Potential Dependence of the Electrochemical Transfer Coefficient agreement that one obtains from such a simple theory with only one adjustable paramenter per salt, and in some cases, per ion. Acknowledgments. We are deeply indebted to Professor H. L. Friedman for numerous comments and his critical reading of the manuscript, and to Drs. C. V. Krishnan and A. H. Narten for their very useful suggestions. One of the authors also acknowledges the support of Oak Ridge Associated Universities (Contract No. S-1358)for part of this work.
References and Notes (1) P. Debye and E. Huckel, Phys. Z.,24, 185,334 (1923). (2) E. A. Guggenheim, Phil. Mag., (7) 19,588 (1935); G. Scatchard, Chem. Rev., 19,309 (1939); M. H. Lletzke and R. W. Stoughton, J. Phys. Chem., 66,508 (1962). (3) K. S. Pitzer, J. Phys. Chem., 77,2268 (1973); K. S. Pitzer and G. Mayorga, ibid., 77, 2300 (1973). (4) .J. E. Mayer, J. Chem. Phys., 18, 1426 (1950);H. L. Friedman, “Ionic Solution Theory Based on Cluster Expansion Methods”, Interscience, New York, N.Y., 1962. (5) A. R. Allnatt, Mol. Phys., 8, 533 (1964).
1861 (6) J. C. Rasaiahand H. L. Friedman, J. Chem. Phys., 46,2742 (1968); 50,3965 (1969); P. S. Ramanathan, C. V. Krishnan, and H. L. Friedman, J. Solution Chem., 1, 237 (1972). (7) J. L. Lebowitz and J. K. Percus, Phys. Rev,, 144, 251 (1966). (8) E. Waisman and J. L. Lebowitz, J. Chem. Phys., 56, 3086, 3093 (1972). (9) H. C. Andersen, D. Chandler, and J. D. Weeks, J. Chem. Phys., 57,2626 (1972). (10) J. S. Hoye, J. L. Lebowitz, and G. Stell, J. Chem. Phys., 61, 3253 (1974). (11) G. Stelland K. C. Wu, J. Chem. Phys., 63,491 (1975). (12) L. Blum, J. Chem. Phys., 61, 2129 (1974), and unpublishedwork. (13) L. Blum, Mol. Phys., 30, 1529 (1975). (14) J. R. Grigera and L. Blum, Chem. Phys. Lett., in press. (15) R. A. Robinson and R. H. Stokes, “Electrolytic Solutions”, Butterworths, London, 1959. (16) P. W. Gurney, “Ionic Processes in Solutlon”, Dover, New York, N.Y., 1953. (17) H. L. Friedman and C. V. Krishnan, “Thermodynamics of Ion Hydration” a chapter in “Water, a Comprehensive Treatise”, F. Franks, Ed., Plenum Press, New York, N.Y., 1973. (18) G. Stell, J. Chem. Phys., 59, 3926 (1973). {19) J. L. Lebowitz, Phys. Rev., 133, A895 (1964). (20) H.L. Friedman, J. Solution Chem., 1, 387, 413, 419 (1972). (21) J. C. Rasaiah, J. Solution Chem., 2, 301 (1973). (22) For an excellent discusslon see H. L. Friedman and W. D. T. Dale, “Electrolyte Solutions at Equilibrium” in “Modern Theoretical Chemistry”, Vol. IV, B. J. Berne, Ed., Plenum Press, New York, N.Y., 1976.
Potential Dependence of the Electrochemical Transfer Coefficient. Further Studies of the Reduction of Chromium(ll1) at Mercury Electrodes Michael J. Weaver and Fred C. Anson* A. A. Noyes Laboratory,’ California lnstitute of Technology, Pasadena, California 9 I125 (Received March 15, 1976) Publication costs assisted by the National Science Foundationand the U.S.Army Research Office (Triangle Park)
The electrochemical reduction rates of three complexes of Cr(II1) have been measured over an unusually large potential range in order to provide a stringent test of the theoretical prediction of Marcus that the electrochemical transfer coefficient, a , should exhibit a potential dependence. The three complexes studied, Cr(OH2)50S03+, Cr(OH2)5F2+,and Cr(OH2)63+,follow outer-sphere reaction mechanisms and bear differing charges which lead to diffuse double-layer corrections of varying magnitudes. This allowed the reliability of the diffuse-layer corrections to be established. Within experimental error, no potential dependence of a was observed with the sulfato complex under conditions where an easily detectable dependence is predicted by the Marcus theory. Some potential dependence of a was observed with the fluoro and aquo complexes but it was of the opposite sign and much smaller than that predicted by the Marcus theory and is attributed to uncertainties in diffuse-layer corrections. The present results are compared and contrasted with previous attempts to detect a potential dependence of the transfer coefficient.
