Chapter 1: An Introduction to Electrode Reactions Free

Electrode reactions are a class of chemical reactions that involve the transfer of a charged species across an interface, most commonly an electron between a conducting solid and an adjacent solution phase. The electron is negatively charged and the transfer is driven by a voltage gradient across the interface created by applying a potential between the conducting solid (an electrode) and a second electrode.

Electrode reactions are central to numerous diverse technologies (Table 1.1), many of which are contributing to a healthier and cleaner environment and their importance will grow as the pressure increases to provide clean water, reduce CO₂ emissions and to use resources more efficiently. They also underlie several natural phenomena. In addition, laboratory electrochemical techniques provide approaches to understanding a broad range of chemical systems related to synthesis, biology, materials, component fabrication, energy conservation and engineering.

A simple type of electrode reaction involves only electron transfer between an inert metal electrode and an ion or molecule in solution. At an anode, the electron passes from the solution phase to the electrode and the electroactive species in solution is oxidized, e.g.:

Conversely, at a cathode the electron passes in the opposite direction so that the species in solution is reduced, e.g.:

Clearly, such electron-transfer reactions are going to be possible only when the electroactive species is within molecular distances of the electrode surface (maybe, even a bond between the reactant and the surface is necessary). Hence, even with these simple electrode reactions, the continuous conversion of reactant into product must be a multistep process involving the supply of reactant to the surface and the removal of the product in addition to the electron-transfer event. The overall conversion of reactant, A, into product, B, must occur in a minimum of three steps:

The rate of conversion of reactant in the bulk solution into product in the bulk solution must always be determined by the slowest of these three steps. As a result, to develop an understanding of electrode reactions, one must first discuss the physical chemistry of both electron transfer and mass transfer as well as explaining the way in which the two processes interact. These themes will be developed later in this chapter.

First, however, it is important to emphasize that most electrode reactions are more complex. This can be seen from the examples of both cathode and anode reactions in Table 1.2.

Inspection of these reactions shows that several other fundamental steps, in addition to electron transfer and mass transfer, may be important. These include:

1. Adsorption: The simplest model of electron transfer would envisage that the electron transfers as the reactant ‘bounces’ on the electrode surface. For many electrode reactions, it is more likely that there is a chemical interaction between the reactant (and/or product) and the electrode surface, i.e. adsorption occurs. The interaction may be of varied strength and either electrostatic (between an ion or dipole in solution and the charged electrode surface) or a covalent bond. Hydrogen evolution, Reactions (1.6) or (1.7), is the classical example of a reaction that involves adsorbed hydrogen atoms as intermediates.

2. Chemical reactions: Commonly, the initial product of electron transfer is not stable in the environment of the electrode reaction and the conversion of reactant into product involves chemical reactions. Thus, Reactions (1.6), (1.7), (1.10–1.13) and (1.16–1.20) all involve the cleavage or formation of chemical bond(s). The chemistry may be either homogeneous in solution as an intermediate moves away from the electrode or heterogeneous with the intermediate adsorbed on the electrode surface. An example of the former is Reaction (1.19) which involves cationic intermediates (the cation radical of toluene and later the benzyl cation) and these react with the nucleophilic solvent, methanol. In contrast, in Reaction (1.6) the H–H bond is formed by coupling of adsorbed H atoms.

3. Phase formation: Some electrode reactions involve the formation of a new phase, a metal lattice in the case of Reaction (1.9), an oxide layer in Reaction (1.16), the interconversion of two solid phases in Reactions (1.10) and (1.18) or the formation of a gas in Reactions (1.6), (1.7) and (1.12). The characteristics of the electrode reaction will depend on the properties of the new phase (e.g. is it conducting?) and by the need for nucleation of the new phase and thereafter by the way it grows.

4. Multiple electron transfer: Many of the reactions in Table 1.2 involve the transfer of more than one electron. In most cases, the electrons will not be transferred consecutively. Rather the electrode reactions are a cascade of events with the electron transfers separated by other steps such as chemical reactions.

Clearly, a complete description of an electrode reaction may need to include a statement of the number and type of fundamental steps, the order in which they occur and the thermodynamics and kinetics of each step. While this is seldom possible for complex reactions, we shall wish to identify the key steps in the overall sequence, especially the rate-determining step and how the rate may be changed by modification of the electrolysis conditions (choice of electrode material, solution conditions, temperature, etc.). In particular, when discussing experiments, emphasis will be placed on experimental tests or data treatments that allow the identification of key steps.

Table 1.2 also illustrates the wide range of chemistry possible at an electrode. Notably, the reactant/product may be:

• inorganic, organic or a biomolecule;

• a solid (even the electrode itself), a gas, the solvent or a dissolved species;

• at either anode or cathode, the reactant may be a cation, an anion or a neutral species.

Moreover, the electrode may be a metal, another material with metal-like conductivity (a carbon or conducting polymer) or a semiconductor. It may also be a bulk material, a coating on an inert, conducting substrate or a three-dimensional structure such as a foam or a gas diffusion electrode. It is also interesting to note the range of currents and current densities that may be met. In the laboratory experiment, the currents measured may commonly range from 1 nA to 1 A while in the applications of Table 1.1, the currents of interest may range from 1 μA to 10⁶ A. Overall, current densities of interest may range from 1 pA cm⁻² to 1 A cm⁻².

Figure 1.1 shows a sketch of a typical electrolysis cell, in fact a modern membrane cell for the manufacture of chlorine and sodium hydroxide. The electrode reactions are (1.7) and (1.17) so that the net chemical change in the cell is:

Many of the characteristics of an electrolysis cell may be understood by noting the need for charge transport throughout a complete circuit consisting of the cell and external connections between the two electrodes. Moreover, it is necessary to avoid charge accumulation at any point in the system. For each molecule of chlorine formed, two chloride ions will be oxidized and two electrons will cross the anolyte/anode interface. To maintain charge neutrality throughout the system, several consequences follow:

1. The negative charge lost from the solution at the anode must be replaced by the transfer of two electrons from the cathode to the solution. Hence, in terms of electrons, the amount of oxidation at the anode and reduction at the cathode must be the same – the formation of 1 mole of chlorine at the anode will be accompanied by the formation of 1 mole of hydrogen and 2 moles of hydroxide at the cathode.

2. To maintain charge neutrality throughout the solution, ions must move between the electrodes – anions towards the anode and/or cations towards the cathode. Since the only requirement is charge balance, it is not necessarily the same ion moving throughout the interelectrode gap. In the cell shown in Figure 1.1, the charge will be carried by sodium and chloride ions within the anolyte and sodium and hydroxide ions in the catholyte (the fraction carried by each ion depending on their transport numbers, see Chapter 2). The cell chemistry, however, is dependent on a membrane between anolyte and catholyte that allows only the transport of sodium ions.

3. To avoid accumulation of charge on the electrodes, two electrons must pass through the external circuit from anode to cathode.

