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CHAPTER 3

Extensions, Electron Transfer at Electrodes, Applications

After the fundamental results of the wonder year 1955 (published ten months later in 1956), M. developed his theory and applied it to a variety of chemical and electrochemical systems. This chapter is devoted to the description of results mainly reported in Refs. [1–7], which span the years 1956–1964. I have also freely resorted to the reviews in Refs. [8–13], altogether a real “Guide of the Perplexed” [14]. Among the early applications are comparisons of calculated and experimental data, examples of how to use the theory to investigate reaction mechanisms and of how to systematize and correlate experimental data. These last characteristics have been typical of chemical theories like, say, the theory of the periodic system of elements or that of electrode potentials.

3.1.Theoretical Equations

In Chapter 1, the Landau–Zener–Stueckelberg–Majorana Equation was introduced, giving the probability P of a nonadiabatic transition in the narrow avoided crossing region.

If a transition from the R curve on the left of the col in Fig. 1.4a to the R on the right occurs, the ET does not occur because the system remains on the reagents R set of configurations. The probability that the system remains on the lower potential energy surface (PES) passing, through the col of the surface, to the products P set of nuclear configurations is γ = 1 − P. Marcus denotes with κ the nuclear velocity weighted average of γ [11, p. 181]. The formula for k_rate is written by M. as:

The same equation can obviously be written as

where ΔF^* refers now to a mole of ET reactions.

The M. theory in its early formulation is adiabatic, that is, κ is supposed equal to 1. Note that even if κ were 0.01 the value of would only increase by kT ln 100, that is, 2.5kcal mol⁻¹ at 0^∘C. The symbol ρ is “the ratio of the root mean square fluctuations in separation distance in the activated complex to the root mean square fluctuations of a coordinate for motion away from the intersection surface... and should have a value of the order of unity.” [11, p. 178]

M. introduces in Ref. [1] the symbol λ:

in terms of which

which shows very neatly the two components of the activation free energy:

(i) the reorganization free energy m²λ from an equilibrium state at distance of approach R and (ii) the Coulombic interaction at R, with the usual meaning of the symbols. R is the average distance between the centers of the reactants in the activated complex [6]. Note that in the following the symbol R is also used for the ideal gas constant.

The energy of activation of a bimolecular reaction is defined in chemical kinetics as:

ΔS^* is expressed in thermodynamics in terms of ΔF^* and T as:

Introducing Eq. (3.1; with ) into the defining equation of E_a and neglecting the small dependence of Z on T one gets (using the symbol k_bi for k_rate and R instead of k):

The terms ΔS^* and ΔF^* are not the same as the usual free energy and entropy of activation ΔF^‡ and ΔS^‡ as in, for example, Ref. [15, p. 195 ff.]. There we find ΔF^‡ and ΔS^‡ defined by the Eyring relation:

At room temperature, the term kT/h equals 10¹³ sec⁻¹ while Z, the collision number in solution, is about equal to 10¹¹ liter mole⁻¹ sec⁻¹ [16]. M. will later use the more precise value 10¹² [16a].

Marcus calls ΔS^* and ΔF^* excess entropy and excess free energy of activation. The excess for ΔF^* is the free energy of formation of the intermediate state from the reactants in excess of what it would be if the state were simply the usual transient collision complex at the thermally averaged geometry of two neutral, nonreactive (i.e., not forming or breaking chemical bonds) molecules in a solution. ΔS^* and ΔS^‡, ΔΔF^* and ΔF^‡ are related in Note 13 [1, p. 868, M19, Part II of the Theory].

Equations (3.1) and (3.7) express the rate constant of the redox step in terms of ΔF^* and ΔS^*, and Eqs. (3.4) and (3.6) relate ΔF^* and ΔS^* to ΔF⁰, the standard free energy change of the redox step, and to the polarizing radii (vide infra) of the reactants a₁ and a₂. Defining now and as:

from Eqs. (3.4) and (3.6) we get:

In Eq. (3.9), the last term is generally very small and we then have as an approximate equation for ΔS^*:

3.2.Frequency Factors, Activation Energy, and Transition Probability Factor κ

Equation (3.1) is also written [4, p. 156] as:

and the usual Arrhenius expression for k is:

The preexponential coefficient A is generally known as the frequency factor. The name derives from the interpretation of the expression in terms of the energy barrier to reaction: the exponential states the probability of surmounting the barrier E_a while A is related to the frequency of attempts [17, p. 87]. Writing E_a as E_a = ΔF^* + T ΔS^* we see that:

The calculated and experimental frequency factors provide the most direct measure of the probability factors κ [4, p.160]. M. found a good agreement between experimental and calculated frequency factors for the and for the systems, supporting — at least for these systems — a probability of adiabatic reaction of the order of one (vide infra).

3.3.The Dependence of ΔF^* on ΔF0 for a Series of Compounds at Constant λ

From Eqs. (3.2) and (3.4), a simple dependence of ΔF^* on ΔF⁰ can be deduced for the reaction of a reagent with a series of compounds chemically similar to each other in the sense that the charged part of the molecule remains the same throughout the series and the molecules differ only in some substituent.

Two such series will be discussed later. The polarizing radius is that of the charged part of the molecule which causes the electrical polarization of the medium. For each member of a series, it will be taken to be the same, since each has the same charged group with the same radius which is then a constant for all members of the series. It will therefore be seen from Eq. (3.2) that λ is the same for each member of the series. Accordingly, we are interested in the dependence of ΔF^* on ΔF⁰ at a given λ, that is, in (∂ ΔF^*/∂ ΔF⁰)_λ. We find, from Eqs. (3.3) and (3.4) that:

This predicts that when is very small, a plot of ΔF^* versus ΔF⁰ should be a straight line with a slope of When is large, a plot of ΔF^* versus ΔF⁰ will be approximately linear for small changes in ΔF⁰; the slope will be about unity if is as large as unity.

3.4.Theoretical Equations for Isotopic Exchange Reactions

For such reactions, the expression for ΔF^* simplifies because and

Moreover, “since, in the present case, reacting particle 1 as a product is the same as 2 as a reactant, and conversely, we have a₁ = a₂ and will denote these by a. Since R is taken to equal a₁ + a₂ it therefore equals 2a.” The expression for ΔF^* simplifies then to:

We immediately see that the reorganization term increases as a decreases and Δe increases, as expected from the discussion in Chapter 2.

As D_op is approximately independent of temperature, ΔS^* is obtained from Eqs. (3.4) and (3.6):

When only one electron is transferred, the term (Δe)²/4 will generally be appreciably less than and we would then have:

The first term of Eq. (3.15) is the entropy change which occurs when two ions approach each other under electrostatic equilibrium conditions. The lowering of ΔS^* for ions of the same sign when they get close to each other, that is, the entropy change which results when two ions are brought together is due to the enhanced local electric field causing an ordering of the solvent molecules, that is, when and are like charges, ΔS^* < 0. This is demonstrated in Section 3.5 because it may well not be immediately evident. In the opinion of J. Randles, later seen to be incorrect, for instance, in a discussion in Ref. [5, p. 246]: “with like charges the solvent molecules do not know which way to turn: I would expect that this would represent an increase in entropy.” A correct interpretation is given there by Marcus’ answer to Randles’ remark.

The second term in the equation is the entropy change resulting from the reorganization of the solvent molecules which accompanies the formation of the nonequilibrium state from the equilibrium one at distance R. The smallness of the second term relative to the first is due to a cancellation effect, because accompanying the solvent reorganization there is a decrease of entropy of solvation around one ion which approximately cancels the increase around the other: the solvent near the more highly charged reacting ion becomes less oriented while it becomes more oriented near the less charged ion.

3.5.*Demonstration That the Solvation Entropy of Ions of Like Sign Decreases with Their Distance [5, p. 246]

The free energy of Coulombic repulsion between two ions of like sign and charge is ΔF = q²/D_s R.

Moreover, the free energy difference between two states of a system at the same temperature is ΔF = ΔU − T ΔS. If we now show that for our system the ratio we will have demonstrated that ΔS^* < 0 and that ΔS depends on R as ΔF does.

Recalling that ΔS = −(∂ΔF/∂T) we have:

and for water at room temperature we have

Marcus’ comment: “D decreases with increasing T so the RHS of your equation above is positive, for water at room temperature it is 1.1. This makes physical sense, ΔS will get smaller with increasing T (less ordered) and ΔF will get larger with increasing T (smaller D), so the net result for the LHS of eq 1 is positive.”

3.6.The Work Terms w and wp (w^* and w, wr, and wp)

The work term is “the free energy change when the reactants are brought together to the separation distance R” [18, p. 690], it is the work required to bring the reactants from infinity to their separation distance R. It was initially taken as e₁e₂/D_s R, which is the Coulombic work to bring the reactants together at the distance R at infinite dilution. M. subsequently considered the possibility of polar (e.g., electrostatic) and nonpolar contributions to the work, and a more realistic situation in which the reactants were in a solution with some electrolyte concentration. For a discussion of the most appropriate value of R, see Section 2.9 of Chapter 2. One should also consider that “When R becomes large κ tends to zero and when R is small the van der Waals’ repulsion makes F^*(R) large” [18, p. 685]. The symbol w was then used for the earlier work term “in the prevailing medium,” and the symbol −w^p was used for the work required to separate the products from R to infinity in that same medium.

Using the new symbols, Eq. (3.4) is written as:

and Eq. (3.3) as:

with m equal to:

Eq. (3.14) for ΔF^* for isotopic exchange reactions can be written as:

and Eq. (3.15) becomes:

I have used here the symbols w and w^p instead than the symbols w* and w used earlier. A certain confusion may arise because in terms of the older symbolism Eq. (3.15) is written as:

Moreover, w^* is also indicated by Marcus as w^r. One finds then the following couples of symbols for the work terms in Marcus’ papers:

In the following only the symbols w^r and w^p will be used. M. uses also sometimes the symbol F and sometimes the symbol G for the free energy.

3.7.Inner and Outer Contributions to λ

In the first formulation of the theory (1956), the ion was treated as a sphere inside of which no changes in interatomic distances occurred during the reaction. This assumption was later eliminated when the theory was extended to include the effect of changes in bond distances and bond angles in the inner coordination shell of each reactant [6, p. 853]. In the earlier equations, λ will then be in general equal to:

where λ_o (λ outer) is what we have until now indicated with λ, that is:

and λ_i (λ inner) is given by:

where k_j and denote the force constants of the jth vibrational coordinate in a species involved in the reaction when that species is a reactant and when it is a product, respectively. The summation is over both reactants in the homogeneous case and over the one reactant in the electrode case (vide infra). When (ΔF⁰ + w^p − w^r)/λ is small, say 1/4, then:

ΔF^* is then linear in ΔF⁰ with a slope 0.5. When w^r and w^p are small and when ΔF⁰ = 0 we have λ = 4(ΔF^*)₀ (the subscript indicates of ΔF⁰ = 0). Thus, the earlier condition for linearity in ΔF⁰ can be written as

a condition often fulfilled in practice. More generally, the instantaneous slope of a plot of ΔF^* versus ΔF⁰ is, according to Eqs. (3.17) and (3.19), ¹/₂[1 + ΔF⁰/4(ΔF^*)₀] when the work terms are small.

