1454
J . Phys. Chem. 1990, 94, 1454-1463
Nonlocal Electrostatic Effects on Electron-Transfer Activation Energies: Some Consequences for and Comparisons wlth Electrochemical and Homogeneous-Phase Kineticst Donald K. Phelps, A. A. Kornyshev,' and Michael J. Weaver* Department of Chemistry, Purdue University, West Lafayette, Indiana 47907 (Received: June 29, 1989)
The combined role of field penetration into the metal electrode and solvent spatial correlations (nonlocal electrostatics) upon the outer-shell free energy of activation, AG*,, for electrochemical exchange reactions as described by a recent treatment (ref 4 ) are explored in relation to the activation free energy, AG*h,of homogeneous self-exchange processes by means of illustrative numerical calculations and by comparisons with experimentally derived barriers. The latter are obtained from ~) exchange and homogeneous solvent-dependent kinetic data for cobaltocenium-cobaltocene ( C ~ , C O + /electrochemical self-exchange, along with optical electron-transfer energies for biferrocenylacetylene cation (BFA+). The predicted values of AG*, in relation to AG*h are examined for systematic variations in spatial and solvent parameters. In contrast to the expectations of the conventional dielectric continuum treatment, consideration of metal field penetration yields a predicted destabifiration of the electrochemical transition state (Le., increase in AG*,) so that typically AG*, k AG*h. The inclusion of solvent spatial correlations increases the degree of divergence from the continuum predictions. Comparison with hG*h values derived from the BFA' optical electron-transfer data, AG*h(op),indicate the significance of solvent spatial correlations, especially in hydrogen-bonded media. Experimental estimates of AG*, in seven solvents are derived from the C ~ , C O + / ~ electrochemical rate constants, the required preexponential factors being estimated from the knowledge, based on a solvent dynamical analysis (ref 8c), that essentially adiabatic pathways are followed. These AG*, estimates lie in the range 5 f 0.5 kcal mol-', which when compared with AG*,,(op) as well as the calculated AG*, values supports the predicted absence of stabilizing metal-reactant image interactions. Values of (AG*, - AGlh) for CP,CO+/~exchange are also extracted in nine solvents by combining the electrochemicaland homogeneous rate data; this procedure exploits the additional detailed information that has been obtained recently for the self-exchange preexponential factors (ref 9d). These experimentally derived (AG*, - AG*J estimates are uniformly small and positive ( 5 2 kcal mol-'), in harmony with the theoretical predictions based on the combined effects of metal field penetration and nonlocal solvent electrostatics.
Given the central role of the solvent in determining the energetics of electron-transfer reactions both in solution and at electrodesolution interfaces, the theoretical description of activation energies for these processes is of obvious importance. The dielectric continuum model embodied in Marcus theory has long formed the mainstay for this purpose., According to this treatment, the intrinsic outer-shell activation energies (Le., that in the absence of a thermodynamic driving force) for one-electron reactions in homogeneous solution and at electrode surfaces, AG*h and AG*,, respectively, are given by2
and Here, e is the electronic charge, a is the reactant radius, Rh is the homogeneous reactant internuclear separation, R , is the reactant-electrode distance, and eOpand t, are the solvent optical and static dielectric constants, respectively. However, several more sophisticated theoretical approaches have appeared recently that should provide significantly improved descriptions of the solvent reorganization energetic^.^" A nonlocal electrostatic theory' of solvent reorganization has recently been developed and applied to electron transfer in homogeneous solution3 and at electrode ~ u r f a c e s .This ~ approach provides a convenient framework for incorporating the effect of spatial dispersion of the solvent dielectric permittivity ("solvent spatial correlation") upon the reorganization energies. Apart from providing a more realistic description of the reorganization energetics for "isolated" reactants, Le., in the absence of reactant-reactant or reactantelectrode interactions, nonlocal dielectric treatments have been developed that also account for the necessary proximity of the reacting partner^.^,^ The treatment for homogeneous-phase re+ Dedicated
to the memory of Dr. George E. McManis 111. ( I ) Permanent address: The A. N. Frumkin Institute of Electrochemistry of the Academy of Sciences of the USSR, 117071 Moscow, Leninskii Prosp. 3 I , USSR.
