Correlations between Electrochemical and Homogeneous Redox

rate constanrts corrected for Coulomlbic double-layer effects. kEco, as a function of the electrode potential E from the corresponding observed ("appa...
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568

J. Phys. Chem. 1980, 84, 568-576

Correlations between Electrochemical and Homogeneous Redox Reactivity. Quantitative Comparisons of Rate Constants and Activation Parameters for Some Inorganic Outer-Sphere Reactions Mlchael J. Weaver Depatfment of Chemistty, Michigan State University, East Lansing, Michigan 48824 (Received August 15, 1979)

The electrochemicalreactivities of a number of simple spherical transition-metal complexes containing aquo and ammine ligands at the mercury-aqueous interface are compared with the reactivities of the corresponding homogeneous electron-transfer reactions between these complexes. Reactant systems were selected for which outer-sphere pathways are expected and work-term corrections upon the electrochemicalas well as the homogeneous rate constants could be made with confidence. Following these corrections,significant differences were found between the relative free energies of activation for corresponding electrochemical and homogeneous processes and the predictions obtained from the usual form of the Marcus model. For reactions involving aquo complexes, the electrochemical free energies of activation were found to be somewhat larger than expected on the basis of the homogeneous free energies of activation,whereas the opposite was found for reactions involving ammine complexes. Larger differences between the individual enthalpies and entropies of activation and the theoretical predictions were obtained for some reactions involving aquo complexes. Possible explanations for these results are discussed in terms of the Marcus model.

Introduction Contemporary theories of solution-phase electrontransfer processes1 predict that a number of simple relationships should hold between the kinetics of different types of redox reaction^.^^^ These correlations are of two main t y p e d (a) between the rate constants for homogeneous self-exchange (“hom~nuclear”~) reactions and the corresponding cross- (“heter~nuclear”~) reactions, and (b) between the rate constants for corresponding homogeneous and heterogeneous (Le., electrochemical) reactions. Most experimental kinetic data have been examined by using the following two relations derived by Marcus2”vbthat refer to correlations a and b, respectively: kh12 = (hhllkh22Kl&’/2

(1)

where

and

where khlZis the rate constant for a homogeneous crossreaction with an equilibrium constant, KI2,khll and hhZ2 are the rate constants for the corresponding homogeneous self-exchange reactions; Z h and Ze are the homogeneous and heterogeneous (electrochemical) collision frequencies, respectively, and he,, is the standard electrochemical rate constant corresponding to khll. The derivation of eq 1 and 2 involves the assumption that the reactants are spherical and structureless, the reactions are adiabatic (i.e,, the probability of electron transfer in the activated state is unity), and yet the thermal activation of each species is unaffected (or affected equally) by the presence of the other reactant5 or the electrode surface. Such reactions can be termed “weakly adiabatic” processes. Equations 1and 2 are therefore most likely to be applicable to reactions following outer-sphere pathways, where the coordination shells of each reactant (including the electrode’s “coordination layer” of solvent molecules) remain intact during electron transfer. For eq 1 it is ad0022-3654/80/2084-0568$0 1.OO/O

ditionally required that the vibrations of the reactants’ coordination shells and the fluctuations of the surrounding solvent be harmonic. Equation 1has received extensive experimental scrutiny for reactions involving transitionmetal complexes in aqueous solution.6 In contrast, relatively few attempts have been made to assess the validity of eq 2.’*t7-’ This latter situation is due in part to the paucity of quantitative electrochemicalrate data for simple outer-sphere, chemically reversible redox couples for which the kinetics of the corresponding homogeneous self-exchange reactions have also been determined. However, we have recently described a simple procedure by which the kinetics of homogeneous cross-reactions can be directly compared with the kinetics of the two constituent electrochemical half- reaction^.^ Thus, by making the same assumptions that were required in order to derive eq 2 (see the Appendix), it can also be shown that3 ke12/Ze = (kh12/Zh)liz (3) where ke12is the heterogeneous rate constant a t the intersection of the two (cathodic and anodic) rate-potential plots for the constituent pair of electrochemical half-reaction^.^ Since the available rate data for inorganic cross-reactionsare much more numerous and span a wider range of reactants than those available for self-exchange processes, the comparison of homogeneous and heterogeneous rate constants using eq 3 in addition to eq 2 should allow the similarities and differences between these two types of redox processes to be explored more extensively than hitherto. However, a major obstacle to the quantitative testing of eq 2 and 3 is that the experimental rate constants need to be corrected for the work required to form the “collision complexes” from the separated reactants (or reactant and electrode surface).2 Since the large majority of outersphere homogeneous reactions between metal complexes that have been studied involve di- or tripositive reactants, these work terms will often approximately cancel in eq 1. However, the magnitude or even the sign of the work terms for the corresponding electrode reactions will be sensitive to the electrode potential where the kinetics are monitored and to the ionic composition of the supporting electrolyte 1980 American Chemical Society

