J . Phys. Chem. 1990, 94, 8608-8613
8608
An Alternatlve Approach to the Intetcomparlson of Electron-Transfer Reactlvitles at Metal Surfaces and In Homogeneous Solutiont Michael J. Weaver Department of Chemistry, Purdue University, West Lafayette, Indiana 47907 (Received: April 9, 1990)
A procedure is outlined for relating and comparing the outer-sphere electroreduction (or electrooxidation) kinetics for a given half-reaction with the rates for corresponding homogeneous-phase reactions involving reversible one-electron reagents. This involves evaluating the work-corrected electrochemical rate constant, k&, at an electrode potential equal to the formal potential of the homogeneous coreacting couple and correcting the corresponding homogeneous-phase rate constant, kkr, for any inner-shell reorganization of the coreacting redox couple. The influence of the disparate geometries of the electrode and homogeneous-phase 'coreactants" upon kzorand k k r can be removed conveniently by transforming kLr into an "equivalent second-order" rate constant, kgr (M-' s-l). The Occurrence of large disparities between k z r and kk, signals the presence of effects exerted by the electrochemical and/or homogeneous-phase environments that are beyond those expected from conventional adiabatic continuum treatments. This procedure applied to the electroreduction of transition-metal ammine and aquo complexes at mercury in aqueous media in comparison with their homogeneous reduction by Ru(NH,)?+, Cr(bpy)?+ (bpy = 2,2'-bipyridine), and related complexes shows that the former are markedly (up to 104-fold)more facile. These enhanced electrochemical reactivities are speculated to be due primarily to the occurrence of more efficient electron tunneling at the metal surface compared to the intermolecular redox environment. Illustrative applications to oxygen reduction in aqueous solution and to electron-exchange reactions are also briefly outlined. The particular applicability of the present procedure to the analysis of chemically irreversible redox reactions is emphasized.
the corresponding theoretical rate ratios can yield much insight into particular additional factors that influence experimental systems. This is because various components of the theoretical expressions can thereby be arranged to be held constant and hence will cancel, when suitable relative rates are considered. We discuss here alternative strategies for providing relative rate comparisons for electrochemical with respect to homogeneousphase electron-transfer reactions, and outline a new formalism which offers some significant advantages, especially for chemically irreversible processes. An illustration of its usefulness is furnished in the identification of some unexpected differences between outer-sphere reaction energetics in these two types of redox environment.
Introduction The study of environmental factors in electron-transfer kinetics at metal surfaces has long been a topic of fundamental significance in electrochemistry, as has the examination of corresponding factors for homogeneous-phase redox processes in conventional chemical kinetics. Given the close relationship between the energetics of electron transfer in heterogeneous- and homogeneous-phase environments, the comparison of reaction rates involving a given redox couple in these markedly different types of reaction environment is clearly of interest in this regard. Such experimental comparisons have so far utilized primarily the following simplified relations extracted from the weak-overlap adiabatic approach of Marcus:' k:,/AC = ( k : x / A h ) t / 2
Equation 1 relates the rate constants for electron exchange of a given redox couple at a metal electrode (Le., at the standard potential) and for self-exchange in homogeneous solution, k:, and k:,, respectively, where Ae (cm s-l) and Ah (M-l s-l) are the corresponding electrochemical and homogeneous-phase frequency factors. Equation 2 refers to the rate constants for a pair of electrochemical reduction (or oxidation) reactions, kf and k;, at the Same electrode potential in comparison with the rate constants kt and k$ for the Corresponding processes in homogeneous solution involving a fixed reducing (or oxidizing) agent R. Broadly speaking, these and allied relations have been found to be roughly in accordance with experimental data,* although significant and even substantial disparities are not uncommon. Such relationships can be viewed as examples of "relative" theory-experiment comparisons, as distinct from "absolute" comparisons which involve the theoretical calculation of kinetic parameters for individual reactions in a given e n ~ i r o n m e n t . ~ , ~ While the latter can provide a demanding test of the underlying theoretical models, the former are less likely to exhibit deviations between theory and experiment. This is due to the cancellation of terms in the theoretical expressions when the kinetics of closely related processes are compared, as in eqs 1 and 2. On the other hand, examination of the relative rate parameters for judiciously chosen series of allied reactions, or as a function of system state (such as temperature or electrode potential), in comparison with 'Dedicated to the memory of Dr. George E. McManis 111.