3.8.An Outline of the Theory of Electron Transfer Processes at Electrodes—Introduction

The Marcus theory of the rates of electron transfer reactions at electrodes follows in the steps of the ET theory in solution because chemical and electrochemical redox reactions have many characteristics in common from the point of view of experimental results and theory.

I shall follow here mainly an abridged very clear qualitative description of the theory given by M. in Ref. [5]. The theory had been originally developed in two ONR Reports [19, 20] later published in Ref. [13]. The original theory of 1956 appears here already extended considering not only the effect of the nonequilibrium solvent polarization in inducing ET processes but also the effect of molecular vibrations and of ionic atmospheres.

The electrochemical processes are characterized by various levels of complexity. In the simpler case, they consist of only one elementary reaction, the redox step. But the process is usually more complicated because the elementary step may be preceded or followed by various chemical reactions and equilibria involving the electrochemically active species. It is then essential to analyze the experimental data in sufficient detail so that the electric current is known as a function of the overpotential of the redox step, and not simply of the more usual overpotential of the overall reaction sequence. “When such an analysis has been performed for a complicated electrode reaction, attention can then be focused on the actual redox step itself,” that is, the one dealt with in the M. theory.

The redox step can be visualized in the way usual in reaction rate theory: the atomic motions in the electrochemical systems happen on PESs which are functions of the coordinates of all atoms in the system, which means that to the usual atomic coordinates of solvent and solute we should add the coordinates of the atoms making up:

(i)The electrical double layer at the electrode/solution interface [17, 21–23]

(ii)The electrode itself

By a suitable fluctuation, that is, by a suitable concerted motion of the atoms, the system moves from a region of the many dimensional PES “where the electrochemically active species exists in one valence state to a region where it exists in the other valence state, with the electrode having undergone a corresponding change.”

Figure 1 from Ref. [5] represents the simplest schematic potential energy diagram for the electrochemical process;

The diagram is very similar to those already seen in Chapter 1 and the electrode appears here as a “giant central ion” [21, p. 774] but with at least two important differences from those of the usual ions. First, the electrode has numerous energy levels and then connecting it to an external potential source it is possible to control its potential, charge, and energy levels. If the curves in Fig. 1 refer to a situation of electrodic equilibrium potential E_eq, changing the external potential making the electrode more positive the curve to the left is lowered and is instead shifted to higher energies setting the potential to a level more negative [17, 21, 22, 24, pp. 148, 149]. This gives the possibility of changing the activation energies for the cathodic and anodic currents at the electrode, as described in Refs. [17, 21, 22, 24].

Fig. 1. Cross section of two intersecting electronic energy surfaces in N-dimensional atomic configuration space. Electrochemical process: A_ox + ne⁻ (metal) → A_red + (metal). The intersecting dashed lines indicate zero overlap of the electronic orbitals of A and metal (From Ref. [5]).

The qualitative description of an electrodic ET using Fig. 1 from Ref. [5] is very simplified because the potential energy diagram represents only a mean of the electronic energy levels of the electrode taking part in the ET process. For a metal piece of finite size, a more realistic representation is given in Fig. 2 from Ref. [11] — another Marcus’ logo. Each surface there is “a many-electron energy level of the entire reacting system and is a function of the nuclear coordinates.” The different R and P surfaces differ in the distribution of the electrons among the one-electron quantum states in the metal piece. As one sees there are many intersections among the different R and P surfaces and consequently the calculation of the velocity averaged adiabatic transition probability κ cannot be dealt with by a simple straightforward application of the Landau–Zener equation.

The use of a single surface of the PESs in Ref. [5] makes sense because the electron from the metal comes from levels in a neighborhood of width kT around the Fermi level E_F [11].

For our purposes, it is possible to classify the electrode reactions into at least three classes.

To a first class belong electrode reactions which happen with chemical bonds broken or formed, as, for instance, in the famous reaction:

where M is the electrode. To this class of bond rupture atom transfer type belong also the electrode processes involving deposition of metal cations [17].

In a second class are reactions in which no chemical bonds are broken or formed, such as:

The ions in the above reactions are “tightly knit” (M.), that is, the interatomic distances in the oxidized and reduced forms are about the same (vide infra). In these reactions, the changes of atomic configurations consist primarily of a large reorientation of the solvent molecules about the species. Such a reorientation, which arises from the change in ionic charge, naturally occurs in reaction (3.28) as well, but is of particular importance in the cases (3.29) and (3.30) in determining the reaction kinetics. To this class also belong reactions in which bonds are considerably stretched [25] but not broken, such as the Co–N bond in:

To a third class might belong electrode reactions in which bonds between ions and electrode surface are formed to facilitate ET and then broken. Such a class of “bridged activated complexes” occurs in certain electron transfers in solution where an anion may form a bridging species between two cations [26]. One may consider a reaction in which the ET between cation and electrode happens through a bridging anion adsorbed on the electrode.

From the point of view of Marcus theory, it is then convenient to classify the ET reactions according to whether:

(a)Chemical bonds are broken or formed during ET.

(b)No bond rupture or formation occurs, but merely an ET.

(c)Bonds are formed to facilitate ET and then broken.

Fig. 2. Same plot as in Fig. 1 but for an electrode reaction. The finite spacing between levels, reflecting the finite size of the electrode, is enormously exaggerated. Only three of the numerous electronic energy levels of this system are indicated. The splitting differs from level to level, and the spacing decreases as the size of the metal increases. (From Ref. [11]).

The Marcus theory of ET at electrodes deals with (b) processes. In these reactions, the reactant approaches the electrode close enough to effect those electronic interactions between its orbitals and those of the electrode which are necessary to induce ET. After ET, the product recedes from the electrode.

3.9.The Theory

In the electrochemical case, the rate of the reaction is measured by the electric current passing through the electrode and that equals the total probability of reaching the saddle point connecting the valley of the reagents and that of the products multiplied by an appropriate frequency factor for passage through and by the concentration of active ions in the solution. “This probability is calculated by means of equilibrium statistical mechanics and a knowledge of the potential energy surface in the valley of the reactants and at the passes (saddle points). The assumption normally made for this purpose is that the reaction hardly disturbs the equilibrium between systems in these two regions.” [17]

In describing the PESs of Fig. 1, “let us first examine the hypothetical case of zero electronic interaction.” There are then two electronic states to consider, “each having its own many-dimensional PES. In the first of these states, the ion is in an oxidized form A_ox and the electrode has the electronic charge distribution appropriate to the metal-solution potential difference. In the second state, the ion is in the reduced form A_red, the electrode has lost to it n electrons and again has a charge distribution appropriate to the same potential difference.

The valley of one surface is centered at quite different atomic coordinates from that of the other surface. The difference in ionic charge between reactants and products results in differences in:

(i)Average stable orientation of the solvent molecules outside the coordination shell

(ii)Ionic atmospheres about the ion and electrode

(iii)Interatomic distances inside the inner shells

If these surfaces are plotted versus N atomic coordinates of the entire system, they will intersect on some (N − 1)-dimensional surface. At the intersection surface, the atomic coordinates have values which represent a compromise configuration of solvent molecules, atmospheric ions, and coordination shell. The compromise is between the stable atomic configurations of the reactants and those of the products. Because of the assumed absence of electronic interactions between ion and electrode, there will be no electron transfer if a system moving on one diabatic PES reaches the intersection region. The system merely stays on the surface appropriate to its electronic configuration. During the reverse fluctuation, the system again passes through this intersection region, always staying on the initial PES, and reverts to the stable atomic configuration of the latter.

Let us consider next the case of a weak electronic interaction between ion and electrode. Such a weak interaction hardly affects the two PESs, but it removes the degeneracy at the intersection, as indicated in Fig. 1, so that we no longer have an intersection and the system is now described by adiabatic PESs. The quantum mechanical reason for this is briefly the following. If ψ_I denotes the electronic wave function for one of the two electronic states in the hypothetical zero-interaction case of Fig. 5^*(a) of Chapter 1 and if ψ_{I I} denotes the other, ψ_I and ψ_{I I} are functions of the atomic coordinates. They describe not only the motion of electrons of the central ion but also those in the molecules outside this ion. They describe for instance the electronic polarization P_e of the solvent by a central ion in each of the two valence states. Because of the ionic charge difference in the two states, this electronic polarization is quite different. The weak interaction may be shown from quantum mechanics to produce two new wave functions which in the quantum mechanical adiabatic case, the one now considered by M., are essentially ψ_I + ψ_{I I} and ψ_I − ψ_{I I} for atomic coordinates corresponding to the intersection surface. Because of the weak interaction, these wave functions have energies slightly different from those of ψ_I and ψ_{I I} , as indicated in Fig. 1.

The wave functions ψ_I ± ψ_{I I} describe the state better than ψ_I or ψ_{I I} alone, one of them will have an energy lower than either ψ_I or ψ_{I I} , lower by the resonance energy in the usual manner familiar to chemists, as indicated in the figure.

The behavior of the system on passing through the avoided crossing region of the PES will depend on the magnitude of this resonance energy. If it is large enough, about 1 kcal mole⁻¹, that is, of the order of 1/100 of the energy of a normal chemical bond, a system passing through this region and initially residing on the lower surface, on the left side of Fig. 1, will remain throughout on the lowest surface and end up on the right side of the lowest surface.

If we consider next the case of an extremely small resonance energy, that is, of an extremely small electronic interaction between ion and electrode, we see that “On passing through the intersection region, the system would tend to stay on the same surface that it stays on in the case of zero interaction. That is, if the system is started on the lower surface in the left of Fig. 1, the atomic motion would tend to carry it to the upper surface on the right and conversely during the return fluctuation. We see that when the interaction is extremely weak, the system ‘jumps’ from the lower surface to the upper one at the intersection. Actually, in the intersection region the concept of PESs and of the Born–Oppenheimer approximation on which it is based breaks down to this extent when the interaction is so extremely weak. During a passage through this region there is, however, a certain probability that the system will not ‘jump’ and hence a certain probability that the system will end up on the lower surface on the right in Fig. 1, and so have effected an electron transfer.” [5]

M. calls electron transfer mechanisms involving weak interactions and extremely weak interactions the quantum mechanically adiabatic and nonadiabatic mechanisms respectively, according to customary terminology. At the time of writing of Ref. [5], the few experimental data on preexponential factors tended to favor the adiabatic mechanism.