This last statement leads to several consequences. Firstly, the number of electrons/second passing through the external circuit must be the same as the number of electrons/second crossing the anode/solution and cathode/solution interfaces. In consequence, measurement of the number of electrons/second through the external circuit (i.e. the current, I, when the flux of electrons is converted into amperes) gives a direct measure of the rate of chemical change at the electrode surfaces. In electrochemistry, the current or current density, j(current per unit area of electrode surface), always provides a trivial way of monitoring the instantaneous rate of chemical change within the cell. Secondly, the total amount of chemical change within a period of time must be related to the total number of electrons passed through the external circuit during this time period. In other words, it is necessary to integrate the current with respect to time, t, to obtain the total charge passed, Q, and then to apply Faraday's law:

where m is the number of moles of reactant consumed or product formed, n is the number of electrons required to convert the reactant into product and F is the Faraday constant (96485 C mol⁻¹). Notably, Faraday's law is really just a statement of mass balance. In the reaction:

the conversion of 1 mole of O into 1 mole of R requires n moles of electrons and the Faraday constant is the charge on 1 mole of electrons (in fact, the Avogadro constant, 6.02 × 10²³ electrons per mole, multiplied by the charge on a single electron, 1.60 × 10⁻¹⁹ C). It might also be noted that the rate of conversion of reactant into product at the electrode surface is proportional to the current density and a current density of 0.1 A cm⁻² corresponds to the formation of 1.8 mmol cm⁻² h⁻¹ for a 2e⁻ reaction.

So far it has been assumed that only one reaction occurs at each electrode. In reality, there are often competing reactions. For example, in the cell of Figure 1.1, the formation of chlorine, Reaction (1.17) is actually accompanied by a small amount of oxygen evolution, Reaction (1.12). Then the current efficiency, ϕ, must be defined:

The current for the reaction of interest is ϕI and the charge consumed by this reaction is ϕQ.

As noted above, the overall chemical reaction in the cell of Figure 1.1 is given by Equation (1.21). The Gibbs free energy change, ΔG, associated with the reaction may be estimated from tables of thermodynamic data; as written, the free energy change is 419 kJ mol⁻¹ of chlorine. Clearly, the reaction is very unfavourable, in accord with our knowledge that a solution of sodium chloride is highly stable. As a result, for the electrolysis to be driven, energy must be applied by applying a voltage between the two electrodes. The magnitude of the essential equilibrium cell voltage, E$cell e$, may be calculated from:

Such a calculation leads to an equilibrium cell voltage of −2.17 V. This cell voltage could also be calculated from:

where the equilibrium potentials of the cathode and anode, E$ce$ and E$ae$, respectively, may be calculated from the Nernst equation (see the next section).

In practice, applying the equilibrium cell voltage would not be sufficient for electrolysis to occur. To achieve electrolysis at a significant rate, it is necessary to apply an overpotential, η, to each electrode to increase the rate of electron transfer (again, see the next section) and also apply additional voltages to drive the ions through the electrolytes and membrane. These voltages will depend on the cell current and the resistance of the components and will be equal to IR_soln and IR_mem, respectively. Overall, the cell voltage required to have an electrolysis current, I, will be given by:

where the modulus symbols are used to ensure that the last four terms all make E_cell a larger more negative voltage, thereby increasing the energy consumption for the electrolysis. The overpotentials and IR terms are inefficiencies and the associated energy ends up as heat. In technology, the cell design and choice of materials will seek to minimize these terms.

Notably, in laboratory experiments to investigate mechanism and kinetics it is desirable that the cell response is determined by a single electrode (the working electrode) and that IR terms do not distort the measurements. This can be achieved in a two-electrode cell provided the second electrode does not influence the response in any way; this is normally when the second electrode is a reliable reference electrode and the cell current is low; the low cell current also ensures that the IR drops are insignificant. More commonly, however, a three-electrode cell and a potentiostat are employed. In this situation the current is passed between the working and a counter electrode (usually much larger than the working electrode) and the potential of the working electrode is controlled versus a reference electrode. The potentiostat is a feedback circuit based on an operational amplifier that ensures that the current through the reference electrode is effectively zero. If the current between working and reference electrodes is zero, then the IR drop must also be zero and the potential of the working electrode can be measured directly with a digital voltmeter and the value will not be affected by IR drop. The design of laboratory cells is discussed in more detail in Chapter 6.

In this section, we consider the thermodynamics and kinetics of a simple electron-transfer reaction:

Throughout we shall consider a cell with an inert working electrode and a reference electrode in a solution containing low concentrations of both O and R, c_O and c_R, respectively, and a high concentration of an inert electrolyte (most importantly, to give the solution conductivity). The only electrode reaction that can occur at the electrode, at least in the potential range under consideration, is the interconversion of O and R.

Initially it is helpful to specify the situation at equilibrium. The simple experiment set out in Figure 1.2 can be used as illustration. The figure shows the two electrodes dipping into the solution specified above and the two electrodes are connected by a circuit containing a high impedance digital voltmeter; this allows the potential difference between the electrodes to be monitored but prevents the passage of current through the external circuit.

In the absence of a current, the concentrations of O and R cannot change and the working electrode will take up the equilibrium potential for the couple in solution, E_e. Hence, the equilibrium potential can be read with the digital voltmeter. It could also be calculated from the Nernst equation:

where E$eo$ is the formal potential for the couple O/R; it is clearly the equilibrium potential when the concentrations of O and R are equal. The formal potential reflects the ease of addition of an electron to O and removal of an electron from R and is determined by the chemistry of O and R in the particular solution under investigation. For example, the addition of a complexing agent that stabilizes O more than R will make O more difficult to reduce and lead to a negative shift in the formal potential.

Although seldom carried out in practical electrochemistry, the thermodynamic equation, (1.29), should strictly be written in terms of activities. For the general electrode reaction:

the Nernst equation in the more precise form would be written:

where E$e o$ is the standard potential (the equilibrium potential when all the reactants and products are in their standard states). While it is wise always to consider the approximations involved in using concentrations rather than activities (e.g. it would be unwise if the solution contained a high concentration of electroactive species and no excess of electrolyte), the discussion throughout this book will use ‘concentration’ and the ‘formal potential’. This is partly because for dilute solutions of reactant and product in the presence of a large excess of inert electrolyte the activity coefficients of O and R are likely to be similar and therefore cancel in the Nernst equation. In addition, notably, for the reaction:

the Nernst equation may be written:

since the activity of a metal is 1 (the measured potential is independent of the amount of the metal used). Also by definition, the activities of elements and gases at a pressure of 1 atmosphere are 1, often allowing further simplification of Nernst equations.

Standard electrode potentials are readily available in textbooks and handbooks of physical chemistry and are the starting point for precise thermodynamic calculations. More often, however, they are used for qualitative assessments of cell potentials and then they should be regarded only as guidelines as they apply to standard state conditions and not necessarily to the experimental conditions. Table 1.3 presents a short list of standard potentials; in accordance with general practice, they are quoted versus the standard hydrogen electrode (SHE). This is seldom the reference electrode in the laboratory but conversion into the experimental reference electrode is straightforward (Chapter 6).

Returning to the experiment of Figure 1.2, no current is passing through the cell and therefore the composition of the solution cannot change. It has been noted that the inert electrode will take up the equilibrium potential for the O/R couple with the particular solution concentrations of O and R. Now, it needs to be recognized that, in common with other chemical systems at equilibrium, a dynamic situation will prevail at the electrode surface – both reduction of O to R and oxidation of R to O will be occurring but these reactions will take place at the same rate. Hence, in terms of current densities, one can write:

where the partial anodic and cathodic current densities, j_a and j_c, respectively, have opposite signs since during oxidation and reduction at the electrode the electrons pass in the opposite direction across the electrode/solution interface. By convention, anodic currents are positive and cathodic currents are negative. The magnitude of these partial current densities at equilibrium turns out to be a useful kinetic parameter for the O/R couple and hence it is given a name ‘the exchange current density’, j_o:

The exchange current density is a measure of the electron-transfer activity at the equilibrium potential. A large value indicates that there is extensive oxidation and reduction occurring while a low value reflects only a small amount. The value also depends on the concentrations of O and R since both oxidation and reduction are first-order reactions in their reactant.

