8 Phenomenological treatment of electron-transfer reactions

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1 8 Phenomenological treatment of electron-transfer reactions 8.1 Outer-sphere electron-transfer Electron-transfer reactions are the simplest class of electrochemical reactions. They play a special role in that every electrochemical reaction involves at least one electron-transfer step. This is even true if the current across the electrochemical interface is carried by ions since, depending on the direction of the current, the ions must either be generated or discharged by an exchange of electrons with the surroundings. In general electron-transfer reactions can be quite complicated, involving breaking or forming of chemical bonds, adsorption of at least one of the redox partners, or the presence of certain catalysts. So far our understanding is limited to the simplest possible case, so-called outer-sphere electron-transfer reactions, in which from a chemist s point of view nothing happens but the exchange of one electron as we shall see later, the simultaneous transfer of two or more electrons is highly unlikely. In the course of such a reaction, no bonds are broken or formed, the reactants are not specifically adsorbed, and catalysts play no role. If one of these conditions is not fulfilled, the reaction is said to proceed via an inner-sphere pathway. Unfortunately, there are not many examples for outer sphere reactions; here are two: [RuNH 3 ) 6 ] 2+ [RuNH 3 ) 6 ] 3+ +e [FeH 2 O) 6 ] 2+ [FeH 2 O) 6 ] 3+ +e 8.1) In aqueous solutions these reactions seem to proceed via an outer-sphere mechanism on most metals. Typically such reactions involve metal ions surrounded by inert ligands, which prevent adsorption. Note that the last example reacts via an outer-sphere pathway only if trace impurities of halide ions are carefully removed from the solution; otherwise it is catalyzed by these ions.

2 72 8 Phenomenological treatment of electron-transfer reactions 8.2 The Butler-Volmer equation In this chapter we treat electron-transfer reactions from a macroscopic point of view using concepts familiar from chemical kinetics. The overall rate v of an electrochemical reaction is the difference between the rates of oxidation the anodic reaction) and reduction the cathodic reaction); it is customary to denote the anodic reaction, and the current associated with it, as positive: v = k ox c s red k red c s ox 8.2) where c s red,cs ox denote the surface concentrations of the reduced and oxidized species, and k ox and k red are the rate constants. Using absolute rate theory, the latter can be written in the form: ) k ox = A exp G oxφ) k red = A exp G red φ) ) 8.3) The phenomenological treatment assumes that the Gibbs energies of activation G ox and G red depend on the electrode potential φ, but that the preexponential factor A does not. We expand the energy of activation about the standard equilibrium potential φ 00 of the redox reaction; keeping terms up to first order, we obtain for the anodic reaction: G oxφ) = G oxφ 00 ) αf φ φ 00 ), 8.4) with α = 1 G ox F φ The quantity α is the anodic transfer coefficient; the factor 1/F was introduced, because Fφ is the electrostatic contribution to the molar Gibbs energy, and the sign was chosen such that α is positive obviously an increase in the electrode potential makes the anodic reaction go faster, and decreases the corresponding energy of activation. Note that α is dimensionless. For the cathodic reaction: φ00 G red φ) = G red φ 00)+βF φ φ 00 ), 8.5) with β = 1 G red F φ where the cathodic transfer coefficient β is also positive. One would expect that higher terms in the expansion of the Gibbs energy of activation will become important at potentials far from the standard equilibrium potential φ 00 ; we will return to this point in the next chapter. The Gibbs energies of activation are related by: φ00

3 8.2 The Butler-Volmer equation 73 potential energy αfφ φ 00 ) Fφ φ 00 ) reaction coordinate Fig Potential energy curves for an outer-sphere reaction; the upper curve is for the standard equilibrium potential φ 00; the lower curve for φ > φ 00.Potential energy curves for an outer-sphere reaction; the upper curve is for the standard equilibrium potential φ 00; the lower curve for φ > φ 00. G oxφ) G red φ) =G ox G red 8.6) to the molar Gibbs energies G ox and G red of the oxidized and reduced state; in particular: G oxφ 00 )= G red φ 00) = G ) When the electrode potential is changed from φ 00 to a value φ, the Gibbs energy of the electrons on the electrode is lowered by an amount F φ φ 00 ), and so is the energy of the oxidized state. If the reactants are so far from the metal surface that their electrostatic potentials are unchanged when the electrode potential is varied, then the Gibbs energy of the reaction is also changed by F φ φ 00 ). This condition is generally fulfilled for outersphere reactions in the presence of a high concentration of an inert electrolyte which screens the electrode potential; it is not fulfilled when the reactants are adsorbed as in inner-sphere reactions. When it is fulfilled we have: G oxφ) G red φ) = F φ φ 00) 8.8) By differentiation we obtain for the sume of the two transfer coefficients the relation: α + β = 1 8.9) Since both coefficients are positive, they lie between zero and one; we can generally expect a value near 1/2 unless the reaction is strongly unsymmetrical. The transfer coefficients have a simple geometrical interpretation. In a one-dimensional picture we can plot the potential energy of the system as