0022- 3654 19012094-1454S02.50I O
actions predicts that AGZhshould usually diminish with decreasing Rh, thereby in qualitative (but not quantitative) accordance with eq 1 . 3 Interestingly, however, the opposite situation can apply commonly to electrochemical reactions in that AG*, is often predicted to increase with decreasing This latter "antiMarcusian" prediction arises from a consideration of field penetration into the metal along with solvent spatial di~persion.~ As a consequence, it appears that eq 2 can at least seriously overestimate the effects of reactant-electrode imaging in diminishing AG*,, and may even be qualitatively incorrect in this respect. Such effects might be expected to have a significant and perhaps striking influence on the kinetics of electrochemical reactions. A general difficulty, however, in extracting effective barrier heights from electrochemical rate data is that independent information is required on the preexponential factor. We have recently been interested in extracting information on the latter component from solvent-dependent rate data for electrochemical*and homogeneous electron-exchange reactions: especially for metallocene redox (2) Marcus, R. A. J . Chem. Phys. 1965, 43, 679. (3) Kornyshev, A. A.; Ulstrup, J. Chem. Phys. Lett. 1986, 126, 74. (4) Dzhavakhidze, P.G.;Kornyshev, A. A.; Krishtalik, L. I. J . Elecfroanal. Chem. 1981, 228, 329. (5) (a) Kuznetsov, A. M. Faraduy Discuss. Chem. SOC.1982, 74,49. (b) Kuznetsov, A. M. Chem. Phys. Lett. 1982, 91, 34. (6) (a) Wolynes, P. G. J. Chem. Phys. 1987, 86, 5133. (b) Rips, I.; Klafter, J.; Jortner, J. J . Chem. Phys. 1988,88, 3246; J . Chem. Phys. 1988, 89, 4188. (7) (a) Kornyshev, A. A. Electrochim. Acta 1981,26, 1. (b) Kornyshev, A. A. Theory of Solvation in The Chemical Physics of Solvation, Part A; Dogonadze, R. R., Kilman, E., Komyshev, A. A., Ulstrup, J., Eds.; Elsevier: Amsterdam, 1985; Chapter 3. (8) (a) Weaver, M. J.; Gennett, T. Chem. Phys. Lett. 1985,113, 213. (b) Gennett, T.; Milner, D. F.; Weaver, M. J. J. Phys. Chem. 1985,89,2787. (c) McManis, G. E.; Golovin, M. N.; Weaver, M. J. J . Phys. Chem. 1986, 90, 6563. (d) Nielson, R. M.; Weaver, M. J. J. Electroanul. Chem. 1989, 260, 15. (9) (a) Nielson, R. M.; McManis, G. E.; Golovin, M. N.; Weaver, M. J. J . Phys. Chem. 1988,92,3441. (b) Nielson, R. M.; McManis, G. E.; Safford, L. K.; Weaver, M. J. J . Phys. Chem. 1989, 93, 2152. (c) Nielson, R. M.; McManis, G. E.; Weaver, M. J. J . Phys. Chem. 1989, 93, 4703. (d) McManis, G . E.; Nielson, R. M.; Gochev, A,; Weaver, M. J. J . Am. Chem. SOC.1989, 111, 5533.
0 1990 American Chemical Society
Nonlocal Electrostatic Effects in Electron Transfer couples, specifically to explore the role of solvent friction in the reaction dynamics. Although such analyses do require some assumptions to be made regarding the solvent dependence of the barriers,*v9collectively they enable at least approximate preexponential factors to be extracted (vide infra). Some of these electrochemical and homogeneous-phase kinetic data, especially for the cobaltocenium-cobaltocene redox couple (Cp2Co+/O; C p = cyclopentadienyl) are utilized here with the objective of ascertaining to what extent the nonlocal electrostatic treatment may provide a better description of the electrochemical barrier energetics than eq 2. One strategy adopted here is to compare the kinetics in corresponding electrochemical and homogeneous reaction environments so to yield information on the differences between AG*, and AC*h. Especially when combined with homogeneous electron-transfer barriers derived from optical data for related biferrocene cations,1° at least semiquantitative information on the electrochemical barriers may be obtained. Also included here are representative numerical results of electrochemical versus homogeneous-phase activation barriers calculated by using the above nonlocal electrostatic approach for systematic variations in solvent and spatial parameters (cf. ref 3) so to illustrate the likely practical consequences of such effects.
