Chapter 1: Fundamentals of Electrode Reactions Free

Electrochemistry is essentially based on the relationships between chemical changes and flows of electrons (i.e. the passage of electricity). In this connection it is well known that electron-transfer processes play an essential role in many physical, chemical and biological mechanisms and several such examples will be illustrated in the text. Perhaps in no other field of chemical reactivity can one look for and find so many relationships between theory and experimental measurements.

Two disciplines cover the majority of the theoretical and practical aspects of the mechanisms through which electron transfers proceed: electrochemistry and photochemistry. This book considers only mechanisms relating to electrochemistry.

In a purely formal manner the description of an electron-transfer event, such as the reduction in solution of Fe(III) ion, can be written in two ways, depending on whether the reduction is operated by a chemical agent or by an electrode:

• through a reducing agent (redox reaction in a homogeneous phase):

• through an electrode (redox reaction in a heterogeneous phase):

In both cases, the adopted symbolism only gives a picture of the overall process. In fact, mechanistically, the redox reactions (as with any other type of reaction) proceed by a series of intermediate steps involving phenomena such as:

• diffusion of the species through the solution;

• interaction between reagents, in the case of reactions in a homogenous phase, or interactions between reagents and electrode, in the case of reactions in a heterogeneous phase;

• formation of short- or long-lived intermediates due to variations in electronic configurations, to the eventual substitution of ligands, etc.

Commonly, oxidation–reduction reactions in a homogenous phase are classified as:

• outer-sphere reactions;

• inner-sphere reactions.

In inner-sphere reactions, the process involves a ‘transition state’ in which a mutual strong penetration of the coordination spheres of the reagents occurs (and, therefore, strong interaction between reagents), whereas in the outer-sphere reactions there is no overlap of the coordination spheres of the reagents (and, therefore, there is weak interaction between reagents).

The classical example of inner-sphere mechanism is the reduction of the Co(III) salt [Co(NH₃)₅Cl]²⁺ by Cr(II) ions ([Cr(H₂O)₆]²⁺):

The fact that from a chloro-cobalt complex a chloro-chromium complex is formed, suggests that the reaction must proceed through an intermediate state that enables the transfer of a chlorine atom from cobalt to chromium. The proposed mechanism for this reaction is shown in Scheme 1.

It is assumed that the intermediate product is [(NH₃)₅Co^III–Cl–Cr^II(H₂O)₅]⁴⁺ (Structure 1 ), in which there is a clear overlap of the coordination spheres of the two reagents. It follows that the electron transfer can take place only after such an intermediate has formed.

As an example of a reaction that involves an outer-sphere mechanism the following reaction can be considered:

In this case one may assume that the charge transfer takes place as soon as the two reagents collide, without the occurrence of any exchange of ligands (which would imply breaking of old or formation of new bonds in the reaction intermediate).

As a consequence, the mechanism with which a homogeneous reaction proceeds is conditioned by the rate of either the ligand exchange or the electron transfer. An outer-sphere mechanism is certainly active when the exchange of ligands between reagents is slower than the exchange of electrons between reagents.

According to this picture, the electron-transfer processes mediated by metallic electrodes (redox reactions in a heterogeneous phase) can also be classified as proceeding according to outer-sphere or inner-sphere mechanisms (obviously, considering the electrode surface as a reagent).

One can define as outer-sphere electrode processes those in which the electron transfer between the electrode and the active site occurs through the layer of solvent directly in contact with the electrode surface. The electrode and electroactive species are, therefore, separated such that the chemical interaction between them can be considered as practically nil (obviously, apart from their electrostatic interaction) (Figure 1).

Inner-sphere electrode processes are defined as those in which the electron exchange occurs between the electrode and the electroactive species (the metal core or its ligand) that are in direct contact with the electrode surface (Figure 2).

It should be emphasized that most electrochemically induced redox processes in inorganic chemistry proceed (or are assumed to proceed) through outer-sphere mechanisms.

As we shall be considering the electrochemical characterization of chemical systems, it is useful at this point to make clear a few fundamental concepts inherent in electrochemical processes.^1–10 

An electrode reaction is always a heterogeneous chemical process, in that it involves the passage of an electron from an electrode (metal or semiconductor) to a chemical species in solution, or vice versa.

As illustrated in Figure 3, one can depict the electrode/solution system as being partitioned roughly into four regions:

1. electrode

2. double layer

3. diffusion layer

4. mass (or bulk) of the solution.

The electrode/solution interface represents a discontinuous plane with respect to the distribution of the electrical charge. This is the result of the electrode possessing an excess of charge of a given sign (e.g. negative in the figure) in immediate contact with an excess of charge of opposite sign, due to the electrostatic attraction. This situation generates the so-called double layer, which, as we shall see, has important consequences on the electrochemical events.

The so-called diffusion layer is still a region dominated by an unequal charge distribution (i.e. in such a zone the principle of electro-neutrality is not valid) due to the electron-transfer processes occurring at the electrode surface. In fact, the electrode acts as an electrostatic pump for species of a certain charge, resulting in a flow of these charged systems from the mass of the solution (i.e. the bulk of the solution where the principle of electro-neutrality is fully valid) towards the electrode, or vice versa.

