10 Theoretical considerations of electron-transfer reactions

Transcription

1 10 Theoretical considerations of electron-transfer reactions 10.1 Qualitative aspects Chemical and electrochemical reactions in condensed phases are generally quite complex processes, but outer-sphere electron-transfer reactions are sufficiently simple that we have reached a fair understanding of them in terms of microscopic concepts. In this chapter we give a simple derivation of a semiclassical theory of outer-sphere electron-transfer reactions, which was first systematically developed by Marcus [40] and Hush [41] in a series of papers. Several of the concepts that we develop here play also a role in electrocatalysis. We begin with qualitative considerations. During the course of an outersphere electron-transfer reaction, the reactants get very close, up to a few Ångstroms, to the electrode surface. Electrons can tunnel over such a short distance, and the reaction would be very fast if nothing happened but the transfer of an electron. In fact, outer-sphere reactions are fast, but they have a measurable rate, and an energy of activation of typically ev, since electron transfer is accompanied by reorganization processes of atoms and molecules that require thermal activation. While the reacting complex often has the same or similar structure in the oxidized and reduced form, metalligand bonds are typically shorter in the complex with the higher charge, which is also more strongly solvated. So the reaction is accompanied by a reorganization of both the complex, or inner sphere, and the solvation sheath, or outer sphere (see Fig. 10.1). These processes require an energy of activation and slow the reaction down. A natural question is: In which temporal order do the reorganization processes and the proper electron transfer take place? The answer is given by the Frank-Condon principle, which in this context states: First the heavy particles of the inner and outer sphere must assume a suitable intermediate configuration, then the electron is exchanged isoenergetically, and finally the system relaxes to its new equilibrium configuration. A simple illustration is given in Fig. 10.2, where we have drawn potential energy surfaces for the reduced

2 98 10 Theoretical considerations of electron-transfer reactions metal solution + + Fe Fe 3+ Fig Reorganization of inner and outer sphere during an electron-transfer reaction. energy red ox + e q 2 saddle point of reaction hypersurface q 1 Fig Schematic diagram of the potential energy surfaces for the reduced and the oxidized state. and the oxidized state as a function of two generalized reaction coordinates representing the positions of particles in the inner and outer sphere. During the course of an oxidation reaction, the system first moves along the surface for the reduced state till it reaches a crossing point with the surface for the oxidized state; at this configuration the electron can be transferred, and then the system moves to its new equilibrium position. Generally, the reaction will proceed via the saddle point of the intersection, since such transitions require the smallest energy of activation. The same diagram can also be used to illustrate the concept of adiabaticity: If the electron transfer takes place every time that the system is on the intersection surface, we speak of an adiabatic,

3 10.2 Harmonic oscillator with linear coupling 99 otherwise of a nonadiabatic reaction. Of course, Fig is highly simplified: In reality, we must plot the potential energy as a function of the positions of all the heavy particles involved, so that we obtain multidimensional potential energy surfaces. Fortunately, for most purposes the dimensionality does not matter, and a one-dimensional model, which we will present below, suffices Harmonic oscillator with linear coupling In an electron transfer reaction, we distinguish two different electronic states, an initial i and a final state f, which interact differently with the surroundings. The simplest model for a mode which interacts with the electron transfer is a harmonic oscillator. This amounts to expanding the energy of that mode about its equilibrium position and keeping terms up to second order. Thus, we may write for the energy of one such mode in the initial state: E i = 1 2 mω2 x 2 or E i = 1 2 αx2 (10.1) The former notation is natural for a real vibration, the latter for a more general case like a solvent mode. To first order, the interaction with the transferring electron is linear, i.e. proportional to x, and its strength is determined by a coupling constant g. It is convenient to define g in such a way that we may write for the final state: E f = 1 2 αx2 + αgx (10.2) This can be rewritten as: E f = 1 2 α(x + g)2 λ with λ = 1 2 αg2 (10.3) Thus, the origin of the harmonic oscillator has been shifted to x = g, and its energy been lowered by an amount λ. Later we shall identify λ with the contribution of this mode to the energy of reorganization of the electron transfer reaction. It is convenient to simplify the notation by introducing a normalized coordinate q = x/g. Thisresultsin: E i = λq 2 E f = λq 2 +2λq = λ q 2 +2q (10.4) In general, there is also a change in electronic energy E e between final and initial state. If only a single mode is reorganized, the total change in energy between the equilibria of the final and initial states is: E = G = E e λ (10.5) This change in energy may be identified with the change in the Gibbs free energy, because in this model there is no change in the entropy. If several modes couple, λ is the sum over all the contributions to the energy of reorganization. In the electrochemical case, we may write: G = e 0 η,whereη is the overpotential. With these preliminaries, we are ready to consider electron transfer in a systematic manner.