Theoretical Framework It is appropriate at the outset to provide a brief discourse on the nonlocal electrostatic treatments for the electrochemical and homogeneous-phase environments utilized here in relation to the simple models embodied in eq 1 and 2. Since the principles of nonlocal electrostatics are detailed elsewhere,’” we shall outline only briefly the necessary analytical results together with some physical explanation. Taking into account the solvent nonlocal polarization, AG*h can be expressed as3J1 AG*h =
-1ez
2*
0
dk
(top’- [r(k)]-I)
where e(k) is the solvent static dielectric function. When the medium is considered to be entirely structureless such that r ( k ) = e,, eq 3 reduces to eq 1; otherwise AG*h is sensitive to the character of e(k). Some consequences for the energetics of optical electron transfer within mixed-valence complexes have been d i s c u s ~ e d . ~ The * ~ ~simplest * ~ ~ approximation for e(k) of a polar liquid corresponds to an exponential decay of the spatial correlation of the molecular orientations, with a characteristic decay length A. No account is made of spatial correlations for higher frequency solvent modes, reflecting the anticipated lack of intermolecular coupling on librational and especially vibrational motions. In this case where e. is the so-called “infinite frequency” dielectric constant in the Debye spectrum (usually measured in the microwave/ far-infrared region), which is considered to separate regions lying toward lower and higher frequencies, where spatial correlation of solvent molecules does and does not occur, respectively. Combining eq 3 and 4 yields hG*h = (e2/r)(€&’ - €*-‘)(U-’ - Rh-’) 4(e2/r)(c.-I - e[I)(a-I4(2a/A)- Rh-’[ 1 - y eXp(-Rh/A)]) (5) where 4(x) = 1 - ([I - exp(-x)l/xl (6) and y = ( L ~ ~ / ~ U ~ ) [ -C 11 ~ ( ~ U / A ) (7)
The Journal of Physical Chemistry, Vol. 94, No. 4, 1990 1455 This result for the case where C. = top was first obtained and analyzed in ref 3. For A + 0, eq 6 reduces to the “Marcus limit” eq 1. While the first term in eq 5 is Marcusian in its dependence upon a and Rhrthe behavior of the second term is quite different when A 2 a,Rh. Overall, under the latter conditions AG*h is diminished with respect to that when A = 0. This reflects the reduced ability of the “structured” polarization to respond to local alterations in the electric field, thus diminishing the energy of solvent repolarization required for electron transfer. Although AG*h is predicted to increase monotonically with increasing reactant separation Rh, this increase is more marked than in the Marcus limit (vide infra). When A >> 2a, the second term in eq 5 can be simplified, so that3
As pointed out in ref 4, the predicted nonlocal electrostatic effects upon the activation energies of electrochemical reactions are notably different from the homogeneous-phase case. Inspection of eq 1 and 2 shows that AG*, is anticipated to diminish with decreasing electrode-reactant distance (Re) in a similar fashion as AG*h decreases for smaller reactant pair separations (Rh) for the homogeneous case. While the inclusion of nonlocal effects upon AG*h, as already noted, does not qualitatively alter this picture, for electrochemical reactions AG*, is often predicted to increase with decreasing Re.4 This striking finding, in qualitative disagreement with the simple reactant-electrode imaging model leading to eq 2, arises from the nonlocality of screening in the metal, yielding finite penetration of the static electric field into the metal phase. The fundamental physics involved has been described in detail.l3-I9 The crucial feature is a “pancake”-shaped countercharge induced in the metal by the presence of the nearby solvated ion. While the thickness of the pancake region is of the order of the Thomas-Fermi screening length, K ~ ~ - its I , characteristic radius, rd, is markedly larger, (C,/CM)KTF-’. Consequently, when the ion-electrode distance Re is noticeably smaller than rd, the contribution of the reduced polarizability to the field strength near the ion is similar to that applied at the boundary between two dielectrics e, and eM. The latter “effective metal” dielectric constant approaches unity (ca. 1-2) due to the low polarizability of the metal ionic skeleton. This situation creates a repulsive image potential for such short ion-metal distances, where Re < rd, with a crossover to the classical electrostatic attraction only at larger separations. As demonstrated in ref 4 and enumerated further here, corresponding effects are predicted upon the solvent reorganization energy for species in the vicinity of metal-solution interfaces. Moreover, inclusion of spatial dispersion effects of the solvent dielectric constant (nonlocal electrostatics) acts to magnify the influence of metal field penetration, so that the combined effects of these presumably more realistic physical treatments yield Re-dependent AG*, values that are markedly different from the conventional dielectric continuum/imaging expression, eq 2. The relevant expressions for AG*, utilized here, combining the metal field penetration and nonlocal electrostatic aspects, are based on the results of ref 4. In view of their cumbersome nature, we summarize them in the Appendix (eq A1-4). (Reproducing them here provides us an opportunity to correct several misprints in the formulas appearing in ref 4.) These results are based on the so-called “sharp boundary approximation”.I*15 While the general
Chem. 1989, 93,1133.
