The non-stationary transport equation is reached calculating the divergence of the Nernst-Planck equation:

In the supporting electrolyte case which is overwhelmingly the most common in electrochemistry, \( \nabla\phi \approx\frac{i}{\kappa}=0 \). Convection only concerns in hydrodynamic methods that will be discussed later. In the absence of convection, Equation (3.53) reduces to Fick’s second law:

An electrochemical problem constitutes the solution of the above equation with the boundary condition (3.48) as well as other pertinent initial and boundary conditions, depending on the experimental method. These issues are addressed in Chapter 7.

Electrochemical cell consists of an electrolyte solution, electrodes and an external circuit as defined in Chapter 1.1. Electrochemical cells can be divided into two main types depending on the spontaneity of the cell reaction. The cell reaction in galvanic cells is spontaneous, and chemical energy converts directly into electric energy. In other words, the \( \Delta \)G of the reaction is negative and therefore the cell potential according to equation (1.6) is positive. In electrolysis cells, electrical energy is converted to chemical energy. The cell reaction is hence forced towards the non-spontaneous direction and the \( \Delta \)G of an electrolysis cell is positive and cell potential negative.

Galvanic cells are used in everyday life as primary and secondary batteries and fuel cells. An unfortunate side effect of galvanic cell is electrochemical corrosion. Examples of electrolysis are, for example, charging of batteries, production of chlorine gas and sodium hydroxide from electrolysis of sodium chloride solution, and different electrochemical analysis methods. The most important analysis methods and the requirements for the electrochemical cell used in connection with them are introduced in Chapters 7 and 8.

We know from thermodynamics that the \( \Delta \)G of a reaction gives the maximum work available from a particular reaction. In practice, electrochemical cells are also more or less non-ideal, and efficiencies of 100% are not achieved. Deviations from ideal behavior stem from irreversible processes that take place when electric current flows, e.g. ohmic losses due to restricted electrolyte conductivity (Figure 4.1) and possible slowness of electrode reactions, which is seen as overpotentials (Chapter 6.3).

In this chapter, we discuss the description of electrochemical cells, the conventions associated with them and the formation of the cell potential. Applications of cells in equilibrium, potentiometry and pH-potential equilibrium diagrams are also briefly introduced. Finally, one of the most important cell structures for electrochemical analysis, the three-electrode set-up, is presented.

A number of conventions are needed in the presentation of electrochemical cells. First, reduction will always take place at the cathode and oxidation at the anode. Electrodes are called cathodes and anodes in both galvanic and electrolysis cells. Chapter 1 states that it is impossible to measure the potential of any particular electrode: one always measures or controls the potential difference between two electrodes, i.e. the cell potential. It is not appropriate to call electrodes cathodes and anodes in electrochemical analysis. The reaction of interest takes place at the working electrode and the other electrode in the system is called the counter electrode. The purpose of the counter electrode is to maintain the electroneutrality of the solution: the reaction occurring at the counter electrode is opposite the one taking place at the working electrode (oxidation or reduction). The potential difference between these two electrodes, the cell potential, is easily measurable. As seen in the simplified scheme in Figure 4.1, the cell potential is the sum of different potential differences. In Figure 4.1, the cell potential is composed of the electrode potentials due to a potential change as a result of an electric double layer (Chapter 5) close to the electrode surface. We will show later how this potential difference controls the faradic reactions taking place at the electrode. In addition, an ohmic iR drop affects the cell potential when current flows in the system. This iR drop is caused by the limited conductivity or the resistivity of the electrolyte solution, which is a function of the ionic strength, the dielectric constant of the solvent, and the distance between the electrodes. Only if the iR drop (or loss) can be assumed to be negligible can the measured cell potential be taken as the potential difference between the electrodes. Furthermore, different liquid junctions caused by the structure of the cell (Chapter 4.4.) can complicate the situation.

Figure 4.1. Potential profile in an electrochemical cell.

In electrochemical potentiometric methods, the aim is to accurately measure the potential of the working electrode. In electrochemical amperometric methods, the aim is to accurately control the potential of the working electrode and to measure the current. Because the experimental quantity to measure or control is the cell potential, it is clear that in order to accurately know the potential of working electrode, the potential of counter electrode needs to be constant and the iR drop contribution has to be negligible or known.

Figure 4.2. Galvanic cell with a standard hydrogen electrode as an anode and a gold electrode in Fe³⁺/Fe²⁺-solution as a cathode.

An example of a galvanic cell is presented in Figure 4.2. The standard hydrogen electrode is an anode; there is an agreed convention that it is always presented on the left-hand side of the scheme. The standard hydrogen electrode is a platinum black (very fine platinum powder) plated platinum disc. Hydrogen gas (at pressure of 1 bar) is passed over the disc in solution where the H⁺ ion concentration is 1 M. A gold electrode is immersed in a solution containing Fe³⁺ and Fe²⁺ ions. The acidic solution in the anode compartment and Fe³⁺/Fe²⁺ solution in the cathode compartment are separated from each other with a porous diaphragm. The purpose of the diaphragm is to prevent the direct mixing of the electrolyte solutions as this can lead to undesired side reactions and/or make the theoretical treatment of the cell difficult. Ions will pass through the diaphragm and hence current flows. The connection between the two solutions can also be achieved by using a salt bridge (Chapter 4.4.1).

A cell diagram shows the parts of a (galvanic) cell in a concise manner. The different phases in the cell are written from left to right. A boundary between two phases is represented with | and a salt bridge (or any other liquid junction with a small junction potential) with ||. The cell diagram for the cell in Figure 4.2 is:

                                         Cu | Pt | H⁺, H₂ || Fe³⁺, Fe²⁺ | Au | Cu

The copper phases on both ends are the conductors, which connect the cell to the potentiometer.

The electrochemical reactions that take place in the cell in Figure 4.2 are:

Electric current is a property of a closed circuit. In this cell, electrons move from the hydrogen gas oxidising at the platinum anode through the external circuit to the cathode. At the cathode, electrons move back to the solution through the reduction of iron(III). In the solution, the current is carried by charged components, ions.

Combining reactions (4.1) and (4.2), the total cell reaction is:

When considering electrochemical net cell reactions, it is advisable to keep in mind that the reaction equation represents a virtual reaction. The reaction species are not in direct contact through collisions in the solution, but the half-reactions take place heterogeneously on their corresponding electrodes. The equilibrium depicted by the reaction equation could only be realized if the electrons could move freely in the external circuit. If the cell is in equilibrium, the net current in the cell would be zero and the cell potential would be also zero (equation (1.6), at equilibrium \( \Delta \)G = 0). In practice, the external circuit holds a high-impedance potentiometer (or load if a galvanic cell is used as a source of voltage), which restricts the flow of current in the circuit. The use of a potentiometer in the circuit therefore results in a situation in which very little current flows in the system. This virtual equilibrium is discussed in the next chapters. 

Let us consider the measurement of the potential of the cell in Figure 4.2. According to an agreement, the potentiometer (voltmeter) is connected to a galvanic cell in such a manner that the potential is defined as the potential of the right-hand side electrode with reference to the left-hand side electrode *:

The electrode with reference to which the potential is measured and therefore the value of the potential is defined is the reference electrode. In the cell shown in Figure 4.2, the reference electrode is the standard hydrogen electrode on the left.  The other electrode is the working electrode. It would often be useful to know the potential of a half-reaction. The measurement gives the total cell voltage, i.e. the difference between the half-cell potentials. In order to access the potential value for a half-reaction, another agreement is needed to divide the measured potential between the two half-reactions. This agreement and the measurement of standard potentials is presented in Chapter 4.3.