I shall now report the results of the Marcus treatment of ET processes at electrodes. The demonstration of how they were obtained, in Refs. [19] and [20], will be outlined in Chapter 4.

In order to understand these results, it is necessary to anticipate that M. uses the method of electrostatic images to determine the ion-electrode potential and so the electron released by, say, the reactant to the electrode is formally released by the reactant to its electrostatic image. The use of images permits the condition of zero electric field at a metal electrode to be satisfied. If the ion lies in solution at a distance R/2 from the electrode surface, its image lies inside the electrode also at a distance R/2 from the surface. Consequently, the distance between the ion and its image will be R but the electrostatic image is just a fictitious ion and so the energy λ for the process will be one half of the value for the homogeneous case where two real ions exchange an electron.

The electrostatic method of images is described, for example, in Refs. [27–30].

The current density i as a function of the activation overpotential η_a is:

where

In these equations, w^r and w^p are the works required to bring the reactant and the product to the electrode, respectively, m and λ are defined by Eqs. (3.34) and (3.35), n is the number of electrons transferred, e is the unit electronic charge, F is the Faraday, a is the ionic radius, R is twice the distance between the electrode and the center of the ion, D_op and D_s have the usual meaning, A′ is of the order of 1 × 10⁴ to 5 × 10⁴ cm sec⁻². The activation overpotential is:

where E is the electric potential, E⁰′ is the value the latter would have under equilibrium condition if the reactant and product concentrations were equal (thereby E⁰′ is the formal standard potential in the prevailing medium, that is, at given salt concentration [11, 17]).

If the activation overpotential is small relative to the free energy barrier prevailing at zero overpotential, on expanding m² in Eq. (3.33) about its value at η_a = 0 one obtains a linear dependence of ΔF^* on η_a:

One sees from this equation that for a salt concentration suffi-ciently large for w^p and w^r to be small, the transfer coefficient, ∂ ln k/∂(neη_a), the slope of the Tafel plot [17, 21, 23], should be equal to 0.5.

The reason why, at sufficiently high salt concentration, the electric work necessary to bring the ion to the electrode or away from it is close to zero, is that in the presence of a sufficiently high salt concentration the transference number of the electroactive ion decreases practically to zero and the electrical resistance of the solution and thus the potential gradient throughout is made small. In such conditions, the electroactive ion plays only a negligible part in the electrolyte conduction and its motion is affected by the diffusive force alone, see, for example, Refs. [17, 31–33].

As in the case of homogeneous ET, the Franck–Condon principle plays here an essential role as it was first recognized by J. Randles [4]. Even now a nonradiative ET becomes possible with little or no change of electronic configurations and momenta only at the intersection region of PESs. The electron can be released from the electrode to the oxidized form A_ox of the ionic or neutral electroactive molecule only if a suitable previous thermal fluctuation of polarized solvent molecules, of ionic atmospheres and of inner shell vibrations have brought A_ox in an X^* configuration of energy equal to that of an X configuration of A_red [22].

3.10.The Vibrational Motion within the Reactants

After having extended the use of PESs to the description of ET reactions at electrodes, I shall now show how potential energy curves (PECs) can describe the vibrations of the electroactive species inside the inner shell which, in the first formulation of the theory, was approximated as a rigid sphere. The description is here very simple and follows a formulation to be found in Ref. [13].

A simple ET reactions at an electrode can be written as

while a simple ET between species A₁ and A₂ in solution is represented as

with by now evident meaning of the symbols. In Fig. 1 of Ref. [13] curve R represents the PEC that describes how the potential energy of the reactant system, A₁(ox)+M(ne), varies as function of a single vibrational coordinate q varied in A₁(ox). The P curve likewise represents the products system A₁(red) + M, when a single vibrational coordinate is varied in A₁(red), all the other coordinates remaining fixed. Considering the simple but realistic case of harmonic vibrations, the figure represents the potential energies of R and P as two parabolas crossing at a point. Several facts are noted:

(i)The minima of the two curves occur at different values of q, because the equilibrium bond length in A_ox is different (and usually shorter in the case of transition metal ions and a metal–ligand bond) from that in A_red.

(ii)The relative height of the two minima ΔU^∘ depends on the electrostatic potential, the P curve being lowered vertically relative to the R curve by making the electrode more negative, that is, by decreasing e(ϕ_M − ϕ_S), where the ϕ’s represent the Galvani potentials [21, 24] of the metal and solution.

(iii)For a given ΔU^∘, the barrier height ΔU^* is lower when the difference Δq^∘ of equilibrium bond lengths is smaller.

(iv)When the P curve is lowered relative to the R, the height ΔU^* of the barrier between the minimum of the R curve and the intersection of the R and P curves, is reduced. This is not true in the “inverted region,” see Chapter 5.

Fig. 3. A plot of the potential energy U of the system consisting of reactants plus solvent (R), along some coordinate q, and of the system consisting of products plus solvent (P), holding all other coordinates fixed, for reaction [3] or [4]. (From Ref. [13]).

The two curves have properties which are by now familiar. When the reactant is far from the electrode, the curves in Fig. 3 merely cross in the intersection region (dotted lines there, see also Fig. 5^*(a)). When the reactant is close to the electrode, the electronic interaction of the orbitals of the reactant and the electrode perturbs the dotted line curves in Fig. 3 in the intersection region.

At the intersection, the unperturbed electronic R and P quantum states are degenerate, and the degeneracy is broken by the interaction. The new curves are the solid curves. The energy at the maximum of the lower solid curve is less than that at the intersection by an electronic interaction energy ε₁₂, the resonance energy. In this case we see that the fluctuation needed for ET to occur is that of the vibrational coordinate q which may induce the ET, of course, only if the reactant—reactant distance is sufficiently small for the electronic overlap to occur. Here the estimate of the barrier height in Fig. 3 is very simple. It will be assumed that the resonance energy ε₁₂ is small enough that the height of the crossing point is approximately that of the maximum of the lower solid curve in the figure. We let the potential energy of the R and P curves to be approximated by [34, p. 7]:

Where and are the equilibrium bond lengths for A(ox) and A(red), respectively. ΔU⁰ is a linear function of ne(ϕ_M − ϕ_S). At the intersection of the dotted curves we have:

and q = q^‡. From the earlier condition one gets:

where

Introducing q^‡ into Eq. (3.39) one obtains:

where

And we see that λ_i/4 is the barrier height when ΔU⁰ = 0.

Figure 3 suffices to describe also the vibrational motion within the reactants in the ET reaction in solution:

with completely analogous results.

The rate constant k_r is given by the collision frequency Z (per unit area of electrode and unit time in the electrode case, per unit concentration and unit time in the homogeneous case) multiplied by the Boltzmann factor exp(−ΔU^*/RT) and by a Boltzmann factor exp(−w^r/RT), where w^r is the work term, if any, required to bring the reactants together to some separation distance R, and finally by κ:

When such work terms occur, the ΔU^∘ in Eq. (3.40) is replaced by that is, the ΔU⁰ at separation distance R. Thus Eq. (3.42) is replaced by:

where ΔU⁰ is the ΔU at infinite separation and w^p is the work to bring the products together to the separation distance R.

The earlier equations refer to the contribution given by λ_i to the total λ = λ_i +λ_o. The equations found here in a very simple way considering a cross section of the PES where only simple harmonic forces are involved are the same as those found using the nonequilibrium dielectric polarization theory for the complicated atomic motions and interactions of the solvent, that is,

This “method of the intersecting parabolas” is used, for instance, in Ref. [35] in a simple description of Marcus theory.

The above calculation of ΔU^* is classical and indeed classical mechanics is commonly used nowadays to treat reactive collisions but for some problems a quantum mechanical treatment is needed as, for example, for treating a proton vibration in the equation:

A quantum treatment is given in Refs. [36, 37].

The results of Eqs. (3.45–3.47a) may be extended to all vibrations of the reactant(s) in reactions 3.38 or 3.44. The PECs of Fig. 3 are replaced by PESs plotted as a function of all the q’s in the system rather than just one. The coordinate q in the figure represents then in this case some path in the N-dimensional q-space and the R and P curves are profiles of the actual PESs plotted along that path. λ_i becomes a sum of terms of the type in Eq. (3.43), summed over all vibrations, that is,

k_i is related to the force constants of a bond, and in the oxidized and reduced form for Eqs. (3.38) or (3.44). A more sophisticated treatment of λ_i is given in Ref. [18], leading to k_i being a certain average of and More on this in Chapter 6.

“There remain the solvent fluctuations outside of the inner coordination shell of the reactant in Eq. (3.38a) or reactants in Eq. (3.38b). Here, the potential energy functions do not depend on the solvent coordinates (orientations, translations) in the simple quadratic fashion in Eqs. (3.39) and (3.40) of Fig. 1. The treatment of the solvent coordinates is correspondingly more complicated. However, one feature is immediately clear: Just as a thermal fluctuation of vibrational coordinates was needed to reach the intersection region in Fig. 1, a suitable thermal fluctuation of solvent orientation coordinates or reactant’s vibrations also permits the system to reach the N − 1 dimensional hypersurface (the intersection region). A statistical mechanical treatment of the free energy associated with these fluctuations is given in Ref. [18]. Dielectric continuum theory also permits an estimate of the latter to be made.” [13, p. 166]

In the very instructive review of Ref. [13], M. gives a simple and elegant derivation of the free energy change needed to reach the intersection region by the fluctuations of solvent dielectric polarization.

I shall report it now almost verbatim with explanatory notes.

3.11.*The Nonequilibrium Polarization Expression by a Two Steps Charging Process—A Simplified Derivation

Let us consider the homogeneous reaction system in Eq. (3.44) and the modification for the electrode case in Eq. (3.38a). The charges of the reactants are denoted by e_i and their radii by a_i for reactants 1 and 2 (i = 1, 2). A superscript p to the e_i denotes the charges of the products. Let D_s and D_op be the static and optical dielectric constants of the solvent. The separation distance of the centers of the reactants is denoted by R. Marcus describes here in a simplified way his famous method [38] of calculating the nonequilibrium dielectric polarization of the medium. He produced the state of nonequilibrium polarization P = P_u +P_e by a two stages reversible charging process. “Since each step is reversible, the free energy of formation of this non-equilibrium system, i.e. the free energy of this polarization fluctuation, can be calculated in a relatively straightforward manner.”

The two-step charging process at a given separation distance R is the following:

(i)The charge of each reactant i is changed from e_i to where is so chosen as to produce the desired orientational–vibrational dielectric polarization P_u

(ii)The charge of each particle i is changed back from to e_i holding the earlier orientational–vibrational dielectric polarization P_u fixed. After step (ii), the desired electronic polarization P_e is also obtained because P_e is now in equilibrium with the electrostatic potential, that is, it is determined by ψ(r) which is, on its turn, determined by the ionic charges and by P_u (see Eq. 2.11b). Note then that, once back to the initial charge e_i this charge is not alone in determining P_e because P_e is also determined by P_u.