0022-3654/90/2094-8608$02.50/0
Theoretical Background The observed rate constants for one-electron-transfer processes, kob, can be related to the activation free energy, AG', for the overall reaction by4,' kob = K,v,K,,r, exp(-AG*/RT)
(3)
where Kp,is the equilibrium constant for forming the precursor state, v, IS the nuclear frequency factor, K ~ is , the electronic transmission coefficient, and rnis the nuclear tunneling factor. At least for outer-sphere processes, it is generally useful to consider ( I ) (a) Marcus, R. A. J . Phys. Chem. 1%3,67,853. (b) Marcus, R. A.; J. Chem. Phys. 1965, 43, 679. (c) Marcus, R. A. In Special Topics in Electrochemistry; Rock, P. A., Ed.; Elsevier: Amsterdam, 1977; pp 161, 180, 210. (2) (a) Forno, A. E. J.; Peover, M. E.; Wilson, R. Trans. Faraday SOC. 1970,66, 1322. (b) Peover, M. E. In Reactions of Molecules at Electrodes; Hush, N. S., Ed.; Wiley: New York, 1971; p 259. (c) Hale, J. M. Ibid. p 229. (d) Endicott, J. F.; Schroeder, R. R.; Chidester, D. H.; Ferrier, D. R. J. Phys. Chem. 1973, 77, 2579. (e) Kojima, H.; Bard, A. J. J. Am. Chem. SOC.1975, 97, 6317. (f) Saji, T.; Yamada, T.; Aoyagui, S.J. Electroanal. Chem. 1975,61, 147. (9) Saji, T.; Maruyama, Y.;Aoyagui, S.J. Electroanal. Chem. 1978,86,219. (h) Endicott, J. F.; Taube, H. J. Am. Chem. Soc. 1964, 86, 1686. (i) Satterberg, T. L.; Weaver, M. J. J. Phys. Chem. 1978.82, 1784. 6)Weaver, M. J. J. Phys. Chem. 1980,84,568. (k) Weaver, M. J.; Tyma, P. D.; Nettles, S. M. J . Elecrroanal. Chem. 1980, 114, 5 3 . (I) Weaver, M. J.; Hupp, J. T. ACSSymp. Ser. 1982, 198, 181. (m) Li, T. T.-T.; Weaver, M. J. Inorg. Chem. 1985, 24, 1882. (3) Hupp, J. T.; Weaver, M. J. J . Phys. Chem. 1985,89, 2795. (4) (a) For a review, see: Weaver, M. J. In Comprehensioe Chemical Kinetics; Compton, R. G., Ed.; Elsevier: Amsterdam, 1987; Vol. 27, Chapter 1. (b) Sutin, N. Prog. Inorg. Chem. 1983, 30, 441. ( 5 ) (a) Hupp, J. T.; Weaver, M. J. J. ElectroanaL Chem. 1983, 152, I . (b) Sutin, N. Prog. Inorg. Chem. 1983, 30, 441.
0 1990 American Chemical Society
Electron-Transfer Reactivities at Metal Surfaces
The Journal of Physical Chemistry, Vol. 94, No. 23, 1990 8609
so-called "work-corrected" rate constants, k,, that would be observed in the absence of the free-energy changes associated with assembling the precursor and successor states ("work terms"). These can be expressed as4vS k,,, = KOu,KJn exp(-AG*/RT) (4) where KO is the statistical part of K , (Le., that describes the probability that the precursor state will form in the absence of work terms) and AG* is the activation free energy for the elementary electron-transfer step, also corrected for work terms (see eq 7a of ref 4a). Let us consider a corresponding pair of electrochemical and homogeneous processes (i.e., those involving a common redox couple):
+ e-(electrode) Red, Oxl + Red2 Red, + Ox2
Oxl
F?
F?