This chapter shows the derivation of cell potential for the cell in the previous chapter at virtual equilibrium. At equilibrium, the cell cannot be called a galvanic or electrolysis cell. Equilibrium is an important starting point for the examination of cell potentials, as the function of an electrochemical cell can be considered to approach thermodynamic equilibrium when the current flowing through the cell is close to zero. Potentiometric measurements are in practice very close to this situation. The cell potential (or cell voltage) is derived in two ways, first with Gibbs free energy change and equation (1.6) and second by considering the different equilibria at the phase boundaries through the concept of electrochemical potential.

The Gibbs free energy change for reaction (4.3) is:

and taking into account the relation between potential and Gibbs free energy change in equation (1.6) (n = 1, as seen from reaction equation) we get:

Let us derive the potential for cell in Figure 4.2 via an equilibrium treatment. At equilibrium, all the phases in an electrochemical cell that are in direct contact with each other are in thermic, mechanical and chemical equilibrium. An equilibrium between two phases that are in contact does not require the presence of all components and species in all phases, or the chemical potential of each component to be the same in all of the phases. For example, the equilibrium between a metallic gold electrode and Fe³⁺ and Fe²⁺  ions in the solution does not contain gold in the solution phase or iron ions in the gold phase. The equilibrium takes place through electrons that are able to move freely across the phase boundaries.

It should be noted that the concept of an equilibrium in the treatment of electrochemical cells corresponds in a way to an impeded cell reaction. This equilibrium in an electrochemical cell can be achieved only if the external circuit is open or the impedance in the external circuit is so large that an electrochemical equilibrium is maintained at the phase boundaries despite a small current flowing through the system. It is also assumed that the system does not have irreversible phenomena such as corrosion, or the system does not exhibit temperature, pressure or concentration gradients.

The phases of the example cell are depicted as Greek symbols in the cell diagram below:

Cu    |  Pt   | H⁺, H₂   ||   Fe³⁺, Fe²⁺  |  Au   |Cu’              

 \( \alpha \)        \( \beta \)           \( \sigma \)                 \( \varepsilon \)           \(\gamma \)       \( \alpha \) .’

At equilibrium, the cell exhibits equilibria across  the following phase boundaries: phase \( \alpha \) | phase  \( \beta \), phase  \( \beta \) | phase  \( \sigma \), phase  \( \sigma \) | phase  \( \varepsilon \), phase  \( \varepsilon \) | phase  \( \gamma \) and phase  \( \gamma \) | phase  \( \alpha \)’. The chemical potential of an ion is an ill-defined quantity and hence the derivation uses the equality of the electrochemical potential (equation (2.7)) as the condition for equilibrium. The equilibria in the cell are as follows:

Phase \( \alpha \) | phase \( \beta \) i.e. the contact between copper wire and platinum electrode:

Phase \( \beta \) | phase \( \sigma \) i.e. the equilibrium at platinum electrode (reaction equation 4.2):

Phase \( \sigma \) | phase \( \varepsilon \), the interface between two liquid phases, in practice often a salt bridge. In this simple derivation, we will assume that the inner potential of the solution is the same in both sides of the interface. Some cases where this assumption is not done will be covered later on (junction potentials, chapter 4.4). 

Phase \( \varepsilon \) | phase \( \gamma \) i.e. the equilibrium at the gold electrode (reaction equation 4.1)

Phase \( \gamma \) | phase \( \alpha \)’ i.e. the contact between copper wire and gold electrode:  

By using equation (2.7), these equilibrium conditions can be presented as:

Combining equations (4.12)-(4.15) and taking into account (4.9) the cell potential i.e. the galvanic potential difference between the two copper wires is obtained as (note that the chemical potentials of the wires are equal because the phases are identical):

which naturally corresponds to equation (4.5) that was obtained earlier. Let us consider equation (4.17) further, keeping in mind the relationship between the Gibbs free energy change and the potential (equation (1.6)). If the virtual total cell reaction (4.3) of our cell is exergonic (\( \Delta \)G < 0), hydrogen gas can, as described in the reaction equation, reduce Fe³⁺. This reaction produces work, and if the potentiometer in Figure 4.2 is replaced by a load, the work of the reaction can be utilized. This work corresponds to the transfer of electrons from the gold electrode to the platinum electrode (in solution), and in an ideal case the value of this work is –FE_cell.

Quantity E⁰ in equations (4.5) and (4.17) is the standard value for cell potential.  Thermodynamics defines certain standard conditions in order to enable reliable comparison of quantities and preparation of tables. In the example cell, the standard cell potential can be divided in two parts. These two parts are thought to be the contributions of the potential that each electrode reaction produces:

where

Combining the half-cells gives the E⁰ of the example cell:

The standard potential of the cell gives the work needed to transfer one electron between the gold electrode and the redox couple (Fe³⁺/Fe²⁺) in the solution, on a potential scale that is defined by the hydrogen electrode on the left side of the cell in standard conditions. Standard potentials of different half-cells are tabulated and combining these gives the standard cell potential of galvanic cells. The tabulated values of E⁰ also enable the estimation of spontaneity of different reactions. As seen from (4.19), the larger the value of standard reduction potential, the stronger the driving force for a particular reaction (as the more negative the value of Gibbs free energy change). This allows conclusions concerning half-reactions, e.g. choosing which half-reaction in galvanic cell is oxidation and which is reduction.  

One has to keep in mind that the standard potential for a half-cell is not the same as the “absolute” electrode potential i.e. the galvani potential difference. The standard electrode potential for the half-cell Pt | H⁺,H₂ based on equation (4.13) is

whereas \( E^0_{\text{H}^+/\text{H}_2} \) is obtained from (4.17)

Example 1: Why is FeCl₃ used to dissolve copper in the manufacture of circuit boards? Justify your answer with the standard reduction potentials below.

\( E^0_{{\text{Fe}^{3+}}/{\text{Fe}^{2+}}}=0.77\text{V} \)   \( E^0_{{\text{Cu}^{2+}}/{\text{Cu}^{+}}}=0.34\text{V} \)

4.3.1 Standard potential values

The standard potential E⁰ of a cell can be obtained in four ways:

·    from the standard Gibbs free energy change of the virtual total cell reaction  

·    from the tabulated standard reduction potentials of the half-reactions

·    from the equilibrium constant K of the virtual total cell reaction

·    extrapolating from experimental results

Quite often one needs to measure the standard potential of a half-reaction. As the standard hydrogen electrode defines the zero of the potential scale, the desired half-cell can be determined by constructing a cell that has this as the other electrode. A silver-silver chloride electrode is a typical reference electrode, which can be prepared by electrolyzing a silver wire in a solution containing chloride, thus forming a solid layer of silver chloride at the surface of the silver wire. In order to find the standard potential of the silver-silver chloride electrode, the following cell can be constructed in standard conditions:

Half-reactions are now written in terms of reduction, like the tabulated values:

The cell potential is obtained from the corresponding Nernst equations and equation (4.4):

Let us assume that the silver electrode is made of pure silver and, therefore, its activity is one. In addition, equation (4.22) defines that the standard reduction potential of H⁺/H₂ is zero, and equation (4.25) becomes

At the surface of the right-hand side electrode of cell (4.23), the equilibrium is achieved with chloride ions. Silver chloride is a sparingly soluble salt, and the chloride ion activity can be taken into equation (4.26) with the help of the solubility product, K_s = 1.77×10^–10. It is the equilibrium constant of the reaction:

Hence, because AgCl is a pure, solid compound with the unit activity,

Combining (4.26) and (4.28), and replacing the activity of the gas with its fugacity ratio at the reference pressure (according to the definition of activity) gives:

Constant terms in (4.29) form the standard reduction potential of the Ag/AgCl electrode. The value of this standard potential can be calculated from the standard reduction potential of silver and the solubility product of AgCl:

If the fugacity of the hydrogen gas is kept constant at 1 bar (standard conditions), the third term on the right-hand side of equation (4.29) is zero. In this case, the cell potential of (4.22) is a direct function of HCl concentration. Measuring the cell potential as a function of the concentration and extrapolating to infinite dilution, where the activity coefficient can be estimated with unity, the experimental value for standard reduction potential is obtained:

A hydrogen electrode with bubbled gas supply is not the most practical choice. As a matter of fact, the Ag/AgCl electrode in the previous example is currently the most popular reference electrode. Another popular choice is the calomel electrode, although it contains mercury salts (e.g. Hg₂Cl₂). Figure 4.3. presents the measurement of cell potential with Ag/AgCl electrode as the reference electrode instead of the hydrogen electrode.