Let us consider a chemical species that possesses the two different oxidation states oxidized (Ox) and reduced (Red), both of which are stable and soluble in the electrolytic medium (solvent+inert electrolyte). The simplest formulation of the electrode reaction which converts Ox into Red:

in reality hides a sequence of elementary processes. In fact, to maintain a continuous flow of electrons:

• first, the electrode surface must be continually supplied with reagent (Ox);

• then, the heterogeneous electron-transfer process from the solid electrode to the species Ox must takes place (through an inner- or outer-sphere mechanism);

• finally, the reaction product (Red) must be removed from the electrode surface to allow the access of further amounts of Ox to the electrode surface.

Consequently, we can rationalize the process represented by Equation (1) as involving at least the following three elementary steps:

Clearly, the overall rate of the reduction process will be conditioned by the slowest elementary step, which can be associated either with the mass transport (from the bulk of the solution to the electrode surface, and vice versa) or with the heterogeneous electron transfer (from the electrode to the electroactive species, or vice versa).

As pointed out, the electrode process (1) can be described by the mentioned sequence of ‘at least’ three elementary stages. In reality, quite often other phenomena complicate the electrode reactions. These consist of fundamentally three types:

• Coupled chemical reactions It is possible that the species Red generated at the electrode surface is unstable and tends to decompose. It may also be involved in chemical reactions with other species present in solution while it is moving towards the mass of the solution (homogeneous chemical reactions) or while it is still adsorbed on the electrode surface (heterogeneous chemical reactions). Furthermore, the new species formed during such reactions may be electroactive too. These kinds of reactions are called following chemical reactions (following, obviously, the electron transfer). In addition, though less common, there are cases of preceding chemical reactions (preceding, naturally, the electron transfer). In this case, the reagent Ox is the product of a preliminary chemical reaction of a species that is not itself electroactive. For example, the reduction of acetic acid proceeds through the two microscopic stages:

• Adsorption In the sequence of reactions (2)–(4) it was assumed that the electron exchange takes place without the interaction of the species Ox and Red with the electrode surface. However, it is possible that the exchange of electrons does not occur unless the reagent Ox, or the product Red, is weakly or strongly adsorbed on the electrode surface. It is also possible that the adsorption of the species Ox or Red might cause poisoning of the electrode surface, thus preventing any electron-transfer process.

• Formation of phases The electrode reaction can involve the formation of a new phase (e.g. electro-deposition processes used in galvanizing metals). The formation of a new phase is a multi-stage process since it requires a first nucleation step followed by crystal growth (in which atoms must diffuse through the solid phase to then become located in the appropriate site of the crystal lattice).

An electrode reaction always implies a transfer of electrons. If we consider again the reaction:

Clearly, if, as in this case, the process occurring at the working electrode is a reduction half-reaction then there will be an oxidation half-reaction at the counter-electrode.

Faraday's law states that if M mol of reagent Ox is reduced, the total charge spent is given by:

The variation of charge with time, i.e. the current, i, will be equal to:

The variation of the mol number with time, dM/dt, reflects the variation of concentration per unit time, or the reaction rate, v (mol s⁻¹).

Since we are considering heterogeneous processes, the rate of which is commonly proportional to the area of the electrode, one can normalize with respect to the electrode area, A, so that:

This expression shows that the current flowing in the external circuit of Figure 4 is proportional to the rate of the electrode reaction:

This type of current, which originates from chemical processes that obey Faraday's law, is called a faradaic current, to distinguish it from non-faradaic currents that, as we shall see in Section 1.5, arise from processes of a strictly physical nature.

During an electrochemical experiment the experimental conditions are carefully controlled to minimize the onset of non-faradaic currents as much as possible.

According to band theory, the electrons inside a metal populate the valence band up to the highest occupied molecular orbital, which is called the Fermi level. The potential applied to a metallic electrode governs the energy of its electrons according to Figure 5.

If the electrode potential is made more negative with respect to the zero-current value, the energy of the Fermi level is raised to a level at which the electrons of the metal (or, the electrode) flow into the empty orbitals (LUMO) of the electroactive species S present in solution (Figure 5a). Thus, the reduction process S+e⁻→S⁻ takes place.

In an analogous way, the energy of the Fermi level can be decreased by imposing an electrode potential more positive than the zero-current value. A situation is now reached in which it is energetically more favourable that the electroactive species donates electrons from its occupied molecular orbital (HOMO) to the electrode (Figure 5b). The oxidation process S→S⁺+e⁻ has been activated.

The critical potential at which these electron-transfer processes occur identifies the standard potential, E°, of the couples S/S⁻ and S⁺/S, respectively.

Let us consider, for example, the case of the reduction process:

By raising the electrode potential towards more and more negative values a threshold value will be reached: above this value the reduced form S⁻ is stabilized at the electrode surface, whereas below this value the oxidized form S is stabilized at the electrode surface. This threshold value is defined as the standard potential of the S/S⁻ couple.

Since the potential regulates the energy of the electron exchanges, it also controls the rate of such exchanges and, hence, the current.

This biunique correspondence between current and potential implies that if one of the two parameters is fixed the other, consequently, also becomes fixed.

