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CHAPTER 4

Theory of Electrochemical Electron Transfer

The first and fundamental formulation of Marcus theory for homogeneous ET reactions (redox reactions) was summarized in Chapter 2. In this chapter, an abridged version is presented of its extension to electrode systems from Refs. [1, 2, and 2a]. The two Office of Naval Research reports were written by Marcus in 1957 but were published in 1977 in a book edited by Peter A. Rock. In the following, the symbol M. is for Marcus.

4.1.Electrostatic Free Energy and Nonequilibrium Dielectric Polarization for Electrode Systems

In ET at electrodes, the total dielectric polarization P(r) is, like in the homogeneous case, the sum of two terms:

We remind the reader that the electronic polarization P_e(r) is the portion of the electric polarization which is in electrostatic equilibrium with the electric field strength E(r), that is, the polarization is determined by the field through the electronic polarizability α_e:

The E-type polarization is related to D_op, the square of the refractive index:

An important point is that E(r) depends on the charge distribution and on P_u(r). On the other hand, P_u is the portion of dielectric polarization which is in general not in equilibrium with E. When it is in equilibrium, it can be expressed in terms of E and of the atomic–orientational polarizability α_u:

When it is not, it appears as a solution of a variational equation.

The U-type polarization depends on both D_op and D_s:

4.2.The Electrochemical System

After having briefly reminded the reader of the properties of the polarization vector function, I describe now the simple model of electrode system considered by Marcus.

An oriented, thin, and nonpolarizable solvent layer is supposed to exist at the electrode–solution interface (“inner layer” of the electrical double layer, see, e.g., Ref. [3]). The adsorbed layer produces a fixed potential difference χ across the interface. The magnitude of χ depends on the solvent, on the nature of the electrode and on the temperature.χ is first supposed to be independent of the average electrode charge density. The ions taking part in ET are designated as “central ions.” When the solution is not infinitely diluted, the central ions will be surrounded by an ionic atmosphere of other ions. If a salt is present in solution whose ions do not exchange electrons with the electrode (inert electrolyte), the ionic atmosphere will be made up also from ions of the inert electrolyte.

The picture of the system is now complete: metal electrode, solvent layer, central ions taking part in the ET process, ionic atmospheres surrounding them [1, p. 211].

4.3.Electrostatic Potential

As in the case of homogeneous ET, “each central ion is treated as a sphere having a surface charge density σ(r) equal to the ionic charge divided by its area. The remaining ions are described in terms of a volume charge density ρ(r). There will also be a surface charge distribution on the electrode surface.” As in Chapter 2 (see Ref. [4]), “the potential can be expressed in terms of contributions from the charges and from the polarized volume elements.” The potential arises then from the following contributions:

(i)Volume charges of the ions in ionic atmospheres

(ii)Surface charges on the central ions

(iii)Surface charges on the surface of the M electrode

(iv)Polarized volume elements

(v)Potential difference at the interface

This potential “is the same as the ‘inner potential’ φ(r) described by Parsons [5] and Lange [6].”

“We have:

where ∇_r is the gradient operator and where

The volume integrals are over the entire volume of the dielectric (the solution). The surface integral is over the surface of each central ion and over the surface of the electrode M:

It is convenient to define a function ψ(r) which is identical with the potential employed in Marcus’ Part I (Ref. [4]):

For the electric field strength E(r), we have

E(r) = −∇φ = ∇ψ” [1, p. 212].

Equation (4.1) is like Eq. (2.11a) in Chapter 2 with the addition of the and β terms, characteristic of the electrode surface and of the electrode–solution interface.

4.4.Electrode Charge Distribution and the Method of Electrical Images

The term is expressed by M. using the method of electrical images.

“In systems having equilibrium polarization the method of images [7] can be used to determine the charge distribution” [1, p. 217].

M. briefly discusses the quantum limitations to the image force theory in reference 9 of Ref. [1] deducing from Ref. [8] that the percent error in the energy change computed from the image force theory is small.

“The same method is also suitable for systems having nonequilibrium polarization. For simplicity, consider a planar electrode, that is, any electrode whose radius of curvature is appreciably greater than the thickness of the double layer. Let the electrode–dielectric interface be situated in the plane x = 0 and let the dielectric occupy the semi-infinite region x > 0. As before, the coordinates of any point in the dielectric will be specified by the vector r drawn from any arbitrary origin to the point. The coordinates of the ‘image point,’ having the same y and z and differing only in the sign of x, will be specified by a vector r_im. Mirror image functions and will then be defined in the region of negative x, according to Eq. (4.4). They are undefined in the region of positive x, i.e., in the dielectric.