4 74 8 Phenomenological treatment of electron-transfer reactions a function of a generalized reaction coordinate see Fig. 5.1). The reduced and the oxidized states are separated by an energy barrier. Changing the electrode potential by an amount φ φ 00 ) changes the molar Gibbs energy of the oxidized state by F φ φ 00 ); the Gibbs energy of the transition state located at the maximum will generally change by a fraction αf φ φ 00 ), where 0 < α < 1. The relation α + β = 1 is easily derived from this picture. The current density j associated with the reaction is simply j = Fv. Combining Eqs. 5.2)-5.4) and 5.9) gives the Butler-Volmer equation [25, 26] in the form: where Using the Nernst equation: j = Fk 0 c s red exp αf φ φ 00) Fk 0 c s ox exp 1 α)f φ φ ) 00) ) k 0 = A exp G φ 00 ) 8.10) 8.11) φ 0 = φ 00 + F ln cs ox c s red 8.12) for the equilibrium potential φ 0, and introducing the overpotential η = φ φ 0, which is the deviation from the equilibrium potential, we rewrite the Butler- Volmer equation in the form: [ j = j 0 exp αf η exp where )] 1 α)fη 8.13) j 0 = Fk 0 c s red) 1 α) c s ox) α 8.14) is the exchange current density. At the equilibrium potential the anodic and cathodic current both have the magnitude j 0 but opposite sign, thus cancelling each other. The exchange current density for unit surface concentration of the reactants is the standard exchange current density j 00 = Fk 0, which is a measure of the reaction rate at the standard equilibrium potential. According to the Butler-Volmer law, the rates of simple electron-transfer reactions follow a particularly simple law. Both the anodic and the cathodic current densities depend exponentially on the overpotential η see Fig. 5.2). For large absolute values of η, one of the two partial currents dominates, and a plot of ln j or of log 10 j versus η, a so-called Tafel plot [27] see Fig. 5.3), yields a straight line in this region. From its slope and intercept the transfer coefficient and the exchange current density can be obtained. These two quantities completely determine the current-potential curve.

5 8.2 The Butler-Volmer equation α=0.6 j / j α=0.4 α= η / V Fig Current-potential curves according to the Butler-Volmer equation. For small overpotentials, in the range Fη, the Butler-Volmer equation can be linearized by expanding the exponentials: j = j 0 Fη 8.15) The quantity η/j = /j 0 F is called the charge-transfer resistance. Note that the transfer coefficient does not appear in the current-voltage relation for small overpotentials, and hence cannot be determined from measurements at small deviations from equilibrium, they give the exchange current density only. However, α can be obtained by varying the surface concentrations, measuring the exchange current density, and using Eq. 5.14). We will discuss a few examples of outer-sphere electron-transfer reactions in Chapter 11. We conclude these phenomenological considerations with a few remarks: 1. The transfer coefficient is equivalent to the Broenstedt coefficient well known from ordinary chemical kinetics. Both describe the change in the energy of activation with the Gibbs energy of the reaction. 2. The transfer coefficient α has a dual role: 1) It determines the dependence of the current on the electrode potential. 2) It gives the variation of the Gibbs energy of activation with potential, and hence affects the temperature dependence of the current. If an experimental value for α is obtained from current-potential curves, its value should be independent of temperature. A small temperature dependence may arise from quantum effects not treated here), but a strong dependence is not compatible with an outer-sphere mechanism. 3. For small overpotentials the linear approximations of Eqs. 8.5) and 8.6) should be sufficient, but at high overpotentials higher-order terms are expected to contribute.