As discussed in Section 1.2.3, for an electrode reaction to take place one needs two electrodes: a working electrode, at which the electron-transfer process of interest occurs, and a counter electrode, which operates to maintain the electro-neutrality of the solution through a half-reaction of opposite sign, see Figure 4. Unfortunately, it is not possible to measure rigorously the absolute potential of each of the two electrodes (i.e. the energy of the electrons inside each electrode). The difference of potential set up between the two electrodes is, instead, easily experimentally measured and is defined as cell voltage, V. However, as illustrated in Figure 6, this cell voltage is the sum of a series of differences of potential.

At each of the two electrode/solution interfaces, where an electrical double layer is set up, there are sharp changes in potential. These changes of potential control the rate of the faradaic reactions that occur at the two electrodes. Each of the two jumps in potential is identified as the electrode potential of the respective electrodes. In addition, the cell voltage includes a further term because the solution has an intrinsic resistance, R_s. Therefore, when the current flows through the solution between the two electrodes it gives rise to the so-called ohmic drop, which, being equal to the product i·R, is defined as the iR_s drop. Only when one is able to make iR_s=0 (or at least render it negligible) does the measured cell voltage reflect the difference between the two electrode potentials (or, V=ΔE).

Since we are interested in controlling accurately the potential of the working electrode (in order to condition the rate of the electron transfer between this electrode and the electroactive species), we must work on the difference of potential between the two electrodes. Clearly, however, changing the applied potential between the two electrodes causes unpredictable variations in the potential of either the working electrode, or the counter electrode, or in the iR drop. This implies that it is impossible to control accurately the potential of the working electrode unless one resorts to a cell in which:

• the potential of the counter electrode is invariant;

• the iR drop is made negligible.

A counter electrode of constant potential is obtained making use of a half-cell system in which the components are present in concentrations so high as to be appreciably unaffected by a flow of current through it. The saturated calomel electrode (SCE) is the most common example of such an electrode (see Chapter 4, Section 4.1.2). It consists of a mercury pool in contact with solid mercury(i) chloride and potassium chloride that lie at the bottom of the KCl saturated solution. The aqueous solution is thus saturated with , K⁺ and Cl⁻ ions, the concentrations of which are governed by the solubility of the respective salts.

The eventual current flow through the electrode causes the following reaction to proceed in one of the two directions:

This type of counter electrode is defined as a reference electrode. As we will see in Chapter 4, Section 4.1.2, at 25 °C the saturated calomel electrode (SCE) has a potential of +0.2415 V with respect to the standard hydrogen electrode (NHE), which, although difficult to use, is the internationally accepted standard for the potential scale, having by convention E°=0.000 V.

Returning to the control of the potential of the working electrode in the electrochemical cell, the use of a reference electrode as a counter electrode means that every change in the applied potential difference between the two electrodes is assigned entirely to the working electrode, provided that the iR drop is negligible. In this manner we would be able to control accurately the reaction rate at the working electrode.

In reality, the use of a two-electrode cell (working and reference electrodes) must be considered only as a first attempt to control adequately the potential of the working electrode. In principle, a reference electrode that does function as a counter electrode has the disadvantage that the incoming current can cause instantaneous variations in the concentration of its components, therefore leading to a potential value different from the nominal one. In most cases the relatively large surface area and the high concentration of active species typical of the reference electrodes make such variations in potential negligible. However, there are cases, such as large-scale electrolysis or fast voltammetric techniques in non-aqueous solvents, where the current flow is so high that the effects become non-negligible. Furthermore, there is still the problem of ohmic drop, which, for example, in experiments performed in non-aqueous solvents, is by no means insignificant.

To overcome these difficulties one must use a three-electrode cell, which is shown schematically in Figure 7. Here, a third electrode, auxiliary electrode (AE) (or counter electrode, CE) is inserted together with the working and the reference electrodes.

In principle, the auxiliary electrode can be of any material since its electrochemical reactivity does not affect the behaviour of the working electrode, which is our prime concern. To ensure that this is the case, the auxiliary electrode must be positioned in such a way that its activity does not generate electroactive substances that can reach the working electrode and interfere with the process under study. For this reason, in some techniques the auxiliary electrode is placed in a separate compartment, by means of sintered glass separators, from the working electrode.

In addition, the iR drop can be minimized by positioning the reference electrode close to the working electrode.

As deducible from Figure 7, to apply a precise ‘potential’ value to the working electrode means applying a precise difference of potential between the working and the reference electrodes. Since the electronic circuit to monitor such potential difference, V, is properly assembled to possess a high input resistance, only a small fraction of the current generated in the electrochemical cell as a consequence of the applied potential enters the reference electrode (thus not modifying its intrinsic potential): most current is channelled between the working and the auxiliary electrodes.

Nevertheless, even with this experimental set-up, the iR drop is not completely eliminated. The situation can be improved if the reference electrode is placed very close to the working electrode through a Luggin capillary (Figure 8).

The ideal positioning for the Luggin capillary is at a distance 2d from the surface of the working electrode, where d is the outlet diameter of the capillary.

If one bears in mind this new cell design, the iR drop can be reconsidered according to Figure 9 with respect to that represented in Figure 6.

As already mentioned, since the majority of the current has been conveyed towards the region between the working and the auxiliary electrodes, most of the ohmic drop iR_s has no influence on the cell voltage V between the working and the reference electrodes, thus allowing the condition:

It must, however, be kept in mind that one cannot eliminate the fraction of the non-compensated solution resistance R_nc, which generates the ohmic drop iR_nc. Unfortunately, the positioning of the reference electrode even closer to the working electrode (<2d) would cause current oscillations.