Electrcichemical and Homogeneous Redox Reactivity

as well as the ionic strength.'&13 Consequently, most of the published rate data for the electrode kinetics of metal complexes include large "double-layer" contributions that can typically vary lby factors up to ca. lo4, even for reactants carrying the game charge."-13 Some previous rate c ~ m p a r i s o n shave ~ ~ ~ignored ~ ~ ~ ~this ~ problem, probably because the magnitudes of these corrections were largely unknown. However, we have undertaken a systematic series of studies of the effect of the ionic double-layer structure upon the electrode kinetics of various Co(II1) and Cr(III[) aquo and ammine c o m p l e ~ e s ' ~and - ~ ~the Eu(III)/ I[II)aquo coup1el5J6at the merlcury-aqueous interface. Thest! studies have yielded information on the position of the reaction site for outer-sphere pathways involving aquo and ammine complexes"J2 and provided diagnostic criteria for the distinction between outer-sphere and inner-sphere (anioin-bridged) m e c h a n i ~ m s . ~ ' J ~This J ~ inFormation, coupled with some further experiments that are described below, provide a means by which the required doublelayer-corrected rate constants can be estimated with confidence for these and structurally similar outer-sphere electrode re!actions. In the present paper, these procedures are described, and the resulting work-corrected rate constants are compared with those for a number of corresponding homogeneous self-exchange and cross-reactions to assess the applicability of eq 2 and 3 for spherical inorganic reactants containing aquo or ammine ligands. In addition, the apparent enthalpic and entropic contributions to some heterogeneous and homogeneous free energy barriers are compared. These correlations provide some insight into the special features that characterize the composite activation of the two redox centers in homogeneous processes in comparison with the isolated activation of these reactants at inert mercury surfaces.

Experimental Section Stock solutions of the aquo cations Eu,d+ and Craq3+ were prepared as described in ref 15. Solutions of Eu, 2+ and Cr,? were prepared by exhaustive electrolyses of t i e trivalent cations in a slight excess of perchloric acid at ca. -900 mV and -1000 mV vs. SCE using a stirred mercury pool cathode. Solutions of V a F arid U, 4+ were similarly prepared by electrolyzing solutions of VIV) and U(V1) at ca. -1100 mV vs. SCE, to form VBiq2+and Uaq3+,and reoxidizing at -300 mV and -600 mV, respectively. Solutions of Rut,? in KPF6were prepared by using a method similar to that described in ref 17. All solutions for electrode kinetics experiments were prepared with water that was purifiled by pyr~distillation'~ to remove trace organic impurities. Other experimental details were similar to those recently described elsewhere.12Ja Both dc and normal pulse polarography utilizing a dropping mercury electrode (DME) were used to determine heterogeneous rate constants as a function of electrode potential.'La For some systems, standard rate constants were evaluated by cyclic voltammetry.lg All electrode potentials are reported with respect to the saturated (KC1) calomel electrode (SCE). Resullts and Data Analyses In order to test quantitatively eq 2 and 3 for outersphere reactions, it is necessary to obtain electrochemical rate constanrts corrected for Coulomlbic double-layer effects kEco,as a function of the electrode potential E from the correspondingobserved ("apparent") rate constants kEapp. The relationship between these two quantities can be written as10J3 eq 4, where 4, is the potential at the reaction site with respect to the bulk solution, 2 3 is the charge