(5a) (5b)
The k,,, values for these processes can be expressed as k:,, = KE'A:exp(-AG*,/RT)
(6a)
ktor = K ~ ' AeXp(-AG*h/RT) ;
(6b)
where Kou,r, is now expressed as the combined "nuclear frequency factor" A,, and the superscripts and subscripts e and h again refer to electrochemical and homogeneous-phase processes, respectively. The simplest relationship between AG*, and AG*h involves the comparison between electrochemical exchange and homogeneous self-exchange reactions since both these activation free energies then equal the corresponding intrinsic barrier, AI?*^,, and AG*i,h, respectively, and a chemically identical pair of redox couples are involved in the latter process. The general relation between AG*i,eand AG*,,haccording to the dielectric continuum treatment' can be expressed a ~ ~ 9 2AG*i,, = AG*i,h C (7a)
+
where
Here e is the electronic charge, rh and re are the interreactant and reactant-electrode distances in the precursor states for the homogeneous and electrochemical processes, and cop and c, are the optical and static dielectric constants, respectively. In view of eqs 4 and 7, we can writes
2 log (k:x/K:IAz)
N is Avogadro's number and 6r, and 6rhare the "effective reaction-zone thicknesses" for the electrochemical and homogeneous-phase processe~,~-~ and by noting that u, and r, should be very similar for related electrochemical and homogeneous proc e s ~ e s .We ~ ~ can thereby rewrite eq 9 as log (4*Nrh2kEOr)- log ktOr= log (Kz16fe/Kz'6rh)+ (AG*h - AG*,)/2.3RT (10) The factor 4irNrh2in the first term in eq 10 transforms k&, into an "equivalent second-order" rate ~ o n s t a n thaving ,~ the same units as k:,, and corresponding to the (hypothetical) value of kh, that would be obtained if the electrode had a radius, rh, equal to that of the coreactant in the homogeneous reaction. Considering the electrode as a coreactant in this fashion suggests that an especially direct comparison between rate constants for corresponding electrochemical and homogenous reactions (eqs 5a and 5b) involves evaluating the former at an electrode potential equal to the standard (or formal) potential, EJ, of the coreacting species for the latter process (Le., for the Red2/Ox2 couple in eq 5b). Under these conditions, the free-energy driving force for the two processes involving a common redox couple (Ox,/Red, in eqs 5a and 5b) will be equal. The difference between AG*, and AG*h appearing in eq 10 will then reflect primarily6 differences in the intrinsic barriers, AG*i,cand AG*i,h, for the two processes. The precise relationship between AG*i,eand AG*i,hunder these conditions depends on the contribution to the latter term from the activation energy component, AG*i,h2,attributable to the homogeneous coreacting couple Red2/Ox2. On the basis of the dielectric continuum treatment,' the portion of (AG*i,h- AG*+) arising from outer-shell reorganization, (AC*,,h - AG*,,J, IS ~
~
where a2 is the radius of the coreacting redox couple. The component of (AG*i,h - AG*i,e) due to inner-shell reorganization, (AG*is,h- AG*is,e),will simply equal the contribution to AG*is,h from the Red2/Ox2couple, AG*is,h2.(This is because the component of AG*is,hdue to the Ox,/Red, couple will be identical for the homogeneous and electrochemical reactions and the electrode coreactant does not contribute to the activation barrier.) Equation IO therefore becomes
= log ( k k x / K $ f !-) (C/2.3RT)
(8) where k:, and k$ are now the corresponding work-corrected rate constants for electron exchange and self-exchange, respectively. As noted in ref 5a, eq 8 represents a more general form of the simplified expression eq 1 above, the latter referring to the special case of the former for which C = 0. Another insightful, albeit unconventional, way of comparing the energetics of such electrochemical and homogeneous processes involves perceiving the electrode as a special type of coreactant having a zero intrinsic barrier, continuously variable redox potential, and infinite radius. This suggests that electrochemical rate constants can be compared directly with those for homogeneous reactions involving a single common redox couple by accounting for the differences in KO associated with the different reaction site geometry along with the additional inner-shell activation energy for the latter processes that is associated with the homogeneous reaction partner. This alternative approach forms the focus of the present paper. For a corresponding pair of electrochemical and homogeneous reactions (eqs 5a and 5b) we can relate kC,, and k:or generally by
This unwieldy expression can be transformed and simplified somewhat by assuming that ro= 6re and Kfj = 4TNrh2bfh,where