Suppose that a potential positive to the equilibrium potential is imposed on the inert electrode in the solution containing O and R and an excess of electrolyte. The system will now seek to move to a new equilibrium where the concentrations of O and R are those demanded by the Nernst equation for this potential. At a potential of E+ΔE the equality of the Nernst equation, Equation (1.29), can be met only by the ratio c_O/c_R increasing. This requires conversion of R into O so that it must be concluded that at a potential positive to the equilibrium potential an anodic current is to be expected. The same argument will lead to the conclusion that for a potential E–ΔE, i.e. negative to the equilibrium potential, a net cathodic current will be observed.

The above is only a thermodynamic prediction. The next step is therefore to consider the rate (or current density) at which O and R are interconverted at any potential. Following classical kinetics, it is to be expected that the rates of oxidation and reduction will be the product of a rate constant and the concentration of a reactant at the site of electron transfer, i.e. the electrode surface (denoted by the subscript x=0). Hence we can write:

Notably, the rates of conversion on the surface will have the units moles cm⁻² s⁻¹ and therefore the rate constants have the units cm s⁻¹. The rate of the heterogeneous electron transfer would be expected to depend on the potential gradient at the surface driving the movement of the negatively charged electron between the electrode and solution phases. The gradient, in turn, depends on the potential of the electrode (Chapters 3 and 4). It is, however, not straightforward to predict theoretically the way in which the rate constants vary with potential. On the other hand, it is found experimentally that the relationships are generally of the form:

where α_a and α_c are known as the anodic and cathodic transfer coefficients, respectively. For simple electron-transfer reactions α_a+α_c=1 and commonly both have a value close to 0.5. The interpretation of the transfer coefficients and the form of the potential dependence are discussed in more detail in Chapter 3. Moreover, for now, it will also be assumed the concentrations of O and R at the electrode surface are the same as those in the bulk solution (i.e. those prepared for the experiment). This is equivalent to limiting the discussion to conditions where the surface reactions occur at a low rate. Other situations are discussed in Section 1.5 after mass transport has been introduced in Section 1.4. The rates of oxidation and reduction may be converted into current densities simply be multiplying by nF(which will convert the dimensions from moles cm⁻² s⁻¹ into A cm⁻²). Hence, using Equations (1.36–1.39) and remembering the sign convention for current density, an expression for the current density at any potential may be written:

This equation describes the current density as a function of the electrode potential measured versus a reference electrode. A simpler and more useful form is obtained by introducing the concept of overpotential, η, defined as:

The overpotential is the deviation in the applied potential from the equilibrium potential for the couple O/R. The use of overpotential has the effect of transposing the potential from an arbitrary scale determined by the choice of experimental reference electrode to a scale determined by the chemistry of the O/R couple itself. Remember that the current density at the equilibrium potential is zero and it can then be seen that the overpotential can be envisaged as the driving force for oxidation/reduction (depending on the sign of overpotential).

Substituting Equation (1.41) into (1.40) and then considering the situation at the equilibrium potential (putting η=0) leads to:

since at this potential the partial anodic and cathodic current densities are both equal to the exchange current density. Now, using the equalities of (1.42) and Equation (1.41) in Equation (1.40) leads to the Butler–Volmer equation:

which is central to our discussion of the kinetics of simple electron-transfer reactions.

It can now be seen that because of the opposite signs in the arguments of the exponentials, the net anodic current at a potential positive to the equilibrium potential will arise by an increase in the anodic partial current density and a decrease in the partial cathodic current compared to the situation at the equilibrium potential (Figure 1.3). Conversely, the net cathodic current at negative overpotentials arises by an increase in the cathodic partial current density and a decrease in the partial anodic current compared to the situation at the equilibrium potential. The general form of the current density vs overpotential response predicted by Equation (1.43) is illustrated in Figure 1.4; in fact, the responses for three values of the exchange current density are shown and it can readily be seen that the overpotential required to drive the electron-transfer reaction at a particular rate (current density) increases with decreasing exchange current density.

Equation (1.43) has two limiting forms commonly used in experimental electrochemistry. Firstly, only close to the equilibrium potential are both exponential terms in the equation significant. For a moderate positive overpotential (say, η > 26 mV), j_a ≫ j_c and the equation becomes:

It can be seen that the current density depends on the amount of electron-transfer activity at the equilibrium potential and a term containing η that increases strongly with increasing η and effectively acts as a driver for reactions that have poor kinetics, i.e. a low value of j_o.

It is sometimes valuable to discuss the kinetics of electrode reactions in terms of a rate constant rather than an exchange current density. The standard rate constant, k_s, is the rate constant for both oxidation and reduction at the formal potential(in contrast, the exchange current density is the equal anodic and cathodic partial current densities at the equilibrium potential) and is given by:

Equation (1.44) can also be written:

and it is evident that for a reaction with n=1 and α_a=0.5 the current will increase by a factor of ten for every increase in overpotential of 120 mV; overpotential is a very strong driver! In fact, for more complex reactions, the factor of ten increase may be obtained with an increase in overpotential of only 60, 40 or 30 mV. An overpotential of a volt can easily lead to an increase in rate between 10⁸ and 10³⁰– this is massive compared to, for example, the influence of temperature on homogeneous chemical reactions and allows very slow reactions to be driven. A similar argument for negative potentials leads to:

Since a cathodic current will be observed at these overpotentials, –j is a positive quantity. Equations (1.46) and (1.47) are known as Tafel equations; Figure 1.5 illustrates current density vs overpotential in this form. At both positive and negative overpotentials there is a range where the log | j| vs η plot is linear and both extrapolate to log j_o at η= 0. Moreover, the transfer coefficients may be obtained from the slopes and the slopes should be related since α_a+α_c=1. For a reaction where n=1 and α_a=α_c=0.5, the Tafel slopes will be 1/120 mV.

The second limiting form of the Butler–Volmer equation, equation (1.43), applies at low vales of overpotential, i.e. very close to the equilibrium potential. Then, expanding the exponential terms as a series and neglecting squared and higher order terms, for α_a=α_c=0.5 one obtains the very simple equation:

Very close to the equilibrium potential, there is a linear relationship between current and overpotential but the range of overpotential where the relationship is exact is limited. This is illustrated in Figure 1.6, where j/j_o is calculated from Equation (1.43) over a limited range of overpotential. Usually, the linear approximation is really precise only for η<10 mV.

When current density vs overpotential data are available for both oxidation and reduction, and over a wider range of overpotentials, the data can conveniently be presented on a single line using a rearranged form of Equation (1.43):

Again, the exchange current density is found from the intercept at η=0 and the transfer coefficient from the slope.

In summary, the rate of electron transfer or current density for a reaction is determined by:

• the exchange current density (or by the standard rate constant and concentrations),

• the transfer coefficient,

• the applied overpotential,

• temperature, largely through its influence on the exchange current density.

A major obstacle to understanding the kinetics of electron-transfer reactions is the terminology; as a further aid, Table 1.4 summarizes the important parameters used in this section.

The introduction to this chapter emphasized that the supply of reactant and the removal of product from the electrode surface are essential to a continuing chemical change and current. In general, there can be contributions from three forms of mass transport.