It should be emphasized that this design of the three-electrode cell gives good results in most cases. However, as mentioned, in fast electrochemical techniques in non-aqueous solvents, iR_nc can assume values that compromise accurate control of the potential of the working electrode and hence the achievement of reliable electrochemical data. In such cases one must employ electronic circuits that compensate for the resistance of the solution.

Nevertheless, it is important to appreciate that this type of three-electrode arrangement of the electrochemical cell usually enables one to control easily the potential of the working electrode by forcing it to assume all the desired values and hence to control either the start of electrode processes or their rate.

It was mentioned in Section 1.2.2 that even in the case of a simple electrode reaction one must take into account both heterogeneous electron transfer and mass transport processes. Let us therefore examine the mathematical relationships that govern the two processes.

Before examining the electrode reaction kinetics it is necessary to recall a few basic aspects of chemical kinetics. Consider the following elementary process:

Since, for an elementary step, reaction order and molecularity coincide, one can write:

The overall rate of transformation of A into B can, therefore, be expressed by:

At equilibrium, the rate of conversion will be zero (v_f=v_r). Hence:

Thus, chemical kinetics predicts that under equilibrium conditions the ratio of the concentrations of the products and reagents is constant, as demanded by chemical thermodynamics. The agreement between kinetic and thermodynamic data is the ultimate test of any kinetic theory.

It is known that in chemical kinetics one can determine how the free energy of the system varies as a function of the reaction coordinate, i.e. the progress of the reaction.

Let us consider a simple faradaic process (i.e. accompanied neither by chemical complications nor by significant molecular rearrangements) of the type:

If a potential value corresponding to the equilibrium (zero-current) is applied to the working electrode so that both S and S⁻ are stable at the electrode surface, the process can be represented as in Figure 10.

The curves relative to the half-reactions intersect at the point corresponding to the formation of the so-called activated complex. The height of the energy barrier of the two redox processes (oxidation, h_Ox; reduction, h_Red) is inversely proportional to the respective reaction rates. Since in this case h_Ox=h_Red, it is immediately apparent that these conditions identify the equilibrium conditions.

If one now sets the potential of the working electrode more positive than that of equilibrium, the oxidation process is facilitated (as seen in Figure 5). Thus, the profile of the free energy curves becomes that illustrated in Figure 11, in which the energy barrier for the oxidation is lower than that of reduction.

On the other hand, if a potential more negative than that of equilibrium is applied to the working electrode, as indicated in Figure 12, the reduction process is favoured.

This being stated, it is now possible to examine the kinetic aspects of the electron-transfer processes.

Consider the general electron-transfer process:

Under equilibrium conditions the Nernst equation holds:

γ=activity coefficient;

C*=concentration of the active species in the bulk of the solution;

E°′=formal electrode potential of the couple Ox/Red. It differs from the thermodynamic standard potential E° by a factor related to the activity coefficients of the two partners Ox and Red:

As a consequence of the Nernst equation it follows that, when C*_Ox=C*_Red:

Since we have preliminarily stated that any kinetic theory must involve agreement between kinetic and thermodynamic data it follows that, under equilibrium conditions, kinetic theory must afford relationships that coincide with the Nernst equation.

In the electrode process under consideration there is either the reduction path [Ox → Red] or the (inverted) oxidation path [Ox ← Red]. Expressing the concentration of a species, at a distance x from the electrode surface and at the time t, as C(x,t), it follows from Section 1.2.3 that the reaction rate for the reduction reaction is given by:

In a similar manner, for the oxidation reaction one obtains:

The overall reaction rate can therefore be written as:

This means that the current generated at the electrode is expressed by:

The rate constants of the electron transfers vary with the electrode potential. In particular, in their Arrhenius form, they are expressed by:

k°=standard rate constant, which expresses the value of k_Red or k_Ox when the applied potential E is equal to E°′;

α=transfer coefficient (0<α<1);

n=number of electrons (simultaneously) transferred per molecule of Ox.

Upon substituting in the preceding relationship:

This current-potential relationship, also known as the Butler–Volmer equation, governs all the (fast and single step) heterogeneous electron transfers.

At this point a better understanding of the two factors k° and α is called for.

The heterogeneous standard (or conditional) rate constant k° measures the intrinsic ability of a species (say, Ox) to exchange electrons with the electrode in order to convert into its redox partner (say, Red). A species with a large k° will convert into its redox partner on a short time scale; a species with a small k° will convert into its redox partner on a long time scale.

The largest values for the standard rate constant k° (expressed in m s⁻¹) range from 0.01 to 0.1 m s⁻¹, and commonly characterize redox processes that do not involve significant molecular reorganizations.

The smallest values for the standard rate constant k° are around 10⁻¹¹ m s⁻¹.

We must, however, note that even if one partner of a redox couple does not possess a high propensity to exchange electrons with the electrode (i.e. low k°) we can force it to increase its electron transfer rate by applying to the working electrode a potential value E higher than the formal electrode potential E°′ of the couple itself (i.e. more negative for a reduction process; more positive for an oxidation process). In fact, as seen:

The transfer coefficient α is generally an index indicative of the symmetry of the energy barrier for a redox half-reaction. The significance of this definition is the following.