Equation 12 suggests a novel yet relatively straightforward means of comparing rate constants for corresponding electrochemical and homogeneous reactions. It indicates that the rate constant for the electrochemical reaction, k:,, when evaluated at a potential E4 is linked closely to k k , once the differences in the precursorstate probabilities and, primarily, in the additional inner-shell activation energy for the latter attributable to the coreacting redox couple, AG*i,h2,are taken into account. Note that the standard potential and intrinsic barrier for only Red2/Ox2 are required rather than additionally for the common redox couple, Ox,/Red,. Consequently, the analysis is applicable to chemically irreversible processes for which @and AG*i for Ox,/Red, are unknown. This important property is exploited in the illustrative applications described below. Two special cases of eq 12 are worth noting. Firstly, for coreactant couples for which the inner-shell bond distortions are very small, it f o ~ ~ o wthat s AG*i,h2= 0. If in addition 2a2 = rh, and (6) Strictly speaking, the contribution to AG*, and AG*h from the thermodynamic driving force (at a given AGO) will be slightly unequal since this term also depends upon the intrinsic barriers, AG*i,eand AG*i,h, which will usually differ for corresponding electrochemical and homogeneous reactions.' However, this difference should be small or entirely negligible for the systems considered here: in most cases we anticipate that AG*ie AG'1.h and/or the driving force -AGO is insufficient for the effect to be significant.
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The Journal of Physical Chemistry, Vol. 94, No. 23, 1 9'90
4r, >> rh (which should often be approximately the caseb), then the bracketed terms on the right-hand side of eq 12 disappear entirely, yielding
For adiabatic pathways, when kf16re= k,h,6rh,eq 13 takes the extremely simple form 4rNrh2kLr= ktor
(14)
Equation 14 emphasizes the close connection between the energetics of corresponding electrochemical and homogeneous reactions anticipated when the coreactant for the latter as well as for the former process (i.e., the electrode) does not contribute to the inner-shell barrier. These relations can also be applied to homogeneous-phase processes for which AG*i,hz # 0 by employing in lace of k:or a "coreactant inner-shell corrected" rate constant, P obtained from k:or by using
This presumes, of course, that the appropriate structural data are available so that the coreactant inner-shell barrier can be calculated with sufficient accuracy. A second, more familiar, special case of eq 12 concerns the relative rate constants for a pair of electrochemical reactions, kf and k5, in comparison with the rate constants for corresponding homogeneous reactions employing a fixed coreactant, ki and k!. Since the AG*j,h2 term in eq 12 will thereby remain constant and as the following, outer-shell, term should be small (or at least invariant), we can recover eq 2 providing that the electronic transmission coefficient terms also cancel when such rate ratios are considered. It should be noted, however, that such rate ratios tend to cancel any underlying disparities in energetics, etc., between the electrochemical and homogeneous-phase reactions, only exposing such factors when they vary substantially within the series of reactions examined. Equations 12-14, on the other hand, provide a direct comparison of the electron-transfer energetics for a giuen redox couple in allied electrochemical and homogeneous-phase environments. I t is also worth commenting on the relationship between the present simplified expression eq 14 and the well-known relation eq I . The latter, of course, refers specifically to electron-exchange processes. The square root in eq 1 arises from the presumption that 2AG*i,e= AG*i,h. According to eq 7, this equality will be valid when the reactant directly contacts the electrode surface, so that 2re = rh, whereby C = 0. If the reaction site lies further from the electrode so that reactant-metal imaging can be neglected (vide infra), then the outer-shell components of AG*i,eand AG*i,h become equal. In the absence of significant inner-shell activation, then as for eqs 13 and 14 above AG*i,e= AG*i,h,whereby (cf. eq 1)
kE,/Ae = k),/Ah
This relationship, derived originally by Hush,' is also wellknown.2a- In essence, it represents a special case of eq 14 for exchange reactions when the preexponential factors Ae and Ah are expressed in terms of the preequilibrium formalism embedded in eq 4. The practical virtue of the relationships embodied in eqs 12-14 is that they provide interesting tests of the extent to which the differing reactivities of a given redox couple in related electro( 7 ) Hush, N . S. Elecrrochim Acta 1968, 13, 1005. (8) Patel, R . C.; Endicott, J . F. J . Am. Chem. SOC.1968, 90, 6364. (9) Fan, F-R.F.; Gould. E. S . Inorg. Chem. 1974, 13, 2647. ( I O ) Candlin, J . P.;Halpern, J.; Trimm, D.L.J . Am. Chem. SOC.1964,
86. 1019. ( I I ) Weaver, M . J . J . Electroanal. Chem. 1978, 93, 23 1. (12) Sahami, S . ; Weaver, M. J . J . Elecrroanal. Chem. 1981, 124, 3 5 . (13) Gennett. T.;Weaver, M. J . Anal. Chem. 1984, 56, 1444. (14) Tyma, P. D.;Weaver, M. J . J . Elecrroanal. Chem. 1980, I l l , 195.