The last three expressions relate the components of the vector to those of P. In Eq. (4.4), σ refers only to the surface charge density on the central ions and not to that on the electrode.”

Figure 1 shows schematically the electrical image of a central ion, represented by a small dashed circle. The dashed shell around it represents the image of its ionic cloud. The region between the solid line representing the electrode–solution interface and the dashed line parallel to it is the region of the electrode where the images of most of the ions of the electrical double layer are to be found.

Note that because both charge and volume in the image charge density are opposite to those in the charge density, that is, dV = dxdydz in the dielectric but dV = (−dx)dydz in the metal. Notice that not only the charge but also the polarization vector function has an image.

Fig. 1*.

“Applying the method of images, the electrode charge distribution which satisfies the condition of constant potential on the electrode M and in the body of the solution obeys the following equation:

In Eq. (4.5), r and r′ denote any points on the electrode’s Msurface and in the dielectric medium, respectively. The first surface integral is over M and the second over the mirror images of each central ion. The volume integrals are over the entire mirror image of the volume of the dielectric.”

It appears in this way very clearly that the charge induced on the surface of the electrode can be expressed in terms of the dielectric images: “The total charge in the dielectric is equal and opposite to the total image charge, which in turns equals the total electrode charge.”⁽¹⁾

The integrals extended to the ions can be evaluated treating the ions as spheres with surfaces of uniform charge densities. “It can be readily shown (see Chapter 2) that:

Where r is the distance from the ion to the field point, see Fig. 2, and q is the ionic charge.

Fig. 2*.

Similarly, it can be shown that:

where r_im is the distance from the field point to the center of the electrical image of this ion” [1, p. 218].

4.5.Mixing and Electrostatic Entropy and Free Energy

I report here the Note added in proof in p. 983 of Ref. [10]: “The charge distribution at the end of Stage I of the charging process, denoted by ρ⁰ and σ⁰, is a fictitious distribution used to produce the specified U-type polarization P_u(r). By contrast, the charge distribution at the end of Stage II is the actual distribution of charges on the ions of the system. The complete charging process is performed at fixed configurations of these ions, though they may be uncharged in Stage I and charged up in Stage II. Thus the reversible work given by Eq. (4.6)

was also performed at fixed configuration of these ions. There is therefore an additional free energy term which should be considered, namely the entropy term associated with the preliminary formation of this given uncharged configuration of the ions from a random configuration. Let e_i, c_i(r), and denote the charge, the concentration, and the average concentration of ions of the ith species which make up the continuous volume charge distribution ρ(r); Then, this additional entropy term associated with the formation of the given configuration from a random one is the well-known excess entropy of mixing and is given by Eq. (4.6a):

(typo in the original). The total electrostatic contribution to the free energy includes this term and F given by Eq. (4.6). It is the sum of these two terms, rather than F alone, which is the electrostatic free energy, F_e, say

where F is given by Eq. (4.6). In a very dilute solution c_i equals and F_e equals F.

Similarly, the total electrostatic contribution to the entropy is the sum of the term in Eq. (4.6a) and of S given by Eq. (4.7) in the following section

Altogether the total electrostatic entropy will be:

F is computed by the two stages charging process. The work done during each stage can be calculated from the equation:

where λ is a charging parameter which is increased from 0 to 1 during each charging stage and where φ^λ denotes the value of φ at any λ (Notice that dρdV is a charge and that φdρdV is then a product potential × charge, that is, electrical work. The same for φdσdS, compare Eq. (4.9) in Ref. [10]). Introducing Eqs. (4.2) and (4.3) in this expression for W, that is, going from treating ET in solution to the parallel treatment of ET at the electrode, Eq. (4.9) becomes:

Eq. (4.10) differs from the corresponding expression used for the redox case [10] only in the last term. For this reason, the formula deduced for F by the two stages charging process in Ref. [10], see Chapter 2, will differ from the one for the electrode system only in last term:

where E_c(r) is the electric field which the given ionic and electrode charge distribution would exert in a vacuum:

On the electrode, this σ(r) arises from all the induced charges, including those induced by the polarized dipoles. M. defines then a function E_v(r) which “depends only on the ionic charges in the solution and on the surface charge density σ_v(r) which they would induce in a vacuum.” The potential ψ_v(r′) in that system must be a constant on the electrode.

