# 快速时时彩: Theory of electrochemical outer sphere reactions.pdf

Theory of electrochemical outer sphere reactions E. Leiva and C. S′anchez Unidad de Matem′atica y F′?sica, Facultad de Ciencias Qu′?micas, INFIQC, Universidad Nacional de C′ordoba, Argentina 1 ELEMENTARY MODELING The theory of electrochemical outer sphere reactions is deeply rooted in the theory of homogeneous outer sphere electron reactions, to such an extent that practically all textbooks or review articles start with a discussion of the homogeneous problem. This area abounds with work, which causes any suitable referencing of the most relevant one to outnumber that of several chapters of the present vol- ume. Instead, we adopt at this point a recursive approach. The recent book by Kuznetsov and Ulstrup [1] contains a comprehensive view of several aspects of electron transfer, providing at the same time a basic theoretical background for the understanding of electron transfer and extensive reference to advanced work. In the more specific field of surface electron transfer, it is worth mentioning Ref. [2] where several authors discuss the state-of-the-art in the area in the mid 1990s. As a beginner’s introduction to the field, we recommend the didactic exposition given in Schmick- ler’s book on interfacial electrochemistry. [3] In the case of the “classical models” referred to in Outer sphere reactions, Volume 2, the reaction probability is based on transition state theory (TST). The simplest representation of an electron transfer in homogeneous phase is illustrated in Figure 1. In this figure, the free energy status of the reactants and products is represented by parabolic curves along a reaction coordinate, q. As we have seen in Understanding electrocatalysis: From reaction steps to first-principles calculations, Volume 2, the reaction rate should be given by: k f = k B T h e ? Delta1G 0negationslash= RT (1) where Delta1G 0negationslash= represents the activation free energy of the reaction and is given and by: Delta1G 0negationslash= = G r (q negationslash= ) ? G r (2) G r (q negationslash= ) denotes the Gibb’s energy of the reactants and the environment at the configuration of the transition state, excluding the kinetic energy contribution for motion along the coordinate, q. On the other hand, G r represents the free energy of the reactant system involving all coordinates. The vibrational contribution along q can be subtracted from G r in equation (2) to obtain a new version of equation (1): k f = νe ? Delta1G ? RT (3) where ν is the vibrational frequency along q and Delta1G ? is given by: Delta1G ? = G r (q negationslash= ) ? G r (q r )(4) G r (q r ) has now the same number of coordinates as G r (q negationslash= ). An elementary but powerful theory can be constructed from equation (3) and the following assumptions [4] can be made: ? Energy is conserved for the whole system in a radia- tionless process. ? The heavy particles of the system remain still during the electronic transition. (Frank–Condon principle). ? An electrostatic continuum model can be used to cal- culate the polarization of the system. Edited by Wolf Vielstich, Hubert A. Gasteiger, Arnold Lamm and Harumi Yokokawa. ? 2010 John Wiley ε 0 and ε ∞ are the static and optical dielectric constants, respectively. The term λ is the so-called energy of reorganization and is typically of the order of 0.5–1.5 eV. An alternative way to derive equation (6) has been didactically exposed by Schmickler. [3] In this approach, the potential energy surfaces, U, of oxidized and reduced species (denoted by a = ox, red) are expanded in the harmonic approximation: U a = e a + summationdisplay i 1 2 m i ω i (q i ? y a i ) 2 (8) where m i and ω i denote the effective mass and frequency of the mode i and q i and y a i are sets of normal coordinates reorganized during the electron transfer and at equilibrium positions, respectively. The saddle point of the reaction hypersurface is determined by minimizing equation (8) with the restriction U ox (q i ) = U red (q i ), yielding an analogue of equation (6), but with λ given by: λ = 1 2 summationdisplay i m i ω 2 i (y ox i ? y red i ) 2 (9) The meaning of λ is illustrated in Figure 2 in the case of uni-dimensional reaction surfaces. For this simple case, λ corresponds to the difference between two points of the curve ordinates of the reduced state: that corresponding to the oxidized state at equilibrium and that corresponding to its own equilibrium. Going back to equation (6), a remarkable feature is the parabolic dependence of Delta1G ? on Delta1G 0 . This equation, com- bined with equation (3), leads to the conclusion that in principle, reaction rates for reactants exhibiting similar λ values and different Delta1G 0 s should reach a maximum for increasing Delta1G 0 and subsequently drop. The latter case, where the rate constant decreases for increasing Delta1G 0 is G q y j ox y j red λ Figure 2. Illustration of the meaning of the solvent reorganization energy λ for the case of uni-dimensional potential energy surfaces of the oxidized (ox) and reduced (red) states. y ox and y red j denote equilibrium positions of the ox and red species, respectively. 