(13) Kornyshev, A. A.; Rubinshtein, A. I. Vorotyntsev, M. A. Phys. Srarus Solidi B 1977, 84, 125. Kornyshev, A. A.; Vorotyntsev. M. A. J. Phys. C 1978, 1 1 , L691. (14) Rubinshtein, A. I. Phys. Srarus Solidi B 1983, 120, 65. (15) Vorotyntsev, M. A.; Kornyshev, A. A. Z h . Eksp. Teor. Fiz. 1980. 78. 1008;SOU.Phys.-JETP (Engl. Transl.) 1980, 78, 509. (16) Vorotyntsev, M.A.; Isotov, V. Yu.; Kornyshev. A. A. Porerkhnosr 1983, .Yo. 7. 97. (17) Vorotyntsev, M. A.; Holub, K. Elekrrokhimiya 1984, 20, 256. (18) For a review, see: Vorotyntsev, M.A. In The Chemical Physics of Solcation, Pan C; Dogonadze, R. R., Kilmln. E., Kornyshev, A. A,. Ulstrup.
(1 1) Dogonadze, R. R.; Kornyshev, A. A.; Kuznetsov, A. M. Teor. M a r . Fir. 1973, I S , 127; Theor. Math. Phys. (Engl. Transl.) 1973, IS, 401. (12) Ulstrup, J. J. Phys. Chem. 1987, 91, 5153.
I.
(10) McManis, G.E.; Gochev, A.; Nielson, R. M.; Weaver, M. J. J . Phys.
J.. Eds.: Elsevier: Amsterdam, 1988; Chapter 7. (19) Kornyshev, A. A.; Schmickler, W. J. Elecrroanal. Chem. 1986. 202.
1456 The Journal of Physical Chemistry, Vol. 94, No. 4, 1990
treatment has been extended to account for a more realistic “smooth decay” of the surface electronic p r ~ f i l e , lthe ~ * approach ~~ used here is deemed satisfactory for the present approximate purposes (vide infra). We also restrict attention here to the estimation of reorganization energies and their comparison with experiment and do not therefore consider electrostatic work terms (or “double-layer” effects) upon the electron-transfer kinetics. The influence of metal image interactions, including field penetration effects, upon the work terms was included in ref 4; however, this contribution is liable to be relatively small, especially in the presence of supporting electrolyte. It is nonetheless desirable to consider the effects of the ionic atmosphere on AG*h and AG*, at the relatively high ionic strengths ( p = 0.1-0.5) commonly utilized experimentally. For the case of homogeneous self-exchange reactions, the contribution to AG*h due to the ionic atmosphere, AG*ia,is given byZo(at least in the absence of spatial dispersionsz1)
Phelps et al.
i
I
\
r3
id 0
3 4.0 4
a
/
I
//5
3 . 0 1 /I
where K D is the inverse Debye screening length. However, the magnitude of this term is relatively small, typically 5 0 . 3 kcal mol-’ even for moderate ionic strengths (k = 0.1) in polar solvents, and need not be considered further for the present approximate purposes.