The Journal of Physical Chemistry, Vol. 84, No. 6, 1980 569

2arried by the reactant, and aI is the intrinsic transfer coefficientzo("symmetry factor"). The plus/minus signs in eq 4 are appropriate for one-electron oxidation and reduction reactions, respectively. Although 4, is often equated with the average diffuselayer potential &, these two quantities may differ imarkedly depending on the position of the reaction site relative to the outer Helmholtz plane (oHp). In addition, 41, may differ from the average potential on the reaction plane qhrp because of discreteness-of-charge effects.21 We have obtained kinetic data for the outer-sphere electroreduction of various Co(II1) ammine and Cr(II1) aquo and ammine complexes at the mercury-aqueous interface in the absence of specific adsorption of the supporting electrolyte, where the application of eq 4 is most ~traightforward.ll-'~lJnder these conditions, $fpis related simply to the relative distances of the reaction plane and the oHp from the electrode 5urface.l' Differences between 4, and can still occur because the reacting species may significantly interact with its image charge in the metal electrode, and the ionic atmosphere in the double layer may differ from that in the bulk solution.21 However, we can deduce that &, 4, from the following evidence. If such discreteness-of-charge effects are important, then $, will depend on 2, because the self-image potential f? (= (4, - &,)) should depend linearly upon Z,.21 However, the quantity (a4,/@),, which can be estimated from the slopes (of experimental log k, - E plots at a constant electrolyte composition p, has been found to be approximately independent of 2, for the electroreduction of a given series of CO"'(NH~)~X and Cr'I'(OH,),X complexes, where X = NH3,OH2,F,OS032-.11712J4 This indicates that (af[/aE), 0 and hence (a&/aE), (a&,/aE), for these outersphere reactions. This in turn suggests that 0 since {; is expected to be markedly dependent on the electrode potentialS2lIt has been shown that the absolute values of 4Fare markedly larger for the outer-sphere electroreduction of Co(1II) and Cr(II1) ammines in comparison with otherwise similar Cr(II1) aquo c~mplexes.ll-~~ These differences have been attributed to the presence of additional, secondary hydration surrounding the aquo cations at, their distance of closest approach which prevents these reactants from penetrating as deeply into the double layer as the less hydrated ammine complexes.11J2 The usual assumption that 4, = 4 d is therefore inadequate for at least some of these reactions. However, 4, should become small at the potential of zero charge (pzc) in the absence of specific ionic adsorption,22so that from eq 4, kappPZC = kc,,,PzC.13 Consequently, the following two methods were employed ta obtain reasonable estimates of k,, for the various aquo and ammine reactants that were found to be suitahlle for study at the mercury-aqueous interface. (i) The electroreduction (or oxidation) kinetics were monitored in 0.1 M KF and/or 0.1 M KPF6 (acidifiied to pH 2 with HPF6). The former electrolyte exhibits no significant specific anion adsorption at the pzc (-435 rnV).% Although the latter electrolyte exhibits slight specific adsorption of PFC- at the pzc (-440 mV),13J5it provides a noncomplexing, acidic medium which is suitable for monitoring the electrode kinetics of both aquo and ammine complexes. The measurement of log kappat the P:GC(or close to the pzc where 4 d = 0) provided one point on the required log k, vs. E (Tafel) plot. There is strong evidence that aI = 0.5 for the electroreduction of Cr(II1) and Co(II1) aquo and ammine complexes,11J2as well as Euaq3+,l!jover a wide range of overpotentials,26although it appears that

-

-

-

-

-

570

The Journal of Physical Chemistry, Vol. 84, No. 6, 1980

Weaver

TABLE I : Kinetic Parameters for Some Outer-Sphere Reactions at the Mercury-Aqueous Interface, Coulombic Double-Layer Corrections E,mVvs. E f , d mV reactant medium SCE kaPP,a cm s-’ ~z,,,,b cm s-l CY VS.SCE hSCOl: cm s-’ Craq2+ CraQ3+ EUaq:: EUaq

Vaq2+ Vaq3+

Ruaq3+I2+ Ul”4+/3+ a..-

CO(NH,),~+ Co(NH,),OH

3t

Ru(NH,),~+/~’

0.1 M KPF, 1 M NaCl0,g 0.1 M KPF,f 1 M hTac10,g 0.1 M KPF,f 1 M NaC10,g 0 . 2 M KPF, 0 . 5 M NaClO. -I 0 . 5 M HCld, 0.1 M KPF,,? 0.1 M K F 0.1 M KPF6f 0 . 0 2 M NaFh

1.0x l o 4i,P 3.5 x 1o5jIr 2 x 10-3O

- 440 - 800 - 440

2.5 X 10-,’jvr 2 x 10-3ilp 8x lopd,m

- 800 - 440

-700 - 20 -875

-440

2.5

-440 -180

m1.5jpk >2X

X

10-2jlk

cj

x 10-3i,~

W0.7i9~

~ 0 . 4 5 ’ 3 ’ - 655m 0 . 5 ’ ~ ~ -655m ~0.45”’ -625m 0.5-!$m - 625m -470n ~0.5‘3’ ~ 0 . 5 ~ -470n ) ~ - 15n - 875k

2 x 10-6q 2 x 10-64 7 x lo-sQ 8 X 10-5Q L O X 10-34 L O X 10-34 W0.02f 5 x 10-3‘

0.5’ -0.5O - 180

2lou

a Observed (apparent) rate constant at given electrode potential E. Obtained from h”,, = i / F C b , where i is current density corrected for diffusion polarization and back-reactions, and Cb is bulk reactant concentration. t~ Rate constant corrected for Coulombic double-layer effects at given value of E , obtained from hEap, by using eq 4. Intrinsic transfer coefficient a t 2 ” ,obtained from o r I = i 2 . 3 R T / F (a log hcor/aE)p (eq 5). Formal electrode potential of redox couple obtained in given electrolyte. e “Standard” rate constant corrected for Coulombic double-layer effects, obtained at E = E f . Acidified to pH - 2 . 5 with HPF,. g Acidified to pH - 2 . 5 with HClO,. pH - 7 . ’ For electrooxidation. For electroreduction. Data sources: this work, ref 27, ref 15, ref 17, O ref 12. Obtained from hEap, by assuming that kPzccor = kPzcapp, since E a proximately corresponds to pzc for KPF, or K F electrolyte used (see text for details). 4 Obtained from listed values of Obtained from listed value of k”,,,by using eq 4 and assuming that @r = 0 . 5 @dGCS and by using eq 5. (see text for details). Determined at E f by using cyclic voltammetry.19 Obtained from hEap by assuming that doublelayer correction is identical with that for E u 2 +oxidation in same electrolyte. Obtained from hs,, by assuming that double-layer correction is identical with that for CO(NH,),~’reduction in same electrolyte.