2 from Ref. 4. 2000 the American Chemical Society, 2000.) ? Theory of electrocatalysis 30 20 10 01 ??G 0 (eV) In k obsd 2 Figure 3. Gibb’s free energy relations extending to the inverted free energy range. The data correspond to photoinduced intramole- cular electron transfer in binuclear Ir(I) complexes. [45] (Repro- duced from Kuznetsov and Ulstrup (1999) the so-called inverted Gibb’s free energy region. Since the region close to the maximum corresponds to the larger reac- tion rates, diffusion control impeded early confirmation of these predictions. However, experiments with intramolec- ular reactions where the reactants are rigidly kept at their positions, or intermolecular reactions where the reaction rate is slowed down by separating the reactants in different phases, made this region reachable. An example is shown in Figure 3. Further examples can be found in the book by Kuznetsov and Ulstrup. [1] A parabolic relation between ln(k) and Delta1G 0 is found over three orders of magnitude on either side of the maximum, providing a strong backup to the theory. 2 SIMPLE ELECTRON TRANSFER AT THE ELECTROCHEMICAL INTERFACE The extension of the present framework to consider electron transfer at the electrochemical interface from a phenomeno- logical point of view is relatively straightforward. This can be done taking into account the following points. ? Consideration of the interaction of an ion with its image charge leads to an equation formally identical to equa- tion (6), [4] where Delta1G 0 becomes a linear function of the overpotential η. This is so because in the oxidized state the energy of the electrons in the metal is lowered in the quantity ?eη with respect to the equilibrium state. In the case of an oxidation reaction, we have: Delta1G ? ox = (λ ? eη) 2 4λ (10) Note the quadratic dependence of the activation energy on the applied overpotential. ? Electron exchange not only takes place with those states at the Fermi level of the metal but also with other states. For example, in the case of an oxidation reaction, electrons are in principle transferred to all unoccupied states in the metal. If we label the energy states of the metal referred to the Fermi level as ε, the activation energy for electron transfer to the state ε will be: Delta1G ? ox (ε) = (λ + ε ? eη) 2 4λ (11) ? Electron exchange with the metal is slow enough so as not to perturb the equilibrium distribution of electrons in the metal. This allows us to write the occupa- tion probability f (ε) of a given state in terms of the Fermi–Dirac distribution function: f(ε) = 1 1 + exp(ε/kT) (12) Moreover, the interaction of the redox system with the metal is assumed to be small, so that the density of states in the metal, say ρ(ε) remains that of the pure metal. Thus, following with the example of the oxidation reaction, the number of states available for electron transfer to the metal at the energy state ε will be ρ(ε)(1 ? f(ε)). Consequently, the contribution to the reaction rate by electron transfer from the redox system to the energy state ε in the metal will be given by: k(ε) = Aρ(ε)(1 ? f(ε)) exp parenleftbigg ? (λ + ε ? eη) 2 4λ parenrightbigg (13) where A is an energy-independent preexponential factor. The total oxidation (anodic) density, j ox , by the reduced species in concentration, c red , will be given by the integral: j a = Ac red integraldisplay dερ(ε)(1 ? f(ε)) × exp parenleftbigg ? (λ + ε ? eη) 2 4λkT parenrightbigg (14) where the integration is performed over the conduction band of the metal. Analogously, the reduction (cathodic) current can be obtained assuming a current flow from the occupied 3 John Wiley (b) Fe +2 /Fe +3 , contours are 8 ler (1986) Elsevier, 1986.)? drawn in the range from ?1 to 1 eV in steps of 0.2 eV. (Reproduced from Schmickler (1995) Elsevier, 1995.)? Theory of electrocatalysis Let us first analyze the case of the iodine/iodide system. Far from the surface, Delta1 is small and three solutions exist for 〈n(q ν )〉 that make the energy stationary. One with 〈n(q 0 ν )〉≈1 corresponding to the atomic state on the right of Figure 5(a), one with 〈n(q ν )〉≈0 corresponding to the ionic state on the left of Figure 5(a) and one with a partial occupancy belonging to the transition state. When the redox system approaches the surface, the increase in the interaction with the electronic system yields a single adsorbed state, the realm of which is painted in black at the bottom of the figure. While the atomic state reaches the adsorbed status just decreasing its energy, the ion passes through a maximum in the region where it loses an important part of its solvation sheath. However, the interaction of the iodide ion with the solvent is relatively weak, and the activation barrier it must surmount is in the order of 0.4 eV, making ion transfer feasible. The situation