E_v(r′) and ψ_v(r′) are given by:

where ρ_v(r) = ρ(r) and where σ_v(r) and σ(r) are equal on the surface of each central ion but differ on the electrode. They differ there by an amount equal to the surface charge density induced in the electrode by the polarized dielectric, that is, by P(r).

M. shows that Eq. (4.12) can be rewritten in terms of E_v:

For electrode systems, Eq. (4.8) will be:

where stands for and the internal energy will be U = F_e + TS_e.

4.6.Evaluation of the Fe of an Equilibrium System

Equations (4.11) and (4.12) give the general expressions for F_e valid for equilibrium and nonequilibrium dielectric polarization systems. F_e will have its minimum value at equilibrium [1, p. 215]. Fluctuations δP_u and δc_i carry the system away from equilibrium and moves to corresponding nonequilibrium values In order to get the particular expression for F_e at equilibrium, M. minimizes F_e setting δF_e = 0 and considering two equations of restraint. For the variation δF_e, we have from Eq. (4.6b):

In δF the term χ ∫_M dσdS will appear.

In the minimization process, one should consider two equations of constraint of immediate physical interpretation:

(i)∫ δc_i(r)dV = 0 for each i

[2.215] because the total number n_i of ions of species i is fixed in solution ⁽²⁾ and:

(ii)

[2.220] which means that if the charge density on the electrode varies by σ, the charge in the solution must vary by an equal and opposite amount because of the electroneutrality of the whole electrode and solution system.

Solving the variational problem for δF_e subject to the earlier constraints, M. shows that at equilibrium P_u(r) is determined by E(r)(as it was intuitively to be expected):

and that c_i(r) depends on ϕ^eq(r) around a central ion:

[1, p. 215], that is, the concentration c_i(r) of volume charges of ions of species i around a central ion depends on the potential ϕ^eq(r) and is Boltzmann weighted by the ratio:

Notice, moreover, that the integral in the denominator has the dimension of a volume and so there is a concentration in both members of Eq. (4.15).

The equilibrium electrostatic free energy (4.16a) can be deduced directly from Eqs. (4.6b) and (4.10) [2, p. 215]

An equivalent expression is:

[2.222] where The formula is of transparent physical meaning: the equilibrium free energy of the electrodic system is the energy necessary to charge up all of the surfaces present in it. If the system is supposed to be made up of

(i)A central ion

(ii)Mobile ions

(iii)An electrode M

(iv)A medium of dielectric constant D_s the potential ϕ^eq is built up from the contributions of (i), (ii), and (iii) [2, pp. 196, 197]:

“where r and r_i denote the distances from the field point to the center of the ion and to the center of its electrical image, respectively. is the contribution from the mobile ions and from the electrode charges they induce. On the electrode surface, ϕ^eq is a constant, β = χ. On the surface of the central ion of radius a, see Fig. 2, and surface 4πa², the surface density dσ equals dq/4πa².” It can be shown that (see Ref. [4, p. 974] with note on the Born charging formula):

where R is the distance from the center of the ion to its electrical image as in Fig. 2.

We have then:

is, on its turn, the sum of three independent contributions, when the ion is outside the double layer region:

(1) arising from a spherically symmetric ionic atmosphere about the central ion,” see Fig. 1.

(2)“A contribution due to the electrode charge density induced by this atmosphere. It is symbolized by a dashed spherical shell in Fig. 1. Since the atmosphere is concentric with the central ion, spherically symmetric, and has a total charge of −q, it can be shown that the same electrode charge density would be introduced by a point charge −q situated at the center of the central ion. The image of this charge is q and its contribution to the potential is therefore q/D_s R.” This contribution to the potential will give a contribution q²/2D_s R to that will cancel the contribution −q²/2D_s R in Eq. (4.16b). This means that the contributions of the image of the central ion and of the image of its ionic atmosphere cancel each other.

(3)“φ_S arising from the ions of the electrical double layer together with the electrode charges they induce,” see Fig. 1.

“Remembering that both and φ_S are constant over the surface of this ion, we obtain:

Since where γ_q is the electrostatic contribution to the activity coefficient of the ion of charge q in the body of the solution, Eq. (4.17) becomes:

4.7.Theory of Overvoltage for Electrode Processes Possessing ET Mechanism

4.7.1. Introduction

Among the mechanisms for electrode processes, there may be simple ET or more complicated ones in which the ET process “is preceded or followed by chemical equilibrium or by other chemical reactions. From such studies similarities have been noted between the rate with which certain species are oxidized or reduced by electrochemical and by chemical means.” Randles [11] and Vlcek [12] “interpreted the rates of electrochemical electron transfers in terms of concepts related to those used for electron transfer in chemical reactions. A brief qualitative comparison of the relative rates of several isotopic exchange reactions and of the corresponding electrochemical systems” was given by M. in Ref. [13] and is here reported in Chapter 3 [2, pp. 181, 182].