a1 falls significantly below 0.5 for the electrooxidation of

Cra:+ and Eua:+ a t anodic over potential^.^^ These estimates of q ,coupled with the experimental values of k a p p ~ , allowed the construction of the desired cathodic and anodic Tafel plots from the following r e l a t i ~ n : ’ ~ J ~ log kElcor= log kEZco,f aIF(E2 - El)/2.3RT (5) where cyI is the average value of the intrinsic transfer coefficient between the electrode potentials El and E2. (The plus/minus signs in eq 5 again recer to electrooxidation and electroreduction reactions, respectively.) Table I summarizes the resulting values of log kcorPZCobtained in 0.1 M KPF, and/or 0.1 M KF for the various reactions for which this analysis was found to be suitable, along with the corresponding estimates of aIand the work-corrected “standard” rate constants kscor(Le., these obtained at the formal potential E,) for couples for which Ef can be measured. (ii) In order to provide a check of method i, values of ksm,were also estimated for the aquo couples Craq3+iZ+ and Euaq3+12+from measurements of log k,, in 1 M NaC104 at sufficiently negative electrode potentd’s (-E 2 700 mv) so that the effect of perchlorate specific adsorption could be neg1e~ted.l~ Under these conditions, it has been shown for these aquo couples that $r = 0.5 $dGCS, where $dGCS is the Gouy-Chapman-Stern estimate of &.15 With this assumption, estimates of kEm, and hence Pm,were obtained from eq 4 and are also given in Table I. Inspection of Table I reveals that these two independent methods of estimating ksm, yielded values in good agreement. Method ii was also used to estimate kscorfor the Uaq4+/3+couple; this reaction is sufficiently rapid so that its kinetics could only be monitored at negative potentials close to Ef(= -875 mV) at ionic strength p = 1.0 by using cyclic voltammetry. An estimate of ksc,,,was also obtained for Ru,3+Iz+from the experimental value of Papin 0.2 M KPF6by assuming that $, is the same as that oitained for Eu,? oxidation in this electrolyte a t the same electrode potential and by applying eq 4. Both of these estimates of ksco,are also listed in Table I. Comparison of Heterogeneous and Homogeneous Rate Parameters. Table I contains electrochemical rate data

corrected for Coulombic double-layer effects for five aquo couples and three ammine reactions. These rate data can be compared with homogeneous outer-sphere reactivities for various combinations of these oxidants and reductants, using eq 2 and 3 as a convenient starting point. Table I1 lists the relevant homogeneous rate data that are available from the literature. It is also necessary to correct these homogeneous rate constants khapp for the Coulombic work of forming the collision complex from the separated reactants to yield kh,. Although relatively few experimental attempts have been made to assess the magnitude of this correction, the dependence of the rates of a number of outer-sphere cation-cation reactions upon ionic strength p can be successfully fitted to the extended Debye-Huckel-Bransted expres~ion,~*~~

where khois the rate constant when p ---* 0, Z A and Z B are the ionic charges carried by the two reactants, A and B are the conventional Debye-Huckel parameters, and A and C are adjustable constants. (Physically, A can be viewed as the distance between the reactants in the collision complex, and C a term accounting for any “specific” effects of the supporting electrolyte ions on the reaction rate.29) The success of eq 6 suggests that the extended Debye-Huckel model should provide a reasonable estimate of khcor. The appropriate expression is29331eq 7, where e is the electronic log khcor= log khapp+

Ne2zAzB 2.3RTcsA(1 + K A )

(7)

charge, N is Avogadro’s number, cs is the static dielectric constant, and K is the reciprocal Debye length. Equation 7 was used to obtain values of khcorfrom khapp,which are also listed in Table 11. Estimates of A are available for some of these reactions from fitting the ionic stren t h dependence of kha,,, to eq 6; they fall in the range 8-12 .30 For reactions where 2 is unknown, it was taken to be 8-10 A, depending on the sizes of the reactants.30 For reactions

81

Electrochemical and Homogeneous Redox Reactivity

The Journal of Physical Chemistry, Vol. 84, No. 6, 1980 571

TABLE 11: Kinetic Parameters for .Acid-Independent Pathways of Some Homogeneous Reactions Involving Aquo and Ammine Complexes at 25 C. Comparison with Heterogeneous Rate Parameters ionic strength (AGh*)’2cor,C (AGe*)”cond oxidant reductant (p), M khaPP? M - ’ s - ’ khcor,b M - l s - ’ kcal mol‘’ kcal mol-’