4 Theoretical considerations of electron-transfer reactions Fig Density of states of a reactant undergoing reaction 10.6 in the initial, the activated, and the final state, which, at equilibrium, in turn correspond to values of q =0, q = 1/2, and q = 1 of the solvent coordinate Adiabatic electron transfer Outer sphere electron reactions at bare metal electrodes are usually adiabatic. In this case the electron exchange between the reactant and the metal is faster than the motion of the inner or outer sphere modes, and the system is always in electronic equilibrium. We present a simplified version of the theory proposed by one of us [42]. To be specific, we consider an electron transfer of the type: A B + + e (10.6) Before the electron transfer, the valence level of A is filled and lies below the Fermi level. As discussed in chapter 6 for an adsorbed species, the interaction of the reactant with the metal broadens the valence level. It is therefore described by a density of states (DOS) of a certain width ; however,since the interaction of a non-adsorbed species is quite weak, the width is small, typically of the order of 10 3 ev. In the final state, this valence level is empty and thus lies above the Fermi level (see Fig. 10.3). The ion B + interacts with the solvent, and the concomitant solvation energy makes it stable; without solvation, taking an energy from the reactant would only cost energy and would not result in a stable reactant. Electron transfer proceeds in the following way: A thermal fluctuation of the solvent lifts the valence level of A above the Fermi level, so that it is emptied. Then solvation sets in and lifts the valence level to its final state. The energy of activation is the energy required to lift the valence level from its initial state to the Fermi level.

5 10.3 Adiabatic electron transfer 101 For quantitative purposes we need expressions for the electronic energy and the solvation. For the former, we introduce the occupation n of the valence orbital, which is unity when the orbital lies below the Fermi level, and zero when it lies above. Neglecting the small finite width for the moment, we write the electronic energy simply as a n,where a is the energy of the valence level in the absence of the solvent. For a full description of the reorganization of the outer and inner sphere modes, we would really have to consider the multidimensional potential energy surfaces sketched in Fig Thus, we would have two different surfaces for the initial and the final states, each with a different minimum corresponding to the equilibrium respective configuration. Fortunately, as long as only classical modes are reorganized, a one-dimensional model suffices, and gives the same results as a multi-dimensional one; the proof for this is relegated to Section So, we consider a single reaction, or solvent, coordinate q, which in terms of Fig passes from the minimum of the initial surface via the saddle point to the minimum of the second surface. We can normalize this coordinate such that the resulting expression is as simple as possible. As a consequence, the calculated energy takes on the meaning of a free energy, since the other degrees of freedom that are not considered, have effectively been averaged out. To simplify matters further, we follow the original papers of Marcus and Hush and use the harmonic approximation. Thus, in the initial state the energy of solvent 1 is represented by a simple parabola. As shown in the previous section, we can conveniently write this as λq 2,whereλ is called the energy of reorganization of the reaction. When the reactant is ionized, it interacts with the solvent. To first order, this interaction is proportional to the charge on the ion, and will depend of the state of the solvent given by q; it can be written in the form 2λq(1 n). That this expression has this simple form, is a consequence of the convenient normalization. In terms of the notation of the previous section, the term (1 n) simply switches between initial and final state. Thus, the energy of the solvent is: E sol = λq 2 +2λq(1 n) (10.7) In the initial state, when n = 1, the energy is a parabola with minimum at q = 0, and in the final state, when n = 0, a parabola with minimum at q = 1. Also, identifying (1 n) with the charge number, eq. (10.7) suggest an intuitive interpretation of the solvent coordinate: When the state of the solvent is characterized by q, it would be in equilibrium with a reactant of charge number q. Adding the electronic and the solvent terms gives the total energy as a function of the solvent coordinate q: E(q) =λq 2 +2λq +( a 2λq)n + 2π ln ( a 2λq) a + 2 (10.8) 1 really this includes the inner sphere as well, but for brevity we shall subsume it under the general term solvent