In the detailed following calculations, the electrostatic potential in the solvent medium at every point r is denoted by ψ(r).

Step 1

At any stage ν of the charging process, the values of e_i and ψ(r) are denoted by and ψ^ν(r).

They are given by:

r_i is the distance from the field point r to the center of ion i. ν varies between 0 at the beginning of the charging process an 1 at the end, ψ^ν(r) can then be written as in Eq. (3.50). The potential at the surface of ion 1, due to the medium and to ion 2, is obtained replacing r₁ by a₁ in Eq. (3.50). the potential there minus the self-potential, is obtained by subtracting (typo in the original) from Eq. (3.50)

For the potential at the surface of a spherical conducting ion of charge e in a solvent medium of static dielectric constant D_s, compare, the Born equation in Ref. [21].

The average of over the surface of ion 1 is denoted by and is found to be:

where R is the distance between the centers of the two ions.

The average leading from Eqs. (3.51) to (3.52) is the electrostatic result that the average value of a 1/r₂ potential from a uniform distribution over a sphere is 1/R [39, 40].

On multiplying by an increment of charge where:

integrating over ν from 0 to 1, performing the same integration for ion 2 and summing both terms, we obtain the work term W_I required in charging step I:

Eqs. (3.52, 3.53) yield:

where

When the initial charges e₁ and e₂ are both zero, the 1/R term becomes the usual Coulomb repulsion the 1/a₁ term becomes the well-known Born charging term for ion 1, and the 1/a₂ term the Born charging term for ion 2.

Step 2

The charges are given by Eq. (3.49b), where ν goes from 0 to 1.

For ν = 0, (starting point of step 2), for ν = 1, (the initial charge to which one goes back in step 2).

Let ψ_I (r) denotes the potential at the end of step 1 and ψ^ν(r) the potential at any state ν of step 2. The change of potential during step 2 is, for any ν, ψ^ν(r) − ψ_I (r). Since the medium responds now to the change of charge only via the optical dielectric constant D_op (remember that P_u is supposed fixed while P_e is following the charge) during step 2 we have:

compare Eq. (3.50). Writing and δψ^ν = ψ^ν − ψ_I we have:

Where

Keeping now in mind the process that brought from Eqs. 3.50 to 3.52, we see that the average potential on the surface of ion 1 minus the self-potential, is obtained by subtracting from Eq. (3.58) and from the last term but one in Eq. (3.57), then replacing r₁ in those equations by a₁ and averaging the 1/r₂ in those equations over the surface of ion 1, thereby replacing 1/r₂ by 1/R” we thus have:

The charging work of both ions done during this step is W_II

The total work ΔG^r done is the sum of W_I and W_II and is the free energy of this fluctuation. It is:

with e_i given by Eq. (3.55). “If denotes the free energy change for unit concentration of reactants at a separation distance R, it is related to the same quantity at infinite separation ΔG^o′ by an equation similar to Eq. (3.47a),” namely:

Reactants and products have the same distribution of configurations (the same set of values of q^‡) on the intersection hypersurface of Fig. 3, and the same potential energy, U^r = U^p, averaged over a distribution of such configurations. Being the distribution of configurations and momenta, the same for reactants and products on the intersection hypersurface, the entropy is also the same and consequently the free energy of the reactants is there the same as that for the products.

We can write:

where ΔG^r is given by Eq. (3.61) and ΔG^p has the same expression with e_i replaced by

In order to find and that is, the fictitious charges of the activated complex that would be in electrostatic equilibrium with the correct and real nonequilibrium P_u, M. minimizes ΔG^r in Eq. (3.61) subject to the constraint imposed by Eq. (3.63):

Following the recipe of the calculus of variation, one multiplies the second equation by a Lagrange multiplier m and adds the equation so multiplied to the first, introducing then expressions such as Eq. (3.64) for δG^r and δG^p into Eq. (3.65). Setting finally the coefficients of and equal to zero, one finds:

Introducing this result into Eqs. (3.54) and (3.56), one obtains:

and

where

and the charge transferred. Solving for m from Eq. (3.68), we have:

The free energy barrier to reaction ΔG^* consists of two terms: the work term w^r to bring the reactants together and ΔG^r. Introducing m in Eq. (3.67), one obtains:

Turning now to the electrode case, the electrostatic potential in step 1 is given by Eq. (3.50) with e₂ replaced by the image charge of reactant 1 inside the electrode, that is, e₂ → −e₁. “The image charge ensures that the potential given by Eq. (3.50) is constant on the surface of the electrode where r₁ = r₂.” The derivation parallels the one above, where R denotes now the distance from ion 1 to its image, namely twice the distance to the electrode. W_I and W_II are computed considering that in the electrode case, only ion 1 needs to be charged, the image being only a fictitious ion (cf. Ref. [27], p. 124). Expressions similar to Eqs. (3.67)–(3.71) are ultimately found, but now we have:

and

that is, one obtains for the electrode case:

where F is the Faraday.

This type of simplified derivation of the earlier results was given earlier in Ref. [10] and made more readily available in a review which is also an excellent review of the theoretical literature up to its time [41]. In order to consider simultaneously these fluctuations in solvent polarization and those in reactants’ vibrations, one could do so: add to the right side of Eq. (3.61) the term and add to the corresponding expression for ΔG^p. One proceed then as before using Eqs. (3.64), (3.65) but now Eq. (3.64) contains an extra term (because

and δΔG^p contains an analogous extra term. One finds, in addition to Eq. (3.66), the result that:

When this and the again given by Eq. (3.66), are introduced into the expression for ΔG^r, Eq. (3.67) follows again but with λ₀ replaced by λ₀ + λ_i, λ₀ being given by Eq. (3.69) and λ_i by Eq. (3.48), with similar remarks for the electrode reaction case.

3.12.Additive Properties of the Reorganization Term λ

For the reorganization term λ, there is an additivity property which is important to establish for subsequent correlations. Moreover, the λ’s for homogeneous and electrochemical systems are also related with each other.

Let us consider the electron exchange reaction:

and let us indicate its λ₀ as

where a₁ and a₂ are the radii of the oxidized and of the reduced form respectively.

Defining as and likewise we have that

This equation is more simply written as:

If the distance R between the ions is very large, so large that the force field from one reactant does not influence the other, the fluctuations around each reactant are independent [18] and one can write:

and will be simply indicated in the following as λ₁ and λ₂.

If, moreover, and we indicate with a their mean, we have:

Considering now the reaction:

with parallel reasoning and symbols we get:

Considering finally the cross-reaction of the earlier electron exchange reactions:

always in the hypotheses we have:

For R large:

so that finally a cross relation follows among the λ’s of reactions (3.76), (3.79), and (3.80):

At finite distance, we have the approximate relation

with R = a + b.

For the electrochemical reaction:

λ is given by Eq. (3.35) so that:

More simply:

and we see that for large R:

where λ_ex and λ_el are the λ’s for the exchange and electrode reactions, respectively.

The equation agrees with the fact that “in the electrochemical case there is only a contribution” to λ “from one ion” [18].

3.13.A Synopsis of Equations for Chemical and Electrochemical ET Reactions

3.14.Correlation between the Rate Constants of Two Electron Exchange Reactions and That of Their Cross-Reaction

It follows from Eqs. (3.1, 3.78, 3.25) that when condition (3.26) is fulfilled, the forward rate constant k₁₂ of the cross-reaction (3.80) is correlated to the rate constants k₁₁ and k₂₂ of the isotopic exchange reactions (3.76) and (3.79).

The cross relation is:

and are the work terms for the isotopic exchange reactions, denote the equilibrium constant and the work terms of Eq. (3.80).

When like in the case of charge interchange among aquo ions, the formula simplifies to:

For a somewhat more accurate comparison, k₁₂ may be estimated from k₁₁ and k₂₂ using the complete Eqs. (3.1, 3.17, 3.19) and noting that λ₁₂ = (λ₁₁ + λ₂₂)/2. When the work terms are negligible, we have:

where ln f = (ln K₁₂)²/4 ln(k₁₁k₂₂/Z²) [6, p. 856].

The cross relation was studied very extensively by N. Sutin and coworkers [42, 43].

In the words of Henry Taube: “The Marcus correlation is a powerful one, leading as it does to a calculation in most instances of a specific rate to an order of magnitude or so” [44]. It is “a touchstone of normal behavior” for outer sphere activated complexes [45].

3.15.Correlation between Isotopic Electron Exchange Rate and Corresponding Electrochemical Rate Constants

“For an isotopic exchange reaction between ions differing only in valence state, ΔF⁰ = 0, w^r = w^p, and hence m = −0.5 in Eq. (3.19). In the ‘exchange current’ of the corresponding electrochemical system η_a = 0 by definition, and m = −0.5, if the work term w^r − w^p is small. The λ₁’s and λ₂’s in Eq. (3.78) are all equal” [6, p. 854]. Considering Eqs. (3.78) and (3.83) “It then follows that:

(= or < according as the reactant can or cannot penetrate the solvent layer adjacent to the electrode). From a physical viewpoint, the factor of two enters in the exchange system because two ions and their solvation shells are undergoing rearrangement in forming the activated complex, while in the electrochemical system there is but one such a particle.”

*Demonstration by Marcus: “In solution and if a₁ = a₂

When r = 2a

For the electrode

Where R = distance from the ion to its charge image. So

(equal 2a when the ion is in contact with the surface of the electrode). For the case that R = 2a we have

which is 1/2 of the value in Eq. (1).”

Marcus’ comment: “Let’s pick the simplest mode, namely that you have that dielectric continuum, there are no subtleties about saturation, and things like that. Now, if the ion can get up and touch the electrode, then the equality sign would prevail, just looking at the equations. If the ion can’t get that close, then that means that it doesn’t interact much with its image charge, and its image charge would cancel some of the field, and that’s all part of this inequality here, so the image charge is cancelling less if the ion can’t come up close, because of some absorbed layer, and then in that case, because of the image effect, for R bigger than 2a the λel is bigger.”

“It thus follows that:

when w^r and w^p are small in both the ex and el experiments. From Eq. (3.1), we then expect that:

where k_ex and k_el are in units of liter mole sec⁻¹ and cm sec⁻¹, respectively. Another factor tending to favor the ‘>’ sign is the existence, if any, of inactive sites due, say, to any strongly absorbed foreign particles. More recently it has been concluded theoretically that under conditions neither the earlier deduction of this relation nor that of the 0.5 slopes of the ΔF^* plots should be affected if one or both of the reactants form relatively weak complexes with other ions. The ΔF^*’s are then corresponding to the pseudo-rate constants, ‘constants’ which depend on the concentrations of these other ions.” [6, pp. 854–855].