___I___

~~~

~~

~

Vaq3+

Vaq’+

EuaqZ+

’

Uaq +

Ru(NH,),’+ CraqZ Ru( NH,),’

Euaq3+

+

+

Qraq

3

Uaq3+

+

RU(NH,),~+ Ruaq2+

Ruaq C~O(NH,),’+ +

va, +

CragZ Euaq2 Ru(NH&’+ +

+

CID(N H ,) OH

+

Vaq2+

Euaq2+

Ru( NH,),

’+

Ru(NH,),’+ Ru (NH,), ’+

2.0 2.0 2.0 0.18 0.5 1.0 2.0 0.2 1.0 1.0 1.0 0.4 0.2 1.0 0.4 0.22 0.1

0.014e 9x 10-~f 852 1.9 x 10-2hsq 2 x io+ f 7 10-5i,q

I -

6.2 X l O - ’ j 2 x 10-6kpq 60‘ 1x 1x 10-3m 1.7 x 1 0 - 3 n 1.1x 10-Zk 0.53’ 7.5 x 1 0 - ’ n 3.0’ -3 x 1 0 3 ~

0.025 0.016 150 0.25 -8 X 10.’ 3x 7.5 x 2x ,150 4x 4 x 10-3 8x 0.2 2

h

0.4 30 - 5 x 104

16.9 17.2 11.7

9.1 9.2 6.4

20.3

11.8

16.2

9.3

11.7 16.6 18.0 17.5

7.4 8.2 9.1 8.3

14.3

7.1

15.2

7.1

8.3

23.5

Rate constant corrected a Observed (apparent) second-order homogeneous rate constant for pH-independent pathway. Standard free energy of activation for homogeneous reaction, obtained for Coulombic work terms using eq 7 (see text). Standard free from eq 11 by using corresponding value of hhcor and assuming that zh= 6 X 10’’ M-’ s - ’ (see text). energy of activation for pair of electrochemical reactions corresponding t o homogeneous reaction. Calculated from eq 10 by using 2, = 5 x l o 3 cm s-’, and the value of kEcor a t the electrode potential E where the two cathodic and anodic (Tafel) plots of log k,,, vs. E: intersect (see text). Data taken from Table I. Sources of homogeneous rate data: e K. V. Krishnamurty and A. C. Wahl, J. Am. Chem. SOC.,80, 5921 (1958). f A. Adin and A. G. Sykes, J. Chem. SOC.A , 1230 C. A. Jacks and L. E. Bennett, A. Ekstrom, A. B. McLaren, and L. E. Symthe, Znorg. Chem., 14, 1035 (1975). (1966). Znorg. Chem., 13, 2035 (1974). M. Faraggi and A. Feder, Znorg. Chem., 12, 236 (1973). j A. Ekstrom, A. B. McLaren, J. F. Endicott and H. Taube, J. Am. Chem. SOC.,86, 1686 (1964). and L. E. Smythe, Znorg. Chem., 16, 1032 (1977). T. J. Przytas and N. Satin, J. Am. Chem. SOC., W. Btjttcher, G. M. Brown, and N. Sutin, Inorg. Chem., 18, 1447 (1979). 95, 5545 (1973). J. Doyle and A. G. Sykes, J. Chem. SOC.A , 2836 (1968). O P. H. Dodel and H. Taube, 2. Phys. Chem. (Frankfurt a m Main), 44, 92 (1965). P Reference 31. Obtained from rate of reverse reaction, coupled with equilibrium constant calculated from appropriate formal potentials listed in ref 1 7 .

’

where the qluoted value of khappwas measured a t ionic strengths p > 1 where the C p term in eq 6 becomes significaint, a small additional correction to kh was made by using the “typical” value C = 0.1 M-1.30aPP In order to compare these heterogeneous and homogeneous rate parameters, it is also necessary to estimate values of the electrochemical and homogeneous collision frequencies 2, and z h . These quantities are usually calculated by using the following expressions that can be derived from simple statistical consideration^:^^^^^

z,= (kBT/ 2 r m )‘1’ 2, = N(8rkBTm;1)1/z X 10-3r2

(8)

(9) where m is the mass of the heterogeneous reactant, kB is the Boltzmamn constant, and m, and r are the effective reduced mass and the distance between the metal centers, respectively, for the homogeneous collision complex. Inserting the typical values (for the present reactants) of N m = 200, m, = 100, and r = 7 X lo4 cm into eq 8 and 9 yields 2, = 4 5 X lo3 cm s-l and 2, = 5.7 X 1O1O M-l Although the actual absolute values of 2, and are uncertain, only the relative values are required for the comparison of rate constants using eq 2 and 3. Fortunately, the ratio z,/& will be affected only slightly by physically reasonable variations of r in eq 9. Also, the use of eq 9, coupled with experimental rate data, has been shown to yield activation free energies for some simple homogeneous Ru(III)/Ru(II) self-exchange reactions that are in good agreement with the theoretical prediction^.^^ Since the theoretical model leading to eq 2 and 3 is based on activation free energies,2instead of comparing hecorand

khcordirectly it is more revealing to compare the corresponding work-corrected activation free energies th.at are obtained from kecorand 2, and khcorand Zh by using the relations2 (AGe*)cor = -RT In (hecor/Ze)

(10)

(AGh*)cor = -RT In (khcor/Zh)

(11)

-

(12)

We shall compare the homogeneous reactions Oxl

+ Redz

Redl

+ Ox2

having the activation free energy (AGh*)12corwith the constituent electrochemical half-reactions Oxl + e-(El) Red2

-

-

Redl

(134

Ox2 + e-(Ez)

l(13b)