/
i 3.0
I
1
1
7.0
11.0
15.0
19.0
Re, A
Numerical Consequences of Theoretical Treatment Prior to considering the experimental data, it is instructive to examine illustrative numerical predictions of AG*, in comparison with AG*h as a function of the relevant solvent and spatial parameters. Even though some aspects of this behavior for AG*, have been considered previously,4 the comparative examination of AG*, and AG*h is emphasized here. Figures 1-4 consist of plots of AG*, and AG*h as a function of the reactant-electrode and reactant-reactant distances, Re and Rh, respectively, for various trial solvent parameters, extracted from eq 5-7 and eq A1-4. In each case, KTF-’ was taken as 0.5 and eM (the “dielectric constant” of the ionic skeleton cores) as 2.0; these are typical values, appropriate for mercury (vide infra).** (The small values, typically 1-3, for t M reflect the absence of free electron polarizability in this quantity.zz) The reactant radius, a, was set at 3.8 A, appropriate for the C P ~ C O + / ~ system considered be10w.~~*~ (The smallest R, and Rhvalues shown in the figures, 3.8 and 7.6 A, respectively, correspond to the closest approach of the “reactant pair”.) Curves 1-4 in Figure l a are AC*,-Re plots for increasing values of the solvent correlation length A, specifically 0, 2, 3.5, and 5 A. In each case, top, e*, and t, are fixed at the “typical” values 2.0, 3.0, and 40, respectively. Curve 5 is that obtained for the “Marcus” limit where KTF-I and A equal zero, Le., in the absence of metal field penetration and solvent correlations, whereupon eq 2 will apply. Comparison of curve 5 with curve 1 illustrates the substantial muting of the reactant-electrode image effect, Le., AC*, decreasing for smaller Re, brought about by consideration of electrode field penetration. Moreover, the introduction of any significant solvent correlations, obtained by setting A > 0 (curves 2-4), entirely eliminates any vestige of such Marcusian behavior, AG*, values that increase sharply with decreasing Re being obtained instead (cf. ref 4). Figure 1b shows plots of AG*h upon Rh for the same sequence of A values ( A = 0, 2, 3.5, 5 for curves 1-4, respectively). Even though the AG*, values at a given reactant internuclear distance are predicted to decrease substantially as A increases, in contrast to the corresponding AG*,-R, behavior the morphology of the L?&*h-Rh curves does not differ greatly from the Marcus limit ( A = 0, curve l ) , corresponding to eq 1 . (20) German, 8. D.; Kuznetsov, A. M. Elekrrokhimiya 1987, 23, 1671. (21) Accounting for solvent spatial dispersion may enhance AG*is if K ~ - ’ I;A; however, eq 9 should strictly be valid when K ~ - ’ >> A. (22) For a discussion of cM values for various metals, see the Appendix in: Vorotyntsev, M. A,, Kornyshev, A. A. Efektrokhimiya 1984, 20, 3
-
d
4
0
E
i
(d
0
3 f
C
(3
a
I‘ 7.0
,
I
I
I
11.0
15.0
19.0
23.0
Figure 1. (a) Illustrative plots of the outer-shell free energy of activation for electrochemical exchange, AG’,, versus the metal-surface-reactant separation, Re, as extracted from eqs A2-A4 for various values of the solvent “correlation length”, A, as follows: curve I , A = 0; curve 2, 2 A; curve 3, 3.5 A; curve 4, 5 A. Assumed values of other parameters: K+I = 0.5 A, a = 3.8 A, c M = 2.0, L = 2.0, tr = 3.0, ts = 40. The dashed ?P ~ A -equal l zero, curve 5 refers to the ‘Marcus limiting” case where ~ ~ and whereupon eq 2 will apply (see text). (b) Illustrative plots of the outer-shell free energy of activation for homogeneous self-exchange, AG*h, versus the reactant pair internuclear distance, Rhrfor various values of the solvent correlation length, A, as determined from eq 5. Parameters for curves 1-4 as noted in a.