1. Diffusion is the movement of a species due to a gradient in concentration. In other words, it is the physical process whereby nature minimizes differences in concentration, and diffusion will always occur from regions with high concentration of a species to regions that are more dilute until the concentration is uniform. Diffusion is inevitable with all electrode reactions since the electron transfer leads to a lowering of reactant concentration and an increase in product concentration at the surface compared to the bulk solution.

2. Convection is the movement of a species due to external mechanical forces. Convection may be introduced deliberately by shaking the cell, sparging the solution with gas, stirring the solution or moving the electrode. In many circumstances, however, it is desirable to be able to describe the convective regime quantitatively; this is possible only for systems with simple hydrodynamics and these include the rotating disc electrode and flow of the solution past a flat plate electrode. Normally, if the experiment involves forced convection, the rate of transport of species by convection totally dominates that by diffusion. The practical electrochemist must also be aware of ‘natural convection’ in unstirred solution. This arises from causes such as random vibrations in the laboratory or the small density gradient resulting from both concentration and temperature changes within the layer adjacent to the electrode associated with the electron-transfer reaction itself at the electrode surface.

3. Migration is the movement of charged species due to a potential field. In all electrochemical cells, current is driven through the solution between two electrodes and this requires the existence of a potential gradient. Migration is then the process whereby charge passes through the solution (see the discussion of Figure 1.1). Migration is purely an electrostatic phenomenon and is not necessarily an important mode of transport for either the reactant or product of the electrode reaction. Although, in all electron-transfer reactions, either the reactant or the product (or both) must be an ionic species, migration may not be an important form of transport for these species. In systems with a large excess of an inert electrolyte it will be largely the ions from this electrolyte that carry the charge through the solution. Indeed, this is another reason why most laboratory experiments are carried out with a high concentration of an electrolyte in solution. In contrast, in industrial electrolysis cells, the reactant may be charged and present in high concentration (e.g. a chlor-alkali cell, Figure 1.1) and then migration of the reactant must be taken into account.

Before proceeding further, it is important to recognize the principles underlying the interpretation of data from electrochemical experiments. Conclusions from practical electrochemistry are always based on a comparison of the experimental responses with predictions based on a model, as illustrated in Figure 1.7. These may be qualitative or quantitative but both are based on an understanding of the mass transport regime. Indeed, the latter requires that the regime may be described by a set of equations that may be solved; such experiments are therefore limited to a few mass transport regimes. The electrochemical experiment is introduced into the model via the surface concentrations and how they change with time; these are calculated from either the Nernst equation or the Butler–Volmer equation (Section 1.6). Fortunately, in general, the mathematics have been done earlier and the solutions are in the literature. The qualitative interpretation, however, greatly enhances understanding of our experiments and the use of equations from the literature requires us to be certain that the experimental mass transport regime is appropriate to the equations used. Again, in contrast, in industrial cells the objective is usually only to enhance the rate of mass transport, which can be achieved with a wide variety of approaches.

At this stage, it is useful to be very clear as to the objectives of laboratory experiments. As well as defining the overall chemical change, the aim must be to determine as much as possible about the mechanism and kinetics of the electrode process. The first stage will be to identify the key steps, including the rate-determining step; for a simple electrode reaction, this may be electron transfer or mass transport but it will also be necessary to develop tests to identify when adsorption, coupled chemical reactions and/or phase growth are involved and to identify features of the experimental responses that will disclose roles for such steps. Then, the next stage will be to determine parameters quantitatively, for at least the rate-determining step. In support of electrochemical technology, laboratory experiments usually have two clear goals, (a) to increase the selectivity of the reaction of interest when, for example, there are competing electron-transfer reactions or coupled chemical processes and (b) to modify the rate (current density) for the reaction of interest – usually the aim will be to increase the rate of the reaction of interest (the exception is the inhibition of corrosion).

In practice, laboratory experiments are carried out using one of two types of mass transport regime:

1. Diffusion only systems – The experiments are carried out with a still solution in a thermostat with a large excess of an inert electrolyte. Diffusion is the only form of mass transport that need be considered, at least until natural convection begins to interfere after some 10–100 s.

2. Convection as the predominant form of mass transport. In this book, such experiments will be illustrated by the rotating disc electrode. Migration as an influence on the transport of electroactive species will again be avoided by the addition of an inert electrolyte. Diffusion close to the electrode surface will still occur but commonly the introduction of convection increases the rate of transport by more than a factor of ten so that it is clearly the predominant mode of mass transport. In such experiments the regime is often called convective-diffusion.

The simplest model to describe such experiments is that of linear diffusion to a plane electrode. It assumes that the electrode is completely flat on a molecular scale and also of infinite dimensions so that concentration differences arise only in the direction perpendicular to the electrode surface. At first sight, this seems a very idealistic model since the electrode will not be completely flat and is certainly of finite size. Moreover, the common electrodes in the laboratory are discs, spheres, wires and spades. In fact, however, linear diffusion to a plane electrode is a very satisfactory model and several real electrode geometries may be shown by rigorous mathematics to be adequately approximated by it in experimental conditions.

The theoretical treatment of many electrochemical experiments is developed through calculating the way that concentrations of reactants and products change with distance from the electrode through a layer close to the surface that is disturbed by the electrode reaction – the so-called concentration profiles. Moreover, our understanding of our experiments is greatly enhanced by considering in a qualitative way how these concentration profiles change with time during experiments.

Diffusion is described quantitatively by Fick's laws (Figure 1.8). In the context of the model, linear diffusion to a plane electrode, these may be written in one dimension, namely, that perpendicular to the surface. Fick's first law discusses the rate of diffusion through a plane parallel to the electrode and at a distance, x, from the surface.

The rate of diffusion is known as the flux and it has the units mol cm⁻² s⁻¹. Fick's first law states that the flux is proportional to the concentration gradient, dc/dx at the distance, x, from the electrode or:

where the proportionality constant is known as the diffusion coefficient, D(cm² s⁻¹). The minus sign ensures that the species diffuses from concentrated to dilute regions of the solution. In aqueous solutions, a typical value for a diffusion coefficient is 10⁻⁵ cm² s⁻¹. The second law considers the change with time of the concentration of the diffusing species at the centre of a volume element bounded by two planes parallel to the surface. The concentration changes result from diffusion into the element through one plane and out through the other. Fick's second law states:

The concentration is now a function of both distance and time, i.e. the concentration profiles, c=f(x,t) will change with time as diffusion seeks to minimize differences in concentrations throughout space. The solution of this partial differential equation, together with initial and boundary conditions appropriate to the experiment, is the approach to developing a precise theoretical description of experiments. Generally, this is achieved through Laplace transform procedures although, in general, the solutions are readily available in textbooks.

An important application of Fick's law is to the situation at the electrode surface (Figure 1.9). The electrode reaction leads only to the interconversion of O and R and, since the law of conservation of matter must apply, the fluxes of O to the surface and R away from it must be equal. In addition, the conversion of one O into one R must be accompanied by the transfer of n electrons. Hence, the fluxes of O and R at the surface may be related to the flux of electrons within the electrode:

This shows the relationship between the current density and fluxes of O and R at the surface and leads to expressions for the current density from the solutions to Equation (1.51):

In the steady state, diffusion profiles will always be linear; if the profiles are not linear, there will be some points in space where the concentration differences have not reached a minimum value and diffusion will continue until the concentration differences have been minimized everywhere.