having the activation free energies (AG:)lcor and (AG0*)2COr, respectively. Since the conventional Marcus correlation for exchange reactions (eq 2) is really a special case of eq 3: in view of eq 10 and 11these relations can be rewritten in terms of activation free energies in the general foirm of eq 14 where (AGe*)12co,is the electrochemical free energy (14)

2(AGe*)12cor = (AGh*)12cor

of activation a t an electrode Dotential such that (AG-*)l,.-. . = (AGe*)2cor= (AGe*)12cor. In Table I1 are listed values of (AGh*)lz,.nv that were obtained from the corresponding values of-khcor for the various homonuclear and heteronuclear aquo-aquo reactions by using eq 11 and assuming that z h = 6 X 10’” M-’ r

I

CVI

572

The Journal of Physical Chemistry, Vol. 84, No. 6, 1980

Weaver

I ,' 60

'-1,,/ I,

IO 5

I 6

1

01

20

:I

I

I

[I

I

I

10

8

12

-8

Flgure 1. Comparison of the standard free energies of activation (AGh1)12cor and (AG,')I2, for homogeneous and electrochemical electron transfer, respectively, involving various aquo metal complexes and Co(NH3)3+,Co(NH,),OH,3+. Values corrected for the Coulomblc work of formlng the collision complexes from the separated reactants. Data taken from Table I1 (see text and footnotes to Table I1 for details). Key: Open circles, aquo-aquo reaction; closed cirdes, aquo-ammine reactions. (1) RU;', 4- Ru,?, (2) V, 3+ 4- U, '+, (3 Cr,? V 2+ ;4) V4: V,?, (5) V'T 4- Eu *,(61Eu,? Cr,)', (7) C o ( N H p Vaq , (8) Co(NH -I-,:Eu, ( 9 ) CO(NH,),~+ Cr,;' (10) Co(NH,),OH;+ V!, , (1 1) Co(NH,),OH;+ EL;,I+. Dashed line is theoretical predictlon from eq 14.

+

+

!!-

+

4 0

-4

(AGz)&r

+

+

+

s-l. In the adjacent column are listed the corresponding standard electrochemical free energies of activation (AGeS)12cor. These were obtained from the values of kEcor at the intersection of the constituent anodic and cathodic (Tafel) plots for the appropriate pairs of half-reactions by using the data in Table I. For correlations with homogeneous self-exchange reactions, the appropriate value of log k,,, will be the standard rate constant log kscor. For cross-reactions, the electrode potential at the intersection of the Tafel lines will differ from the formal potentials of both redox couples and will partly depend upon the kinetics of each half-rea~tion.~ The reactions for which values of (AG:)12cor are given in Table I1 are those for which little or no extrapolation of the cathodic and anodic Tafel lines is required in order to find the intersection point. Inspection of the corresponding values of (AG;),,, and (AGh*)corin Table I1 reveals that for each of the two homonuclear and four heteronuclear aquo-aquo reactions, 2(AGe*)12cor> (AGh*)12cor,in contrast to the predictions of eq 14. This result is also illustrated graphically in Figure 1, which is a plot of (AGh*)12corvs. (AGe*)12cor.It is seen that the experimental points for the aquo-aquo reactions (open circles) have values of (AGh*)12c,, that are 1-3 kcal molt1 below the dashed straight line predicted by eq 14. Table I1 and Figure 1 also contain values of (AGh*)12corand (AGeS)12co, for the reduction of C O ( N H ~ ) and ~ ~ +Co(NH3)50H23+ by divalent aquo complexes. In contrast to the behavior of the aquo-aquo reactions, for these ammine-aquo reactions it is seen that 2(AG,*),,, 5 (AGh*)mr. In view of these discrepancies between the experimental results and eq 14, it is desirable to extend such rate correlations to include a wider range of homogeneous redox reagents to provide clues regarding their origin. Weak

2

-2

[ log k:orr-(iog

h

NH

kcorr)

31

Flgure 2. The logarithm of the Coulombic work-corrected rate constant for electrochemicalreduction log km at -600 mV vs. SCE for various oxidants expressed relative to log kaW for Co(NH,):+ reduction, [log (log k-m,)Ny, plotted against the corresponding quantities - (log kh,)N for homogeneous reductions of the same [log k series of oxldants by various reductants. Data taken from Table 11. Homogeneousreductants are labeled for each experimental point. Key to oxidants: (0) Va;+, (A) ELI,?, ('7) Cr,;' (0)Co(NH,),OH;+. Dashed straight line is plot expected on the bask of eq 15 (see text).

-

adiabatic theories of electron transfer that predict that eq 2 and 3 should be obeyed also predict that the ratios of the (work-corrected) rate constants for homogeneous reduction (or oxidation) of a series of reactants using a fixed reagent will be independent of the reagent chosen and will also be the same as the rate ratios for the corresponding (work-corrected) electrochemical reactions when the latter are measured at a fixed electrode potential,2 i.e., (A log kecor)E = (A log k h c o r ) ~