The theory presented in this chapter for ET in electrode processes is a continuation of that for homogeneous ET reactions. Like the theory for homogeneous (redox) ET reactions, even this theory “proceeds from first principles plus assumptions that appear reasonable on a priori grounds and is free from arbitrary assumptions and adjustable parameters. As before, the theory is not applicable to atom transfer mechanisms (hydrogen overvoltage, for example)” [2, p. 182].

4.8.The ET Rate Constants

“Equations describing any overall electrochemical process may include terms for

(i)The transport of ions to the electrochemical double layer region at the electrode for any chemical reaction of the electrochemically active species

(ii)The work required to penetrate the double layer (if necessary)

(iii)The actual ET

Only under certain conditions can these facts be disentangled in a relatively simple way and simple ET rates defined: The double layer region should be sufficiently thin that

(1)No chemical reaction occurs in it

(2)Diffusion across it is sufficiently rapid so that the concentration of an ion at any point in the double layer is related to its concentration just outside it by the work required to transport it to that point.

At appreciable salt concentration, the double layer thickness appears to be only of the order of several Ångstroms [14]. Under the earlier conditions, one can describe the entire electrochemical process by force-free differential equations [14] (containing any chemical reaction terms [16] if necessary) outside of the double layer region, while the boundary condition, at a boundary surface S drawn just outside the double layer, contains rate constants which depend only on the ET process itself. This boundary condition is that the flux of any ion through this surface S equals its net rate of disappearance by ET. If A and B represent the electrochemically active ion before and after its electron transfer with the electrode, the ET step may be written as:

where n is the number of electrons lost by the electrode.

If and denote the concentrations of A and B just outside the double layer, then the ET constants, k_f and k_b, will be defined by the relation:

Net ET per unit area = [2, pp. 182–183].

4.9.Basic Assumptions

The basic assumptions used in this extension of the Marcus theory to the electrode systems are analogous to those made in the oxidation–reduction theory for homogeneous systems [4].

The assumptions are

(a)“In the activated state of reaction (4.19), the spatial overlap between the electronic orbitals of the two ‘reactants’ is assumed to be small. The ‘reactants’ are now the electrode and the discharging ion or molecule (to be referred as ‘central ion’).” This assumption was previously discussed in Chapter 1.

(b)The central ion in this 1957 early form of the theory is treated as “a sphere within which no changes in interatomic distances occur during reaction (4.19) and outside of which the solvent is treated as a dielectrically unsaturated continuum. For complex ions or hydrated monoatomic cations, the sphere includes the first coordination shell of the central atom [4, 17].” [2, p. 183] The concept of “effective size” for unsymmetric organic molecules has been described in Chapter 3. In a refinement of the theory, changes, if any, in interatomic distances during ET were considered by Marcus, and were briefly mentioned in Chapter 3 and will be further dealt with later in Chapter 6.

4.10.Nature of the Transition State

As in the case of homogeneous ET, a successful ET between the central ion and the electrode proceeds via two successive intermediate states, X* and X, both participating in the ET process. X* represents the state of the reactants just before ET, X that of the products just after ET while a linear combination of the two wave function of the states X* and X represents the TS. “These states have the same atomic configuration and the same total energy but in X* the electronic configuration in the central ion is that of A and in X that of B. The electronic configuration of the electrode undergoes a corresponding change” because of the Franck–Condon principle as was already observed in Chapter 1. “By arguments similar to those employed in the redox theory” dealt with in the preceding Chapters, “it then follows that the actual ET, that is, the formation of X from X*, must be preceded by a reorientation of the solvent molecules in the vicinity of the discharging ion and nearby area of the electrode. For similar reasons, a change in ionic atmosphere in this region also precedes ET. The new configuration of the solvent and ionic atmosphere, which is the same in X* and X, will prove to be intermediate between that of the initial state and that in the final state. As such, it is not that which is predictable by the charge distribution in X* or by that in X. That is, it is not in electrostatic equilibrium with either charge distribution, and its properties cannot be described by the usual electrostatic expressions. Instead, expressions which take this nonequilibrium behavior into account must be used. Once again [4] there are an infinite number of pairs of thermodynamic states, X* and X, satisfying the constant energy-atomic configuration restriction and we are interested in determining the properties of the most probable pair, that is, the one with the minimum free energy of formation from the initial state. This is done by minimizing the expression for the free energy of formation subject to the restriction that X* and X have the same atomic configuration and the same energy” [2, pp. 183–184].