Figure 4.3. A galvanic cell for the determination of the standard potentials using the Ag/AgCl reference electrode.

This cell can be represented by the following cell diagram:

The cell potential is

Combining the constant terms in equation (4.33) (activity of Cl⁻ is constant because the solution is saturated) gives

where \( E^{\text{0,AgCl/Ag}}_{\text{Fe}^{3+}/\text{Fe}^{2+}} \) is the standard reduction potential for the Fe(II)/Fe(III) redox pair on the scale of the Ag/AgCl reference electrode with the saturated KCl solution. As with the measurement with the standard hydrogen electrode, cell potentials are measured at different iron ion concentrations and the results extrapolated to infinite dilution to access the standard potential at the working electrode.

         Standard potentials can also be determined with other experimental methods. Chapter 7 describes the use of cyclic voltammetry to determine the standard potential of a half-reaction. In addition, standard potentials can be resolved by combining electrochemical methods with spectroscopic methods.

4.3.2 Standard potentials on absolute scale

Electrochemical methods are used to measure the ionization potentials of conducting polymers or to determine the HOMO and LUMO orbital energies of organic molecules. The values typically obtained with cyclic voltammetry and the Ag/AgCl reference electrode need to be converted into what is known as the absolute scale and presented in electron volts (eV). The reference point of the absolute scale is a vacuum and the scale is also called the ‘vacuum scale’. The link between the standard hydrogen scale and the absolute scale is derived next. A more detailed account can be found in the literature[2].

         Let us consider a case in which the standard potential of a redox couple in a solution is measured with the standard hydrogen electrode as the reference electrode. Let us assume that the electrostatic effect due to the surface potential of the electrolyte can be omitted (c ^s =0).

\( E^{0, \text{abs}}_{\text{red/ox}} \) is the standard reduction potential of the redox couple on the absolute scale,  \( \alpha^0_{\text{H}^+} \) is the true standard chemical potential of the proton in the solution (i.e. the electrochemical potential in a non-charged environment) and \( \mu^0_{\text{H}_2} \) is the standard chemical potential of the gaseous hydrogen molecule. In order to find the value of equation (4.35), it is very important to use the same reference point for all the quantities in the equation, the reference point being a vacuum. The standard chemical potential for a gaseous hydrogen molecule can be estimated from the ionization energy of the hydrogen atom and the covalent bond energy of the hydrogen molecule:

         H⁺(g) + e⁻(g) \(\ce{ <=> }\)  H(g)                       \( \Delta \)G⁰ = −13.613 eV

         H(g) \(\ce{ <=> }\) ½ H₂(g)                               \( \Delta \)G⁰ = −2.107 eV

The real standard chemical potential of the proton in an aqueous solution is

         H⁺(g) \(\ce{ <=> }\)  H⁺(aq)                              \( \alpha^0_{\text{H}^+} \) −11.276 eV

The standard potential of the hydrogen electrode on the absolute scale in Volts is:

The potential values measured with reference to the standard hydrogen electrode (SHE) can be converted to the absolute scale with

If the potential values have been determined using another reference electrode, the potential values are first converted to the SHE scale and then to the absolute scale. Potential values for different reference electrodes can be found in the literature (taking into account the different concentrations of the inner solutions of the electrodes) at different temperatures.

The assumption up until now in this chapter has been that there is a heterogeneous equilibrium on each phase boundary. A cell reaction will proceed reversibly in both directions if the equilibrium is disturbed slightly. If liquid junctions are present in the cell, they cause a particular, non-measurable potential loss in the cell. A typical case where this kind of potential losses is found is a cell in which two solutions with different compositions are in direct contact with each other. Concentration gradients will force the different species in the system to transfer from one phase to another. If these species involve ions, a diffusion potential is formed between the solution phases. If a diaphragm or a membrane separates the phases, ionic transport creates a membrane potential between the phases. In practice, the cells in Figures 4.2 and 4.3 contain liquid junctions. In chapters 4.3-4.4 this potential loss was assumed to be insignificant and, therefore, was not taken into account. In those exemplary cells, the liquid junctions would form over the diaphragm in Figure 4.2 and the porous diaphragm of the Ag/AgCl electrode structure in Figure 4.3.

4.4.1 Liquid junctions

A liquid junction potential between two solutions is formed when:

·         solutions are in direct contact

·         solutions contain different ions at different concentrations

·         ions have different mobilities

A very common example of a liquid junction is the calomel (or the Ag/AgCl) electrode. It is used as a ‘universal’ reference electrode as it can be immersed in almost any aqueous solution. In many cases, no reference electrode is truly fully reversible with respect to the ion/s present in the solution. Furthermore, reference electrodes are often either also prone to interference by other ions or do not work reliably in concentrated solutions – a fact that often limits the use of ion-selective electrodes.

When a calomel electrode is immersed in, for example, a phosphate buffer solution at pH 7, a liquid junction is formed across the glass frit of the electrode. Inside the electrode there are K⁺ and Cl⁻ ions, whilst H⁺, H₂PO₄⁻   and HPO₄^{2  −} ions are present in the measurement solution. In the frit, there is a mixture of all these ions and their concentrations will vary as a function of position in an unknown manner. In principle, the solution of the liquid junction potential follows equation (3.49) there is no electric current present, i.e. i = 0. Since the transport numbers are functions of the ionic concentrations, there is no exact solution of the problem in a multi-ionic system as the exact concentration profiles are unknown. Assuming that all the ions have a similar concentration profile across the glass frit, namely

where g(x) is an undetermined function that has the property g(0) = 0 and g(h) = 1 (h is the frit thickness), the integration of equation (3.49) leads to the following useful form:          

 Equation (4.38b) is known as the Henderson equation. Let us assume that the inner solution is 1 M KCl and the outer (measurement) solution is a 0.1 M phosphate buffer at pH 7 with the following ionic concentration (in mol/dm³): H⁺, 10⁻⁷; Na⁺, 0.158; H₂PO₄⁻   0.042; HPO₄^{2  −} 0.058. The contribution of H⁺ can be omitted, as its concentration is orders of magnitude lower than that of the other ions. By using the diffusion coefficient values given in Appendix 2 and presuming that the diffusion coefficients of the phosphate species are approximately 10⁻⁵ cm²/s, equation (4.38b) gives \Delta f ≈ −2 mV. This calculation shows that junction potentials are very small and are, therefore, usually omitted from the cell potential. Although, there are more elaborate treatises of the junction potential in essence they do not change the overall picture, except in some special cases.

4.4.2 Concentration cells

A concentration cell has identical electrodes and electrode reactions. In principle, if two identical electrodes are immersed in the same solution, the cell potential will be zero (equation 4.4). However, if the electrodes are immersed in solutions which differ only in their concentrations, a cell potential will form. The solutions have to include an ion with respect to which the electrode is reversible.