6 Theoretical considerations of electron-transfer reactions 0.4! = 0! = 0.5 ev! = 1.1 ev energy / ev initial state 0.8 final state solvent coordinate q Fig Free energy curves for three different overpotentials. The energy of reorganization was taken as λ =1.0 ev. The last term is a small correction caused by the finite width of the density of states. It can be neglected in most circumstances, but is qualitatively important at high overpotentials. The energy in the initial state, with q =0,n=1 is simply a ; in the final state, with q = 1, n =0itis λ. Changing the electrode potential changes the level a with respect to the Fermi level of the metal, which we take to be E F = 0 as always. For a = λ the reaction is in equilibrium, so that we can write: a = λ+e 0 η,whereη is the overpotential. The energy as a function of the solvent coordinate q can easily be plotted from eq. (10.8); a few examples for different overpotentials are shown in Fig Neglecting the last term, the formula gives two different parabolas for n = 1 and n = 0; for each value of q, the lower of the two energies is the correct adiabatic energy. For η <λthe crossing of the two parabolas gives the activated state with coordinate q s and a corresponding energy of activation, taken with respect to the initial energy a = λ + e 0 η: q s = λ e 0η 2λ E act = (λ e 0η) 2 4λ for η <λ (10.9) If we replace e 0 η by the free reaction energy G, the formula for the energy of activation is identical to that derived by Marcus and Hush for homogeneous electron transfer. For η>λthe initial state is no longer stable, and electron transfer is immediate and without activation (see Fig. 10.4). The resulting energy of activation is shown in Fig. 10.5; it starts from a value of E act = λ/4 at equilibrium and vanishes for η λ. In contrast, the Marcus-Hush formula predicts an increase of the energy of activation with η for η>λ. For homogenous electron transfer, a decrease of the rate constant for very large values of G has indeed been observed. But this does not occur at metal electrodes, neither in the adiabatic nor in the non-adiabatic case, which will be treated in the next section.

7 10.3 Adiabatic electron transfer 103 E Fig Energy of activation as a function of the overpotential. Full curve: adiabatic theory; dotted curve: Marcus-Hush formula. λ = 1.0 ev. At equilibrium, eq. (10.9) predicts a transfer coefficient of α =1/2: indeed, experimental values for outer sphere reaction always lie close to this value. With increasing overpotential, α decreases, and for η >λ it vanishes, as does the energy of activation, and the current becomes constant. The full curve will be displayed Fig in the next section for non-adiabatic transfer, where it has exactly the same shape. In order to obtain the rate constant from the energy of activation, we require an estimate of the pre-exponential factor. As long as the reaction is adiabatic, it is independent of the strength of the interaction between the reactant and the electrode, but is solely governed by the dynamics of the reorganization. For water, typical times for the reorientation are of the order s, and the typical frequency is the inverse of this time. Adiabatic electron transfer requires the reactant to be no further than about 5 10 Å from the electrode. Converting the bulk concentration to the number of particles in this range, we obtain a rough estimate of A 10 3 cm s 1,anumber that is quite compatible with experimental data for fast outer sphere reactions (see chapter 12). So the rate constant for the oxidation is now defined; the rate for the reverse reaction is obtained from the usual relation: k red =exp e 0η kt k ox (10.10) Finally we note, that larger values of the interaction lead to a significant reduction of the energy of activation. We shall return to this effect when we consider electrocatalysis.