chemical and homogeneous-phase reaction environments can be accounted for in terms of adiabatic continuum outer-sphere models. They are clearly most useful when the inner-shell barrier for the homogeneous coreacting couple is negligible or small; fortunately, this is often the case. The presence of other factors not included in the theoretical treatments, such as variations in precursor stabilities or in the degree of reaction nonadiabaticity between the different reaction environments, will be manifested as deviations of the experimental reactivities from the relative values predicted by these relationships. Some illustrative reactivity comparisons along these lines will now be presented. Illustrative Examples Reactions between Inorganic Metal Complexes. One-electron reactions between pairs of metal complexes in aqueous solution have played a central role in the development of our understanding of outer-sphere redox reactivityS6Studies involving metal complexes at electrodes have also contributed significantly to analogous developments in electrochemical kinetic^.^ Not surprisingly, then, such systems have been featured in a detailed "absolute" comparison between theory and experiment on a unified basis for electrochemical and homogeneous-phase reacti0ns.j Such examinations are restricted inherently to reactions involving chemically reversible redox couples, Le., for which formal potentials are available. Many reactions of fundamental interest, however, involve chemically irreversible couples. Interesting examples are high-spin Co(III)/(II) couples for which extensive rate data for irreversible Co(11I) reductions have been gathered at electrodes as well as in solution. As noted above, the present analysis is applicable to such processes providing that E{ and AG*i,h2for the homogeneous coreactant are known. Table I contains a comparison of rate constants, kkr, for 19 homogeneous-phase reactions involving transition-metal aquo, ammine, or bipyridine complexes in aqueous solution, including nine Co(II1) ammine reductions, with corresponding electrochemical rate constants, kLr, obtained at the mercury-aqueous interface (see Table I and ref 3 for details of work-term corrections and data sources). The latter values are determined at the formal potential, E{, for the coreacting redox couples. This coreactant is portrayed uniformly as the reductant in Table I, so that the electrochemical reactions involve in each case the homogeneous-phase oxidant (Le., are electroreductions). The far right-hand pair of columns consist of rate constants kEr and kzr derived with eqs 13 and 14 in mind. The former quantity was obtained from the corresponding kLr values by using eq 15. The required values of AG*, were estimated as outlined in ref 3; these are15C r ( b ~ y ) ~ ~and + / ~O+~ ( b p y ) , ~ + (bpy / ~ + = 2,2'-bipyridine), =O kcal mol-'; R u ( N H ~ ) ~ ~ +0.35 / ' + , kcal mol-'; R u ( O H ~ ) ~ ~ + / ~ + , 1.8 kcal mol-'. The quantity k z r is the corresponding equivalent second-order electrochemical rate constant, equal to 4?rNrhZkLr. The required values of rh were chosen so as to be appropriate for the homogeneous-phase reaction with which kLr is to be compared. Thus rh is taken to be ca. 6.5 A for reactions involving only aquo and/or ammine complexes, and 10 A for those involving aquo/ ammine and bipyridine complexes. Comparison of the corresponding k z and kZr values listed in Table I reveals that the latter are systematically ca. 10-104-fold larger than the former, in contrast to the prediction that k g = k z r (eq 14). The degree of disparity between these homogeneous and electrochemical rate parameters is apparently dependent on the nature of the ligands, and possibility also on the metal cation. Thus for reactions involving ammine complexes (mostly involving Co(II1) ammines), k t r (102-104)kEr,whereas for those involving aquo complexes the disparities between kzr and kEr tend to be somewhat smaller, ca. IO-fold (Table I). This comparison is also provided in graphical form in Figure 1 (see the figure caption for explanation of symbols).