It has been shown previously that an electrode reaction can be depicted through ΔG°/reaction coordinate plots. Reconsidering the simple half-reaction:

If we apply to the working electrode a potential value different from E°′, based on the well-known relationship ΔG=–n·F·ΔE, the free energy of the electrode process will vary by

For instance, following the dashed line, if the potential of the electrode is made more positive than E°′, we increase the activation barrier for the reduction process by , while the activation barrier for the oxidation process is lowered by an amount equal to the residual variation of the overall energy:

As far as the definition of α as a measure of the symmetry of the activation barrier is concerned, as shown in Figure 14, let us focus on the apical region of the intersection between the free energy curves illustrated in Figure 13.

If θ measures the slope of the curve of the reduction half-reaction and ϕ the slope of the oxidation half-reaction, simple trigonometric rules afford:

As illustrated in Figure 15, it follows that if the slope of the reduction curve equals that of oxidation (i.e. if θ=ϕ) then α=0.5. Otherwise, α can assume values:

• between 0.0 and 0.5, if θ<ϕ,

• between 0.5 and 1.0, if θ<ϕ.

It has already been remarked that the ultimate test of any kinetic theory is that, under equilibrium conditions, the kinetic equations must coincide with the thermodynamic equations.

The fundamental relationship for a heterogeneous charge transfer (Butler–Volmer equation) is:

Thus, when the electrode, depending upon the concentrations of Ox and Red, assumes an equilibrium potential, E_eq (keep in mind that E°′ is the ‘particular’ equilibrium potential set up when the concentrations of Ox and Red at the electrode surface are identical), it will reach the zero current condition (i=0), so that:

However, if there is an equilibrium between the two species at the electrode surface, such an equilibrium must also exist in the bulk of the solution. Hence:

In reality such an expression is nothing more than the exponential form of the Nernst equation seen previously (Sections 1.3 and 1.4.1):

It has been shown that, under equilibrium conditions (i.e. when both forms of a redox couple are present in solution), the faradaic current is zero. Such a result must be seen, however, in a dynamic context: the current is zero because the cathodic current (i_c) generated by the reduction process equals the anodic current (i_a) generated by the oxidation process.

This current, equal in both directions and exchanged under equilibrium conditions, is defined as the exchange current, i₀.

As will now be discussed, the exchange current i₀ is proportional to the standard rate constant, thus resulting in the common practice of using i₀ instead of k° in kinetic equations.

Recall that under equilibrium conditions one has:

The terms on either side of the equation are, in fact, now equal to i_c and i_a, respectively, under equilibrium conditions and each of the two terms represents i₀. Considering one of the terms (e.g. that on the left of the equation) one can write:

Under equilibrium conditions it was also found that:

Raising this expression to the power –α one obtains:

Indeed, this is just the general expression that one uses to express the exchange current.

Sometimes, the exchange current is expressed as the exchange current density, J₀:

Like the standard rate constant, k°, the exchange current, i₀, characterizes the rate of the electron-transfer process inside a redox couple.

As k° can range from about 0.1 m s⁻¹, for very fast processes, to about 10⁻¹¹ m s⁻¹, for very slow processes, analogously the exchange current density varies from a few hundred kA m⁻² to a few tenths of nA m⁻² on passing from very fast to very slow electron exchanges.

As noted above, often the kinetic equations are written as a function of i₀ rather than k°. One of the advantages of using i₀ is that the faradaic current can be described as a function of the difference between the potential applied to the electrode, E, and the equilibrium potential, E_eq, rather than with respect to the formal electrode potential, E°′, (which, as previously mentioned, is a particular case of equilibrium potential [C_Ox(0,t)=C_Red(0,t)], and at times may be unknown). In fact, dividing the fundamental expression of i by that of i₀ one obtains:

Since it has been shown above that:

By indicating with η, defined as overvoltage, the difference between the applied potential and the equilibrium potential (η=E – E_eq), one obtains:

This important equation can be qualitatively interpreted in the following way. When the two components Ox and Red are present in solution at certain concentrations, the working electrode will spontaneously find its equilibrium potential (imposed by the Nernst equation) and there will be no overall current flow. For Ox to be reduced or Red oxidized, the system must be moved from equilibrium. This can be achieved by setting a potential different from that for equilibrium. The process of oxidation or reduction will be favoured depending on whether the potential is moved towards more positive or more negative values, respectively, compared to the equilibrium potential. Moreover, the more the potential is removed from equilibrium the greater will be the current and, hence, the faster the faradaic process.

In fact, the above equation quantifies these considerations. The first term describes the cathodic component of the current, whereas the second term describes the anodic component.

As an example, Figure 16 shows how, for the generic process Ox+ne ^–=Red, the current varies as a function of the electrode overvoltage.

Before commenting on it, it is useful to consider that the potential/current signs in the IUPAC convention follow the criterion: ‘positive currents for anodic processes (which usually occur at positive potentials); negative currents for cathodic processes (which usually occur at negative potentials)’ (see Chapter 2, Figure 4). The American convention still adopts the opposite criterion: ‘positive currents for cathodic processes; negative currents for anodic processes’. Nevertheless, remember that in electrochemistry the signs are conventional.