6 76 8 Phenomenological treatment of electron-transfer reactions 2 α=0.6 lnj / j 0 ) 1 0 α=0.4 α= η / V Fig Tafel plot for the anodic current density of an outer-sphere reaction. 4. The transfer coefficient determines the symmetry or lack thereof of the current-potential curves; they are symmetric for α = 1/2. For this reason the transfer coefficient is also known as the symmetry factor. 5. The surface concentrations are generally not known, and may vary with time as the reaction proceeds. One way to circumvent this problem is to work under conditions of controlled convection, so that the surface concentrations can be calculated from the bulk concentrations. Another technique consists in the use of potential or current pulses, which allows an extrapolation back to the time of the onset of the pulse when surface and bulk concentrations are equal. These techniques will be discussed in detail in Chapters 18 and Inner-sphere electron-transfer reactions are not expected to obey the Butler-Volmer equation. In these reactions the breaking or formation of a bond, or an adsorption step, may be rate determining. When the reactant is adsorbed on the metal surface, the electrostatic potential that it experiences must change appreciably when the electrode potential is varied. 8.3 Double-layer corrections When the concentration of the inert electrolyte is low, the electrostatic potential at the reaction site differs from that in the bulk and changes with the applied potential. This results in two effects [28]: 1. The surface concentrations c s ox and c s red differ from those in the bulk even if the surface region and the bulk are in equilibrium. Using the same

7 8.4 A note on inner-sphere reactions 77 arguments as in the Gouy-Chapman theory, the surface concentration c s of a species with charge number z is: c s = c 0 exp ze ) 0φ ) kt where c 0 is the bulk concentration, φ 2 the potential at the reaction site, and the potential in the bulk of the solution has been set to zero. 2. On application of an overpotential η, the Gibbs energy of the electrontransfer step changes by e 0 [η φ 2 η)], where φ 2 η) is the corresponding change in the potential φ 2 at the reaction site. Consequently, η must be replaced by [η φ 2 η)] in the Butler-Volmer equation 5.13). These modifications are known as the Frumkin double-layer corrections. They are useful when the electrolyte concentration is sufficiently low, so that φ 2 can be calculated from Gouy-Chapman theory, and the uncertainty in the position of the reaction site is unimportant. Whenever possible, kinetic investigations should be carried out with a high concentration of supporting electrolyte, so that double-layer corrections can be avoided. 8.4 A note on inner-sphere reactions There is no general law for the current-potential characteristics of inner-sphere reactions. Depending on the system under consideration, various reaction steps can determine the overall rate: adsorption of the reacting species, an electron-transfer step, a preceding chemical reaction, coadsorption of a catalyst. If the rate-determining step is an outer-sphere reaction, the current will obey the Butler-Volmer equation. A similar equation may hold if an innersphere electron transfer, for example, from an adsorbed species to the metal, determines the rate. In this case, application of an overpotential η changes the Gibbs energy of this step only by a fraction of Fη; furthermore, the concentration of the adsorbed species will change with η. These effects may result in phenomenological equations of the form: k ox = k 0 exp αf η, k red = k 0 exp βf η ) 8.17) with apparent transfer coefficients α and β, but α and β may depend on temperature. If the rate-determining step is the adsorption of an ion, the reaction obeys the laws for ion-transfer reactions see Chapter 12), and again a Butler- Volmer-type law will hold. Problems 1. Derive Eq. 5.13) from Eqs. 5.10) and 5.12).

8 78 8 Phenomenological treatment of electron-transfer reactions 2. The reduced species of an outer-sphere electron-transfer reaction is generated by a chemical reaction of the form: A red Denote the forward and backward rate constants of this reaction by k a and k b. When the reaction proceeds under stationary conditions, the rates of the chemical and of the electron-transfer reaction are equal. Derive the currentpotential relationship for this case. Assume that the concentrations of A and of the oxidized species are constant. 3. The Gibbs energy of activation in Eq. 5.4) can be split into an enthalpy and an entropy term: G ox = H ox T S ox. Define two transfer coefficients α H = 1 F H ox φ φ 00), αs = 1 F S ox φ φ 00) and derive the corresponding current-potential relations. Note: For outersphere electron-transfer reactions α S seems to be negligible; it has, however, been used to explain a temperature dependence of the apparent transfer coefficients in some inner-sphere reactions.

surface c, c. Concentrations in bulk s b s b red red ox red

surface c, c. Concentrations in bulk s b s b red red ox red CHEM465/865, 26-3, Lecture 16, Oct. 13, 26 compact layer S c ox,red b c ox,red Note, that we explicitly distinguish concentrations at surface bulk b red c, c from those in s red b ox s ox c, c. Concentrations