Figure 2a illustrates the effect of altering the static solvent dielectric constant, e,, upon the AG*,-Re dependence. Curves 1-3 are obtained for increasing e, values, 20,40, and 80; the solid and dashed curves (1-3 and 1’-3’, respectively) refer to A values of 0 and 5 A, respectively. The other parameters are as in Figure la. These results show clearly that the deviation from the AG*,-R, dependence predicted from eq 2 increases markedly with increasing
The Journal of Physical Chemistry, Vol. 94, No. 4, 1990 1457
Nonlocal Electrostatic Effects in Electron Transfer a
I
I
I
I
\
i
\i \
I. \
\
i
\
\
\
\
-
\
6.0
1-
3
(d
\
0
5.0
“I 3.0
5.0
9.0
13.0
Re
I
b
3
(d
0
24
c
6.0
I
I
3.0
17.0
13.0
9.0
5.0
17.0
Re, A
9
I
i 7.0
I
1
I
I
11.0
15.0
19.0
23.0
R,,
A Figure 2. (a) Plots of AC*, versus Re as calculated from eqs A2-A4 for various solvent static dielectric anstants: curves 1 and l’, 4 = 20; curves 2 and 2’, 40; curves 3 and 3’, 80. The solid and dashed (Le., primed) curves refer to A values of 0 and 5 A, respectively. Other parameters are as in Figure la. (b) Plots of AG*h versus Rh as calculated from eq 5, for various solvent static dielectric constants, as in a. Even in the complete absence of solvent spatial correlations (Le., A = 0), increasing c, to 80 (curve 3) virtually removes any depression of AG*, due to electrode imaging; moreover, substantial anti-Marcusian behavior is observed under these conditions for A > 0 (curves 1’-3’). This again demonstrates how the inclusion of solvent spatial correlations enhances markedly.the extent of the deviations in the AG*,-Re dependence from eq 2. The solid and dashed traces in Figure 2b (curves 1-3 and 1’-3’, respectively) show the corresponding AG*h-Rh dependencies under these conditions. Interestingly, in contrast to the electrochemical case, AG*,, is insensitive to t8,especially for small Rhand/or larger A. These marked behavioral differences with Figure 2a reflect the sensitivity of AG*, to e,, being due primarily to the dependence t,.
Figure 3. (a) Plots of AG*, versus Re as calculated from eqs A2-A4 for various values: 2.5, 3.5, 4.5 (curves 1-3), with A = 5 A. Other parameters as in Figure l a . (b) Plots of AG* versus Rh as calculated from eq A I , for various C. values and A = 5 as in a. €0
.k,
of the ‘induced image” charge radius, rd, upon t, (vide s ~ p r a ) . ~ It is also of interest to examine to what extent variation in t. alters such profiles in the presence of solvent spatial correlations. Parts a and b of Figure 3 display representative AG*,-Re and AG*h-Rh curves, respectively, for three t. values, 2.5, 3.5, and 4.5 (curves 1-3), with A set at 5 A. The other parameters are as for Figure la,b. Once again, marked (and even qualitative) behavioral differences are observed in the electrochemical and homogeneous-phase environments. Thus, while AG*hdecreases with diminishing toat all Rhvalues (Figure 3b), for small R,values ( R e < 9 A) AG*, exhibits the opposite dependence (Figure 3a). This behavior could be anticipated in part from Figure la,b, in that decreasing t. at a given A is tantamount to expanding the frequency range over which the various solvent motions that contribute to the dielectric polarization are spatially correlated.
