A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 33
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Obviously outer-sphere reactions are less dependent on electrode materialthan inner-sphere ones. 13Homogenous Electron TransferOuter-sphere3+Cr(bpy) 3 2 + ->Co(NH 3 ) 2 + -.Inner-sphereCo(NH 3 ) 5 CI 2 +Cr(bpy)33+> (NH3)5Co Cl Cr(H2O)*Homogenous Electron TransferOuter-sphereSolvent13Inner-sphereFigure 3.6.1 Outer-sphere and inner-spherereactions. The inner sphere homogeneous reactionproduces, with loss of H2O, a ligand-bridgedcomplex (shown above), which decomposes toCrCl(H2O)^+ and Co(NH 3 ) 5 (H 2 O) 2+ .
In theheterogeneous reactions, the diagram shows a metalion (M) surrounded by ligands. In the inner spherereaction, a ligand that adsorbs on the electrode andbridges to the metal is indicatedin a darker color. An example of the latter is the+oxidation of Сг(Н 2 О)з at a mercury electrodein the presence of Cl~ or Br~.Even if there is not a strong interaction with the electrode, an outer-sphere reaction can depend on theelectrode material, because of (a) double-layer effects (Section 13.7), (b) the effect of the metal on the structureof the Helmholtz layer, or (c) the effect of the energy and distribution of electronic states in the electrode.3.6 Microscopic Theories of Charge Transfer117Outer-sphere electron transfers can be treated in a more general way than innersphere processes, where specific chemistry and interactions are important.
For this reason,the theory of outer-sphere electron transfer is much more highly developed, and the discussion that follows pertains to these kinds of reactions. However, in practical applications, such as in fuel cells and batteries, the more complicated inner-sphere reactions areimportant. A theory of these requires consideration of specific adsorption effects, as described in Chapter 13, as well as many of the factors important in heterogeneous catalyticreactions (56).3.6.1The Marcus Microscopic ModelConsider an outer-sphere, single electron transfer from an electrode to species O, to formthe product R.
This heterogeneous process is closely related to the homogeneous reduction of О to R by reaction with a suitable reductant, R\0 + R ' ^ R + O'(3.6.1)We will find it convenient to consider the two situations in the same theoretical context.Electron-transfer reactions, whether homogeneous or heterogeneous, are radiationlesselectronic rearrangements of reacting species.
Accordingly, there are many common elements between theories of electron transfer and treatments of radiationless deactivation inexcited molecules (57). Since the transfer is radiationless, the electron must move from aninitial state (on the electrode or in the reductant, R') to a receiving state (in species О oron the electrode) of the same energy. This demand for isoenergetic electron transfer is afundamental aspect with extensive consequences.A second important aspect of most microscopic theories of electron transfer is the assumption that the reactants and products do not change their configurations during the actual act of transfer. This idea is based essentially on the Franck-Condon principle, whichsays, in part, that nuclear momenta and positions do not change on the time scale of electronic transitions.
Thus, the reactant and product, О and R, share a common nuclear configuration at the moment of transfer.Let us consider again a plot of the standard free energy14 of species О and R as afunction of reaction coordinate (see Figure 3.3.2), but we now give more careful consideration to the nature of the reaction coordinate and the computation of the standard free energy. Our goal is to obtain an expression for the standard free energy of activation, AG^\as a function of structural parameters of the reactant, so that equation 3.1.17 (or a closelyrelated form) can be used to calculate the rate constant. In earlier theoretical work, thepre-exponential factor for the rate constant was written in terms of a collision number (37,38, 58, 59), but the formalism now used leads to expressions like:kf = K?,ovnKelexp(-AG}/RT)(3.6.2)where AGjf is the activation energy for reduction of О; К? о is a precursor equilibriumconstant, representing the ratio of the reactant concentration in the reactive position atthe electrode (the precursor state) to the concentration in bulk solution; vn is the nuclear frequency factor (s" 1 ), which represents the frequency of attempts on the energybarrier (generally associated with bond vibrations and solvent motion); and к е 1 is theelectronic transmission coefficient (related to the probability of electron tunneling; seeSection 3.6.4).