8 Theoretical considerations of electron-transfer reactions 10.4 Non-adiabatic electron-transfer reactions For non-adiabatic reactions the electronic interaction is much weaker, so that the system can pass the saddle point without an electron transfer, and subsequently return to its initial state. Therefore the interaction strength will enter into the pre-exponential factor. Also, we now have to consider into which electronic level the electron is actually transferred, a question that makes no sense for adiabatic reactions, since reactant and electrode simply share their electrons. The basic ideas about solvent reorganization remain valid, but the equations for the energy require a few modifications. Firstly, in eq. (10.8) we drop the last term, since is even smaller than before. Further, we have to consider to which electronic level on the metal the electron is being transferred. We denote this energy by and, as always, measure it with respect to the Fermi level. The free energy of the reaction is reduced by, which has the same effect as replacing a by a. Therefore we obtain for the free energy curves for transitions to a particular level : E(q, ) =λq 2 +2λq +( a 2λq)n (10.11) Again, we obtain two different parabolas for n = 1 and n = 0, but now we cannot argue, that the system is always on the surface with the lower energy, since the reaction is no longer adiabatic. Therefore, each parabola describes a redox state, and the energy of activation is obtained by calculating the intersection point. Equation (10.9) is now valid for all η, but again we have to replace a by a : E act () = (λ + e 0η) 2 4λ (10.12) From this we can define a rate constant for the contribution that passes to an energy ; bearing in mind that the transition can only take place if there is an empty level of energy on the electrode, we write: k ox () [1 f()] exp (λ + e 0η) 2 4λkT (10.13) where f() is the Fermi-Dirac distribution. To obtain the total rate constant, we have to integrate over all energies : Thus, the second factor in eq. (10.13) has the meaning of a probability distribution for the transition. However, this distribution is not properly normalized to unity, and an integration would give as an extra dimension of energy. To normalize the distribution, we note that it has the form of a Gaussian, and hence the normalizing factor is (4πλkT) 1/2. The pre-exponential factor can be obtained from first order perturbation theory, and is /. Finally, we again have to convert the bulk to surface concentration, which gives an extra factor of the order of A = cm. So we obtain for the total rate constant:

9 10.4 Non-adiabatic electron-transfer reactions "=0.5 ev ln j/jlim -4-8 "=1 ev "=0.75 ev ! / V Fig The anodic current density as a function of electrode potential according to Eq. (10.14). k ox = A (4πλkT) 1/2 [1 f()] exp (λ + e 0η) 2 4λkT d (10.14) The integral is to be performed over the conduction band of the metal; in practice the limits can be extended to ±, since the integrand is negligible far from the Fermi level. The rate constant for the reduction can be obtained from eq. (10.10); alternatively we can note that we must change the signs of and η, and that the rate for a given must be proportional to f(). This gives: k red = A (4πλkT) 1/2 f()exp (λ + e 0η) 2 4λkT d (10.15) Equations (10.14) and (10.15) are the general relations for the rate constants in the non-adiabatic case. The resulting current-potential curves for the anodic direction are shown in Fig They have been normalized by the constant limiting value at high overpotentials. The corresponding curves for the adiabatic case are very similar, so we do not show them separately. There are two useful approximations: For small η, only the region near the Fermi level contributes; it is sufficient to keep first-order terms in and η in the energy of activation. The integral can then be performed explicitly, resulting in: k ox = A 1/2 πkt exp λ 2e 0η, for e 0 η λ (10.16) 4λ 4kT where the integration limits have been extended to ±. Again, this equation has the form of the familiar Butler-Volmer law with a transfer coefficient of one-half.

10 Theoretical considerations of electron-transfer reactions A good approximation to the current-potential curve is obtained by replacing the Fermi-Dirac distribution with a step function: which results in: where k ox = A 2 erfc λ e 0η (4λkT ) 1/2 (10.17) erfc(x) = 2 exp( y 2 ) dy =1 erf(x) π x =1 2 x exp( y 2 ) dy π 0 is the compliment of the error function erf(x). Equation (10.17) is a good approximation in the region e 0 η kt. In particular we obtain at very large overpotentials a limiting rate: k lim = A 2, for e 0η λ (10.18) which is independent of the applied potential. The corresponding expressions for the reduction are: k red = A πkt 1/2 4λ exp λ+2e0η 4kT, for e 0 η λ (10.19) k red = A e0η+λ 2erfc, for e (4λkT ) 1/2 0 η kt (10.20) For reasons of symmetry, the limiting rates are the same in both directions. The current-potential relation in Fig show Butler-Volmer behavior for small overpotentials, and limiting currents for large overpotentials, and have the same form for adiabatic and non-adiabatic reactions alike. The two kinds of reactions differ principally in the pre-exponential factors, which for an adiabatic reaction are independent of the electrode material, and for a non-adiabatic reaction depend on the strength of the interaction. A direct comparison of the form of the current-potential curves with experiments is not easy. At low overpotentials one always observes Butler-Volmer behavior in agreement with the theory. At high overpotentials is difficult to measure kinetic currents since then the reaction is fast and usually transport controlled (see Chapter 19). The deviations from the Butler-Volmer equation predicted by theory were doubted for some time. But they have now been observed beyond doubt, and we shall review some relevant experimental results in Chapter 12, where we shall also concern ourselves with the question of adiabaticity. The model presented here is simplified in several ways: harmonic approximation, purely classical treatment reorganization. But it does explain the basic features of electron-transfer reactions, relates the observed energies of activation to the reorganization of the inner and outer sphere, and does predict the correct form of the current-potential relationship. In some cases the energy of reorganization can be estimated (see the following), and then quantitative comparisons between theory and experiment can be made.