1458 The Journal of Physical Chemistry, Vol. 94, No. 4 , 1990
Phelps et al. 5.0
I
I
A
1 00
’0
20
30
4 0
_--
5c
A, A Figure 4. Plots of the difference between corresponding outer-shell ac-
tivation free energies for electrochemical exchange and homogeneous self-exchange, AG*, - AGSh, calculated from eqs A2-A4 and eq 5 versus the solvent correlation decay length, A, for e* values of 2.5, 3.5, and 4.5 (curves 1-3, l’-3’). The solid and dashed traces refer to Re values of 4 and 8 A; Rh was taken as 7.6 A (=2a). Other parameters are as in Figure la. Trial AG*,-Re calculations were also performed for differing values of the metal-ion core dielectric constant tM. However, varying t M over the range ca. 1-3 anticipated to be appropriate for sp metals22typically yields changes in AG*, of ca. 10% or less. An important general consequence of the marked differences in the sensitivity of AG*, and AG*h to solvent parameters, especially for small Re and Rh, is that the relative rates of corresponding outer-sphere electrochemical and homogeneous-phase reactions may differ substantially from predictions derived from eq 1 and 2. As an illustration, Figure 4 contains plots of AG*, - AG*h versus A for three t o values, 2.5, 3.5, and 4.5 (CUrveS 1-3, 1’-3’), to and es being fixed as before at 2 and 40, respective1 . The soliJ and dashed traces refer to Re values of 4 and 8 , respectively: these cases correspond to the reactant essentially in contact with the electrode and separated from it by a layer of solvent molecules. While the latter circumstance is often expected for outer-sphere electrochemical reactions,23the former corresponds to the maximum influence of electrodereactant interactions. The Rh value is taken as 7.6 A in both cases, corresponding to the reactants being in contact (cf. ref 9). The corresponding AC*, - AG*h values predicted from eq 1 and 2 are -2.6 and -1.2 kcal mol-’ for Re = 4 and 8 A, respectively. Figure 5 shows the dependence of AG*, upon cs for three A values, 0, 1, and 5 (curves 1-3, 1’-3’), top and t * being taken to be 2 and 3, respectively. As in Figure 4, the solid and dashed traces refer to R, = 4 and 8 A, respectively. Inspection of Figures 4 and 5 shows that AG*, - AG*h values that are markedly larger than the Marcus predictions are generally obtained on the basis of the nonlocal treatment. Not surprisingly from the foregoing, the AG*, - AGZhvalues are seen to increase with increasing A and tsand with decreasing 6.. Significantly (vide infra), small and positive AG*, - AG*h values are obtained under most circumstances. In particular, the presence of even a minor extent of solvent spatial correlations, such that A 5 0.5-1, is sufficient to yield AG*, 5 AG*h, rather than AG*, < AG*h as predicted from eq 1 and 2. In other words, the nonlocal electrostatic treatment predicts that the metal-reactant imaging interactions will usually not yield significantly smaller intrinsic electrochemical barriers than for corresponding homogeneousphase reactions, in contrast to the predictions of eq 1 and 2.
k
(23) For example, see: Weaver, M . J. J . Phys. Chem. 1980, 84, 568
I
I
I
20.0
40.0
60.0
80.0
E Figure 5. Plots of AG*, - AG*,, calculated from eqs A2-A4 and eq 5 versus the solvent static dielectric constant, cs, for A values of 0, 1, and 5 (curves 1-3, 1’-3’). The solid and dashed traces refer to Re values of 4 and 8 A; Rh was taken as 7.6 A (=2a). Other parameters are as in Figure la.
Analysis of Experimental Kinetics It is clearly of interest to examine the solvent-dependent kinetics of corresponding electrochemical and homogeneous-phase processes in light of these findings. As noted above, an obvious complication in inferring effective free-energy barriers from electron-exchange kinetic data is that it is necessary to assume, or preferably have independent information on, the relevant preexponential factors. In general, we can express the work-corrected rate constants for homogeneous self-exchange and electrochemical exchange reactions, kh and k,, respectively, as24325 eXp[-(AG*h
+ AG*h,is - q : ) / R T ]
(10)
k, = KC~K$Y, exp[-(AG*,
+ AG*e,i,- H ; , ) / R T ]
(11)
kh = q K $ ,
and Here, K ! ~ and K : ~ are the corresponding electronic transmission coefficients, Y , is the nuclear frequency factor (s-l), AG*h,isand AG*e,isare the corresponding inner-shell (Le., reactant intramolecular) contributions to the intrinsic barriers, and H;, are the barrier diminutions caused by donor-acceptor electronic coupling (the “electronic coupling matrix elementsngd),and f$ and reflect the statistical probabilities of forming the “precursor complexes” (Le., the interreactant and reactant-electrode geometries most conducive to electron transfer). The latter are given approximately by24325
Khp = 4.rrNRh28rh
(12)
K i = 6re
(13)
and where 6f-h and 6r, are the appropriate “reaction-zone thicknesses” within which electron transfer can occur. Strictly speaking, a range of precursor complex geometries can contribute to both kh and k, so that the overall electron-transfer (24) Hupp, J. T.; Weaver, M. J. J. Electround. Chem. 1983, 152, I . ( 2 5 ) Sutin, N. Prog. fnorg. Chem. 1983, 30, 441.