-
( 1 5) Note that the AG*i,h2values are essentially equal to one half the intrinsic inner-shell barrier for the corresponding self-exchange reaction. This is because only one redox center is involved in the cross reaction, rather than a pair of chemically identical centers as in self-exchange.
The Journal of Physical Chemistry, Vol. 94, No. 23, 1990 8611
Electron-Transfer Reactivities at Metal Surfaces
TABLE I: Comparison of Rate Constants for Reactions Involving Cationic Metal Complexes at Mercury-Aqueous Interface and in Homogeneous Aqueous Solution at 25 O C system no. I 2
E:,"
IO
reductant mV vs S C E Ru(NH3):+ -185 Co(N H3),It R u ( N H ~ ) ~ ~ + -185 CO(NH,)~OH?+ -185 C O ( N H ~ ) ~ F ~ + RU(NHJ)~~+ Co(N H 3 ) 5 0 A ~ Z' t R u ( N H ~ ) ~ ~ + -185 Cr(bpy)32t " -480 Co(NHj)," -480 C O ( N H ~ ) ~ O H ~ ~Cr(bpy)32t + Cr(bpy)32t -480 Co(N H1)5F2t -480 C O ( N H ~ ) ~ O S O ~ + Cr(bpy)32t -480 C O ( N H ~ ) ~ O A C ~ +Cr(bpy)32f ~ Cr(bPy),2t -480 Co(en)33+
11 12 13 14 15
RU(NH&~+ Ru(OH2)$+ Fe(OH2)63t V(OH2)63+ Co(bpy)3'+ "
16 17 18 19
V(OH2)b3+
3 4 5 6 7 8
9
oxidant
~(OHZ)~~' Fc( 0 H v(oH~)~~+
h
b
kcor,
M-I 0.29 5 6 1 .s' O.lt
5.5 x 1 0 3 h 4 x 105h 1.0
x
kk"
kmr: cm s-'
s-I
104h
1 . 1 x 105h 6.5 x 1 0 3 h 2.2 x 1 0 2 h
M-I
4 X IO-j'J 2 x Io-"J 6 X 10-5'-k 6 X IO-" 4 x 10-2'J -1.5'J 6 X 10-2'*k 0.2m 8X 4 x 10-2"
M-l .
s-I
0.45 90 2.7 0.18 5.5 x 103 4 x 105 1.0 x 104
-185 -185 -185 -185 -185
1.9 x 105 1.1 x 105 1 . 5 x 107 0.8' 9 x 104
2.5O 1.2) -5i 5.5 x -0.1q
3.5 x 2.0 x 2.7 x 1.4 1.6 x
Cr(bpy)?+ " Os(bpy)32t " RU(OH2)62t Ru(OH2):'
-480 590 -1 5 -1 5
1.5 X 1.5 x 102' 1.9 x 104 3 x 10-5'
1.0 x I O - 3 i . p ~ - I x 10-5'9 -0.2' 2 x IO-9~9
1.5 x 103 1.5 x 102 4 x 105 6 X IO4
IO-7~9
4.5 x 106 1.5 x 107
1.1 x 105 6.5 x 103 2.2 x 102
Ru(NH3),2+ Ru(NH3):+ Ru(NH3):+ Ru(NH3):+ Ru(NH3)tt
105 105
107 105
s-l .