The solid curve is the sum of the cathodic and anodic components, which are represented by the respective dashed lines.

It is evident that at very negative potential values (compared to E_eq) the anodic component is zero, so that the current is due only to the reduction process. The inverse effect occurs for very positive potentials (compared to E_eq). In contrast, on moving away from the equilibrium potential in either direction, even only slightly, the current rises rapidly as a consequence of the exponential terms in the equation. However, at high values of η the current reaches a limiting value (i_𝓁), beyond which it can rise no more. This happens because the current is limited by the rate of the mass transport of the species Ox or Red from the bulk of the solution to the electrode surface, rather than from the rate of the heterogeneous electron transfer. Hence, one can say that the effect of the exponential factors in the equation is restrained by the ratios and which are directly related to the rate with which the electrode is supplied with electroactive material.

One can obtain a useful approximation of the preceding equation when the concentrations of Ox and Red at the electrode surface are not significantly different from their respective concentrations in the body of the solution (one can, for example, achieve such a condition by stirring the solution continuously). In fact, if C(0,t)=C*, then:

In some texts this relationship is also called the Butler–Volmer equation.

It is common practice to use this equation experimentally in one of the following three limiting forms:

• if one applies a potential that is much more negative than the equilibrium potential, the anodic component of the current will be negligible (as shown previously) and the equation becomes:

• conversely, if one applies a potential much more positive than the equilibrium potential, the cathodic component will be negligible, hence:

• when the difference between the applied potential E and the equilibrium potential E_eq is very small, one can exploit the mathematical expression that if x is small then e^x ≅ 1+x, and the equation reduces to:

This relationship indicates that the current that flows in a faradaic process is proportional to the applied overvoltage only in a small interval of potential values very close to E_eq (less than ±100 mV).

As seen previously (Section 1.2.2), the rate of an electron transfer is also conditioned by the rate with which the electrode is supplied with reagent and cleared of the electrogenerated product.

There are three physical mechanisms by which a redox-active species can move from the mass of the solution to the electrode surface:

1. Convection is the movement of a species under the action of a mechanical force (a gradient of pressure). The convective movements can be fortuitous (resulting from collisions or vibrations of the electrochemical cell) or intentionally forced (through controlled stirring).

2. Diffusion is the movement caused by the presence in solution of regions with different concentrations of the active species (a gradient of concentration). It tends to randomize the distribution of molecules in a system transporting species from regions of high concentration to regions of low concentration. Electrode reactions are an ideal mode to generate diffusive movement. In fact, if a reaction Ox → Red is occurring at the electrode surface it is obvious that in a layer of solution close to the electrode surface (thickness of about 0.0001 m) the concentration of Red will be higher and the concentration of Ox lower than that present in the mass of the solution, respectively. This concentration gradient of the two species makes more Ox move from the bulk of the solution to the electrode, while the species Red moves from the electrode surface (where it was generated) to the mass of the solution. The propulsive force clearly is to establish a uniform distribution of the two species in the whole of the solution.

3. Migration is the movement of an ionic solute under the action of an electric field (a gradient of electrical potential). In fact, it is simply the movement of the ions in solution during an electrode process: the positive ions are attracted by the negatively charged electrode, while the negative ions are attracted by the positively charged electrode.

To derive mathematical equations able to describe the movement of a species towards or from an electrode surface it would be necessary to know the physical laws that govern the three modes of the mass transport.

Apart from a limited number of cases (laminar flows around a rotating disk or through a tubular electrode), it is very difficult to make a rigorous treatment of the convective movements.

It is even more difficult to handle mathematically migration (which depends on the dimensions of the ions, on the form and disposition of the electrodes, on the resistance of the solution, and so on).

Diffusion is the only mode of mass transport for which we possess well-known mathematical treatments.

For this reason one always tries to minimize the effect of migration by adding a supporting electrolyte to the solution (i.e. a salt that produces non-electroactive ions in the potential region of interest) in a ratio of at least 100 : 1 compared to the electroactive species. In this way it is statistically more probable that, under the effect of an applied potential, the inert ions of the supporting electrolyte migrate to the electrodes rather than those of the electroactive species under study. Analogously, to avoid convection, the solution is maintained unstirred (or under accurately controlled stirring).

In conclusion, one always tries to study the electron-transfer process under conditions where mass transport is governed only by diffusion (for which the laws are rigorously known).

Commonly, to deduce the mathematical relationships that govern the diffusion of an electroactive species towards the electrode, one considers an electron-transfer process taking place at a planar electrode, in an unstirred solution, so to make active only the diffusive motion of the redox-active species in a direction perpendicular to the electrode surface (Figure 17).

The diffusive event involves two aspects:

1. variation of the concentration of the active species along the approaching distance to the electrode surface (concentration gradient with space);

2. variation of the concentration of the active species with time (concentration gradient with time).

To discuss these two aspects let us consider the usual electrode process:

Within the solution, let us consider the parallel-to-the-electrode plane at position x, see Figure 17a. The mole number of species Ox crossing the unit area of the plane per unit time is called the flux of Ox, J_Ox(x,t) (mol s⁻¹ m⁻²), and represents the rate of the mass transport.