for the redox couple Fe +2 /Fe +3 , illustrated in Figure 5(b), is quite different, especially when approach- ing the electrode surface. As stated above, far from the surface three solutions exist for 〈n(q 0 ν )〉. At a shorter dis- tance, they also merge into a single solution, but the strong interaction with the solvent produces a steep increase in the potential energy of the system. The energy barrier that must be surmounted to reach the adsorbed state from any of the initial states amounts now to several electron volts. On the other hand, the two ionic states are separated by a comparatively low energy barrier, indicating that electron transfer should be favored over ion transfer. One must bear in mind, however, that the calculation of the reaction rate for this reaction should employ diabatic energy surfaces rather than the adiabatic potential energy surface shown in Figure 5(b), since this reaction is expected to occur in the regions where the coupling is weak. Other applications of the Anderson–Newns model have extended the spinless model to consider the electron/elec- tron interaction in the Hartree–Fock approximation, [16, 26] and more recently to consider the Coulomb correlations between electrons. [23] In the latter paper, Kuznetsov and Medveved introduced the correlation effects on the exactly solvable surface molecule limit of the Anderson–Newns model. These authors determined the range of values of coupling constants (see equation (30)) and the Coulomb repulsion between electrons for which correlation effects are relevant to activation barriers and the shapes of adiabatic Gibb’s energy surfaces. Another recent extension made to the Anderson–Newns model has been the consideration of changes in the frequencies of the inner sphere modes made by Schmickler and Koper. [21] These latter calculations show that in this case the Butler–Volmer transfer coefficient can no longer be identified with the electron occupancy in the transition state saddle point, thus, deviating from the value of 1/2. The problem of the transfer of multiple electrons between an electrode and a multilevel redox center has been addressed by Boroda and Voth [19] within this formal- ism, performing a comparative analysis of sequential and parallel processes. Significantly different solvent activation patterns emerge from this analysis, which show different sensitivities to the applied potential difference. 6 STOCHASTIC ASPECTS A last point that we shall address in this bird’s-eye view of the simplest electron transfer reactions at the electro- chemical interface is concerning their stochastic aspects. In very much the same way as the motion of tiny parti- cles in liquids (Brownian Motion) can be described without knowledge of the detailed dynamic status of the system (particle + solvent), considerable understanding of electron transfer reactions can be gained by thinking of this pro- cess as stochastic. Recent extensive reviewing can be found in Ref. [27] and in Chapter 13 of Ref. [1]. The seminal work in this area is that of Kramers in 1940. [28] Work fol- lowing this line has been extensively reviewed by H¨anggi et al. [29] Instead of considering the deterministic (newto- nian) motion of the reacting particle and its environment, Kramers’s approach describes its motion along a reaction coordinate x according to the following equation of motion: m d 2 x dt 2 =? ?V(x) ?x ? ξv + f(t) (36) where v is the velocity of the particle, m is its mass, ξ is the friction coefficient of the particle with the medium, V(x)is a uni-dimensional external potential describing the energy barrier and f(t) is a random force that averages to zero: 〈f(t)〉=0 (37) Thus, aside from the usual first term on the right-hand side of equation (36), we find a (second) dissipative term and a (third) fluctuating contribution. Equation (36) is known as Langevine equation, and the fluctuating force obeys the fluctuation/dissipation relation: 〈f(t)f(t prime )〉=2kTξmδ(t ? t prime )(38) Within the present approach, the reaction rate of a pro- cess is calculated from the joint probability distribution p(x,v,t)dx dv of the position and velocity of the particle. The time evolution of the probability density follows the Klein–Kramers equation: 9 Theory of electrochemical outer sphere reactions ?p(x,v,t) ?t = bracketleftbigg ? ? ?x v + ? ?v U prime (x) + Mγv m + ξkT m ? 2 ?v 2 bracketrightbigg p(x,v,t) (39) This cannot be exactly solved in the general case. In order to tackle the problem of calculating reaction rates in this context, Kramers made a number of approximations which we shall enumerate in terms of Figure 6. We show that there is an adiabatic potential energy profile for two species (A and C), separated by a potential barrier. k + and k ? denote the forward and backward reaction rates. The main approximations are: (1) The steady state rate is considered. (say from A to C). This is maintained by sources in A and sinks in B. This condition implies the following conditions for equation