Many experiments are carried out under conditions of non-steady state diffusion; with electronic instrumentation it is possible to change the electrode potential, and therefore the surface concentration of reactant, rather rapidly. In comparison, diffusion is a rather slow process and changes to the concentration profiles close to the electrode resulting from the change in surface concentration will occur over several seconds. The argument is best developed using a specific example:

From Equations (1.29) and (1.44) the change in potential is effectively an instruction to the electrode to change instantaneously the ratio of c_O/c_R at the surface to a very high value. This can happen only by the rapid conversion of R into O at the electrode (a high current density will be observed) and the concentration of R at the surface will drop from c_R to very close to zero). But this change in concentration is achieved immediately only at the electrode surface, x=0. However, the change in concentration at the electrode surface has created concentration differences and diffusion will result, causing R to move towards the electrode. If the potential is held at the high positive overpotential, the surface concentration of R will remain close to zero and the concentration profile will develop with time. Diffusion will seek to minimize concentration differences at all distances from the surface. Inevitably, two trends will occur: (a) the flux of R to the surface will decrease with time and, since the observed current is proportional to this flux, the current density will drop significantly with time and (b) the thickness of the layer affected by the experiment (the diffusion layer) will increase as species away from the surface learn about the event at the electrode surface. Figure 1.10(a) shows the development of the concentration profiles during this experiment.

Such a qualitative discussion of the concentration profiles during the experiment is illuminating and leads to the conclusion that the response to the potential step is a transient where the current density decreases with time. However, to predict the exact form of the falling transient it is necessary to solve Fick's second law, Equation (1.51). Effectively, the mathematical procedure will have to carry out three integrations, one with respect to time and two with respect to distance. To evaluate the three integration constants it is necessary to specify the concentration of R at all distances at one time and two distances for all times. Fortunately, this information is available. At the instant the potential is stepped, t=0, no chemistry has occurred and the concentration of R is uniform at the level prepared for the experiment, i.e.:

In addition, following the potential step, the concentration of R at the electrode surface can be written as zero while a long way from the electrode, no chemical change has occurred, i.e.:

The solution of this set of equations (found by Laplace transformation procedures) leads to:

This is known as the Cottrell equation and the theoretical transient is shown in Fig 1.10(b). In the experiment discussed, all information about electron transfer has been lost by the choice of potential and the only parameter that may be determined relates to diffusion, the diffusion coefficient. The discussion leads to (a) the exact form of the falling j vs t transient, Equation (1.56), (b) tests to determine whether an electrode reaction is diffusion controlled at an experimental potential and (c) methods to determine the diffusion coefficient; (b) and (c) can be achieved in several ways:

• Plotting j vs t^1/2 and obtaining a straight line passing through the origin confirms that the reaction is diffusion controlled –D is obtained from the slope.

• Demonstrating that jt^1/2 is a constant also shows that the reaction is diffusion controlled –D is then calculated from this value.

• Plotting Equation (1.56) for various values of D and fitting to the experimental data.

To study the kinetics of electron transfer, it would be necessary to step to a lower overpotential where the surface concentration is not zero. Fick's second law would then have different boundary conditions at x=0 and the j vs t transient would have a different shape (Chapter 7).

Another clear conclusion from this discussion is that the rate of non-steady state diffusion is a function of time; by carrying out experiments at short time, the rate of diffusion to the electrode can be increased substantially. This concept is central to the way electrochemical experiments are carried out. Exploitation of time in chronoamperometry, scan rate in cyclic voltammetry and frequency in ac impedance allows experiments to be carried out with different rates of diffusion and this is central to investigating the kinetics of both electron transfer and coupled chemical reactions. This will become clearer in Chapter 7.

Under these conditions, the dominant form of mass transport will be convection; in the laboratory it is quite possible to design experiments where the introduction of convection increases the rate of mass transport by more than a factor of a hundred. For mechanistic and kinetic studies, it is important that the hydrodynamics of the system can be described precisely. The example used here is the rotating disc electrode.

The rotating disc electrode consists of a polished disc of the selected electrode material surrounded by an insulating sheath of significantly larger diameter. Figure 1.11 illustrates the solution movement introduced by the rotation of the disc electrode. The principal movement is towards the disc, perpendicular to the surface. But since the solution cannot pass through the solid electrode/sheath surface, close to the surface the solution is thrown outwards and then circulates back into the bulk volume. As far as the disc is concerned, the mass transport approximates well to a one-dimensional regime where the critical direction is perpendicular to the surface. It can also be seen that the rate of transport of species to the electrode surface will be determined by the rotation rate of the disc, ω; the higher the rotation rate, the stronger the stirring of the solution and the higher the rate of transport. Chapter 7 covers applications of the rotating disc electrode; here we shall only outline a widely used model for the mass transport to the rotating disc electrode, namely, the Nernst diffusion layer model.

Figure 1.12 sets out the model, where it can be seen that the solution is divided into two regions: (a) the bulk solution that is strongly mixed by the stirring and thereby maintains a constant concentration of the reactant throughout this region and (b) within a boundary layer adjacent to the surface, thickness δ, the solution is totally stagnant and diffusion becomes the only mode of mass transport. Clearly, this is an oversimplification. The hydrodynamics cannot change from a regime of strong stirring to a still solution immediately at x=δ. However, the solution away from the disc is strongly and uniformly pumped towards the surface and the solution decelerates as it approaches the surface until the flow rate in the x-direction is zero at the solid surface. Hence, the Nernst diffusion layer model may be regarded as a useful equivalence model although it must be recognized that the layer thickness, δ, has no physical reality. Fortunately, the conclusions from the model are supported by a more precise mathematical model. The Nernst diffusion layer model, however, allows us to understand the key features of voltammetry at a rotating disc electrode.

Firstly, the response will depend on the rotation rate of the disc. In the model, stronger stirring of the bulk solution equates with a decrease in the thickness of the boundary layer. The precise mathematical model concludes that the thickness of this ‘equivalent’ boundary layer is given by:

where ν is the kinematic viscosity (i.e. viscosity/density) of the solution. It can be seen from Figure 1.12 that increasing the rotation rate will increase the flux of reactant to the surface and hence the current density. Notably, the thickness of this mass transport layer is of the order of microns; the double layer discussed in Chapter 3 is quite different and it is a layer with a thickness of a few molecular dimensions, i.e. nanometres.

Secondly, it is now possible to write simple expressions for the current density. In the steady state, the concentration profiles within the stagnant layer must again be linear and hence the current density (for the oxidation of R to O) is given by:

The surface concentration of the reactant, R, is determined by the potential applied and, at a high enough overpotential, this effectively becomes zero. Increasing the overpotential further cannot further change the surface concentration and the current density will become independent of potential. This limiting current density, j_L, represents the highest rate at which R can be converted into O. The limiting current density is given by:

Thirdly, it can be seen why a voltammogram at a rotating disc electrode is sigmoidal. Figure 1.13(a) shows the way in which the concentration profiles change with applied potential. Only the surface concentration changes with potential; at the equilibrium potential, the concentration at the surface will be the same as the bulk, while as the potential is made more positive the rate of conversion of R into O will increase, thereby reducing the surface concentration of R towards zero when no further change can occur. While the surface concentration is decreasing the current will increase, but once it reaches zero no further decrease is possible and the current will plateau (Figure 1.13b).

In both Equations (1.58) and (1.59) it is possible to use Equation (1.57) to replace δ by measurable quantities. For example, with Equation (1.59), one obtains an expression known as the Levich equation:

It shows that a mass transport controlled current density is proportional to the square root of the rotation rate of the disc.