3.16.Chemical and Electrochemical Transfer Coefficients

When the work terms can be made small by using high electrolyte concentration, or when they are essentially constant, one may draw from Eq. (3.25) the following variations in the plots of ΔF^* versus ΔF⁰ and of ΔF^* versus −nFη_a:

1.In the oxidation—reduction reaction of a given reagent with a series of related compounds such that the reactions’ ΔF⁰ is essentially the only parameter varied, a plot of ΔF^* versus ΔF⁰ and hence of log k versus log K should be linear with a slope of 0.5 for ΔF⁰’s satisfying Eq. (3.26).

2.In the electrochemical case, the corresponding plot of ΔF^* versus −nFη_a (or of −RT/nF ln k versus electrode potential), the electrochemical transfer coefficient, should also be linear with a slope of 0.5.

By analogy, M. calls the slope of the ΔF^* versus ΔF⁰ plot in case (1) the “chemical transfer coefficient” of the reaction [6].

3.17.Taube’s Definition of Inner-Sphere and Outer-Sphere Electron Transfer Reactions

An important classification of ET reactions due to H. Taube, is that of inner-sphere and outer-sphere ET reactions: “The distinction between inner-sphere and outer-sphere activated complex... is fundamentally between reactions in which electron transfer takes place from one primary bond system to another (outer-sphere mechanism), and those in which electron transfer takes place within a single primary bond system (inner-sphere mechanism),” from Ref. [45, p. 28].

Using the former nomenclature, we can say that the Marcus theory generally applies to ET reactions with outer-sphere activated complex mechanism. This kind of mechanism operates, for example, for redox reactions both self-exchange and cross-reactions among the complexes in Table 1 of Ref. [44]. Many reactions to which they apply are reported in Ref. [24].

3.18.Atom versus Electron Transfer

“Isotopic exchange reactions may occur by the alternative mechanisms of atom transfer and of electron transfer.” [1, p. 868]. Let us consider for instance the reaction:

In the case of an electron transfer mechanism, an electron would jump from Cr⁺² to CrCl⁺² and we would directly have the products. In the case of an atom transfer mechanism, a Cl⁻ would transfer from CrCl⁺² to Cr⁺². The Cr⁺²–CrCl⁺² isotopic exchange reaction has been shown to possess an atom transfer mechanism [46]. “On the other hand reasonable evidence was obtained for a bridge-activated complex electron transfer mechanism for the reduction of and of by Cr⁺²” [47]. The Marcus theory does not in general apply to atom or group transfer reactions or to bridged complex electron transfer reactions.

3.19.Three Groups of ET Reactants

M. classifies the reactants “somewhat loosely” in three groups [4, p. 156].

Class I consists of species for which the interatomic distances in the coordination shells of the oxidized and the reduced form are essentially the same.

Class II consists of species in which the bonds of one form are slightly stretched or compressed compared with the other form.

Class III consists of those in which large stretchings or compressions appear.

Examples of Class I: and probably

Examples of Class II: many hydrated metal cations such as

Examples of Class III: and [4, p. 156].

The theory of Marcus in its first formulation applies in general to the “tightly knit covalently bound ions” reactants [3, p. 426] of Class I. The extension of the theory to the consideration of harmonic vibrations in the reactants allows its application to Class II reactants. The theoretical expressions for ΔF^* and ΔS^*of Class III reactants would contain additional terms beyond those considered in the present form of the theory [4, p. 158].

3.20.Interpretation of the Equations

The former set of equations suggests experimental studies on the dependence of the rate constants, for both chemical and electrochemical ET, from parameters such as:

(i)Standard free energy change or activation overpotential

(ii)Ionic structure and ligand field

(iii)Frequency factor and activation energy

(iv)Added salts

(v)Solvent medium

(vi)Electrode material and surface contamination

It also suggests experiments on:

—The extent of parallelism between both rate constants for a series of reactants

—Correlation between rate constants and between chemical and electrochemical transfer coefficients [4, p. 157].

There is a close relation between the theoretical equations of the chemical and electrochemical processes. In each case, ΔF^* decreases with increasing a, decreasing ne, decreasing R, increasing D_op, decreasing D_s (the latter’s effect is perhaps more important on w^r and w^p), and increasingly negative ΔF⁰ or −neη_a. The physical interpretation of this behavior is simple and related to the ease with which the energy restriction can be satisfied [1, 2]. “For example, ion–solvent interactions decrease with increasing ionic radius a, so that there would be a smaller energy difference between the two hypothetical charge distributions in the original atomic configuration of the reactants and therefore a smaller reorganization energy would be needed to equalize the two energies” [4, p. 158].

“Similarly, the smaller the charge transferred ne, or the smaller the distance between the reactants (or between the ion and its electrostatic image in the electrode), the less the medium can discriminate energywise” between initial and final charge distributions “and therefore the lesser will be the reorganization needed for energy equalization. A larger optical dielectric constant, D_op, serves to partially neutralize the electrostatic fields of the charges and therefore to reduce their energy difference for a given atomic configuration. The more D_s approaches D_op, the smaller the dipolar contribution to the electrical polarization of the medium and the less the necessary rearrangement of the atoms (at least for zero ΔF⁰ or zero η_a).” [4, p. 158]

From Eqs. (3.18) and (3.34), we see that the effective driving force of the reaction is not just given by the free energy difference (ΔF⁰ or −neη_a) between reactants and products when the reacting species are far apart but rather when they are in the positions they occupy in the activated complex. By lowering the free energy of the final state of the system relative to that of the initial state, these terms reduce the amount of reorganization of atomic configuration of the initial state necessary to satisfy the energy restriction [1].

We have seen that λ in Eqs. (3.17) and (3.18) has become λ/2 in Eqs. (3.33) and (3.34) for the electrode system. “This difference can be attributed to the absence of one half of the dielectric medium in the electrode case. (Effectively, the charge is transferred from the ion to its electrostatic image, and the metal surface bisects the line drawn between the two).” [4, p. 158]

3.21.Interpretation of the Data

A discussion of data follows on experiments reported in Refs. [4] and [6], from the point of view of the various factors influencing the reaction rate.

3.21.1. “Standard” Free Energy Change or Activation Overpotentials

The influence of this only factor alone on the rate constant is more easily investigated for the electrode processes. As a matter of fact, it is obviously much easier to vary the potential of the working electrode at which a substance reacts, than studying the redox reactions of the substance with a variety of reactants.

“When w^p and w^r are sufficiently small, it can be deduced from Eqs. (3.11), (3.33), and (3.34) that the transfer coefficient is 0.5, a value found in a number of simple electron transfer processes [48–50].” “The quantitative effect of ΔF⁰ on the homogeneous bimolecular rate constant has been measured experimentally for some nonspherical reactants [51, 52]. While ion–solvent interactions in these systems (oxidation of hydroquinone-like compounds) were not as simple as assumed in Eq. (3.17) and an atom transfer mechanism could not be ruled out, reasonable agreement with the experimental rate constant was found [2].” [4, p. 158]

The detailed theoretical description from Ref. [2] is reported in a following section [4, p. 159].

3.21.2. Ionic Structure and Ligand Field Effects

In order to study the effect of ion size on the rates of homogeneous reaction, it is most convenient to study isotopic exchange reactions because there the reactants merely exchange their charges, ΔF⁰ and w^r − w^p are both zero, and m of Eq. (3.85) is simply Wahl and coworkers have extensively investigated the factor of ion size [53, 54]. The rates of large ions, probably of Class I, were found, as expected, to be much larger than those of the smaller ions, and It is interesting that “the latter two systems had comparable rates in spite of the greater Coulombic repulsion in the second case, perhaps because of larger size of the iron cyanide ion (4.5 vs. 2.9Å). Rather good agreement was obtained between experimental and calculated rate constants using the theoretical equations, taking into consideration the absence of adjustable parameters [55].”

Marcus theory incorporates the influence of ligand field effects on the rate constants of oxidation–reduction reactions; they influence, in particular, k_i, and ΔF⁰. Accordingly, the discussions of ligand field effects on kinetic problems is then converted into a discussion of the problem of estimating k_i, and ΔF⁰ [6, p. 67].

“The cobalt–nitrogen bonded complexes probably are members of Class III [25, 56] and the rates are therefore extremely slow. The marked increase in rate [56, 57] in the sequence may be partly a reflection of increasing ion size, and partly a reflection of ligand field [58] effects.

The ferrous–ferric system is probably of class II, and its rate [59] is correspondingly much less than that of the or systems. Nevertheless, its rate greatly exceeds that [60] of a probable member of Class III [56]. Reasonable agreement between calculated and experimental between calculated and experimental results for the was found by adapting the theory to Class II reactants [3]. Thus far [1958. . .] there is no experimental evidence which definitely establishes either an electron or an atom transfer mechanism for these homogeneous isotopic exchange reactions of hydrated cations. The general parallelism of various rates in the two processes discussed in the following suggests, however, a common electron transfer mechanism.” [4, p. 159]

3.21.3. Parallelism between Rates in Solution and at the Electrodes

The theoretical equations suggest a close parallelism between the rates of the two processes, when similar mechanisms are operative, in particular when ΔF⁰ and η_a are zero and the work terms are small, a situation typical of isotopic exchange reactions. “The parallelism becomes very close, therefore, when the rates of isotopic exchange reactions are compared with the corresponding electrochemical exchange currents. The chemical and electrochemical electron transfer rates of the systems and [54, 56, 59, 61] both increase in the order given as do those of the systems and [55, 57, 62]. The absolute rates of various electrochemical processes [60] are in the general range expected from the values of the chemical rate constants, but further work is desirable, both theoretically on Class II and Class III systems as well as experimentally on salt effects and hydrolysis effects.” [4, p. 159]

3.21.3.1.Frequency Factor and Activation Energy

For simple isotopic exchange reactions: “ΔF^* equals w^r + λ/4. Correspondingly, ΔS^* is the sum of two terms. The first, −∂w^r/∂T , is the usual entropy change which results when two ions are brought together. The second, −(dλ/dT)/4, is the entropy of formation of the ‘nonequilibrium’ atomic configurations from the equilibrium ones at the same r in order to satisfy the energy restriction. Its value may be computed from Eq. (3.35). It proves to be very small for reactants of Class I.

Applications of the theoretical equations to Class I reactants by the writer” (M.) “led to good agreement between experimental and calculated frequency factors for the system and (within a factor of 50) for the system, the latter system having an extremely small frequency factor.”

The last frequency factor was measured by A. C. Wahl. “The data were extrapolated to infinite dilution, taking cognizance of the comparative insensitivity of the theoretical activation energy to salt effects.” [4, pp. 159–160]

Recalling Eq. (3.12) A = A′ exp(−ΔS^*/R) = κ Z exp (−ΔS^*/R) relating the excess entropy of activation to the frequency factor we see that the smaller the absolute value of ΔS^* < 0, the smaller A.