4.11.Reaction Scheme

“From the preceding discussion, we may write for the mechanism of reaction (4.19) the following scheme, analogous to that in Ref. [4]” and here reported in Chapter 2:

“Here, a central ion just outside the double layer is denoted, in its initial electronic state, by A and in its final state by B. As noted earlier, step (4.20) involves a suitable reorganization of the solvent and ionic atmosphere and (if necessary) a suitable penetration of the electrical double layer. Step (4.21) is the actual ET itself, and step (4.22) involves a reversion of configuration of solvent and atmosphere to one in equilibrium with the new charge on the central ion. Step (4.22) also involves a motion away from the electrode. As in the redox theory, the reverse of Eq. (4.22) occurs but it needs not be considered in the computation of k_f . Steady-state considerations for c_X* and c_X” (where the c’s are concentrations) “lead to the relation [4, p. 969]:

As discussed later, when the probability of ET in the lifetime of the intermediate state X* (10⁻¹³ sec) is large, k_f is about half of k₁. k₁ depends on the free energy of formation of X* from the initial state in reaction (4.20). We proceed first to the calculation of this free energy change and later to a discussion of the evaluation of the rate constants.”

The treatment given here by Marcus in 1957 is today considered by him only as a formalism that helped him to get started, just a step in an evolution, “a thing that may be interesting from the point of view of how thoughts developed but not for presentation of a way of thinking. That was history, I’m not thinking anymore in those terms.” (M). I shall then not discuss the rate constants of the elementary steps and I refer the reader to Chapter 6 for the modern treatment.

4.12.Free Energy of Formation of State X*

“The electrostatic contribution, ΔF*, to the free energy of formation of state X* from state A will be different from zero because of

(1)The work which may be required to transport the central ion from a point just outside the double layer to some particular point in it (if necessary)

(2)The work required to reorient the solvent molecules and the ionic atmosphere to a nonequilibrium configuration, for this position of the central ion

Let and denote the electrostatic free energy of the electrode system when the system is in the states A, X*, X, and B, respectively (throughout, asterisks will be used to designate the properties of X* and will be omitted in designating the properties of X). We have then for ΔF*:

is given by the general expression (4.24) and F_e is given by a similar equation, minus the asterisks:

where = electric field at point r exerted in a vacuum by all the ionic charges in the state X* and by those electrode charges which they would induce in a vacuum.

E*(r) = electric field in state X*. It equals −∇φ*.

φ*(r) = inner potential in state X* (cf. Eq. (4.1))

α_e = E-type polarizability = (D_op − 1)/4π

α_u = U-type polarizability = (D_s − D_op)/4π

c_i (r) = concentration of ions of type i in states X* and X

= average concentration of ions of type i. It equals their number

n_i in the solution divided by the latter’s volume V

P_u(r) = U-type polarization in states X* and X

χ = potential drop at electrode–solution interface due to an oriented solvent dipolar layer”

Note that c_i(r) and P_u(r) in state X* are the same as those in state X because of the constant atomic configuration restriction.

is obtained from the general Eq. (4.24) introducing in it the equilibrium expressions (4.14) and (4.15) for P_u and c_i.

is given by a similar equation, but the functions in the integrand now refer to state B.

Equation (4.24) and subsequent equations treat χ as being independent of the average electrode charge density.” However, M. shows that “the final equations” (4.31)–(4.33) in the following “are unchanged even ifχ were a function of this quantity” [2, pp. 186–187].

4.13.Constraint Imposed by the Constant Energy — Constant Atomic Configuration Restriction

“The free energy of formation of X* from A consists of the electrostatic contribution ΔF* and of a term, described later, associated with the localization of the center of gravity of the central ion in a narrow region near the electrode. ⁽³⁾ The free energy of formation of X from B contains an electrostatic term ΔF say, and a center of gravity term equal to that noted previously. As in the redox theory, it follows from assumption (a) given earlier that no energy change and no configurational entropy change accompany the formation of state X from state X*. The electronic entropy change arising from any possible change in the electronic degeneracy of the central ion and of the electrode is zero or negligible. Accordingly, X* and X have the same free energy.”

Note that the equality of the free energies of X* and X have been deduced from the equality of their energies and of their entropies.

