A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 34
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Then, AG° is the standard free energy change for the reaction. The picture at the top is ageneral representation of structural changes that might accompany electron transfer. The changes inspacing of the six surrounding dots could represent, for example, changes in bond lengths within theelectroactive species or the restructuring of the surrounding solvent shell.qx.
In keeping with the Franck-Condon principle, electron transfer only occurs at thisposition.The free energies at the transition state are thus given by- qof2- qR)x(3.6.5)+ AG°x(3.6.6)xSince G%(q ) = GR(q ), (3.6.5) and (3.6.6) can be solved for q with the result,2AG°' k(qR -qo)(3.6.7)The free energy of activation for reduction of О is given by(3.6.8)Chapter 3. Kinetics of Electrode Reactionswhere we have noted that G%(q0) = 0, as defined in (3.6.3).
Substitution for cf from(3.6.7) into (3.6.5) then yields:02AG°12(3.6.9)Defining A = (k/2)(qR — q0)2, we have(3.6.10a)or, for an electrode reaction(3.6.10b)There can be free energy contributions beyond those considered in the derivationjust described. In general, they are energy changes involved in bringing the reactantsand products from the average environment in the medium to the special environmentwhere electron transfer occurs. Among them are the energy of ion pairing and the electrostatic work needed to reach the reactive position (e.g., to bring a positively chargedreactant to a position near a positively charged electrode). Such effects are usuallytreated by the inclusion of work terms, WQ and WR, which are adjustments to AG° orF(E — E ).
For simplicity, they were omitted above. The complete equations, includingthe work terms, are 1 6A/4f\ G\-\4A 1 Л\AG°- -WQAFiE-E+ wRy/°)" WQ +A(3.6.11a)V)(3.6.11b)The critical parameter is Л, the reorganization energy, which represents the energynecessary to transform the nuclear configurations in the reactant and the solvent to thoseof the product state.
It is usually separated into inner, Aj, and outer, Ao, components:A = Ai + Ao(3.6.12)where Aj represents the contribution from reorganization of species O, and Ao that fromreorganization of the solvent.1716The convention is to define vv0 and wR as the work required to establish the reactive position from the averageenvironment of reactants and products in the medium. The signs in (3.6.1 la,b) follow from this. In manycircumstances, the work terms are also the free energy changes for the precursor equilibria.
When that is true,w017= -RT In Kpt0 and wR = -RTInKPR.One should not confuse the inner and outer components of Л with the concept of inner- and outer-spherereaction. In the treatment under consideration, we are dealing with an outer-sphere reaction, and Aj and Aosimply apportion the energy to terms applying to changes in bond lengths (e.g., of a metal-ligand bond) andchanges in solvation, respectively.3.6 Microscopic Theories of Charge Transfer < 121To the extent that the normal modes of the reactant remain harmonic over the rangeof distortion needed, one can, in principle, calculate Aj by summing over the normal vibrational modes of the reactant, that is,A(3.6.13)i -Ewhere the k's are force constants, and the g's are displacements in the normal modecoordinates.Typically, Ao is computed by assuming that the solvent is a dielectric continuum, andthe reactant is a sphere of radius ao.
For an electrode reaction,1ло — -е ( 1лЛ (1\£оР1)(3.6.14a)where e o p and e s are the optical and static dielectric constants, respectively, and R is takenas twice the distance from the center of the molecule to the electrode (i.e., 2JC0, which isthe distance between the reactant and its image charge in the electrode).18 For a homogeneous electron-transfer reaction:(3.6.14b)where a\ and #2 are the radii of the reactants (O and R' in equation 3.6.1) and d = a\ + #2Typical values of A are in the range of 0.5 to 1 eV.Predictions from Marcus TheoryWhile it is possible, in principle, to estimate the rate constant for an electrode reactionby computation of the pre-exponential terms and the A values, this is rarely done inpractice.
The theory's greater value is the chemical and physical insight that it affords,which arises from its capacity for prediction and generalization about electron-transferreactions.For example, one can obtain the predicted a-value from (3.6.10b):1 <^fFffi=2+tt=I , F(£ 2A(3.6.15a)E°) - (wo - wR)2A(3.6.15b)or with the inclusion of work terms:F(E-Thus, the theory predicts not only that a ~ 0.5, but also that it depends on potential in aparticular way.
As mentioned in Section 3.3.4, the Butler-Volmer (BV) theory can accommodate a potential dependence of a, but in its classic version, the BV theory handlesa as a constant. Moreover, there is no basis in the В V theory for predicting the form of thepotential dependence. On the other hand, the potential-dependent term in (3.6.15a,b),18In some treatments of electron transfer, the assumption is made that the reactant charge is largely shielded bycounter ions in solution, so that an image charge does not form in the electrode.