According to Fick's first law, the flux of Ox is proportional to the concentration gradient of Ox along the direction of propagation:

The constant of proportionality, D_Ox, is defined as the diffusion coefficient of the species Ox. The units of the diffusion coefficients are m² s⁻¹. The negative sign is conventional.

As the reduction process Ox → Red proceeds, the species Ox, flowing through the plane x at the time t, reaches the electrode surface (x=0) and instantaneously disappears to generate the species Red, which in turn will cross the plane x in the opposite direction. This means that:

Since the amount of species Ox reaching the electrode [or, J_Ox(0,t)] will generate a current, the intensity of which is proportional to the number of electrons exchanged with time (Section 1.2.3):

In the first instance such an expression tells us that the current is a function of the concentration of the active species at any distance from the electrode. However, the concentration of the active species depends either on distance or time. Fick's second law determines how the concentration of species Ox changes with time.

Stated that C_Ox(x,t) represents the concentration of Ox in the infinitesimal volume of solution between the planes x and x – dx in Figure 17a, the time dependence of the concentration will be given by the difference between the flux of Ox that enters the infinitesimal volume of solution and the flux of Ox that leaves the same element of volume, or:

Upon substituting such a relationship in Fick's first law, we obtain:

The same holds for the species Red moving away from the electrode:

If, as might happen, the electrode is spherical rather than planar (e.g. using a hanging drop mercury electrode), see Figure 18, Fick's second law should be integrated by corrective terms accounting for the sphericity, or the radius r, of the electrode:

As previously mentioned, the region close to the electrode surface where the concentrations of Ox and Red are different from the corresponding ones in the bulk of the solution is defined as the diffusion layer.

During an electrode reaction in an unstirred solution, the thickness of the diffusion layer grows with time up to a limiting value of about 10⁻⁴ m, beyond which, because of the Brownian motion, the charges become uniformly distributed. At ambient temperature the diffusion layer reaches such a limiting value in about 10 s. This implies that in an electrochemical experiment the variation of concentration of a species close to the electrode surface can be attributed to diffusion only for about 10 s, then convection takes place.

Graphs that show the dependence of the concentration of a species on distance from the electrode surface and how it evolves with time are called concentration profiles.

To obtain such diagrams, one must mathematically solve Fick's second law:

The resolution makes use of non-elementary mathematical treatments (Laplace transform). Neglecting such treatments, one obtains:

Figure 19 shows a series of concentration profiles at different times for the reaction Ox+ne⁻ → Red. In the experiment, at time t=0 only the species Ox is present in solution; then the electrode potential is suddenly changed from a value more positive than the formal potential of the couple Ox/Red to a value much more negative, so that the reduction process Ox → Red immediately takes place.

The slope of each concentration profile expresses the concentration gradient of species Ox at various times, where . The point at which the concentration gradient becomes zero (i.e. when ) identifies the thickness of the diffusion layer.

The thickness of the diffusion layer (in metres) is approximately

For instance, for we obtain:

Figure 19 shows the accuracy of the calculation for the first two cases.

Clearly, as represented in Figure 20, concomitantly the concentration profiles of the species Red are opposite to those of Ox.

As the current is a function of the flux of species Ox that reaches the electrode surface:

The dependence of the current upon time, once again, can be obtained by solving (in a non-elementary way) Fick's second law. The final result is:

The linear dependence of the current on the square root of the time is used as a diagnostic test for electrochemical reactions controlled by diffusion.

Notably, the above relationship is derived from linear diffusion at a planar electrode. For a spherical electrode of radius, r₀, the above relationship becomes:

It has been calculated, for example, that for an electrode of radius r₀=0.001 m the second term on the right-hand side of the equation becomes negligible (i.e. the simple laws of linear diffusion are valid also for spherical electrodes) if the response is recorded for a time shorter than 3 s from the start of the faradaic process. Obviously, increasing r₀ also increases the time for which linear diffusion remains valid. It has been calculated that to an accuracy of 10%, and for D_Ox=1·10⁻⁹ m² s⁻¹, the following relation holds:

Hence, the time t (in s) in which linear diffusion is valid for a spherical electrode of radius r₀ (in m) is:

In Sections 1.4.1 and 1.4.2, the electron transfer and the mass transport involved in a simple electrode reaction [simple=not complicated by preceding or following reactions, by absorption, or by formation of phases (Section 1.2.2)] have been treated separately. However, it is to be expected that in reality both phenomena act in a concerted manner during a faradaic process. Thus, as seen previously, even the simple electrode process:

In this connection, there are two fundamental types of behaviour:

• if the rate of the electron transfer is higher than the rate of the mass transport (or, if both k_Red and k_Ox are large, and greater than the rate constant of the mass transport), the process is defined as electrochemically reversible;

• if the rate of the electron transfer is lower than the rate of the mass transport (or, if k_Red and k_Ox are not both large, and lower than the rate constant of the mass transport), the process is defined as electrochemically irreversible.

It is commonly accepted that if:

It must, however, be taken into account that the concept of electrochemical reversibility or irreversibility of an electron transfer is relative. In fact, to accelerate the redox processes one can act either on the mass transport (by stirring the solution) or on k_Red and k_Ox (by changing the electrode potential, as seen in Section 1.4.1.1).

As will become evident from an examination of the various voltammetric techniques, the electrochemical reversibility or irreversibility of a process influences the form of the relative current/potential curves.