“The net free energy change in forming B from A is therefore

Independently, this free energy change of reaction (4.19) can be written as the sum of the following terms:

(a)The change in chemical potential of ions A and B and of the electrons in electrode M. This is where n is the number of electrons transferred in reaction (4.21).

(b)The change in free energy due to the transfer of charge e−e* from a metal of inner potential φ_M to a solution of inner potential φ_S, e* and e denoting the charges of ions A and B, respectively. This term is (e*−e)(φ_M −φ_S) which we shall denote by (e* −e) Δφ”.

“If electrode M were in electrochemical equilibrium with the actual concentrations of A and B just outside the double layer, then the sum of these two terms would be zero.”

Denoting the corresponding value of Δφ as Δφ′ we would then have:

For Δϕ ≠ Δϕ′, there wouldn’t be such an equilibrium and the difference:

defines the activation overvoltage.

For the net free energy change ΔF*− ΔF written as sum (a) + (b) we have:

Introducing Δφ′ in the above equation using Eq. (4.26) we have:

“It may be remarked that μ_B − μ_A is related to its standard value and Δφ to its standard value Δφ⁰, by Eqs. (4.28) and (4.29), where the f’s denote activity coefficients and the c^S’s denote concentrations just outside the double layer:

[2, pp. 187–188].

4.14.Minimization of ΔF * Subject to the Constraint Imposed by Eq. (4.27)

The Final Theoretical Equations for ΔF *

We shall not delve in the details of the minimization process of ΔF* which is analogous to the one described in Chapters 2 and 3 for the redox theory. I only remark that Marcus considers now not only the configuration of the solvent molecules in the activated complex, but also the configuration of the ionic atmosphere so that ΔF* is minimized now not only with respect to arbitrary variations δP_u(r) of the orientation–atomic polarization, but also with respect to arbitrary variations δc_i(r) of ionic concentrations in the ionic atmosphere. There is then now a new condition of constraint, that is, besides the condition:

there is the new condition of fixed number of ions n_i in the solution, expressed as:

There are correspondingly two Lagrange multipliers: the m, already met in Chapter 2, and a new one, − ln l_i. In terms of the multipliers, the solvent and ionic configuration of the pair of intermediate states X* and X is obtained so that:

We see that the concentration of ions at distancer from the central ion is obtained by Boltzmann weighting the total number of ions n_i with the expression in the fraction which depends on the inner potential φ* + m(φ* − φ).

Notice that Marcus uses here the condition of constraint (4.30) instead of the condition δF* − δF = 0 which was used in the redox case but the two conditions are really the same because ΔF* = and

Marcus evaluates then ΔF* introducing what he will later name in Ref. [20] the equivalent equilibrium system (e.e.s). Such a system is introduced here as a hypothetical system having fictitious charges on the central ion and on the electrode which are in equilibrium with the configuration of solvent and ionic atmosphere characterized by the same P_u(r) and c_i(r) as those of states X* and X and which were not in equilibrium with the real charges of X* and X. The electrostatic free energy of this hypothetical system is determined and used to calculate the final theoretical equations for ΔF*. The hypothetical equilibrium system is characterized by having an inner potential φ^† = φ* + m(φ* − φ), ionic surface charge σ^† = σ* + m(σ* − σ) and, likewise, E^† = E* + m(E* − E), α_u(E* +m(E* −E), and e^† = e* +m(e* −e), where e^† is the charge on the central ion in the hypothetical system.

Note that it is not

The final theoretical equations for ΔF* are:

“where m satisfies the equation

and where

In these equations, Δe = e* − e, the charge transferred to the electrode in reaction (4.19); a denotes the effective radius of the central ion, discussed earlier; R/2 is the distance of the ion to the electrode surface in the state X*; w* and w denote the work required to transport the central ion from the body of the solution to a distance R/2 from the electrode surface when the ion is its initial and final state, respectively; the remaining quantities have been defined previously” [2, pp. 190–191].

I want to stress that ΔF* = ΔF*(R), i.e., ΔF* depends, through λ, on the distance of the central ion from the electrode.

“The term w* can be evaluated by solving the usual Poisson–Boltzmann equation [3] when the central ion is in the body of the solution, and then introducing these solutions for the electrostatic potential into the appropriate equations for the electrostatic free energy of the system (Eqs. (4.16) and (4.18)). The difference in electrostatic free energy is w*. Similarly, w can be computed from analogous equations for the central ion B.