Notably, under all mass transport regimes, a universal expression for the mass transport controlled current density is:

where k_m is a rate constant describing the rate of mass transfer. It is known as the mass transfer coefficient and can be determined experimentally from measurements with a known mass transport controlled reaction. At a rotating disc electrode, comparison of Equations (1.60) and (1.61) shows that:

So far we have largely treated electron transfer and mass transfer as independent processes but at the beginning we noted that the overall rate of the sequence:

will be determined by the slowest step. Hence, understanding a complete current density vs potential response requires the recognition of how the steps interact. Again, we shall consider a particular experiment:

For the oxidation of R to O, four situations may be recognized:

1. At the equilibrium potential the current density will be zero – no net chemical change will occur. Because of the ratio of concentrations in the solution, the equilibrium potential will be negative to the formal potential, in fact it will be [E$e o$−(60/n)]mV [Equation (1.29)].

2. The solely electron-transfer controlled regime may be recognized experimentally; the current density:

   1. varies strongly with potential;

   2. is independent of the mass transport regime (tested by varying the rotation rate of the disc, bubbling gas or even shaking the cell).

3. At very high overpotential the rate of electron transfer must increase to a very high rate and mass transfer will become the rate-determining step. The surface concentration of R will drop to zero and Equations (1.59–1.61) will apply. Pure electron transfer may also be recognized experimentally; the current density:

   1. is independent of potential;

   2. varies strongly with the mass transport regime.

4. Intermediate overpotentials, where the rates of electron transfer and mass transfer are similar. The simple equations will not be obeyed and this situation corresponds to the case where the surface concentration is significantly different from c_R but has not yet reached zero. This range of overpotentials extends from where electron transfer is clearly much slower than mass transfer to where the reverse is true. This requires a change in overall rate, say of 25 if a ratio of rates of 5 is considered enough to give a single rate-determining step. Hence one might expect to see mixed control over the range j_L/25 < j < j_L. It can be seen that the whole of the steeply rising portion of the voltammogram corresponds to mixed control; obtaining information about electron transfer or mass transport from this region will always be more complex than from potential regions with a single rate-determining step, i.e. the very foot of the wave and the plateau region. Experimentally, the current density in the region of mixed control:

   1. Varies with potential but less strongly than for pure electron-transfer control. With increasing overpotential, there will be a gradual change from an almost exponential dependence on overpotential to no variation.

   2. Varies with the mass transport regime but less strongly than for full mass transfer control. Again, with increase in overpotential, the variation will be gradual from no dependence on the mass transport regime to that for full mass transfer control.

Figure 1.14 shows the full current vs potential characteristic for the solution where c_R=10c_O both as a plot of j vs E and log j vs η. We have discussed above only oxidation but the same arguments will apply to the reduction portion of the responses. Because of the differences in reactant concentrations, the rate of mass transfer of R and O will differ by a factor of 10 and, in consequence, the mass transfer limiting current densities will also differ by a factor of 10. The regions of electron-transfer control, mixed control and mass transport control are marked on the two curves. The separation of the two waves in the j vs E response is a function of the kinetics of electron transfer; the waves separate with decreasing values of j_o(or k_s) since the overpotential to drive both oxidation and reduction will increase.

In electrochemistry, the terms ‘reversible’ and ‘irreversible’ are used in the tradition of thermodynamics. They are used to discuss whether the electron-transfer reaction at the surface is in thermodynamic equilibrium.

So far, we have discussed only the case where the electron transfer is slow compared to mass transfer. Then the electron-transfer reaction is not in equilibrium; the surface concentrations of O and R are determined by kinetic equations and an overpotential is required to drive the reaction at a particular rate. Electrode reactions where this treatment is appropriate are termed ‘irreversible’. There is, however, another possibility. If the electron-transfer step is inherently fast (high k_s and j_o), then it is possible that, at all potentials and under the prevailing mass transfer regime, the electron-transfer reaction on the surface remains in equilibrium. Then, the surface concentrations may be calculated from the Nernst equation, which is purely a thermodynamic equation; no overpotential is necessary to obtain any rate of conversion between O and R. Such electrode reactions are termed ‘reversible’.

Figure 1.15 shows the voltammograms obtained at a rotating disc electrode for both types of electrode reaction. As noted previously in the discussion of Figure 1.14, the voltammogram for an irreversible couple has two well-separated waves, one for oxidation and one for reduction, with the separation depending on the kinetics of the couple O/R. In contrast, the voltammogram for a reversible couple shows a single wave, the response crossing the zero current axis directly between oxidation and reduction because no overpotential is necessary to drive the electron transfer. The wave for a reversible couple is also much steeper.

Notably, however, whether a reaction appears reversible or irreversible will depend on the mass transport regime. For reactions with intermediate kinetics, it is possible that the electrode reaction appears reversible with experimental conditions where the mass transport regime is relatively poor but a substantial increase in the rate of mass transport (e.g. by increasing the rotation rate of a disc electrode or working on a shorter timescale) may lead to the reaction becoming irreversible. This is the concept underlying methods to study fast electron-transfer reactions (Chapter 7). More quantitatively, the distinction between reversible and irreversible depends on the relative values of k_m and k_s, the mass transfer coefficient for the experimental conditions and the standard rate constant for the couple O/R, respectively.

Adsorption is the interaction of species from the solution phase with the electrode surface. The species may be the reactant, an intermediate or a product of the electrode reaction or even another species added to change the rate or mechanism of the electrode reaction (usually by itself adsorbing on the surface). Both organic and inorganic species as well as both ions and neutral molecules may be adsorbed on the electrode. Certainly, the solvent and ions of the inert electrolyte will adsorb under many conditions.

The nature and strength of the interaction vary widely. The strongest arises from the formation of a covalent bond, e.g.:

but there is also a range of interactions resulting from electrostatic forces. At each applied potential, the electrode surface will have a characteristic surface charge that is also dependent on the electrode material and the solution composition. This surface charge leads to the attraction of ions, dipoles and π-electron systems. Moreover, dipoles may be induced by the electric field at the electrode/solution interface. All adsorption processes may be reversible or irreversible and Reactions (1.64) and (1.65) are examples of these two possibilities.

Species may be adsorbed from solution onto a surface at open circuit but the richness of the effects observed is enhanced by the variation of charge with applied potential; the surface charge may readily be made positive or negative and the charge density varied from low to high. The potential where the surface charge changes from positive to negative is known as the potential of zero charge.

The extent of adsorption, often expressed as a fraction of the surface covered by the adsorbate, θ, is best understood in terms of two competitions: (a) between the surface and the solution for a potential adsorbate – thus, for example, organics are less likely to adsorb from an organic medium than water; (b) between all species in the solution (reactant, solvent, ions from the electrolyte, additives and impurities) for the finite number of sites on the electrode surface. The tendency to adsorb is expressed as a Gibbs free energy of adsorption but this depends on both electrode material and the solution medium. The coverage by adsorbate also depends on the potential of the electrode; ions are most likely to absorb on a surface of opposite charge while neutral molecules tend to adsorb most strongly at the potential of zero charge where there is no competition from ions.