          Let us consider a cell that has two identical Ag/AgCl electrodes immersed in KCl solutions of different concentrations:

 The cell potential is formed as the sum of the galvanic potential differences between the phases:

 In equation (4.40), (\( \phi_\text{Ag/AgCl'}-\phi_\text{KCl(II)} \)) and (\( \phi_\text{KCl(I)}-\phi_\text{Ag/AgCl} \)) are the electrode potentials and (\( \phi_\text{KCl(II)}-\phi_\text{KCl(I)} \)) is the potential difference between solution phases I and II. The electrodes are reversible with respect to the chloride ion. Electrochemical equilibrium gives:

 Taking into account the similarity of the electrodes, which makes the chemical potential of chloride ion equal in both electrodes, gives:

 Potential difference between KCl solutions of different concentrations is calculated using the equation of diffusion potential, Equation (3.49). This gives in a one-dimensional case, which when added up over all the ions in the solution is:

 The cell potential is obtained by inserting (4.44) into (4.43):

 Because transport numbers are constants in a binary solution, equation (4.45) can be integrated to give

 where the relation \( t_{\text{K}^+}+t_{\text{Cl}^-}=1 \) and equation (2.20) have been applied.

As is clear by now, standard potentials can be used to deduce the direction of redox reactions. Corrosion of metals often involves acid-base reactions coupled with redox reactions. The thermodynamic treatment of corrosion often involves the dependence of reactions on the equilibrium potential and pH. This is presented in a graphic form known as the Pourbaix diagram. It is an isothermic equilibrium diagram, usually presented at 25 °C. The vertical axis is the thermodynamic equilibrium potential on the standard hydrogen electrode scale and the horizontal axis is the pH of the solution in contact with the metal phase. Ionic concentrations have usually been fixed to 10^-6 M. Figure 4.4 presents the Pourbaix diagram of iron. The thermodynamically stable species is marked on each zone of the diagram. The lines are defined by electrochemical or acid-base equilibria of the species present. The upper dashed line is the equilibrium potential of the oxygen electrode and the lower dashed line that of the hydrogen electrode, i.e. they represent the potential limits at which the oxygen and hydrogen evolution reactions begin to occur.

 Figure 4.4. Pourbaix diagram of iron. T = 25ºC and ionic concentrations 10^–3 M. Source: https://chem.libretexts.org/

The horizontal lines describe pH independent equilibria. For example, line 1 is the equilibrium  Fe²⁺ + 2 e^–  \(\rightleftharpoons \)  Fe  which is dependent on potential according to the Nernst equation

Inserting values into the Nernst equation gives E = –0.53 V, which is the expression corresponding to line 1.

Vertical lines represent pH-dependent equilibria with no electron transfer (i.e. they are independent of potential). For example, line 3 represents equilibrium

 2 Fe³⁺ + 3 H₂O   \( \rightleftharpoons \)       Fe₂O₃(s) + 6 H⁺

with equilibrium constant

and the expression for line 3 is pH = 0.1 because logK = 0.90 and [Fe³⁺] = 10^-3 M.

         All sloped lines in the diagram represent equilibria, which are dependent on both potential and pH. For example, line 4 represents equilibrium

Fe₂O₃(s) + 6 H⁺+ 2e^–  \( \rightleftharpoons \)     2 Fe²⁺ + 3 H₂O    

The Nernst equation for this equilibrium is

 The expression for line 4 is E/V = 0.91 − 0.18·pH.

         The Pourbaix diagram includes the equilibrium potentials of hydrogen and oxygen electrodes as dashed lines. The difference between these lines is 1.23 V, which is the thermodynamic dissociation potential of water. Above the oxygen electrode line, water is spontaneously dissociating and oxygen is evolving, below the hydrogen electrode line water is dissociating and hydrogen gas is evolving. Most corrosion reactions take place in the domain between these lines where water is stable.

         Assuming that all solid oxides or hydroxides will form a passivating film on the metal surface, the Pourbaix diagram gives the zones where metal should be immune, active (corroding) or passive (not corroding or corroding only very slightly).

         A Pourbaix diagram of a metal shows the conditions for the metal to remain passive or immune to the corrosive effects of the environment. The information from the diagram has to be used with caution as it is based purely on thermodynamics. The most important restrictions are

·   The assumption of equilibrium between the metal, its ions, and its ions and different solid corrosion products. In practice, equilibrium may not be reached but the kinetics of the corrosion reactions dictates the reaction and the structure of the products.

·   The pH-value in the diagram is a surface value. In practice, pH can vary locally and is usually lower on the anode surfaces and higher on the cathode surfaces compared to the value measured from the bulk solution.

·   Thermodynamics does not tell the structure of the solid corrosion products, or how well they actually protect the metal from corrosion. Passive zones in the diagram need to be confirmed experimentally.

·   The Pourbaix diagram does not provide any information on the rate of corrosion, which is dependent on the kinetics of the corrosion reaction

Potentiometric methods provide an accurate means to measure the cell potential. In an elementary measurement, the Nernst equation is applied to find the relation between the concentration and the measured cell potential. Potentiometric measurements can be used to determine, for example, standard potentials, concentrations of electroactive species, mean activity coefficients of electrolytes, acid and base constants, stability constants of complexes, and equilibrium constants of redox reactions. An example of determining the standard potential of the Ag/AgCl electrode was presented in chapter 4.3.1.

The best-known example of a widely used potentiometric measurement is the determination of pH. A pH meter typically uses an Ag/AgCl as the reference electrode and what is known as the glass electrode as the working (or indicator) electrode. The glass electrode is an example of ion-selective electrodes, which are used in various chemical environments to determine concentrations. The heart of an ion-selective electrode is a semipermeable membrane; the potential difference across the membrane depends only on the concentration of the species of interest.

The tip of the glass electrode is made of glass, and the inside of the electrode holds HCl solution at the constant concentration. The electrodes (the reference and indicator electrodes are usually made into one structure) are immersed in the studied solution. The potential change across the glass membrane is dependent on the activity difference between protons in the inner liquid and the studied solution. The Ag/AgCl reference electrode detects this potential. According to the Nernst equation:

where a is the activity of proton. Changing to the Briggs logarithm and setting the temperature to 25 °C gives

The pH meter has a very high input impedance, i.e. it does not allow practically any current to flow across the glass electrode, which ensures the quasi-equilibrium condition. The constant in equation (4.51) is ill defined and typical for each electrode. Usually, it is determined by calibrating the pH meter with known buffer solutions. If temperature changes, the slope of the line E – pH will change. In digital pH meters, calibration is carried out automatically with two buffer solutions and the calibration line according to (4.51) is calculated.

The operation of the glass electrode is based on ion-exchange properties of the silicate matrix of the membrane. Glass is made of a three-dimensional, irregular network of silicon and oxygen. Some of the oxygen atoms are negatively charged, and this charge is balanced in the structure with, for example, sodium or lithium ions. These small cations can move in the structure and hence carry the electric charge. When a glass electrode is immersed in water, the outer layer of the glass will exchange some of the alkali metal ions for protons in the water; it will work as an ion-exchanger. This ion-exchange equilibrium is affected by the pH of the solutions. The activity of a proton will therefore set to a particular value on the hydrated layer on the surface of the glass. This will form a pH-dependent potential difference across the membrane. The proton concentration inside the electrode is kept constant and the pH of the studied solutions can be determined through calibration. By changing the type of glass used, sodium, potassium or silver ion concentrations can also be determined.