Often, #cel is taken as unity for a reaction where the reactant is close tothe electrode, so that there is strong coupling between the reactant and the electrode14See the footnote relating to the use of standard thermodynamic quantities in Section 3.1.2.118 • Chapter 3. Kinetics of Electrode Reactions(see Section 3.6.4).15 Methods for estimating the various factors are available (48), butthere is considerable uncertainty in their values.Actually, equation 3.6.2 can be used for either a heterogeneous reduction at an electrode or a homogeneous electron transfer in which О is reduced to R by another reactantin solution. For a heterogeneous electron transfer, the precursor state can be considered tobe a reactant molecule situated near the electrode at a distance where electron transfer ispossible.
Thus KFO = Co,Surf/Co> where C0,SUrfis a surface concentration having units ofmol/cm2. Consequently Kp o has units of cm, and kf has units of cm/s, as required. For ahomogeneous electron transfer between О and R\ one can think of the precursor state as areactive unit, OR', where the two species are close enough to allow transfer of an electron. Then KpO = [OR']/[O][R'], which has units of M~x if the concentrations are expressed conventionally.
This result gives a rate constant, kf, in units of M ^ s " 1 , again asrequired.In either case, we consider the reaction as occurring on a multidimensional surfacedefining the standard free energy of the system in terms of the nuclear coordinates (i.e.,the relative positions of the atoms) of the reactant, product, and solvent. Changes in nuclear coordinates come about from vibrational and rotational motion in О and R, and fromfluctuations in the position and orientation of the solvent molecules. As usual, we focuson the energetically favored path between reactants and products, and we measureprogress in terms of a reaction coordinate, q. Two general assumptions are (a) that the reactant, O, is centered at some fixed position with respect to the electrode (or in a bimolecular homogeneous reaction, that the reactants are at a fixed distance from each other) and(b) that the standard free energies of О and R, GQ and GR, depend quadratically on the reaction coordinate, q (49):G°0(q) = (k/2)(q - q0)2&= (k/2)(q - qR)2+ AG°(3.6.3)(3.6.4)where qo and qR are the values of the coordinate for the equilibrium atomic configurations in О and R, and к is a proportionality constant (e.g., a force constant for a change inbond length).
Depending on the case under consideration, AG° is either the free energy ofreaction for a homogeneous electron transfer or F(E - E°) for an electrode reaction.Let us consider a particularly simple case to give a physical picture of what is impliedhere. Suppose the reactant is A-B, a diatomic molecule, and the product is A-B~. To afirst approximation the nuclear coordinate could be the bond length in A-B (qo) and A-B~(gR), and the equations for the free energy could represent the energy for lengthening orcontraction of the bond within the usual harmonic oscillator approximation. This pictureis oversimplified in that the solvent molecules would also make a contribution to the freeenergy of activation (sometimes the dominant one).
In the discussion that follows, theyare assumed to contribute in a quadratic relationship involving coordinates of the solventdipole.Figure 3.6.2 shows a typical free energy plot based on (3.6.3) and (3.6.4). The molecules shown at the top of the figure are meant to represent the stable configurations of thereactants, for example, Ru(NH 3 )6 + and Ru(NH 3 )6 + as О and R, as well as to provide aview of the change in nuclear configuration upon reduction. The transition state is theposition where О and R have the same configuration, denoted by the reaction coordinate15The pre-exponential term sometimes also includes a nuclear tunneling factor, Г п . This arises from a quantummechanical treatment that accounts for electron transfer for nuclear configurations with energies below thetransition state (48, 60).3.6 Microscopic Theories of Charge Transfer « 1190(3+)Figure 3.6.2 Standard free energy, G , as a function of reaction coordinate, q, for an electrontransfer reaction, such as Ru(NH3)6 + e —>• RU(NH 3 )J5 + .
This diagram applies either to aheterogeneous reaction in which О and R react at an electrode or a homogeneous reaction in which Оand R react with members of another redox couple as shown in (3.6.1). For the heterogeneous case,the curve for О is actually the sum of energies for species О and for an electron on the electrode at theFermi level corresponding to potential E. Then, AG = F(E — E°). For the homogeneous case, thecurve for О is the sum of energies for О and its reactant partner, R', while the curve for R is a sum forR and O'.
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