11 10.5 Gerischer s formulation 107 electronic energy electronic energy W ox W ox Fermi level! Fermi level -e 0 " W red W red Fig Distributions W ox and W red at equilibrium (left) and after application of a cathodic overpotential Gerischer s formulation The equations for the rate constants in the non-adiabatic case derived above have a suggestive interpretation proposed by Gerischer [43]. In the expression for the oxidation rate the term [1 f()] is the probability to find an empty state of energy on the electrode surface. If one interprets: W red (, η) =(4πλkT) 1/2 exp (λ + e 0η) 2 (10.21) 4λkT as the (normalized) probability of finding an occupied (reduced) state of energy in the solution, then the anodic rate is simply proportional to the probability of finding an occupied state of energy in the solution multiplied by the probability to find an empty state of energy on the metal. The maximum of W red is at = λ + e 0 η; so application of an overpotential shifts it by an amount e 0 η with respect to the Fermi level of the metal. Mutatis mutandis, the same argument can be made for the cathodic direction by defining: W ox (, η) =(4πλkT) 1/2 exp (λ + e 0η) 2 (10.22) 4λkT as the probability to find an empty (oxidized) state of energy in the solution. This has its maximum at = λ + e 0 η; so on application of an overpotential it is shifted by the same amount as W red. Illustrations such as the one presented in Fig offer a useful way of visualizing simple electron-transfer reactions. Unfortunately, the probabilities W ox and W red are sometimes denoted as densities of states of the oxidized and reduced species in the solution. This

12 Theoretical considerations of electron-transfer reactions is a misnomer, since they have nothing to do with the electronic densities of states we have introduced earlier, and can only lead to confusion. Indeed, the two concepts are sometimes confused in the literature. Needless to say, we shall not use this terminology Multidimensional treatment Here we shows that a multi-dimensional treatment of the solvent and inner sphere reorganization gives the same energy of activation. At the same time we prepare a way for estimating reorganization energy. We describe all modes that are reorganized as harmonic oscillators, which interact linearly with the charge on the reactant: H sol = 1 2 α ix 2 i + αx i g i (1 n) (10.23) i For a real harmonic oscillator, we have: α = mω 2,wherem i is the mass of the oscillator, ω i its frequency. However, for the mathematics it is only important that the leading term of the Hamiltonian is second order, and the interaction linear. As we shall see below, the coordinate can have quite a general meaning, and for the outer sphere it is usually taken as the local polarization. As before g i are the interaction constants with the charge on the reactant. For n = 1, the minimum is at x i = 0, for n = 1 at x i = g i. As discussed before, the mathematics can be greatly simplified by normalizing the coordinates and introducing q i = x i /g i. Instead of eq. (10.8) we obtain the generalized form: E(q i )= a n + i λi q 2 i +2λ i q i (1 n) (10.24) λ i = αgi 2 /2 is the contribution of the ith mode to the energy of reorganization. Equation (10.24) defines two different multi-dimensional paraboloids for n = 1, which is our initial state, and n = 0. We proceed to find the saddle point on the intersection. There, the energies on the two surfaces must be equal, which gives: 2 λ i q i a = 0 (10.25) i We determine the saddle point by introducing a Lagrange multiplier µ and minimize the function: F (q i )= a + λ i qi 2 + µ 2 λ i q i a (10.26) i i subject to the constraint of eq.(10.25). This gives: q i = µ µ = a 2λ E act = 2 a 4λ = ( λ + e 0η) 2 4λ (10.27)