1.3 X IO' 7 x 104 2.0 x 103 2.0 x 103 3 x 106 1.2 x 108 6 X IO6 3.5 x 106
8.5 X IO' 4.0 X IO' 1.7 X IO8 20 - I x 107 1.5 x 104
-I
x
13
6.5 X IO6 7 x 102
"Formal potential for coreacting redox couple (Le., that acting as reductant for the forward reaction), for ionic strengths (0.1-1 M) compatible with rate constant measurements; values taken from compilation in ref 3. Work-corrected rate constant for homogeneous reaction, obtained from observed rate constant by means of electrostatic work term corrections by means of the Debye-Huckel-Bronsted equation (eq 19 of ref 3). Literature sources are given in ref 3, unless specifically noted. Work-corrected rate constant for electrochemical reduction at mercury-aqueous interface of oxidant species, evaluated at electrode potential equal to listed formal potential of coreacting redox couple, E:. Values obtained from observed rate constant-potential data by means of Gouy-Chapman-Frumkin equation, as outlined in literature sources indicated. Work-corrected rate constant for homogeneous reaction, also corrected for inner-shell barrier of coreacting redox couple, AG*ih2.by means of eq 15. Values of AG*i,hz taken as follows (see text for details) Cr(bpy)?+/*+, zero; R U ( N H ~ ) ~ ~ +0.35 / ~ ' ,kcal mol-'; R u ( O H ~ ) ~ ~ +1.8 / ~ kcal + , mol-'. e Equivalent second-order rate constant for electrochemical reaction, equal to 4uNrh2k&,. Values of the reactant internuclear distance, f h , chosen so to be appropriate for the homogeneous-phase reaction with which it is compared; rh taken to be ca 6.6 A for reactions involving only aquo and/or ammine complexes, and I O A for those involving aquo/ammine and tris(bipyridine) complexes. /References 2h and 8. $References 9. Reference IO. 'Reference 2k. 1 Reference 2j. Reference 1 1. 'Reference 2m. (Note that the columns in Table I of ref 2m, listing electrochemical rate constants measured at mercury and gold electrodes, are interchanged due to a typesetting error.) Reference 2i. Reference 12. Reference 13. PReference 14. Reference 21. 'Rate constant for forward homogeneous-phase reaction as written refers to energetically unfavorable direction, chosen here for convenience in assigning coreacting redox couple (Le., reductant). Values of kkr obtained from corresponding value quoted in ref 3 for reverse reaction, divided by equilibrium constant obtained from formal potentials given in ref 3. 'Electrochemical rate constant at electrode potential indicated (Le., at E:) refers to reaction in energetically unfavorable direction (see footnote r ) . Values of kLr obtained from corresponding value for reverse reaction from sources auoted. divided bv* eauilibrium constant obtained from formal potential for redox couple. 'OAc = acetate. "bpy = . 2,2'-bipyridine.
From the foregoing discussion, two main factors could be perceived to account for such marked disparities between the expectations of eq 14 and experiment, resulting from the assumptions embodied in the former. Firstly, as noted above, eq 14 presumes that the intrinsic outer-shell free energies of activation for the corresponding electrochemical and homogeneous-phase reactions, AG*,, and AG*,h, are equal. On the basis of the usual dielectric continuum treatment,' this situation is expected if the homogeneous coreactants are in contact (so that 2a2 = tj,) and also that the reactant-electrode imaging term is relatively small (so that 4re >> fh). If the reactant is in contact with the metal surface, however, then re = 0.5rh,so that from eq 7 AG*,,e = (vide supra). Given that AG*qs,h= 4-6 kcal mol-' for the present reactions,j and that there is evidence that the reaction sites for Co(l1l) and other ammine electroreductions lie within the outer Helmholtz plane (Le., close to the metal surface),2i*'6 at least part of the enhanced electrochemical reactivities might be rationalized on this basis. Some evidence, however, against the likelihood that AG*,,e < AG*,,h is forthcoming from a recent treatment of imaging effects upon activation energies that accounts for field penetration into the metal electrode.17a Significantly, consideration of such factors yields repulsioe image potentials for small reactant-electrode surface separations, so that typically AG*,,e 2 AG*,,h.17 The inclusion of solvent spatial correlations (noncontinuum effects) (16) (a) Weaver, M.J.; Satterberg, T. L. J. Phys. Chem. 1977,81, 1772; (b) Weaver, M. J.; Liu, H. Y.; Kim, Y. Can. J. Chem. 1981, 59, 1944. (17) (a) Dzhavakhidze, P. G.; Kornyshev, A. A.; Krishtalik, L. 1. J . Elecrroanal. Chem. 1987, 228, 329. (b) Phelps, D. K.; Kornyshev, A. A,; Weaver, M.J. J . Phys. Chem. 1990, 94, 1454.