More common types of ion-selective electrodes are liquid membrane electrodes or solid membrane electrodes.  Ion-selective electrodes are available at least for the following cations: ammonium (NH₄⁺), barium (Ba²⁺), calcium (Ca²⁺), cadmium (Cd²⁺), copper (Cu²⁺), lead (Pb²⁺), mercury (Hg²⁺), potassium (K⁺), sodium (Na⁺), silver (Ag⁺) and for the following anions: bromide (Br⁻), chloride (Cl⁻), cyanide (CN⁻), fluoride (F⁻), iodide (I⁻), nitrate (NO₃⁻), nitrite (NO₂⁻), perchlorate (ClO₄⁻), sulfide (S⁻) and thiocyanate (SCN⁻). Ion-selective electrodes can also be found for gases. In these electrodes, a gas-permeable membrane separates the inner solution from its exterior. Enzyme electrodes have a gel layer containing a particular enzyme as the selective membrane. When using ion-selective electrodes, one has to keep in mind that the electrode potential is a function of activity, not concentration. As activity is a function of the ionic strength of the solution, in order to obtain reliable results, the ionic strength of the solutions (calibration solutions and the studied solution) must be kept constant.

In electrochemical reactions, there is always an electron transfer step, which takes place at the interface of two phases, for example at the interface of an electrode and electrolyte solution. This interface is studied in greater detail in this chapter.

At the interface of the two phases, there is an interfacial zone, the properties of which are significantly different from those of the inner part of the phase. From an electrochemical point of view, it is important to note that the electrical properties of the interface also differ from those of the bulk phase. If the interfacial region carries a charge, the charge concentrates at the interface. The orientation of the dipole molecules at the interface can also have an effect on the electrical properties of the interface. Furthermore, the adsorption of the dipole molecules or ions on the interface can enhance the accumulation of charges. This charge excess is due to the electrons, ions and dipoles, and generates an electric field. The region in which the charges are located is called the double layer.

When oppositely charged ions and dipoles accumulate on the surface of a charged electrode in an electrolyte solution, a double layer as depicted in Figure 5.1 is formed. The surface properties of the electrode differ from those of the inner phase of the electrode because of the charge excess or deficit with respect to the bulk of the electrode. This interfacial zone on both sides of the phase boundary is called the electrical double layer.

Figure 5.1. Structure of an electrical double layer in an electrolyte solution, when the solution is in contact with a charged electrode. The ions with the solvation layer are closest to the interface at the outer Helmholtz layer (x₂), whereas the specifically adsorbed ions at the inner Helmholtz layer (x₁) are desolvated.

The solution in the region of the double layer is thought to be made up of several layers. The structure of these layers is affected by the finite size of the ions and molecules, electrostatic forces, mixing entropy, and thermal motion, which reduces the level of organization of the species. The layer closest to the interface is called the inner Helmholtz plane (Figure 5.1). The inner Helmholtz plane consists of specifically adsorbed molecules and ions. Species that are specifically adsorbed are affected by not only electrostatic interactions but also chemical interactions between the surface and the species. The specifically adsorbed species are typically anions, solvent molecules or other neutral molecules. Cations, on the other hand, are very seldom specifically adsorbed. The inner Helmhotz plane is located at the electrical centers of the specifically adsorbed ions.

The outer Helmholtz plane (OHP) is thought to be located at the electrical centers of the closest solvated ions near to the electrode surface. These ions are said to be nonspecifically adsorbed, and the interaction of the solvated ions with charged metal surface involves only long-range electrostatic forces. Their interaction is therefore in principle independent of the chemical properties of the ions.

The region extending from the outer Helmholtz plane to the bulk of the solution is called the diffuse double layer, where the species are organized by thermal motion and the electrostatic force, which gets weaker as the distance from the interface increases. The boundary between the diffuse layer and the bulk of the solution is therefore not well defined. The thickness of the diffuse layer depends on the total ionic strength and is of the order of magnitude10-100 \( \mu \)m.

	a
	b
	c

Figure 5.2. Structure of electrical double layer and potential distribution at a) the metal electrode-electrolyte interface, b) the semiconductor-electrolyte interface, and c) the interface of two immiscible electrolyte solutions without specifically adsorbed species. Note that in the latter case there is no layer of organized solvent ions at the interface, according to Verwey and Niessen model.

The structure of the electrical double layer and the potential distribution at a) the metal electrode-electrolyte interface, b) the semiconductor-electrolyte interface, and c) the interface of two immiscible electrolyte solutions without specifically adsorbed species is shown in Figure 5.2. At the surface of a semiconductor, the charges are distributed more widely compared to the surface of a metal electrode. At the interface of two immiscible liquids, there is no layer of organized solvent molecules.

Adsorption on electrode surfaces can take place with or without simultaneous charge transfer between the adsorbing species and the electrode. When charge transfer is involved, it can be either partial or complete charge transfer. In this chapter, we look at adsorption, which takes place without charge transfer. The best tool for this is thermodynamics.

Consider an interface of surface area A separating two phases, \( \alpha \) and \( \beta \). These two phases can be, for example, a metal electrode and an electrolyte solution, or two immiscible liquids in contact with one another. In a system with several components, the (electro) chemical potential of each component has to be equal in both phases and at the interface. Experience has shown that real systems tend to either minimize or maximize the interfacial area. Because of this, the free energy of the system has to depend on the area:

We can give a name to the partial derivative \( (\partial G/\partial A) \): it is the surface tension \( \gamma \). The forces that bind particles together are weaker at the interface because the particles at the surface have fewer neighboring atoms compared to the ones in the bulk. Instead, the particles at the surface interact with particles in the neighboring phase. This changes the equilibrium of forces and can lead to a new force called surface tension. Surface tension is therefore a force at the interface that tends to increase or decrease the growth of the surface. It is significant only in a very thin zone, a few molecules thick, near the interface.

Gibbs energy differentials \( \alpha \) and \( \beta \) are

We will only look at experiments conducted at constant temperature and pressure, and thus the two first terms of the Gibbs equation can be neglected. Since the phases are in equilibrium, the chemical potentials of each species are equal in the whole system:

Subtracting the Gibbs energies of phases \( \alpha \) and \( \beta \), the Gibbs energy of the surface is:

There is either an excess or a deficit of components on the surface compared to the bulk of the solution. The difference in the amounts of substance on the surface and in the bulk is called the surface concentration, or surface excess, \( n_i^{\sigma} \):

Consider a situation in which a surface is formed slowly between two phases from zero to a finite value A, while pressure, temperature, and the composition of the solution are constant. In this case, the surface concentration increases from zero to the value \( n_i^{\sigma} \), but \( \gamma \) and \( \mu_i \) are constant. Equation (5.6) is substituted into Equation (5.5) and integrated

And we obtain

The total differential of Gibbs energy is

Equations (5.7) and (5.10) have to be equivalent, and thus the sum of the last two terms of Equation (5.10) is zero.

It is often convenient to talk about surface excess per unit area. We therefore define the surface concentration \( \Gamma_i=n_i^{\sigma}/A \) , and the previous equation becomes

This equation is called the Gibbs adsorption isotherm. The equation implies that measurements of surface tension will give us information on the structure of the interface.

It is useful to describe the adsorption of other components with respect to one predominant component, e.g. a solvent, in dilute solutions in particular. Here, the main component is given the subscript 1. In the bulk of the solution, the Gibbs-Duhem equation holds. According to it, for any phase at constant temperature and pressure, the following is true:*

The change in the electrochemical potential of the main component is therefore

Using this, the Gibbs adsorption isotherm gives

The bracketed term is called the relative surface excess \( \Gamma^{\sigma} \)

In very dilute solutions n₁ » n_i, and \( \Gamma_i^{\sigma} \approx \Gamma_i \).

Electric charges affect the surface tension of the interface of two phases. If one of these phases is a metal and another is an electrolyte solution, the phenomenon is called electrocapillarity.