13 10.7 The energy of reorganization 109 where λ = i λ i is the total energy of reorganization. So we have obtained the same energy of activation as before. The reaction path is given by the straight line q i = q, whereq runs from 0 to 1 and can be identified with the generalized solvent coordinate. Remember that at equilibrium a = λ; the saddle point lies between the minima for the initial and the final state only for a < The energy of reorganization The contribution of a mode i to the energy of reorganization has been defined above as λ i = αgi 2 /2. During the course of the reaction, the equilibrium value of the coordinate x i changes from x i =0tox i = g i. Therefore, we can also write: λ i = α i ( q i ) 2 /2 (10.28) where q i is the change in the equilibrium position, and α/2 isthecoefficient of the quadratic term. This equation can be used for any type of harmonic oscillator, irrespective of the meaning of the coordinate. Since the energy of reorganization plays a central role in electron-transfer reactions, it is useful to obtain rough estimates for specific systems. As outlined, it contains two contributions: one from the inner and one from the outer sphere. The former is readily calculated from the previous section. As an example, we consider the reaction of the [Fe(H 2 O) 6 ] 2+/3+ couple. During the reaction the distance of the water ligands from the central ion changes; this corresponds to a reorganization of the totally symmetric, or breathing, mode of the complex, and this seems to be the only mode which undergoes substantial reorganization. Let m be the effective mass of this mode, q the change in the equilibrium distance, and ω the frequency. The energy of reorganization of the inner sphere is then: λ in = 1 2 mω2 ( q) 2 (10.29) There is a small complication in that the frequency ω is different for the reduced and oxidized states; so that one has to take an average frequency. Marcus has suggested taking ω av =2ω ox ω red /(ω ox +ω red ). When several innersphere modes are reorganized, one simply sums over the various contributions. The matter becomes complicated if the complex is severely distorted during the reaction, and the two states have different normal coordinates. While the theory can be suitably modified to account for this case, the mathematics are cumbersome. To obtain an estimate for the energy of reorganization of the outer sphere, we start from the Born model, in which the solvation of an ion is viewed as resulting from the Coulomb interaction of the ionic charge with the polarization of the solvent. This polarization contains two contributions: one is from the

14 Theoretical considerations of electron-transfer reactions electronic polarizability of the solvent molecules; the other is caused by the orientation and distortion of the solvent molecules in an external field. The former is also denoted as the fast polarization, since it is electronic in origin and reacts on a time scale of s, so that it reacts practically instantly to the electron transfer; the latter is called the slow polarization since it is caused by the movement of atoms on a time scale of s. To obtain separate expressions for the two components we start with the constitutive relation between the electric field vector E, the dielectric displacement D, and the polarization P: D = 0 E = 0 E + P, or P = 1 1 D (10.30) where is the dielectric constant of the medium 2. If we apply an alternating external field with a high frequency in the optical region, only the electronic polarization can follow, and the optical value of the dielectric constant applies ( = 1.88 for water). So the fast polarization is: P f = 1 1 D (10.31) In a static field both components of the polarization contribute, and the static value s of the dielectric constant must be used in Eq. (10.30). The slow polarization is obtained by subtracting P f, which gives: 1 P s = 1 D (10.32) s The reorganization of the solvent molecules can be expressed through the change in the slow polarization. Consider a small volume element V of the solvent in the vicinity of the reactant; it has a dipole moment m = P s V caused by the slow polarization, and its energy of interaction with the external field E ex caused by the reacting ion is P s E ex V = P s D V/ 0, since E ex = D/ 0. We take the polarization P s as the relevant outer-sphere coordinate, and require an expression for the contribution U of the volume element to the potential energy of the system. In the harmonic approximation this must be a second-order polynomial in P s, and the linear term is the interaction with the external field, so that the equilibrium values of P s in the absence of a field vanishes: U/ V = 1 2 αp2 s P s D/ 0 + C (10.33) where C is independent of P s, and the constant α is still to be determined. For this purpose we calculate the equilibrium value of the slow polarization by minimizing U and identifying this result with the value from Eq. (10.32): 2 We use the usual symbol for the dielectric constant; no confusion should arise with the energy variable employed