I
I
IO'
Io4
106
k:tr, g'i' Figure 1. Logarithmic plot of work-corrected 'equivalent second-order" rate constants, k c r , for electroreduction of various transition-metal complexes at mercury in aqueous solution versus corresponding rate constants, k&, for homogeneous reduction by selected reversible redox couples. See text and Table I footnotes for data sources and explanation. Reaction numbers as listed in Table I. Key to symbols: filled circles, homogeneous reactions between ammine complexes; filled triangles, ammine/polypyridine reactions; half-filled circles, aquo/ammine reactions; open circles, aquo/polypyridine reactions. (Note that reaction 19, Table I, is omitted from the figure for convenience in scaling.)
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The Journal of Physical Chemistry, Vol. 94, No. 23, 1990
further accentuates this departure from the conventional expectations.” The prediction that AG*,,e 2, hG*,,h is consistent with some experimental data.’7b In addition, absolute theoretical predictions of rate constants for a number of these electrochemical reactions featured in Table I yield good agreement with the experimental values only if the imaging effects upon AC*,,, are presumed to be small or negligible.3 Consequently, the present larger electrochemical reactivities than anticipated from eq 14 seem unlikely to be due chiefly to unexpectedly small activation energies. A more likely primary source of the observed deviations from eq 14 lies in the other underlying assumption that @re = KZIdrh. Some evidence that outer-sphere electroreductions of Co( 111) and other ammines typically follow largely adiabatic pathways (Le., I ) , or at least not strongly nonadiabatic pathways, has been obtained by comparison of the kinetics with the reactivities of related electrochemical unimolecular process.I8 In contrast, evidence has been marshalled that suggests that the homogeneous-phase Co(ll1) ammine reductions can proceed via substantially nonadiabatic pathways.I9 In general terms, the occurrence of more substantial donor-acceptor electronic coupling (and hence larger for electrochemical versus related homogeneous-phase reactions has been anticipated on theoretical grounds.20 Some support for this contention has recently been obtained for metallocene exchange reactions2I (vide infra). Consequently, differences in the degree of nonadiabaticity may well be the primary cause of the present observation (Table I, Figure 1) that k::, >> k!ir (cf. ref 3). Indeed, the smaller such disparities observed for reactions involving aquo complexes may be related to the smaller K:ldre values deduced for electroreductions of some aquo versus ammine complexes,Isb consistent with the larger hydrated radii of the former reactants.I6 Reduction of Molecular Oxygen. Another category of chemically irreversible processes of interest in the present context involves nonmetal reductions in aqueous media. Of these, the reduction of dioxygen has received extensive attention. A number of one-electron transition-metal reagents, including Ru( 11) ammines, reduce O2in a manner that kinetic schemes indicate the addition of the first electron (Le., to form 02-) to be rate determining.22 The rates of these homogeneous-phase reductions yield approximately self-consistent estimates of the rate constant for 02/02self-exchange, thereby supporting the notion that oneelectron outer-sphere pathways are uniformly followed.22b*23 While the electroreduction of dioxygen has been examined at a myriad of electrode surfaces with a primary focus on multielectron electrocatalysis, the kinetics at the mercury-aqueous interface are of particular interest here in view of its supposed lack of catalytic properties. At least in neutral or alkaline media at mercury, the O2electroreduction kinetics are essentially pH-independent and indicative of rate control by the first electron transfer.25 Following the above rate comparisons involving Ru( N H3)