However, a considerably simpler though less rigorous procedure has generally been assumed for calculating the work required to transport an ion from the body of the solution to some distance R/2 from the electrode. The assumption is generally made that this work equals the charge of the central ion multiplied by the difference in the value of the electrostatic potential at R/2 and in the body of the solution (i.e., outside the double layer), this potential being computed in the absence of the central ion. This particular potential can be inferred from measurements on the electrical double layer by various methods [14, 21]” [2, p. 192].

4.15.Dependence of χ on Mean Electrode Charge Density

In the earlier theory, “it has been assumed for simplicity that χ, the potential drop due to electrode–solvent and solvent–solvent interactions at the interface, was independent of the mean electrode charge density, say. Recent work suggests, however, that the degree of orientation of the solvent molecules next to the electrode (and hence χ) may vary with this quantity [22, 23].”

Marcus considers then the possible modification of the theory for the case in which χ is not just a constant, but a function and concludes that “when a few electrons are transferred in some process” as in our case “the average electrode charge density is affected only to a negligible extent. In such cases, χ is a constant for all states in the process.” M. then shows that χ considered either constant or depending on does not appear in the final Eqs. (4.31)–(4.33).

“Accordingly, we infer that any effect of a change in degree of orientation in the solvent layer next to the electrode is a more indirect one. It might affect the ‘dielectric constant’ in the vicinity of the electrode, particularly the contribution from orientation polarization. But, we observe from Eq. (4.33), unless D_S is close to D_op, changes in the former have very little effect on ΔF*.

Again, it might affect to some extent the distance of closest approach of the central ion. In some studies of the equilibrium properties of the electrode double layer, Grahame [22, 23] has shown that a self-consistent interpretation of the data can be obtained assuming such an effect. Extremely interesting inferences were drawn about the behavior of the solvent in this region toward ions of different size. It is clear that an analogous study of the electrode kinetics of simple electron transfers at various electrode charge densities should be very interesting. At certain charge densities, it appeared from the equilibrium studies, there is no oriented solvent layer. The interpretation of kinetic data obtained under such conditions would be correspondingly simplified” [2, pp. 204–205].

4.16.Presence of Fixed, Adsorbed Ions in the Electrical Double Layer

If fixed adsorbed ions are present in the double layer [3], one should consider their contribution to the electrostatic free energy

“If q_k denotes the charge of the kth fixed ion and if r_k and denote the distance from the kth ion and from its image to the field point, then the contribution to the potential arising from all fixed ions is, in an equilibrium polarization system,

the summation being over all fixed ions.

This can be shown to add the following terms to in Eq. (4.16):

where a_k is the radius of the kth fixed ion; R_k, r_jk, R_jk, ρ_k, and denote the distance from this ion to its electrical image, to the jth fixed ion, to the latter’s electrical image, to the central ion and to the latter’s electrical image, respectively. The first term in Eq. (4.35) describes the interaction of the mobile ions with the fixed ions.

In Eqs. (4.17) and (4.18), φ_S now represents the potential in the body of the solution due to all ions, fixed and mobile, of the double layer and so includes Eq. (4.34). Accordingly, qφ_S in Eqs. (4.17) and (4.18) now includes the last term of (4.35). The remaining three terms of Eq. (4.35) should be added to Eqs. (4.17) and (4.18)” [2, pp. 203–204].

M. demonstrates that “because of a cancellation of the newly added terms” the equation for ΔF* remains unchanged “except for the new significance of φ_s” [2, p. 203].

4.17.Summary and Final Remarks

At the end of this chapter, it may be useful to highlight some pivotal points of Marcus’ treatment.

M. begins with an extension to the electrode system of his general equation for the electrostatic free energy F_e of systems with nonequilibrium dielectric polarization. In the electrode system, he explicitly considers that the central ions have ionic atmospheres. Moreover, a potential drop χ is present at the electrode–solution interface. The expression for F_e is obtained by the two stages charging process that has been described in detail, for a simplified model, in Chapter 3. The expression for F_e in a nonequilibrium polarization system is then obtained from by a variational calculation considering that F_e obtains from when, because of fluctuations (variations) δP_u and δc_i, F_e departs from its equilibrium value. M. found an expression for in which P_u = α_uE—as expected for a system in electrostatic equilibrium—and an expression for c_i(r) is given as function of φ(r). M. considers here also explicitly for the first time the hypothetical equilibrium system, called equivalent equilibrium system, whose P_u is the same as that in X* and X but in electrostatic equilibrium with fictitious charges on the ions. The treatment of the equivalent equilibrium distribution in Ref. [20] will later extend this concept.