“These results suggest that the probability of adiabatic reaction may be of the order of unity, at least for these systems. The frequency factors of Class I reactants provide the most direct measure of this probability factor.

Many of the hydrated metal cations, which are mainly of Class II, tend to hydrolyze and form other complexes easily. In a careful investigation Silverman and Dodson [59] unraveled the rate constant of the Fe^+3,+2 system . . .The frequency factor was very small.”

NOTE that the term −∂wr/∂T of ΔS^* in Eq. (3.21) can be influenced by the ionic strength, for instance by the high acidity needed to minimize hydrolysis, because w^r is influenced by the ionic strength (see earlier).

“Baker, Basolo, and Neumann [56] have reported a most interesting result of a very high frequency factor, 5 × 10¹⁶ cc mole⁻¹ sec⁻¹ for a Class III system,⁽²⁾

“A similarly high frequency factor has been reported for another Class III system [63],

“The frequency factors of the electrochemical systems. . . Randles and Somerton [61] found them to be typically in the range 3 × 10² to 3 × 10⁴ cm sec⁻¹. The reactants were primarily hydrated metal cations and so were mostly of Class II. In one system, a rather low frequency factor (10cm sec⁻¹) was found. The electrostatic repulsion between ion and electrode was estimated to be large for this system, and the low A-value was attributed to a small transition probability factor because of the increased R.” [4, p. 160]

Marcus indicates now two other parameters which are influenced by R:

“However, we see from Eq. (3.35):

that there is also an image term present. This term would tend to favor small R’s. An alternative explanation for the A-value could be given on the basis of the usual sign of −∂w^r/∂T . When two reactants of like sign approach each other, the enhanced electrostatic field polarizes the solvent more strongly and causes a negative ΔS^*.”.

See above.

“Activation energies for several electrochemical electron transfers have been measured [61]. . .

The results tentatively indicate the activation energies of the electrode reactions to be more than one-half those of the corresponding exchange processes. Should this result prove to be generally true (at least for Class I and Class II reactants) one possible interpretation would be that the distance between the ion in the activated complex and its electrical image exceeds twice the ionic radius.”

We have seen that This is valid in the hypothesis R = 2a. If R > 2a because of the presence of a layer of molecules adjacent to the electrode which the ion cannot penetrate, then λ_el is greater than for R = 2a and recalling the formula for λ_el we can explain the earlier experimental result [4, pp. 159–161].

3.21.4. Salt Effects

“No detailed study of salt effects for simple electron transfers at electrodes appears to have been reported” until 1958. “Therefore it is not yet possible to adequately test the usual assumption (one not made here) that the work required to transport an ion from the body of the solution to the electrode, w^r or w^p, equals the ionic charge multiplied by the difference of potential at the initial and final positions of the ion. This assumption is clearly valid when the ionic charge is so small that it does not perturb the configuration of the remaining ions. It is also clearly invalid at the point of zero charge [17, 20]. At this point, the work term estimated on the basis of the above assumption is zero, whereas it actually equals the free energy of interaction of the ion with the image. For infinite dilution, the latter term is −q²/2D_s R, q being the ionic charge, while for dilute salt solutions, an approximate estimate of it is (−q²/2D_s R) exp(−κ R), κ being the usual Debye κ.” [4, p. 161]

NOTE that the interaction energy is half of what it would be between two real charges [27]. [4, p. 161]

3.21.5. Solvent Effects

“Solvent effects for simple electron transfers will occur, according to the theory, whenever there is a change in dielectric constant, refractive index, ionic radius, or standard free energy of reaction. Changes in composition of the coordination shell, which may include the solvent, naturally alter a, ΔF⁰, and the ease of intramolecular compression or stretching. In addition there are effects which were not incorporated in the theory, effects such as changes in ion pairing, selective solvation of solvent mixtures, and a change of the mechanism itself.”

“Heavy water has been used as a solvent in several studies [64, 65]. Hudis and Dodson [64] have demonstrated the importance, when hydrolyzed species and other complexes are present, of determining the various equilibrium constants so that meaningful electron transfer rate constants for the different species can be obtained and solvent effects evaluated. The rate constant for Fe^+3,+2 was smaller by a factor of two in D₂O [64], while that of was smaller by a somewhat smaller factor [65]. A qualitative interpretation of solvent effects can be given in terms of atom transfer [64, 65] or electron transfer [1] theories. Thus far, [1958], however, comparative studies in the two solvents do not permit either mechanism to be distinguished from the other. This situation results from the fact that ions have different solvation energies in the two media [66], so that water does not behave as an inert solvent.

The rate of the reaction appears to be faster in liquid ammonia [67] than in water, where it was immeasurably slow [57]. The activation energy was high, being in the range found for two other Class III compounds [56, 63]. Possible explanations for the solvent effect include a change of mechanism, such as a dissociation [67] or possibly a hydrolyzed intermediate, [4, pp. 161–162]

3.21.6. Effects of Electrode Material

“A dependence of rate constant on electrode material will occur if there is any change in surface contamination [61] and possible metal–solvent binding, or if electrode charge density at a given η_a changes sufficiently to alter w^r or w^p. The rate constants measured for one system with several solid electrodes underwent no great variation (within a factor of 10) but were much smaller than that found for a mercury electrode [61].” [4, p. 162]

3.21.7. The Image Force Law and Its Implications

“The role of the electrostatic image has been generally ignored in electrochemical theories. A recent investigation [68] of the quantum limitations of the image force law for a vacuum is reassuring. It has been applied by the writer (M.) to dielectric media and to electrochemical theory.

The effect of this image is to partially neutralize the field of the ion and to reduce, thereby, the configurational rearrangement free energy needed to satisfy the energy restriction. It is noteworthy that its calculated effect remains, even at salt concentrations sufficiently large as to make w^r and w^p negligible. This is because it is impossible for the ionic atmosphere in the activated complex to neutralize the ion–image interactions of the two different hypothetical charge distributions at the same time. A compromise configuration results. A similar behavior exists in the homogeneous case, where the 1/r term remains even if w^r and w^p are zero.

At the point of zero electric charge [17, 21], there is a net, shielded Coulombic attraction between the ion and its image, which is quite large for very dilute solutions. According to the theory a positive ΔS^* would result, since the attraction lowers the entropy of solvation.”

See above for the increase of entropy when two ions of opposite charge approach each other.

“A measurement of the frequency factor in this region would permit a determination of this ΔS^*”

“If this prediction is verified for Class I reactants, it will be interesting to compare ΔS^* with the theoretical estimates” [4, p. 162]

3.22.Comparison of Isotopic Exchange Rate and Corresponding Electrochemical Exchange Current

“A comparison of and k_el/10⁴ on the basis of the existing experimental data is given in Table I.” of Ref. [6], “All rate constants are pseudo-rate constants, their use being justified under the conditions cited [6]. The qualitative trend in both k_el and k_ex is seen to be the same, and the values. . .are relatively close to each other, considering the fact that approximations in the theory enter exponentially (a fairer comparison would be of and that stationary electrodes (with their absorption problems) were usually necessary, and that the work terms may not have been negligible.” [6, p. 854]

3.23.Comparison of Chemical and Electrochemical Oxidation–Reduction Rates of a Series of Related Reactants

“In this comparison we shall consider systems in which a constant reagent is used in the chemical system, and a constant electrode potential in the electrochemical one, to oxidize or reduce a series of related compounds. In a series of a given charge type, the work terms are either exactly or roughly constant in each of these two systems. Furthermore, if the ΔF^*’s are in the region where they would depend linearly on ΔF⁰, then according to Eqs. (3.1, 3.78, 3.83, 3.25), the ratio:

should be the same for each member of the series: in both cases, the terms λ₁, ΔF⁰, and, at a constant E, η(= E − E⁰), will normally vary from member to member. λ₂ refers to the constant reagent. However, since ΔF⁰ = −nF E⁰ + const in the series, one sees from Eqs. (3.78, 3.83, 3.25), that these variations in λ₁, ΔF⁰, and E⁰ cancel when one compares values of that is of k_soln/k_el. Vlcek [69] has recently observed (1961) that the electroreduction and the reduction [69] of and has essentially the same k_soln/k_el for both compounds [6]. This experimental result is in agreement with the earlier theoretical deductions. Presumably both E⁰ and λ₁ differed in the two compounds.”

“Similarly the ratio:

for each member of the series oxidized or reduced by two reagents, a and b, should be constant. This result was found experimentally for the Co(NH₃)X compounds reduced by and respectively, with X being NH₃, H₂O, and Cl⁻ [70] (Table II of Ref. [6]).

The restriction to a given charge type will not be important if the work terms are relatively minor.

The comparison involving V⁺² should be accepted with some reserve since the V(II) reaction is not necessarily an ‘outer sphere’ one, as Taube has pointed out.” [6, p. 855]

3.24.Early Comparisons of Theoretical and Experimental Results

(a) Excess Free Energy of Activation

The very first application of the theory consisted in comparisons of experimental and theoretical ΔF^*’s [1, p. 870].

The experimental are known very accurately because they depend only on the logarithm of k_bi and so a factor of 2 in k_bi introduces an error of only 0.4kcal mole⁻¹ in

is found measuring k_bi while Eq. (3.17); (or, more simply, Eq. 3.4) is used to compute

In most instances, M. found an encouraging agreement between experimental and calculated ΔF^*’s, considering that his theory is free from adjustable parameters. The agreement was in particular excellent for compounds of class I (“tightly knit covalently bound ions”), but was not as good, even if reasonable, for small hydrated cations [3, p. 426].

The reasons given to explain differences between experimental and calculated ΔF^*’s are the following:

(i)It is possible that the innermost solvation layer of the ions is not completely dielectrically saturated as assumed. In this case, there would be an additional contribution to ΔF^* arising from any changes which may have to occur in interatomic distances in the innermost solvation layer, that is, within the sphere of radius a, prior to the electronic jump, in order to form the activated state.

(ii)The electron tunneling factor may be somewhat less than unity. Note that this factor is not temperature dependent and therefore does not enter into any comparison between experimental and calculated values of the activation energy.

(iii)For reactions in which there is uncertainty of mechanism, say between two successive one-electron transfer and one two-electron transfer it is necessary to know the mechanism before calculating

From calculations applied to concrete ET cases one finds that the relative magnitude of the two contributions to ΔF^* in Eq. (3.4), the Coulombic repulsion and the solvent reorganization free energy are generally of the same order of magnitude. From this, it may be inferred that no simple correlation between the reaction rate and the size of the Coulombic term would be expected.