(15)

where the subscript "E" refers to a constant electrode potential, and "R' refers to a fixed homogeneous reagent. (Strictly, this correlation requires that the experimental conditions for the heterogeneous and homogeneous reactions correspond t~ aI= 0.5 and f 1, respectively? These conditions are approximately fulfilled for the reactions considered here since the self-exchange rates and/or the thermodynamic driving forces for reaction are small.) Figure 2 is such a relative rate plot for the reduction of Co(NH3):+, C O ( N H ~ ) ~ O HV~ 3+, + , Eu,?, and Cr,? by Ru(NH3)2+,V,q2+, Eu,?, and Era? in comparison with the electroreduction of the same series of oxidants at the mercury-aqueous interface at -600 mV vs. SCE. (This potential was selected because it minimized the extent of data extrapolation that was required; however, choosing other potentials had little effect on the resulting correlation.) Both the homogeneous and electrochemical rate constants are expressed as log k,,, relative to that for Equation C O ( N H ~ )reduction ~~+ [log k,,, - (log k,,,)"3]. 15 predicts that the experimental points will fall on the dashed straight line shown in Figure 2, which has a slope of unity and passes through the origin. However, inspection of Figure 2 reveals that although the three experimental points for Co(NHJ,OH$+ fall on the predicted line derived from the "parent" CO("~)~~+oxidant, ex-

-

Electrochemical and Homogeneous Redlox Reactivity

The Journal of Physical Chemistry, Vol. 84, No. 6, 1980 573

perirnental points ffor reactions involving the aquo oxidants activation parameters corresponding to (AG2)12cor(i.e., at VW3'-,Eu, 3+, and Cr,. all lie significantly below the line. the electrode potential where the cathodic and anodic Tafel This res& is expected in view of Figure 1; both plots plots intersect at the desired temperature), and (h2jTh*)12cor indicate thiat the aquo complexes react somewhat slower and are the homogeneous activation Parameters relative to Co(NH,Jt+ and CO(NI-I~)~OH;+ than expected corresponding to (AGh*)12cor. Equations 16 and 17 emon the basis of their relative reactivities in homogeneous phasize that for weak adiabatic reactions, the enthalpic solution. I t is interesting to note in Figure 2 that the and entropic, as well as the free energy, surfaces for reexperimental points that refer to homogeneous reduction action 12 are expected to be the aggregate of those for reactions 13a and 13b. (Note, however, that although by Ru(NHJ:+ fall consistently close to the corresponding (AG,t)lcor= (AGe*)2corunder these conditions, (AH~)lcor # points for lhe reduction of the same oxidants using aquo reductants. This indicates that essentially the same dis( m e * ) 2 c o r unless (ASe*)lcor = (ASe*)2cor*) Values of (ASe*)lcorhave been determined for the eleccrepancies between the heterogeneous and homogeneous troreduction of Cr,;+, Eu,:+, and V,3+ over a range of reactivities are maintained irrespective of whether Rucathodic overpotentials at the mercury-aqueous interfacela (NH3),2+,V,?, Eu,?, or Cr,? is used as a homogeneous and were found to be essentially independent of eltxtrode reductant. Unfortunately, quantitative electrochemical rate data potential.la Values of ( ASe*)2corfor the reverse (electrowere not obtained for ammine-ammine reactions, because oxidation) reactions can be determined from eq 18 where of the paucity of hexaammine reactions for which elec(ASe*)'cor = (ASe*)'cor - AS,," (18) trooxidation rates can be monitored. Nevertheless, an approximate estimate was obtained for the standard rate AS,," is the reaction entropy for the forward (electroreconstant kseppof the R U ( N H ~ ) ~ system. + / ~ + This couple duction) reaction. Values of AS,," have been determined is substitutionally inert in both oxidation stated7 and is S: using nonisothermal in a manner analogous to that for A expected to have an extremely large value of k",,, since the cells.17 The corresponding values of (AS:)lZcor for desired inner-shell contribution to the activation energy is small.31 combinations of half-reactions can then be determined The tactic employed was to evaluate ksaPpin dilute sodium determined at the appropriate from eq 17a, and (M:)12cor fluoride media; at the formal potential for R U ( N H ~ ) ~ + / ~ potential + = (AGe*)12cor+ from the relation (AHe*)lZcor (-180 mV1") 4dGcsis large and positive in this e l e ~ t r o l y t e ~ ~ T(ASe*)12cor* so that haapp ( M h * ) 1 2 c o r as also seen in Table can be determined to a good approximation from the 111. In fact, the differences between 2(AHe*)12c0rand temperature dependence of the rate constant measured at (mh*)12corare noticeably greater than between 2(AC:,')12,, a constant "nonisothermal" cell potential with the referand (AGh*)12cor, Le., the activation enthalpies show a greater ence electrode held at a fixed temperaturela (the so-called discrepancy with the predictions of eq 16 than the acti"ideal" activation p a r a m e t e r ~ l ~ ? ~ ~ ) . vation free energies with eq 14. Unfortunately, the paucity When eq 14 is valid, it is also to be expected34that the of suitable activation parameter data does not allow this correspondingenthalpic and entropic relations would hold, comparison to be made for a wider range of reactions. Le. Discussion 2(AHe*)12cor= (M[h*)l2cor (16) The results presented in Tables 1-111 and Figures 1and 2 exhibit three types of deviations from the predictions wherie of eq 14-17 which suggest that factors other than those (h'fe*)12cor = 0.51(me*)1c,, + ( m e * ) 2 c o r l (16a) considered in deriving these relations contribute significantly to the observed heterogeneous and/or homogf,ineous and reactivities. Firstly, for reactions between aquo cations, are the homogeneous free energies of activation (AGh*)12cor 2(ASe*)12c0r = (ASh*)12cor (17) typically 1-3 kcal mol-l smaller than predicted from the where corresponding electrochemical reactivities using eq 14 (Figure 1);i.e., the homogeneous rates are up to two orders (ASe*)12cor= 0.5[(ASe*)',, + (ASe*)2cor] (17a) of magnitude larger than expected from eq 2 and 3. and where (AHe*)12corand (AS~)12cor are the electrochemical Secondly, for aquo-aquo reactions, somewhat larger dis-

574

The Journal of Physical Chemistry, Vol. 84, No. 6, 1980

Weaver

TABLE 111: Comparison of Heterogeneous and Homogeneous Activation Parameters for Some Reactions Involving Aquo Complexes at 25 C AS" I Z , ~ oxidant reductant cal deg-' mol"

Vaq3+ Vaq

3+

Euaq3+

Vaq2+ Euaqz+ Cyaq z +

(ASh*)'Zapp,b cal deg-' mol-'

0 - 11

-1

( A s h * )'zcor,c (Affh*)cond 2 ( A S e * ) ' 2cor,p 2(AHe*:)'Zcor,f cal deg-' mol-' cal deg-' mol-' cal deg-' mol-' cal deg-' mol-' -15 - 20 -15

- 25* - 30h - 25'

12.4 11.2 15.8

-6 -12 -6

16.2 21.6 21.6

a Entropy change for net homogeneous reaction, determined from A S o , > = A S o r c , l - A S o r c , z , where ASorc,] and A S o r c , z are t h e reaction entropies for t h e two constituent redox couples. Data taken from ref 17. b Measured (apparent) activation entropies for given homogeneous reaction. Homo eneous activation entropy (corrected for translation entropy6d$3 7 ) ; determined from (Ash*),,, by using the ( A s h . )cor - (ASh*),,, - R In ( h ~ h / h B+~0.5R ) = (ASh*),,, t 10.2 eu Corrected activation enthalpy for homogeneous reaction, determined from for 27 = 6 10" M-'s - ' (see text). + Z'(ASh+)cor. Values of (AGh*)cor taken from Table II. e Sum of electrochemical activation ( A H h )cor - (AGh'),,, entropies for constituent electrochemical half-reactions, corrected for Coulombic work terms. Determined from 2(ASe*)'zcor = (AS,*)'COr + (ASe*)zcor, where (AS,* ,,') and the activation entropies for the cathodic and anodic halfreactions, respectively (eq 17a). Data taken from ref 18. f Sum of electrochemical activation enthalpies for constituent electrochemical half-reactions, corrected for Coulombic work terms. Determined from 2( AH,* )' = 2(AG,*)'Z,0, t K. V. Krishamurty and A. C. Wahl, J. A m . Chem. SOC., 80, 2(AS,*)'Zcor. Values of ( AG,*)12c0r taken from Table 11. 5921 ( 1 9 6 8 ) . At 1.1 = 2. h A. Adin and A. G. Sykes, J. Chem. SOC. A, 1230 ( 1 9 6 6 ) . At 1.1 = 2. I Estimated from measured values of A S * for other aquo-aquo reactions for which A S l 2 ' - 0 . 4 '

g

crepancies are seen between the experimental activation parameters and eq 16 and 17. Thirdly, the substitution of an ammine for an aquo reactant results in values of 2(AG:)'2mr that are at least 1-2 kcal mol-' smaller relative to (AGh*)12cor. As noted in the Appendix, the key assumption made in deriving eq 14 is that the sum of the so-called38"intrinsic" free energies of activation (AG,S)' for the component electrochemical reactions equals the corresponding intrinsic free energy of activation (AGh*)i for the homogeneous r e a c t i ~ n : ~ ~ - ~ O (AGe*)iJ+ (AGe*)i2 = (AGh*)iJ2

(19)

These "intrinsic" barriers are equal to (AGe.*)wror (AG