15 10.8 Adiabatic versus non-adiabatic transitions 111 P eq s = D 1 1, hence = 1 α 0 α 0 s (10.34) During the reaction the dielectric displacement changes from D ox to D red (or vice versa), and the equilibrium value from D ox /α 0 to D red /α 0. Therefore the contribution of the volume element V to the energy of reorganization of the outer sphere is: 1 λ out = s (D ox D red ) 2 V (10.35) The total energy of reorganization of the outer sphere is obtained by integrating over the volume of the solution surrounding the reactant: λ out = (D ox D red ) 2 dv (10.36) 2 0 s The dielectric displacement must be calculated from electrostatics; for a reactant in front of a metal surface the image force has to be considered. For the simple case of a spherical ion in front of a metal electrode experiencing the full image interaction, a straightforward calculation gives: λ out = e2 0 8π s 1 a 1 2d (10.37) where a is the radius of the ion, and d the distance from the metal surface. Because of the use of macroscopic electrostatics, this equation should be viewed as providing no more than an estimate for λ out Adiabatic versus non-adiabatic transitions If a reaction proceeds adiabatically or not, depends on the strength of the electronic interaction between the reactant and the electrode surface, which also determines the width of the density of states of the reacting level. For small interactions, first order perturbation theory holds, and the preexponential factor is proportional to see eq. (10.14). For large, the pre-exponential factor is determined by solvent dynamics. The latter can be described in terms of Kramer s theory, which we cannot treat here in any detail. Briefly, an important factor is the solvent friction, which determines the typical time that the solvent takes to reorient. The higher the friction, the lower is the pre-exponential factor A. As mentioned before, for aqueous solutions a value of A 10 3 cm s 1 seems to be a good estimate. The relation between the rate constant k and the interaction strength is shown in Fig. 10.8, which is based on a computer simulation [44]. For low interactions, the rate follows perturbation theory. Then, solvent dynamics starts to influence the rate, and in a certain region it is independent of. The

16 Theoretical considerations of electron-transfer reactions 6 low friction high friction ln + const ln / ev Fig Dependence of the rate constant on the interaction strength. The black line shows the prediction from 1st order perturbation theory. height of this plateau region depends on the friction: the higher the friction, the lower the plateau. This is the region in which the original theories of Marcus and Hush hold. For very high interactions, the last term of eq comes into play and lowers the energy of activation, so the rate rises again. The latter is a catalytic effect, caused by the electronic interaction with the metal. On bare metal surfaces, outer sphere reactions are typically adiabatic and fall into the Marcus-Hush plateau region. Since they are independent of, they also do not depend on the nature of the metal. On semiconductors, semimetals like graphite, and in particularly on electrodes covered by an insulating film, they can proceed non-adiabatically. The catalytic region does not seem to play a role for outer-sphere reactions they are so fast that they do not need to be catalysed. However, catalysis is very important for inner-sphere like hydrogen evolution, which we will consider in detail later. Problems 1. Consider a one-dimensional system in which the potential energy functions for the oxidized and reduced states are: U ox(q) =e ox mω2 q 2 U red(q) =e red mω2 (q δ) 2 Calculate the intersection point of these two parabolas and define the energy of reorganization. Calculate the energies of activation for the forward and the backward direction. 2. Assume that the current-potential curves of a system are given by Eqs. (10.17) and (10.20). Calculate the effective transfer coefficients defined by:

17 10.8 Adiabatic versus non-adiabatic transitions 113 α = kt e 0 ln j a η β = kt ln j c e 0 η Their values depend on the overpotential. Show that for η =0:α + β = 1. This (small) error arises because the Fermi-Dirac distribution has been replaced by a step function. 3. From Eq. (10.36) calculate the energy of reorganization of a single spherical reactant in the bulk of a solution. Derive Eq. (10.37) for a reactant in front of a metal electrode. 4. Show that for = 1 Eq. (10.36) reduces to the Born equation for the energy of solvation of an ion.

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