The surface tension of an ideally polarizable electrode (see Chapter 1.3.4) always depends on the potential. To understand the dependency of surface tension, composition, and cell potential, consider a situation where one of the electrodes is ideally nonpolarizable. The potential of this reference electrode is therefore constant, and the change in the cell potential has an effect on the interface of the working electrode and the electrolyte. The working electrode must be ideally nonpolarisable.

Let us look at a cell, in which a hydrogen electrode is the ideally nonpolarizable reference electrode and a mercury electrode is the ideally polarizable working electrode. The electrolyte in the cell is an aqueous solution of hydrogen chloride.

As we showed in Chapter 2, the chemical potentials can be replaced by the electrochemical potentials of ions. In the bulk of the solution, the electroneutrality condition cancels the potential terms out, but there is an excess of charge at the interface, in other words a local deviation from the electroneutrality condition. Applying the Gibbs adsorption isotherm (5.12) to the interface of HCl-Hg using the relative surface excesses with respect to water (5.16), we obtain

There is an equilibrium between the metal ions and electrons in mercury and between the ions and molecules in the electrolyte. Thus

The excess of charges at the interface of metallic mercury and the electrolyte, \( \sigma \)^m and \( \sigma \)^s can be expressed as

Substituting Equations (5.19)-(5-22) into Equation (5.18) we obtain

Based on the equilibrium conditions at the phase boundaries, we can write

where superscript w denotes the working electrode and r the reference electrode. On the other hand, the composition of gas in the hydrogen phase of the reference electrode is constant and because of the equilibrium H₂  \(\ce{ <=> }\)  2 H⁺  +  2 e^– we have

Applying the definition of potential E

The last term of Equation (5.23) can be written as

Thus,

Figure 5.3. Electrocapillary curves, i.e. surface tension of mercury electrode as a function of potential in various electrolytes. The potential is plotted with respect to the potential of zero charge for NaF. D.C. Grahame, Chem. Rev. 41 (1947) 441.

Figure 5.3 shows the nearly parabolic shape typical of electrocapillary curves which describe the surface tension as a function of potential for electrolyte solutions in contact with the mercury electrode. If the mercury is pure, \( \mu \)_Hg = 0 and d\( \mu \)_Hg = 0. Also, d\( \mu \)_HCl = 0, if the concentration of the acid is assumed to be constant. In this case, the previous equation becomes 

It is evident that the slope of the tangent to the electrocapillary curve gives the magnitude of the excess charge at any potential.

Even though the curves for each solution are substantially different from each other, we can observe a common feature in all curves: the existence of a maximum surface tension. The corresponding potential is the potential of zero charge (PZC), where \( \sigma \)^m = \( \sigma \)^s = 0. At more negative potentials, the surface of the electrode has a negative excess charge, and at more positive potentials the surface charge is positive. The charge at the surface weakens surface tension because like charges tend to repel each other.

Neutral organic molecules are typically best adsorbed close to the potential of zero charge because at PZC the polar water molecules in the inner Helmhotz plane are less organized and neutral molecules can more easily penetrate the inner parts of the double layer. The surface coverage varies as a function of potential. The surface coverage \( \theta \) is the fraction of the electrode surface that is covered by molecules (0 < \( \theta \) < 1). If the molecules are specifically adsorbed, the previous model is not valid and the surface coverage can be independent of the potential.

The capacitance of the double layer per unit area C is the derivative of the excess charge with respect to the potential at constant composition

The classic Gouy-Chapman model describes the layer between the outer Helmholtz plane and the bulk of the solution. In this layer, the charges are assumed to be point charges and the solution is a medium without a structure described by the relative permittivity \( \varepsilon \)_r. Since the only thing that matters is the potential difference, the potential of the bulk solution can be assumed to be zero and the potential of the Helmholtz plane is denoted \( \phi \)₂. Let us leave the inner layer for a moment and take a closer look at the diffuse layer.

The total charge density r(x) is obtained from electrostatics using the Poisson equation:

On the other hand, the charge density can be expressed as the sum of all ions present in the diffuse layer:

The ions in the double layer follow the Boltzmann distribution:

Combing the two equations, we get the Poisson-Boltzmann differential equation:

Intergration by substitution gives

Thus

The integration constant B can be determined with the help of the following boundary conditions: The electric field must vanish in the bulk solution, and thus this symmetry requirement results in a boundary condition d\( \phi \)/dx = 0. According to the electroneutrality condition, the net charge of the bulk must also be zero, and therefore \( \phi \) = 0. Equation (5.39) therefore becomes

Consider a symmetrical electrolyte, e.g. NaCl (1:1) or CaCO₃ (2:2). Equation (5.40) becomes now

Note that we need to choose the negative root for (d\( \phi \)dx)! The equation can be integrated in a closed form, and after some slightly tedious algebra, the solution becomes

where \( \phi \)₀ is potential at the electrode surface (x = 0) and

is again the Debye length (see also Equation (2.33)). The potential profile (5.43) is not very illustrative and because of that, the tanh terms are usually linearized, tanh(x) ≈ x. The linearized result is simply

Figure 5.4. Simulated potential profiles according to Gouy-Chapman theory, F\( \phi \)₀/RT = ±5: 1:1 electrolyte (blue), 2:1 electrolyte (green), 1:2 electrolyte (red). The black broken line describes the linearized equation (5.44). When \( \phi \)₀ > 0, the curve for the 2:2 electrolyte almost merges with the curve for the 1:2 electrolyte (red) and when \( \phi \)₀ < 0, the curve for the 2:1 electrolyte (green). Note the order of the curves with respect to \( \phi \)₀.

The potential profiles are shown in Figure 5.4, including for the unsymmetrical electrolytes obtained by integrating Equation (5.40) numerically.  In Figure 5.4, \( \kappa \) = (2c^bF²/\( \varepsilon \)_r\( \varepsilon \)₀RT)^1/2.

In aqueous solution at 25 ºC, the Debye length is \( \kappa \)^–1 ≈ 10^–9 (z²c^b)^–1/2(mol/cm)^1/2. A good rule of thumb is that in 1.0 mM solution, for 1:1 electrolyte \( \kappa \)^–1 ≈ 10 nm.

According to Gauss's law, the charge q inside the volume V is

where E is the electric field through a surface S. In our one-dimensional case, the scalar product E·dS is zero at all points except on the electrode surface, where it is (d\( \phi \)/dx)₀dS. The charge density of the double layer is therefore

The charge density of the electrode for a symmetrical electrolyte is

The differential capacitance of the electrode therefore becomes

Figure 5.5. Simulated capacitances according to Gouy-Chapman theory: 1:1 electrolyte (blue), 2:1 electrolyte (green), 1:2 electrolyte (red), 2:2 electrolyte (black).

The scaled capacitances for 1:1 electrolytes are:

For a 2:1 electrolyte:

For a 1:2 electrolyte:

And for a 2:2 electrolyte:

As shown in Figure 5.1, the surface of an electrode includes an inner and an outer Helmholtz layer, which may have a net charge. In the mathematical treatment of the diffuse double layer, \( \phi \)₀ must be replaced by \( \phi \)₂ and x by \( x-x_2 \) in the previous equations; keep in mind that x₂ is the distance between the outer Helmholtz plane and the electrode, and \( \phi \)₂ = \( \phi \)(x₂). From now on, we will treat the Helmholtz layers as one and call this the inner layer.

Continuity of electric flux is one of the central laws of electrostatics. According to it, at the phase boundary \( \alpha \)| \( \beta \), we have

Superscripts \( \alpha \) and \( \beta \) in Equation (5.52) denote the respective phases, and q is the charge density of the phase boundary. If q is zero, and \( \varepsilon_r^{\alpha}=\varepsilon_r^{\beta} \), the continuity condition of electric flux means that the electric field is continuous.