Many adsorption processes are reversible (i.e. rapid and therefore in equilibrium) and then the coverage is conveniently discussed in terms of isotherms. These relate the coverage, θ, at constant temperature, to the concentration of adsorbate in solution and the Gibbs free energy of adsorption. Many isotherms have been proposed and they differ in the extent and method of taking into account the lateral interaction between adjacent adsorbate species on the surface. The simplest isotherm is the Langmuir isotherm and this assumes that there are no lateral interactions (i.e. the Gibbs free energy of adsorption is independent of coverage). It is written:

On the other hand, the Frumkin isotherm is based on the assumption that it becomes more difficult to adsorb further species as the coverage increases – in fact the Gibbs free energy increases linearly with coverage:

When the adsorbate is electroactive or results from an electron-transfer reaction, the coverage can be assessed by recording a cyclic voltammogram and determining the charge under the peak for the electrode reaction. The peak shape for the oxidation/reduction of an adsorbed species is different from that for a solution-free species and the potential for the reaction will also be shifted compared to that for the same reaction with the reactant/product in solution (Chapter 7). For example, adsorption of the reactant on the electrode surface will stabilize the species to electron transfer and will make oxidation/reduction more difficult; oxidation of the adsorbed species will take place positive to the solution species and reduction negative to the solution species. Adsorption of the product will allow the reaction to occur more easily and the opposite shifts are observed. If the adsorbate is not electroactive, at least in the potential range of interest, the surface coverage is normally deduced using AC techniques to determine the capacitance as a function of the concentration of the species in solution and potential (Chapter 3).

When the adsorption process is irreversible, the coverage cannot be discussed in terms of an isotherm. But such reactions are important in fuel cells. Equation (1.65) is the first stage in the oxidation of methanol at a fuel cell anode. It is followed by a series of further steps that involve other adsorbed organic fragments as well as adsorbed carbon monoxide. Each of the adsorbates may be oxidized to CO₂ or decompose in competing pathways so that the coverage by each species depends on the kinetics of several steps. In situ spectroscopic techniques, particularly infrared and mass spectrometry directly coupled to the electrochemistry, have aided our understanding of such reactions but, unsurprisingly, such complex systems cannot yet be fully defined. In simple situations where only one adsorbed species is present, coverage can usually be determined from the charge for oxidation/reduction.

Catalysis is the enhancement of the rate of a reaction by a species that is not consumed in the overall reaction sequence; the role of the catalyst is to provide an alternative, low energy of activation pathway for the conversion of reactant into product. In electrocatalysis, the catalyst is usually the electrode material itself and the mechanism by which it increases the rate usually involves adsorption of reactant or intermediates. Experimentally, the role of the electrocatalyst is to (a) increase the current density at a fixed potential or (b) to reduce the overpotential to support the reaction at a fixed current density. Because of the importance of fuel cells, important reactions requiring electrocatalysis include oxygen reduction and hydrogen or methanol oxidation. These systems are discussed further in Chapters 5 and 8.

The adsorption of molecules not directly involved in the electrode reaction can inhibit the electron-transfer reaction. The best known example is corrosion inhibitors. The role of the inhibitor can be modelled in two ways. Firstly, it can be envisaged that the adsorbate completely covers the surface or, at least, all the active sites, thereby increasing the distance over which the electron must hop between the electrode and reactant. This is a possible mode of operation of inhibitors that slow down iron dissolution, i.e. anodic inhibitors. Secondly, the mode of action may involve competing for sites on the surface with an intermediate in a reaction; for example, cathodic corrosion inhibitors compete with adsorbed hydrogen atoms for sites on a steel surface.

In the electrodeposition of metals, additives are widely used to control the form of the deposit. This may, for example, be its smoothness or brightness, the morphology of the deposit, or the size or shapes of the crystallites. The additives are thought to act by adsorbing on particular sites on the surface.

Additives can direct an electrode reaction down different pathways leading to different products. An example is an industrial process for the hydrodimerization of acrylonitrile to adiponitrile, Reaction (1.11) in Table 1.2. This reaction requires the presence in the medium of tetraalkylammonium ions. In their absence, hydrogen evolution is a major competing electrode reaction and the main organic product is propionitrile. The tetraalkylammonium ions are thought to adsorb on the cathode surface and create a local environment with a low proton donating ability.

It is very common for the product of electron transfer to be unstable and for it to undergo a chemical reaction, e.g.:

where for the moment it will be assumed that P is electroinactive and stable. The chemical step may occur either as a heterogeneous reaction while R is adsorbed on the electrode surface or as a homogeneous reaction in the electrolyte medium while R is moving away from the electrode surface. Only the case of a homogeneous reaction will be considered in this section. In addition, the primary electrode reaction is written as a reduction. Naturally, a similar section could be written with oxidation as the primary step.

Importantly, the chemistry of R will generally be unchanged by its origin at the electrode surface; when present at a similar concentration, the reaction pathways and products will be the same as when it is formed by a chemical reaction in the same medium as used for the electrolysis (although, unlike solutions for chemical reaction, electrolysis media usually contain high levels of electrolyte and this may effect the chemistry of charged intermediates). Commonly, the mechanism and kinetics of the reactions of R are the same, and then electroanalytical techniques provide a way to study the homogeneous chemistry of R. Of course, as always with reactive intermediates, there is no certainty that the chemistry of R will lead to a single product and its decay can often involve competitive pathways. In addition, the rate constant for Reaction (1.70) determines the thickness of a reaction layer at the electrode surface. If the rate constant is high, all the chemistry occurs within a thin layer close to the electrode surface and the chemical reaction will influence experimental data strongly. In contrast, if the rate constant is low, the reaction layer is thick and the intermediate may even survive into the bulk solution; the chemistry then has less influence on experimental data for the electrochemical experiment.

In many systems, the product of the chemical reaction, P, may itself be electroactive and may undergo further reduction at the potential where it is formed or at more negative potentials or, indeed, it may be oxidized (back to O or an entirely different species) at a more positive potential. Such complexities add much to the richness of experimental electrochemistry. For example, cyclic voltammograms are observed with multiple peaks and the dependence of the additional peaks on potential scan rate and potential range provides much information about the chemistry. Another important type of reaction is:

where the product of electron transfer reduces (or oxidizes) an electroinactive molecule with regeneration of the reactant for the electrode reaction; O can be considered as a catalyst for the conversion of Q into P. In the laboratory, these are known as ‘mediated reactions’ while in electrochemical technology they are more usually called ‘indirect electrode reactions’. The mediator couple O/R may be inorganic (e.g. Ce⁴⁺/Ce³⁺, Cr₂O₇²⁻/Cr³⁺, Br₂/Br⁻, Zn²⁺/Zn), organic or an enzyme.

In view of the large variety of mechanisms observed, a nomenclature has been developed to describe mechanisms concisely (Table 1.5). Throughout, ‘e’ stands for an electron-transfer reaction and ‘c’ for a homogeneous chemical step.

Electrolysis has been used for the synthesis of several chemicals, both organic and inorganic – both in the laboratory and on an industrial scale. Commonly, the isolated products result from a complex sequence of chemical steps and also multiple electron transfers. In particular, the electrode reactions of organic molecules almost always involve 2e⁻ steps since the cleavage or formation of bonds involves two electrons. Hence, to form single products in high yield it is necessary to control (a) the electrode potential so that the electrode reaction forms a single intermediate (either R or P could undergo further reduction if the potential is uncontrolled) and with a controlled flux from the electrode surface; this flux (and the current density) will also depend on the reactant concentration and the mass transport regime; and (b) the solution conditions (solvent, electrolytes, pH, the presence of other reactants, temperature, etc.) so that the chemistry of R in solution is selective.