(b) Excess Entropy of Activation

From the carefully studied temperature dependence of the ferrous–ferric reaction, the experimental value of the excess entropy of activation was computed with the aid of Eq. (3.7). It was −23cal mole⁻¹ deg⁻¹ at 0^∘C. “The theoretical value is found from Eq. (3.15) to be −14cal mole⁻¹ deg⁻¹. Considering the experimental errors that always accompany a measurement of ΔS^* and considering the assumptions of the theory, the experimental and calculated values agree reasonably well.” [1, p. 871]

3.25.Reaction Rates in Heavy Water—A Possible Criterion of Mechanism

The way in which the reaction rate changes when D₂O is used as solvent instead of H₂O is rather subtle. Any changes of the interatomic O−H distances in the innermost hydration layer needed to form the activated state are easier for the O−H bonds than for the O−D bonds since the former have a higher zero-point energy and so they span a larger range of interatomic distances. Note that the influence of D₂O on ET can only happen through the differences in atomic polarization in the dielectrically saturated innermost solvation layer since the two solvents do not differ appreciably in their D_s or D_op and so the difference of rates in the two solvents cannot arise from the dielectrically unsaturated part of the medium.

M. proposed [1, p. 871] an experimental method to help distinguishing between a simple outer-sphere ET mechanism and an atom transfer mechanism in a series of reactions. If one measures the reaction rates for the reactions Fe⁺²–Fe⁺³, Fe⁺²–FeOH⁺² and Fe⁺²–FeCl⁺² in H₂O and in D₂O, one should find a comparable D₂O–H₂O isotope effect on the reaction rates if these reactions have a small-overlap electron transfer mechanism, that is, if there is but a small overlap of the electronic orbitals of the two reactants in the activated complex, a basic assumption of the theory. Since the Fe⁺²–Fe⁺³ and Fe⁺²–FeOH⁺² rate constants were twice as great in H₂O as they were in D₂O, this effect would predict an effect of similar magnitude for the Fe⁺²–FeCl⁺² reaction.

If a reaction has, on the other hand, an atom transfer mechanism, then a D₂O–H₂O effect would probably be expected only if the atom were hydrogen. Thus an isotope effect could occur for the Fe⁺²–Fe⁺³ and Fe⁺²–FeOH⁺² reactions, but not for the Fe⁺²–FeCl⁺² reaction if it involves a chlorine atom transfer. This expected absence of isotope effect in a chlorine atom transfer process can, of course, be tested by measuring the rate constant of the atom transfer Cr⁺²–CrCl⁺² reaction in the two solvents (vide supra).

3.26.Two Case Studies in Detail (Verbatim with Glosses) [2]

(a)OXIDATION OF HYDROQUINONES BY FERRIC IONS

Mechanism

“The overall reaction for the oxidation of a hydroquinone by ferric ions is given by Eq. (3.97)

where QH₂ and Q denote the hydroquinone and quinone, respectively. The rate of this reaction has been measured as a function of the concentrations of the various reactants and products. One mechanism consistent with the data involves the ionization of QH₂ to QH⁻, followed by an electron transfer between QH⁻ and Fe³⁺:

Where QH denotes the semiquinone. This step was in turn followed by the ionization of QH to Q⁻ and then by the latter’s oxidation to Q [1]. The four elementary steps of the mechanism are then:

On the right side of the reactions, I have written the symbol of the corresponding equilibrium constants. k₁ and k are symbols of the rate constants.

“We shall be concerned here with the experimental and theoretical value of k, the rate constant for the forward reaction in Eq. (3.99).”

As we know Marcus’ theory applies to the elementary redox step of an overall redox reaction.

“The experimental k can readily be computed from the known overall bimolecular rate constant k₁ and the known first ionization constant of QH₂, K₁.”

For convenience of the reader, I explicitly show how this is done following a note on p. 874 of Ref. [2].

The rate constant of the elementary redox step (3.82) can be written as:

It was also written [51] in terms of a pseudo-rate constant k₁:

Consider reaction (3.98) and the associated chemical equilibrium:

whence:

and

For the pseudo-constant k₁, we have:

and we see that it is inversely proportional to [H⁺].

The various quantities needed for the theoretical calculation of k are computed in the following sections.

3.27.Standard Free Energy Change of the Redox Step, F⁰

The standard free energy ΔF⁰ of reaction (3.99) has not been measured directly but can be estimated from K₁, and from K and K_S, the equilibrium constants of reactions (3.97) and (3.100), respectively,

It is readily verified that ΔF⁰ is given by

The values of K and K₁ have been determined experimentally [51, 70]. Numerous data on the formation constants of semiquinones K_S have also been obtained. When, as in reaction (3.100), all three compounds in this equilibrium are uncharged, or have the same charge, the free energy of this reaction is found to be practically independent of the chemical structure of these compounds (this can be inferred from a detailed analysis of data on the on the pH dependence of the first and second oxidation potential of many hydroquinone-like compounds). The standard free energy change of reaction (3.100), −RT ln K_S, is found to be about 2.3kcal mole⁻¹.

With these values of K , K₁, and K_S, the values of ΔF⁰ given in Table I were calculated from Eq. (3.101).

Table I. Kinetic and thermodynamic data for Fe⁺³ + QH⁻ reaction at 25^∘C^a

^aAll free energy units are in kcal mole⁻¹. ^bx denotes the 2,6-dichloro compounds and y denotes any other compound.

3.28.Effective Radii

“In the derivation of Eq. (3.4) for ΔF^*, each ion was treated as being surrounded by a sphere of radius a inside of which the dielectric medium, that is, the solvent, is saturated and outside of which it is unsaturated. Similar models have been used extensively in calculating the free energy of solvation of the ions,” see Refs. in [2].

As discussed in Chapter 2 “if, as it is usually assumed, the innermost hydration layer around monoatomic ions is largely saturated, it will not contribute to ΔF^*, and a then equals the sum of the crystallographic radius of the ion and of the diameter of a water molecule. Polyatomic ions such as and being rather large, would not be expected to cause dielectric saturation of the solvent as readily as the smaller monoatomic ions, since the electric field of the ion, which is responsible for the saturation, falls off roughly as the square of the distance from the center of the ion. Thus one would expect that a for the cited polyatomic ions would be approximately the actual crystallographic radii. This is also consistent with the assumption that only the first layer of the water molecules about a monoatomic ion is saturated, since a hydrated monoatomic ion has about the same radius as these polyatomic ions.

With a slightly different for an ion when it is a reactant and when it is a product it was suggested that a mean value for a be adopted.” “The crystallographic radii of Fe⁺², Fe⁺³, and H₂O are [2] 0.75, 0.60, and 0.72. The mean of the first two is 0.68Å.”

The refinements of the theory will take into account the effect of the radii variations on the reaction rate.

When organic ions are considered one has to consider two new features. “First, the ion is far from being spherical and, second, it is possible that that the effective radius a could be quite different when this particle is a charged reactant and when it is an uncharged product. This is discussed later, where it is inferred from entropy data that the effective polarizing radius of a hydroquinone-like ion such as HOC₆H₄O⁻ is about the same as that of the hydroxyl ion. It is further suggested that the dielectric saturation around this group when it is a charged particle be neglected as a first approximation, and that a equals the crystallographic radius of the oxygen group 1.4Å. This approximation can be removed by a refinement of the theory.”

3.29.Excess Free Energy of Activation

“Experimental values of ΔF^* were calculated from the rate constant k given in Table 1, using Eq. (3.1) and setting Z equal to 10¹³ liter mole⁻¹ sec⁻¹ in that equation. These values are reported in Table I.

Using the effective radii and the ΔF⁰’s deduced in the preceding section and setting D_op = 1.8 and D_s = 78.5 at 25^∘C, values of were obtained with the aid of Eqs. (3.2)–(3.4), and are given in Table I. These results will be discussed in detail later.”

An important note: M. later [71] used for Z the value 10¹¹ liter mole⁻¹ sec⁻¹ instead of 10¹³ liter mole⁻¹ sec⁻¹ of Ref. [72].

3.30.Excess Entropy of Activation

“Experimental values for the excess entropy of activation of reaction (3.99), were calculated from the data.”

It may be instructive to report, from note 26 in Ref. [2], how this calculation was done.

The pseudo-rate constant k₁ of Eq. (3.97) has an activation energy E₁, say, and a frequency factor A₁ so that k₁ = A₁ exp(−E₁/RT). Values of A₁ were determined experimentally. The rate constant k, Eq. (3.99), has a frequency factor A, say, which can be calculated from the known A₁ [51] and the known [70] entropy of ionization of QH₂, ΔS₁ say. It is found that A = A₁[H⁺] exp(− S₁/R). According to Eq. (3.7), is then found by setting A = Z exp(ΔS^*/R), Z being in Ref. [2] equal to 10¹³ liter mole⁻¹ sec⁻¹ [72]. But see the previous note on the correct value of Z.

“The values of were computed from Eq. (3.9) using the previously determined radii and ΔF⁰ and using the calculated ΔS⁰.” From note 27 in Ref. [2]: “ΔS⁰ = −∂ ΔF⁰/∂T and therefore according to Eq. (3.101), where ΔS and ΔS_S are the standard entropy changes in reactions (3.97) and (3.100), respectively [51, 70]. Since the sum of the translational, rotational, and vibrational entropies of the reactants and of the products of reaction should be about the same, it may be assumed that S_S is essentially zero.”

“In this way was found to be 46, 57, 47, and 44cal mole⁻¹ deg⁻¹, and to be 36, 38, 36, and 31 entropy units, for the 2,6-dichloro, benzo, tolu, and duro hydroquinones, respectively. The average of the former group of values is 49 and that of the latter is 35. Values of were also computed from the approximate equation, Eq. (3.10). They agreed well with the exact calculated values, within about one and a half entropy units.”

(b)AEROBIC OXIDATION OF THE LEUCOINDOPHENOLS

3.31.Excess Free Energy of Activation

“The overall reaction of the leucoindophenols with dissolved oxygen is represented by Eq. (3.102), where QH₂ denotes a leucoindophenol such as HOC₆H₆NHC₆H₆OH, and Q denotes the corresponding indophenol HOC₆H₄NC₆H₄O.

A mechanism consistent with the data [51] involved the ionization of QH₂ to QH⁻, which then transferred an electron to O₂:

This step was followed by reactions of QH and As in the preceding reaction series discussed in Ref. [2], the rate constant k for the forward reaction in Eq. (3.103) can be inferred from the overall bimolecular rate constant k₁ and the first ionization constant K₁ of QH₂ (k = k₁[H⁺]/K₁). The k’s and the values of calculated from them using Eq. (3.1) are given in Table II.