Assume that the charge of the inner layer is zero. It follows that the potential profile, according to the Poisson equation, is linear: 

Because of the partial loss of degrees of freedom of the water molecules in the inner layer, its relative permittivity is low compared to that of the bulk solution. Based on Equations (5.42), (5.48) and (5.52), the continuity condition of the electric flux density between the layers at the boundary at x = x₂ for a symmetrical z:z electrolyte becomes

where \( \varepsilon \)_r is the relative permittivity of the inner layer, and \( \varepsilon \)_r^b   that of the bulk solution. We can now solve this for \( \phi \)₂:

Substituting \( \phi \)₂ into Equation (5.54) and taking the derivative with respect to \( \phi \)₀ results, after some algebra, in the total capacitance of the electrode, C_T:

Equation (5.56) states that the total capacitance is made up of two components, the series combination of the inner layer and the diffuse layer. [1] The capacitance of the inner layer C₂ is independent of potential:

Because C₂ » C_d, C_d dominates the total capacitance. The analysis can even be taken a step further: The charge of the electrode \( \sigma \)^m in Equation (5.32) can be solved by integration:

where E_pzc is the potential of zero charge of the electrode, in other words, the potential at which \( \sigma \)^m = 0. The potential of zero charge is usually taken to be the potential at which the capacitance of the electrode is at the minimum. Without specific adsorption, i.e. the charge of the inner layer, E_pzc ≈ 0. The non-zero minimum of capacitance is therefore considered to be the first signal of specific adsorption. The integral (5.58) must be solved numerically from the experimental capacitance curve because its mathematical form is not known.

Typically, however, there are specifically adsorbed ions on the electrode. In this case, the charge of the electrode \( \sigma \)^m is compensated for the charge of the inner layer

----

Let us consider a redox reaction

where n electrons are transferred to the oxidized species ’O’, forming the reduced species ’R’. At equilibrium, the following equation applies:

where \( \phi^w \) is the Galvani potential in the aqueous phase and \( \phi^{\text{M}} \) that of an electron (in the electrode); the other variables are defined in Chapter 2. The activity of an electron in the solid electrode (most commonly a metal) a_e = 1, and n = z_O - z_R. Hence, rearranging Equation (6.2), the Nernst equation is obtained:

where

When the electrode potential \( \phi^{\text{M}} \) is changed, it follows that

In other words, the change in the electrode potential is converted to a change in the chemical potentials of the species residing in the solution phase.

The Nernst equation contains the activities of the species involved in the reaction, but the estimation of their activity coefficients is not always easy. Fortunately, the ratio of the activity coefficients approaches unity much faster than the individual coefficients. The Nernst equation is therefore most often written in terms of concentrations:

In Equation (6.6) the Galvani potential \( \phi \) is replaced by E to emphasize that the electrode potential is measured against some reference value. The quantity E⁰’ is known as the formal potential. The use of formal potential is also practical because during an electrochemical reaction the concentrations of O and R on the electrode surface change as a function of the electric current, and the formal potential provides one of the boundary conditions of the transport problem. We will look at this in more detain in Chapter 6.2.

The rate of reaction (6.1), r, can be written in the usual form as

where k_red and k_ox are the rate constants (cm s^-1) of the heterogeneous reduction and oxidation reactions respectively and the superscript ’s’ denotes the surface concentrations. Let’s assume that the rate constants follow the Arrhenius formalism:

In Equation (6.8) A_x is the pre-exponential (frequency) factor of reaction x and \( \Delta G_{x,a}^{\ddagger} \)its activation energy. At equilibrium r = 0, viz \( k_{red}c_{\text{O}}^s=k_{ox}c_{\text{R}}^s \), and when E = E^0’*, \( c_{\text{O}}^s=c_{\text{R}}^s \). This means that k_red = k_ox and we call this quantity as the standard rate constant k⁰:

Figure 6.1 shows a scheme of the free energy profiles at equilibrium (left) and after changing the electrode potential (right); the activation energies or the pre-exponential factors do not need to be equal in both oxidation and reduction reactions, although they are shown this way for the sake of clarity.

When the electrode potential is changed, it can be interpreted as the change of the chemical potential or free energy of either of the species, as previously shown. On the right of Figure 6.1 the electrode potential is raised by \( \Delta E \), raising the free energy of R by \( \Delta G=nF\Delta E \), consequently shifting the reaction equilibrium towards oxidation. The energy barrier of the oxidation reaction does not, however, reduce by \( nF\Delta E \) but, as can be seen in the plot, by the fraction that is denoted \( \alpha nf\Delta E \).† Accordingly, the energy barrier of the reduction reaction is increased by the fraction \( (1- \alpha)nf\Delta E \). The quantity a is known as the charge transfer coefficient. Its value depends on the symmetry of the energy barrier and the electrode potential, taking the value 0.5 in the vicinity of E^0’, regardless of the symmetry.

Figure 6.1. Free energy profiles at equilbrium (left) and after changing the electrode potential (right).

where f = F/RT. Equations (6.10) are general in the sense that in all approaches the form of the rate constants remains the same, only the the interpretation of the activation energy and the charge transfer coefficient depend on the model applied.‡

Inserting the rate constants from (6.10) into Equation (6.7) at equilibrium (\( r \equiv0 \)), the Nernst equation is restored, i.e. our kinetic model agrees with thermodynamics:

Inserting the dependence of the rate constants on potential it is therefore possible to sketch a current-voltage curve; the only problem comes from the surface concentrations that change as a function of electric current. We begin the analysis with a reversible reaction, i.e. a reaction that has such a high rate that the surface concentrations follow the Nernst equilibrium at all times and in all currents.

6.2.1 Reversible reaction

We consider only the case of a trace ion (see (3.47)) that gives the boundary condition

In a stationary state (steady state) the diffusion equation (3.53) becomes

making concentration profiles linear. As will be explained in Chapter 7, a diffusion boundary layer (DBL) of the thickness \( \delta \) is formed in the vicinity of an electrode. With the boundary condition \( c_i(\delta)=c_i^b(i=\text{R,O}) \) the following result is reached:

In Figure 6.2, steady-state concentration profiles during an oxidation reaction are depicted (anodic current, i > 0).

It should be noted that the fluxes of O and R are equal but opposite in direction, J_R = -J_O, in order to maintain the mass balance, as Equation (6.13) actually also shows. As can easily be seen from the picture, (and Equation (6.15) the highest possible current is obtained when the surface concentration of R reaches zero. This current is called the limiting current. Accordingly, if we consider a reduction reaction, the highest possible cathodic current (i < 0) is obtained when the surface concentration of O reaches zero. The anodic and cathodic current densities are respectively obtained from Equation (6.15):

The surface concentrations can be written with the help of current density as:

From Equation (6.16), the bulk concentrations can be solved as a function of the limiting current density, as a result of which the following very useful expressions are obtained:

Equation (6.18) or (6.19) cannot naturally be written for a species that has zero bulk concentration. For example, if \( c_{\text{O}}^b=0 \), current is always positive and

Figure 6.3 shows current-voltage curves with \( i_{\text{lim}}^{red}/{i_{\text{lim}}^{ox}}=0 \) or -0.5 and n = 1. The diffusion coefficients are usually rather close to one another, hence \( \xi^2 \approx1 \). Scaled current density \( \chi \) is defined as  \( \chi=i/i_{\text{lim}}^{ox} \). Equation (6.21) provides a diagnostic criterion for a reversible electrode reaction: plotting the electrode potential as a function of the logarithm on the right-hand side of Equation (6.21), a straight line is obtained with a slope of 25.7/n mV, or on the log₁₀ scale, 59.1/n mV per decade.