In this case, R is the distancebetween the center of the reactant molecule and the electrode (24, 39).122Chapter 3. Kinetics of Electrode Reactionswhich depends on the size of A, is usually not very large, so a clear potential dependencyof a has been difficult to observe experimentally. The effect is more obvious in reactionsinvolving electroactive centers bound to electrodes (see Section 14.5.2.).The Marcus theory also makes predictions about the relation between the rate constants for homogeneous and heterogeneous reactions of the same reactant.
Consider therate constant for the self-exchange reaction,O + R^XR + O(3.6.16)in comparison with k° for the related electrode reaction, О + e —> R. One can determinekex by labeling О isotopically and measuring the rate at which the isotope appears in R, orsometimes by other methods like ESR or NMR. A comparison of (3.6.14a) and (3.6.14b),where a0 = a\ = a2 = a and R = d = 2a, yieldsAei = Aex/2(3.6.17)where Ael and Aex are the values of Ao for the electrode reaction and the self-exchangereaction, respectively. For the self-exchange reaction, AG° = 0, so (3.6.10a) givesAGf = Aex/4, as long as Ao dominates Aj in the reorganization energy.
For the electrodereaction, k° corresponds to E = E°, so (3.6.10b) gives AG* = Aei/4, again with the condition that Aj is negligible. From (3.6.17), one can express AGf for the homogeneous andheterogeneous reactions in common terms, and one finds that kex is related to k° by theexpression(£exMex)1/2 = *°/A,l(3-6.18)where Aex and AQ\ are the pre-exponential factors for self-exchange and the electrode reaction.
(Roughly, A d is Ю4 to 105 cm/s and A ex is 10 11 to 10 1 2 Af"1 s" 1 .) 1 9The theory also leads to useful qualitative predictions about reaction kinetics. For example, equation 3.6.10b gives AG^ ~ X/4 at E°, where kf = k^ = £°. Thus, k° will belarger when the internal reorganization is smaller, that is, in reactions where О and R havesimilar structures. Electron transfers involving large structural alterations (such as sizablechanges in bond lengths or bond angles) tend to be slower.
Solvation also has an impactthrough its contribution to A. Large molecules (large ao) tend to show lower solvation energies, and smaller changes in solvation upon reaction, by comparison with smallerspecies. On this basis, one would expect electron transfers to small molecules, such as, thereduction of O 2 to O2~ in 2~aprotic media, to be slower than the reduction of Ar to Агт,where Ar is a large aromatic molecule like anthracene.The effect of solvent in an electron transfer is larger than simply through its energeticcontribution to Ao.
There is evidence that the dynamics of solvent reorganization, oftenrepresented in terms of a solvent longitudinal relaxation time, TL, contribute to the preexponential factor in (3.6.2) (47, 62-65), e.g., vn °c T~l. Since r L is roughly proportionalto the viscosity, an inverse proportionality of this kind implies that the heterogeneous rateconstant would decrease as the solution viscosity increases (i.e., as the diffusion coefficient of the reactant decreases).
This behavior is actually seen in the decrease of k° forelectrode reactions in water upon adding sucrose to increase the viscosity (presumablywithout changing Ao in a significant way) (66, 67). This effect was especially pronouncedin other studies involving Co(IIIAI)tris(bipyridine) complexes modified by the addition of19This equation also applies when the X{ terms are included (but work terms are neglected). This is the casebecause the total contribution to Л j is summed over two reactants in the homogeneous self-exchange reaction,but only over one in the electrode reaction (61).3.6 Microscopic Theories of Charge Transfer < 123long polyethylene or polypropylene oxide chains to the ligands, which cause largechanges in diffusion coefficient in undiluted, highly viscous, ionic melts (68).A particularly interesting prediction from this theory is the existence of an "invertedregion" for homogeneous electron-transfer reactions.
Figure 3.6.3 shows how equation3.6.10a predicts AGjf to vary with the thermodynamic driving force for the electron transfer, AG°. Curves are shown for several different values of A, but the basic pattern of behavior is the same for all, in that there is a predicted minimum in the standard free energyof activation. On the right-hand side of the minimum, there is a normal region, whereAGjf decreases, hence the rate constant increases, as AG° gets larger in magnitude (i.e.,becomes more negative). When AG° = -A, AGf is zero, and the rate constant is predictedto be at a maximum. At more negative AG° values, that is for very strongly driven reactions, the activation energy becomes larger, and the rate constant smaller.
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