It can be shown that the ΔF⁰ of reaction (3.103), needed for the estimation of is given by Eq. (3.104):

Table II. Kinetic and thermodynamic data for O₂ + QH⁻ reaction at 30^∘C^a

^aAll free energy units are in kcal mole⁻¹. ^bThere is an appreciable uncertainty in and therefore in ΔF⁰, of an amount α, where α may be about 0 to +5 kcalmole⁻¹, according to Latimer [73]. This uncertainty introduces a corresponding uncertainty in the which may be about 3 kcal mole⁻¹ less than those reported in this table. ^cx denotes the unsubstituted compound and y denotes the substituted compound.

where is the standard oxidation potential of QH₂ (QH₂ = Q + 2H⁺ + 2e), is that of

Note that M. is using here Latimer’s standard oxidation potentials whose values are opposite of IUPAC’s standard reduction potentials [73].

“F is the Faraday, and K_S is the equilibrium constant of reaction (3.100) for the formation of semiquinones or indophenols. The ΔF⁰’s of Table II were estimated using the known [74] the estimated [73] the value of −RT ln K_S discussed in the previous section (2.3kcal mole⁻¹) and the known [47] K₁.”

Introducing into Eq. (3.4), these ΔF⁰’s and the effective radii later deduced, the values of Table II were computed.

3.32.Excess Entropy of Activation

In this section, M. shows how it is possible to estimate unknown entropy values.

“All the entropy data needed for the estimation of and the experimental and the calculated excess entropy of activation of the electron transfer step, Eq. (3.103), are not available. However, some estimate of the undetermined entropy values may be made.

For example, just as in the case of the ferric-hydroquinone reaction, ΔS^*_expt can be calculated from the frequency factor of the pseudo-rate constant and the entropy of ionization of QH₂, S₁.

Assuming that S₁ equals the value for the hydroquinones and water (which are later shown to be equal) one finds in this way that cal mole⁻¹ deg⁻¹.

The term is seen from the approximate equation, Eq. (3.10), to be about S⁰(1 + ΔF⁰/λ)/2 since e₁e₂ and are each equal to zero in reaction (3.103). In reaction (3.103), it is also expected that the translational, vibrational, and rotational entropies of the reactants are each about equal to those of the products.” Because of the very similar chemical structure of reagents and products.

“Accordingly, if the entropy change S⁰ of reaction (3.103) were appreciable, the main contributions to it would arise from possible differences in the entropy of solvation of QH⁻ and O⁻₂. This would not be expected to be large so that S⁰/2 should be relatively small. Further, ΔF⁰/λ is calculated to be about 0.3, so we conclude that is small, in agreement with the estimated small value of

3.33.Discussion

“A comparison of Tables I and II shows that the rate constants of the redox step in the two series of reactions considered here differ from each other by a factor 10⁹ on the average. This difference stems from the considerable difference in the standard free energy change ΔF⁰ of the redox step in the two cases, one being about—13kcal mole⁻¹, the other perhaps lying between +17 and + 12kcal mole⁻¹, depending on the correct value of This difference in turn is related principally to the differences between standard oxidation potentials of the ferrous ion and of the oxygen molecule ion.

According to the theory developed in Part I, the standard free energy change affects the reaction rate in the following way. During an ET step, there is first a reorganization of the solvent molecules about the reacting ions prior to the jump of an electron from one reactant to the other. Now, for a given S⁰, the more positive ΔF⁰ is, the greater would be the energy of the state of the system just after the jump. Therefore, the energy of the system just before the jump, which is equal to this, would also have to be greater. This simply means that there is a greater energy barrier to forming from the isolated reactants a suitable collision complex in which the electron can jump. Conversely, the more negative ΔF⁰, the less the energy barrier.”

This is pictorially described in Fig. 5^* (c) and (d) of Chapter 1 in terms of PESs.

“In spite of the extremely favorable value of ΔF⁰, the redox step of the ferric ion–hydroquinone ion reaction is seen from Table I not to proceed at every collision.”

As a matter of fact the most rapid reactions in solution are those encounter controlled or microscopic diffusion controlled such as the recombination reaction of H⁺ and OH⁻ in water with a rate constant of 1.4 × 10¹¹ liter mole⁻¹ sec⁻¹ at 25^∘C [75] while the k for the Fe⁺³ — benzo-hydroquinone anion is only 3.3 × 10⁹ liter mole⁻¹ sec⁻¹.

“According to the theory, this is because of the preliminary solvent reorganization prior to the electron jump. However, it is of interest that the rate constant of the redox step in the ferric ion–hydroquinone reaction is much larger, on the average, than those of the isotopic exchange electron transfer reactions having zero standard free energy change considered in the preceding paper [1]. The major reasons for this difference lie in:

(i)The large negative value of ΔF⁰ in the former reaction as compared with the zero value of the latter.

(ii)The Coulombic attraction of the Fe⁺³–QH⁻ reactants, as compared with the Coulombic repulsion of the reactants in those isotopic exchange reactions, and moreover:

“While the general agreement between the calculated and experimental results is satisfactory, the type of agreement obtained in Table II for the absolute value of ΔF^* in the reaction of ferric and hydroquinone ions is partly fortuitous. Two compensating approximations were employed: the a value chosen for the iron ion assumed complete dielectric saturation in the innermost hydration layer of this ion and as in [1] tends to make too small.”

The reason for this statement is that if the dielectric saturation in the innermost hydration layer is not complete, a reorganization energy contribution of this layer to should also be considered.

“The a value for the oxygen group correctly assumed no dielectric saturation around the uncharged reactant but made the same assumption when it was charged. This tends to make too large.”

Because the innermost hydration layer of the ion is supposed completely unsaturated and so it contributes to

“In the oxygen-leucoindophenol reaction, only the second of these approximations was involved and therefore there is no compensation. This may be the reason why the absolute value of ΔF^* is somewhat larger than in Table II.

In Table I, it is observed that the rate constant for the electron transfer step of the durohydroquinone is very high, namely 2.7×10¹¹ liter mole⁻¹ sec⁻¹. This value is about the maximum value that a rate constant can have in solution. The maximum corresponds to the situation in which the probability of reaction per collision is so high that the slow process in the reaction becomes the diffusion of the reactants toward each other” (vide supra). “Using a formula of Debye [76], we estimate that the rate constant for a diffusion-controlled reaction between two ions of charges +3 and −1 is about 5 × 10¹⁰ liter mole⁻¹ sec⁻¹. Within the error of the various determinations, this is about equal to the rate constant k of the electron transfer step for the durohydroquinone reaction.”

3.34. Ionic Radii

“The negative charge on a hydroquinone ion such as HOC₆H₄O⁻ or HOC₆H₄NHC₆H₄O⁻ is largely on the oxygen. Thus it is this atom which polarizes the dielectric. Accordingly, the appropriate polarizing radius a to be used for this charged center may be the same as that for another negatively charged oxygen, such as the OH⁻ ion. It is true that the organic residue will prevent the close approach of some of the solvent molecules and hence reduce their polarization. On the other hand, this residue is itself polarized by the charged oxygen, atomic polarization being induced, although it is less strongly polarized than the solvent. In this way, the organic residue and the solvent play analogous roles.

A similarity between the hydroquinone ion and the hydroxyl ion in their extent of solvation and therefore in their effective polarizing radius a, can be inferred from the standard entropy change of reaction:

In such a reaction, the translational and rotational entropies of each of the two products would be expected to be about the same as those of the corresponding two reactants. Moreover, the sum of the vibrational entropies of the products should be about equal to the sum of those of the reactants.

If there is an appreciable entropy change in the reaction, it would be expected to arise from differences in the ability of the OH⁻ and the HQ⁻ ions to polarize the solvent molecules and therefore to vary in their entropy of solvation”

There is of course a close relation between the ability of solvent polarization by an ion and its entropy of solvation.

“Now the standard entropy change of reaction (3.105) is readily shown to equal the difference in entropy of ionization of water and of QH₂. The entropy of ionization of QH₂ for the various hydroquinones in Table I is [70], in the respective order in which they occur in the table, −26, −32, −29, and −25 entropy units, with an average value of −28. The entropy of ionization of water is [77] −26.7eu. Accordingly, the entropy change of reaction (3.105) is seen to be zero within the experimental error. This therefore provides some basis for assuming that QH⁻ and OH⁻ have about the same polarizing radius a.

Another question discussed in the text concerned the relative abilities of QH⁻ and OH⁻ to dielectrically saturate the neighboring water molecules. The QH molecule, being uncharged, cannot saturate the dielectric. The former, being charged, could. Thus a would apparently be different for QH⁻ and QH, contrary to an assumption made in the derivation of Eq. (3.4) for ΔF^*. However, it is possible that the dielectric saturation by the anion is relatively small. For example, studies [78, 79] of the dielectric constant of aqueous salt solutions indicate less dielectric saturation in the vicinity of anions, as compared with cations. It is of interest, too, though not necessarily significant, that the ‘experimental’ free energy of solvation of anions agree reasonably well [80, 81] with those calculated from the Born formula [82]. The Born formula assumes that the dielectric is unsaturated.

In the interest of simplicity, we shall assume that as a first approximation there is little dielectric saturation in the vicinity of the QH⁻ ion. Accordingly it follows that a will be the crystallographic radius of the oxygen group, which is about 1.4Å.”

M. will later remove the assumption of absence of dielectric saturation.

“The may be regarded as an ellipsoid of revolution whose [83] semimajor axis was found crystallographically to be 2.02 and whose semiminor axis is 1.51Å. As a first approximation, this molecule will be treated as a sphere of radius 1.7Å which has the same volume as the ellipsoid. This ‘crystallographic’ radius will be taken as the value of a for for the same reason given previously for the choice of the crystallographic radius as the a value of the oxygen atom in the hydroquinone ion” [2, p. 877].

NOTES

1.M: “κ is the probability of electron tunneling from one electronic configuration to the other, the rate constant is then proportional to κ. If one wants to phrase that in terms of the Eyring theory, since κ is not an energy dependent factor, that would contribute to the energy independent part of the free energy of activation. The energy independent part is the entropy of activation.”

2.M: “For a typical bimolecular frequency, where we don’t have to worry about charges, the number would be of the order of 10¹⁴, so this is higher than that by a factor of 500, it is a strange because the charges would orient more, would reduce below the entropy, making the number below 10¹⁴. Now, of course, you have all sorts of ion pairing effects. . .”

References

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

2.R. A. Marcus, J. Chem. Phys. 26, 872, (1957).

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

4.R. A. Marcus, Can. J. Chem. 37, 155, (1959).

5.R. A. Marcus, “A Theory of Electron Transfer Processes at Electrodes,” in Transactions of the Symposium on Electrode Processes, p. 239, E. Yeager, Editor, Wiley, New York (1961).

6.R. A. Marcus, J. Phys. Chem. 67, 853, 2889, (1963).

7.R. A. Marcus, “Electron Transfer at Electrodes,” in Encyclopaedia of Electrochemistry, p. 529, C. A. Hampel, Editor, Reinhold, New York (1964).

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