Usually, the conditions for electrolysis are determined by a mixture of experience and a parametric study. Variables to be considered will include:

• electrode potential (although perhaps controlled indirectly through use of a controlled current and knowledge of the relationship between j and E for the particular reactant concentration and mass transport regime);

• electrode material;

• solvent, electrolyte and pH;

• the mass transport regime;

• electroinactive reactants to trap intermediates formed at the electrode;

• additives (adsorbates) to modify the environment at the surface;

• the cell design – divided or undivided, geometry and form of the electrode, mass transport regime.

In practice, any experimental variable may influence several steps in the sequence. For example, temperature will influence the rate of mass transport [D will increase 1–2% per degree Kelvin (K)], the rate of electron-transfer reactions and the rate of all the chemical steps. Since the influence of temperature on the rates of competing reactions will not be the same, its influence on selectivity has to be determined experimentally. In the development of a commercial process, the selection of conditions will also depend on the relative priorities placed upon performance factors such as (a) the rate of conversion (current density), determining the initial cost of purchasing cells and (b) the reaction selectivity, the extraction procedure and energy consumption, determining the running costs. It should also be recognized that in an electrolysis cell the objective is rapid chemical conversion of reactant into product. Hence, the reactant will be present in high concentration and the cell will be designed with a large ratio of electrode area to electrolyte volume and to give a high rate of mass transport. This is quite different from the cell used for experiments to investigate the mechanism and kinetics (Chapter 6).

Returning to the discussion of an ec system, for example Reactions (1.69) and (1.70), it is necessary to consider how the chemical step is introduced into the theoretical description of an experiment. Figure 1.16(a) shows the concentration profiles for both O and R at one instant during an experiment where the reduction of O to R is diffusion controlled, R is stable and the initial solution contains O but not R. The concentration profile for the reactant O was discussed earlier. For the product R, its concentration will be high at the surface where it is being formed and zero in the bulk solution. In fact, the exact form of the concentration profile for R is readily deduced; since both O and R are stable and the only chemistry is the interconversion of O and R at the electrode, the sum of the concentration of O and R must be equal to the initial concentration of O at all distances from the electrode surface. If, however, R is unstable and undergoes a first-order, homogeneous chemical reaction, the concentration of R will be lower at all x because the reaction is taking place with a rate kc_R throughout the reaction layer (Figure 1.16b).

Quantitatively, in a still solution, the change in concentration at any point in the reaction layer will depend on two factors: (a) diffusion into and out of the segment centred on x, i.e. Fick's second law, and (b) the rate of the chemical reaction. Hence:

The electroactive species O is not involved in the homogeneous chemistry and therefore changes in concentration arise only because of diffusion. Hence:

Although it might not be necessary for understanding the electrochemical response, it is also possible to write an equivalent equation for the final product, P:

To complete the description of any experiment, one initial and two boundary conditions for each species will be required to allow integration of Equations (1.73) and (1.74). These will depend on both the solution prepared for the experiment and the nature of the electrochemical experiment.

In practice, the importance of the coupled chemical reaction in determining the response to the electrochemical experiment will depend on the relative values of the half-life of the intermediate R (τ_1/2=(log2)/k) and the timescale of the experiment (e.g. in cyclic voltammetry this is determined by the potential scan rate). If the timescale of the experiment is much longer than the half-life of R, then almost all the intermediate formed at the electrode will have been consumed in the chemical step during the experiment and it is to be expected that the electrochemical response will be strongly changed by the coupled chemistry. In contrast, if the electrochemical experiment is completed in much less time than the half-life, the response will know little of the coupled chemistry. Thus, the concept underlying electrochemical methods to study coupled chemistry is to monitor the response over a range of experimental timescales.

Electrode reactions that lead to the formation of a new solid phase on the electrode surface are surprisingly common. Examples include:

1. cathodic deposition of metals, e.g. Cu, Ni, Cr;

2. anodic deposition of conducting oxides, e.g. PbO₂;

3. deposition of conducting polymers, polypyrrole, polyaniline;

4. anodic formation of metal salts, e.g. AgCl, HgSO₄;

5. anodic formation of passivating oxides, TiO₂, Al₂O₃;

6. deposition of insulating organic films by electropolymerization or electropainting.

The type and extent of conductivity is the major factor in determining the electrochemical response from any experiment but the formation of a solid phase always significantly influences the electrochemical response. For example, the formation of an insulating film will always quickly lead to a very low current density and such processes are usually very irreversible.

A more interesting phase formation process is illustrated by the deposition of copper onto a carbon surface from a sulfuric acid bath:

It involves several distinct stages:

• nucleation of the new phase;

• growth of the individual metal centres;

• overlap of the growing centres, leading to a complete layer;

• thickening of the complete layer.

This is illustrated in Figure 1.17 for the particular case of continuous nucleation of hemispherical centres. Nucleation is the formation of stable centres. Most chemists will know from their attempts to crystallize or recrystallize a salt or compound that the formation of nuclei of a new phase is always an improbable event that must be forced to occur, e.g. by seeding or supersaturating the solution. This is because small centres are always unstable and tend to redissolve; the surface is unstable while stability comes from the interactions of atoms/ions/molecules in the bulk of the material. In the electrodeposition of metal, the nuclei consist of the metal atoms and when the centres are small the ratio of surface area to volume tends to lead to their redissolution. Formation of nuclei large enough to be stable is driven by the application of an overpotential in addition to that required to drive the electron-transfer reaction. There are also two limiting cases of the kinetics of nucleation: (a) instantaneous nucleation, in which nucleation occurs at all available sites, and (b) progressive nucleation, where the number of centres increases with time, following first order kinetics. The number density of nuclei can also vary strongly and it will depend on the electrode material and its pretreatment and the solution conditions as well as the metal being deposited. Once formed as stable entities, the individual centres grow with a characteristic shape such as hemispherical or conical until they are large enough to overlap with other expanding centres on the surface. Eventually, a complete layer will be formed and this will then thicken. At all of these stages, the reduction of cupric ion may be under electron transfer or mass transport control.

Each of these possibilities leads to different responses. In all cases, however, distinct behaviour is observed; at constant potential the current will increase with time early in the deposition process and voltammetry gives rise to responses where the current density vs potential curves are very steep. Such behaviour results from the overpotential required to initiate nucleation; once the nuclei are formed, this overpotential is then available to drive the electron-transfer reaction at a higher rate. In addition, as each centre grows, the surface area available for electron transfer is increasing and, with progressive nucleation, the number of centres will also increase.

The overall chemical change in many electrode reactions involves the transfer of more than one electron. An extreme example is the complete oxidation of azodyes to carbon dioxide, nitrate and possibly sulfate; such reactions can involve in excess of 100 electrons. It is important to recognize that, in general, electrode reactions occur in single electron steps. Indeed, there is a strong case for writing n=1 in all the equations in Section 1.3.2. Almost always, when the overall chemical change involves more than one electron, a complex mechanism is likely. It is probable that chemical steps either on the surface or in the solution close to the electrode will be separate single 1e⁻ events. This was illustrated in Section 1.7 and will be elaborated further in Chapter 5.

The objective of this chapter was to set out the characteristics and physical chemistry of relatively simple electrode reactions and electrochemical cells. It is only possible to expand the discussion to more complex systems when there is recognition of the complications that can arise and the difficulties and possibilities that each introduces. Hence the preliminary discussion of adsorption, coupled chemical reactions and the deposition of solid phases on electrodes. In later chapters many of these topics will be discussed more thoroughly and illustrated further. It should be emphasized that the same concepts underlie both laboratory studies and electrochemical technology.