Nonlocal Electrostatic Effects in Electron Transfer rate is comprised of an integral of reaction sites, each associated with "local" (unimolecular) rate constants, kct.24*2JThe form of the k,,-R integral reflects primarily variations in K , ~ , AG*, and H I 2 . Since H I , and hence K , ~will inexorably decrease markedly with increasing R, however, the spatial range of reaction sites contributing to both kh and k , is usually expected to be quite narrow. In this case, the complex integral can be replaced by the simple "encounter-preequilibrium" formalism embodied in eq 12 and 13, where 6rhand 6r, denote the effective thicknesses of the spherical and linear reaction zones, respectively, reflecting the major portion of the k,,-R integrals. While it is sometimes desirable to utilize instead the full integral form (e.g., for rate-solvent friction ana lyse^^^*^^^), the simpler preequilibrium formulation is adequate for the present approximate purposes. Anticipating the discussion below, while the precursor-complex geometries contributing predominantly to kh should correspond to Rh i= 2a, where AG*h is minimized (as well as H I , and K , ~maximized), k, may well arise chiefly from precursor geometries where Re > a in view of the smaller AG*, values expected under these conditions. The cobaltocenium-cobaltocene ( C ~ , C O + / ~redox ) couple is selected here for detailed consideration for several related reasons. Kinetic data for this system as well as for related metallocene couples have been obtained both in homogeneous solution9 and at mercury electrcdessb$cin a range of polar solvents. Detailed analyses of these solvent-dependent kh and k, values have led to the identification of preexponential factors in both environments that depend upon the solvent dynamics as gauged approximately by 7L-1,where 7Lis the longitudinal solvent relaxation This treatment has proven to be most complete (and reliable) for homogeneous self-exchange, in part since the solvent dependence of AC*h necessary for the solvent dynamical analyses can be extracted from optical electron-transfer energies, Eop,for related biferrocene cations (vide infra).9wv10 Moreover, a recent examination of the solvent-dependent self-exchange kinetics for a series of six metallocene couples, including Cp,Co+/O, has uncovered systematic variations in the degree to which ~ : p ,depends upon 7L-1.9d For the most facile couples, such as CP',CO+/~(Cp' = pentamethylcyclopentadienyl), approximately K$, 0: 7L-1.w This behavior is indicative of reaction adiabaticity, whereupon K , ~ 1 in each solvent, since it is expected for these reactions that u, a T ~ - I . For the least facile systems (e.g., Cp,Fe+/O), in contrast, the combined preexponential factor K:~v, is virtually independent of 7L-1,Windicating nonadiabatic behavior (Le., K : ~ > a; such estimates have been utilized in some analyses of solvent dynamical effects in electrochemical kinetics (vide infra).* Also listed in Table I11 are the corresponding calculated differences in activation free energy, AG*, - AG*h, with AG*h obtained as before, i.e., from eq 5 for Rh = 2a = 7.6 A. This choice of Rh reflects the expectation that the AG*,-Rh profiles generally should favor self-exchange reaction geometries that correspond to close approach of the reaction centers (vide supra). Of central concern here is the comparison between these calculated values of AG*, and AG*, - AG*h in Table 111 with the corresponding "experimental" values extracted from the kinetic data (Table I). Several significant points emerge from a detailed scrutiny of Tables I and 111. The experimental AG*, values are comparable to the corresponding calculated quantities in the majority of solvents, although the former tend to be larger, especially in benzonitrile and TMU. However, given that similar disparities are also noted between AG*h(op) and AG*h(eq 5) in benzonitrile, for example (Table 11), these differences are not disconcerting. Indeed, the experimental AG*, values (Table I) vary with the solvent to a smaller extent than the theoretical predictions, in a manner that mimics the AG*h(op)behavior (Table 11). As might be anticipated from the "model solvent" calculations discussed above, the inclusion of spatial correlations by setting A = rsolrather than A = 0 does not alter greatly the calculated AG*, values in most cases. This is because the presence of metal field penetration tends to offset the decreases in AG*, predicted in the presence of only the solvent correlations. Another significant feature of Table Ill is that while the calculated values of AG*, tend to be larger for Re = 4 than for Re = 8 A, at least in the presence of spatial correlations, the differences are not large (