Based on the way in which an electrode process has been illustrated until now, it would seem reasonable to assume that the only source of electron flow between the electrode and the species in solution might be attributed to faradaic processes of the type Ox+ne⁻=Red. It has already been mentioned in Section 1.2.3, however, that non-faradaic currents exist. Let us discuss their origin.

Recalling the arguments of Section 1.2.1, the electrode/solution interface can be considered to a first approximation as a double layer of the type reported in Figure 21.

A charge distribution of this type is completely analogous to that of a capacitor. Figure 22 illustrates what a capacitor is: two parallel metal plates separated by a dielectric material.

When a difference of potential ΔE is applied to the two plates, an excess of electrons of charge q accumulates on one of them (which is equal to the defect of electrons generated on the other plate) until the following relation is satisfied:

q=the charge on the capacitor (in coulombs),

ΔE=the difference of potential applied between the two plates (in volts),

C=the capacitance of the capacitor (in farads).

In other words, when one applies a difference of potential between the two plates of a capacitor a current flows through the circuit until the capacitor is charged; this current is called the capacitive current.

This is just what happens in an electrochemical cell when a potential is applied between the working and the reference electrodes: the double layer setting up at the working electrode/solution interface generates capacitive currents.

To evaluate the magnitude of capacitive currents in an electrochemical experiment, one can consider the equivalent circuit of an electrochemical cell. As illustrated in Figure 23, in a simple description this is composed of a capacitor of capacitance C, representing the electrode/solution double layer, placed in series with a resistance R, representing the solution resistance.

As a result of the difference of potential ΔE applied between the working and reference electrodes, a capacitive current is generated inside the cell, which flows as a function of time, according to the relation:

This means that during an electrode reaction the capacitive currents decrease exponentially with time, , whereas, as seen in Section 1.4.2.4, the faradaic currents fall off more slowly with time in that they decrease as a function of the square root of time

Consequently, the time dependence of the current arising from an electrochemical response is a diagnostic test to discriminate between faradaic and capacitive (i.e. non-faradaic) currents.

In passing, it has to be underlined that the use of high concentrations of supporting electrolyte is also useful to minimize the capacitive currents (also called charging or residual currents).

The description of the double layer reported in Figures 3 and 21 is only approximate; the composition of the electrode/solution region is somewhat more complex. The double layer has been studied in most detail for a mercury electrode immersed in an aqueous solution. According to the Gouy–Chapman–Stern model there are several layers of solution in contact with the electrode (Figure 24).

The charge on the metallic electrode (q^M, which, depending on the potential applied, can be negative or positive compared to the charge of the solution, q^S) is considered to be concentrated in a layer of about 0.1 Å close to the electrode surface.

By contrast, the charge of the solution, q^S, is distributed in a number of layers. The layer in contact with the electrode, called the internal layer, is largely composed of solvent molecules and in a small part by molecules or anions of other species, which are said to be specifically adsorbed on the electrode. As a consequence of the particular bonds that these molecules or anions form with the metal surface, they are able to resist the repulsive forces that develop between charges of the same sign. This most internal layer is also defined as the compact layer. The distance, x₁, between the nucleus of the specifically adsorbed species and the metallic electrode is called the internal Helmholtz plane (IHP). The ions of opposite charge to that of the electrode, that are obviously solvated, can approach the electrode up to a distance of x₂, defined as the outer Helmholtz plane (OHP).

As the electrostatic interaction between the solvated ions and the metal is indirect, it is virtually independent of the chemical nature of the ions; these latter are said to be non-specifically adsorbed.

The region of solution between the OHP and the bulk of the solution is called the diffusion layer. The majority of the charge present in solution resulting from the applied electrode potential resides in this layer.

If one defines the electrostatic potential of a phase (ϕ) as the work necessary to carry the unit charge from infinity to within this phase, all the layers of the double layer can be assigned a phase potential. Figure 25 shows the variation in the potential profile within the double layer.

The potential difference known as the interface potential, cannot be measured, in that it would need the insertion of another electrode, i.e. another interface. What one can measure is the electrochemical potential of the electrode, E, with respect to a reference electrode; obviously every change of is reflected in a variation of E, or vice versa.

In the treatment of the kinetics of the electron transfer illustrated in Section 1.4.1, it has been assumed that the propulsive force for the electron transfer was the electrochemical potential E (i.e. a quantity directly related to ). However, since the solvated ions cannot enter the inner layer of the double layer (IHP), the true propulsive force should be: or a quantity proportional to the term . With this more precise statement (called the Frumkin correction) both the standard rate constant, k°, and the exchange current, i₀, should become, respectively:

Such relationships allow one to calculate k°_(t) and i_0(t) from the experimental determination of k° and i₀ (through the use of the Butler–Volmer equation).

Notably, although it is possible to calculate ϕ₂ for mercury electrodes (e.g. through the so-called electrocapillary curves), for electrodes of other materials there is not this possibility, in that for such electrodes the composition of the double layer is unknown (one normally assumes it to be similar to that of mercury).

Finally, it must be taken into account that the use of large concentrations of supporting electrolyte minimizes the Frumkin effects. This is important in that we can now realize that high concentrations of supporting electrolyte not only minimize either migration or the capacitive currents, but also allow us to adopt the simple electrode kinetics discussed in Section 1.4.