The ionic charge on the electrode is described in terms of the electrical images of the ions in the body of the solution and at the electrical double layer. With a second variational calculation, Marcus finds the expression for the minimum value of ΔF* corresponding to the activation free energy for the ET process.

Formulas (4.31)–(4.33) for ΔF*, λ, and m are very similar to those for the redox systems and were discussed in detail in Chapter 3. A unifying point of view of the formulas for redox and electrode systems consists in reducing the ET in the electrode case to that between an ion and its electrical image in the electrode.

It is finally demonstrated that the presence of χ can be neglected in deriving the final equations because of cancellations. For the same reason, the presence of adsorbed ions in the double layer can also be overlooked.

The contribution of the ionic atmosphere to the free energy of activation has also been demonstrated to be small. In Chapter 6, this last point will be taken up again and an expression for the contribution of to ΔF* will be given.

NOTES

1.M: “The charge on the electrode is an image charge, it is not uniformly distributed over the electrode. It is on the surface of the electrode but it behaves as though is situated a certain distance into the electrode, that’s the image.”

2.M: “If you have a certain species of ions you may change a local concentration keeping the total number of ions of that species constant.”

3.M: “We have, in the case of a reaction at an electrode, a localization analogous to that considered by Sutin when he considers the reactants to come together and to form a complex.”

References

1.R. A. Marcus, “Theory and Applications of Electron Transfers at Electrodes and in Solution,” in Special Topics in Electrochemistry, p. 161, P. A. Rock, Editor, Elsevier, New York (1977).

2.R. A. Marcus, “Electrostatic Free Energy and Other Properties of States Having Nonequilibrium Polarization. II. Electrode Systems. (ONR Report No. 11, dated 1957),” in Special Topics In Electrochemistry, p. 210, P. A. Rock, Editor, Elsevier, New York (1977); (a) R. A. Marcus, “On the Theory of Overvoltage for Electrode Processes Possessing Electron Transfer Mechanisms. I. (ONR Report No. 12 dated 1957),” in Special Topics in Electrochemistry, p. 180, P. A. Rock, Editor, Elsevier, New York (1977).

3.A. J. Bard, L. R. Faulkner, Electrochemical Methods, Fundamentals and Applications, John Wiley & Sons, Inc., New York, Chichester, Brisbane (1980).

4.R. A. Marcus, J. Chem. Phys. 24, 966–978, (1956).

5.R. Parsons, in Modern Aspects of Electrochemistry, Chap. 3, J. O’M. Bockris, Editor, Academic Press, Inc., New York (1954).

6.Compare Reference 5 for bibliography.

7.M. Mason, W. Weaver, The Electromagnetic Field, Dover Publications, Inc., New York (1929).

8.R. G. Sachs, D. L. Dexter, J. Appl. Phys. 21, 1304, (1950).

9.J. O’M. Bockris, A. K. N. Reddy, Modern Electrochemistry 1, 2nd Edition, Plenum Press, New York and London (1998).

10.R. A. Marcus, J. Chem. Phys. 24, 979–989, (1956).

11.J. E. B. Randles, Trans. Faraday Soc. 48, 828, (1952).

12.A. A. Vlcek, Coll. Czech. Comm. 20, 894 (1955).

13.R. A. Marcus, Trans. N.Y. Acad. Sci. 19, 423, (1957).

14.D. C. Grahame, Chem. Rev. 41, 441, (1947).

15.Compare reviews by (a) J. O’M. Bockris, Modern Aspects of Electrochemistry, Chap. 4, Academic Press, Inc., New York, 1954; (b) P. Delahay, New Instrumental Methods in Electrochemistry, Interscience Publishers, Inc., New York, 1954; (c) A. E. Remick, J. Chem. Ed. 33, 564, (1956).

16.Compare J. Koutecky, Collect. Czech. Chem. Commun. 18, 183, (1953).

17.R. A. Marcus, J. Chem. Phys. 26, 867, (1957).

18.J. J. Lingane, Electroanalytical Chemistry, 2nd Edition, Interscience Publishers, Inc., New York (1958).

19.Compare J. O’M. Bockris, Modern Aspects of Electrochemistry, Chap. 3, p. 167, Academic Press Inc., New York (1954).

20.R. A. Marcus, Discuss. Faraday Soc. 29, 21, (1960).

21.A. Frumkin, Trans. Faraday Soc. 36, 117, (1940).

22.D. C. Grahame, J. Chem. Phys. 25, 364, (1956).

23.D. C. Grahame, J. Am. Chem. Soc. 79, 2093, (1957).