6.2.2 Quasi-reversible reaction

The derived equations above (6.15)-(6.19) are generally valid and can be used in Equation (6.12) to simulate current-voltage curves. Inserting Equation (6.17), for example, we find that

In practice, the curves are preferably presented in terms of measurable quantities. The determination of the thickness of the DBL, \( \delta \),  for example, is not unambiguous, as will be seen in Chapter 7.

Let us consider equilibrium again. Now, k_red,eq c_O^b  = k_ox,eq c_R^b  because in the absence of electric current there is no concentration polarization, and surface concentrations are equal to bulk ones. At equilibrium, the rate of oxidation is equal to the rate of the reduction; they are not zero but the equilibrium is a dynamic process where the net reaction rate is zero. The Nernst equation is valid at equilibrium:

Inserting the above equation into the expression of k_red in Equation (6.10), we find that

Exchange current density, i₀, is defined as the inherent rate of oxidation or reduction at equilibrium:

Exchange current density is the measure of the rate of an electrode reaction, i.e. of its reversibility. A good reference electrode, such as Ag/AgCl has an exchange current as high as 10 A/cm². Such an electrode is ideally non-polarizable, as discussed in Chapter 1.

The definition of overpotential is  \( \eta=E-E_{eq} \). Hence, \( E-E^{0'}=E-E_{eq}+E_{eq}-E^{0'}= \eta+E_{eq}-E^{0'} \). The current-overpotential equation can be derived by applying Equation (6.23) to the last equality:

If the solution is stirred effectively or the current is less than 10% of the limiting current, the concentration polarization can be ignored, and the Butler-Volmer equation is obtained*:

The above equation is traditionally analyzed at limiting values of overpotential. At low overpotentials it can be linearized, giving

The unit of the group of variables RT/(nFi₀) is \( \Omega \text{cm}^2 \); it is known as the charge transfer resistance, R_ct.

At high positive overpotentials, only the anodic term of Equation (6.28) is significant. Hence

Accordingly, at high negative overpotentials only the cathodic term is significant:

Equations (6.30) and (6.31) are called Tafel equations. The graphs of ln| i | vs. \( \eta \) are linear, and their extrapolation to zero overpotential gives the value of i₀. This kind of a graph is called a Tafel plot. The original Tafel equation was empirical and written as

where a and b were experimental constants; b is known as the Tafel slope. Comparing Equations (6.31) and (6.32), the cathodic Tafel slope is†

Very often n = 1 and \( \alpha \approx0.5 \), giving a typical Tafel slope of approximately 120 mV per current decade. If mass transfer to the electrode is inefficient, the linear range of the plot can be rather limited due to concentration polarization, making the determination of Tafel slopes difficult. Measurements are therefore most often carried out with a rotating disk electrode (see Chapter 7.2.1), which makes the explicit elimination of mass transfer possible.

In a complete current-overpotential curve, mass transfer is taken into account. Inserting Equations (6.19) into Equation (6.27) the following expression can be derived:

\( \chi \) is defined similarly to Figure 6.3, \( \chi=i/{i_{\text{lim}}^{ox}} \).

Figure 6.4 presents simulated curves using Equation (6.34). It shows the effect of the exchange current density: the lower it is, i.e. the higher \( \chi_{\text{kin}} \) is, the wider the potential range where hardly any current flows (potential window). At the limit i₀ → 0 an electrode is ideally polarizable and its i-h graph horizontal. As discussed earlier, an ideally non-polarizable electrode has a very high exchange current density and without mass transfer limitations, its \( i-\eta \) graph would be vertical. In Figure 6.4, the curve with \( \chi_{\text{kin}}=0.1 \) is already completely under mass transfer control because with values lower than that the shape of the curve no longer changes.

An ideally non-polarizable electrode naturally cannot exist because solvent decomposes at high overpotentials. A mercury electrode is closest to an ideally polarizable electrode because the overpotential of hydrogen evolution (see paragraph 6.5.2) on mercury is very high, expanding the potential window over 2 V. Immiscible water-organic interfaces are also polarizable, although the potential window usually remains less than 1 V.

Figure 6.4.  Simulated i-\( \eta \) curves with \( \chi \)_kin = 0.1 -10⁴, \( \alpha \) = 0.5, n = 1 or 2 and \( \chi \)_lim = 2.

We have already introduced the term overpotential* and defined this as the deviation from the equilibrium potential, \( \eta=E-E_{eq} \). In electrochemical literature, it is customary to split overpotential into activation (or charge transfer) and concentration overpotentials. If an electrode reaction were reversible, i.e. infinitely rapid (ideally non-polarizable electrode) and mass transfer were also infinitely rapid, the electrode potential would stay at its thermodynamic equilibrium value as determined by the Nernst equation, and no overpotential would exist. Overpotential means that the higher the current we are drawing out of an electrochemical cell, the higher the voltage needed above its thermodynamic cell voltage. In the case of a battery, overpotential means that when current is flowing we cannot utilize its full thermodynamic voltage (terminal voltage), it is instead lowered by the amount of overpotentials.

Tafel equations actually define the activation overpotential \( \eta_a \). Let’s write, for example, Equation (6.30) as

which shows that the lower i₀ is, the higher \( \eta_a \) is. We found earlier that current with \( \chi \)_kin = 0.1 is under mass transfer control; hence it represents pure concentration overpotential. Subtracting the overpotential of this curve from the rest of the curves in Figure 6.4, we obtain the\( i-\eta_a \) curves in Figure 6.5, in other words the activation overpotential is the deviation from the y axis, depending on the current.

The origin of the concentration overpotential is in concentration polarization. Since the surface concentration is lower than the bulk concentration, a higher voltage is needed to draw current out of the cell. Equation (6.27) can be rewritten in the following form:

that shows that the concentration overpotential, \( \eta_c \), is of the form

Figure 6.5. Current-overpotential curves after subtracting for the concentration overpotential. \( \chi \)_kin = 10 - 10⁴ , \( \alpha \) = 0.5, n = 1 or 2.

If an electrode reaction is accompanied by homogeneous reactions (e.g. EC/CE mechanism) they make their contribution, known as the reaction overpotential. When there is a phase formation on the electrode, such as the deposition of metal, the formation and growth of the phase nuclei cause the nucleation overpotential that depends on surface chemical issues, including the geometry of the nuclei. Reaction and nucleation overpotentials are not addressed in this book in a closer detail.

It is necessary to remember that in addition to overpotentials an electrochemical cell always has ohmic losses due to the limited conductivity of electrolytes (see Chapter 3 and 4).

An electric double layer (EDL, see Chapter 5) has an effect on an electrode reaction through two factors. First, in order for an electron transfer to occur, the species involved in the reaction must reside so close to the surface of the electrode that an electron can be tunneled between the species and the electrode.  In the previous chapter it was shown that at this close vicinity of the surface, the concentration of a charged species is different from its bulk concentration according to the Boltzmann distribution:

If mass transfer needs to be taken into account, the bulk concentration is replaced by the surface concentration calculated from mass transfer equations.

Second, at the distance x₂ from the electrode the driving force of electron transfer is not \( E-E^{0'} \) but \( E-\phi_2-E^{0'} \). Taking this into account, as well as Equation (6.38), the expression of k_ox, for example, becomes

The correction factor \( e^{-(\alpha n+z_{\text{R}})f\phi_2} \) is known as the Frumkin correction. If the calculation of \( \phi_2 \) is feasible, the actual k⁰ is denoted as k_t⁰  (’t’ = true). The experimental value of k⁰, calculated from i₀ for example, is therefore

An analogous result is achieved for k_red, taking into account that n = z_O - z_R.