Introduction Despite the considerable recent attention that has been focused on the subject,2-7 an unambiguous experimental test of the predicted8-10 potential dependence of the electrochemical transfer coefficient, a , has remained elusive. The difficulties met in previous studies include those from the following sources: (i) With multiply charged reactants, large corrections are necessary to account for the effect of the diffuse layer on measured reaction rates. As a result, large uncertainties are introduced in extracting the theoretically relevant, intrinsic transfer coefficientll from the apparent transfer coefficient obtained experimentally from slopes of rate-potential plots. (ii) Predicted changes in a are often so small that they are commensurate with experimental uncer-
tainties when the kinetic measurements are made with techniques which do not allow potentials much removed from the standard potential to be explored. (iii) The theoretical treatments were derived for simple, one-electron,outer-sphere reactions of which examples are not abundant, particularly if it is desired to restrict measurements to relatively negative potentials in order to minimize specific adsorption of the ionic components of the supporting electrolytes. It has often been suggested2bf+ that the redox couples most suited to experimental tests of the possible potential dependence of a are those with large standard rate constants because the predicted potential dependence of the transfer coefficient is more marked for larger values of the standard rate constant (i.e,, with lower reorganizational energy barriers for electron transfer). However, the predicted value of da/dE is only twice The Journal of Physical Chemistry, Vol. SO,No. 17, 1976
1862
Michael J. Weaver and Fred C. Anson
as large for a reaction with a standard rate constant of 0.1 cm limited to 30 and 5 mM (10 mM a t -600 mV), respectively, s-l than for one with a constant of cm s-l and considerand in most instances were held well below these values. This ably larger potential ranges can usually be explored with assured that these reactants exerted a negligible influence on reactants having smaller standard rate constants. Thus, unless the diffuse layer structure in the uni-univalent supporting nonconventional kinetic techniques are employed,6 the preelectrolytes. Possible contributions from anodic back reactions dicted potential dependence of the transfer coefficient should were calculated according to the method described in the be observable with slow as well as with fast electrode proAppendix of ref 14 and found to be negligible for all three cesses. complexes even at the least negative potentials employed The present study was prompted by our observation12 that (positive of the standard reversible potentials in some cases). Cr(OH2)50S03+ is almost certainly reduced by an outerThis is because, for this slow reaction, the concentration of sphere mechanism at mercury electrodes.13The single positive reductant (Cr2+)produced at the electrode surface during the charge carried by this reactant leads to considerably lower lifetime of each electrode is much smaller than the concenuncertainties in the diffuse layer corrections compared with tration of oxidant (Cr(II1)) present, so that the oxidation rate the previously studied tri-positive cation C ~ ( O H Z ) ~ ~ +remains . ~ ~ , ~much smaller than the reduction rate even though Furthermore, the electrode reaction is sufficiently irreversible the anodic rate constant is larger than the cathodic rate conthat reduction rates could be measured at potentials signifistant at potentials more positive than the standard potential. cantly more positive than its standard potential without sig(Note that the procedure given in ref 3 for making back-renificant contributions from the anodic back reaction.14 By action corrections is erroneous.14) using a combination of dc polarography and fast chronocouResults lometry, the reduction rate of Cr(OH2)50S03+ can be measured over a potential range of almost 600 mV which correTables I and I1 summarize the experimental apparent sponds to a change in rate constant of almost six orders of transfer coefficients, aaPp,evaluated a t 50-mV intervals durmagnitude. Over this range of overpotential, a change in the ing the reductions of Cr(oH2)s3+, Cr(OH2)5F2+, and electrochemical transfer coefficient of about 25% is predicted Cr(OH2)50S03+. Acidified 1 M NaClQ4 was used as one theoretically.8 supporting electrolyte (Table I) to facilitate comparisons The reduction kinetics of two additional outer-sphere between the present and previous data.3 In addition, the efreactants, Cr(OH2)5F2+and C r ( 0 H ~ ) 6 ~were + , also investifects of specific adsorption of perchlorate anions have been gated. The differing charges of the three reactants produce shown to be minor within the range of potentials covered in quite disparate Frumkin double-layer corrections which is Table I.14 However, recent kinetic studies have indicated that advantageous in separating double-layer effects from any true the coefficient, (d&/dE),, which is required in the conversion potential dependence of the transfer coefficient. The results of apparent to intrinsic transfer coefficients, is significantly of these measurements are summarized in this report. smaller than the value calculated from Gouy-Chapman-Stern (GCS) theory in 1M NaC104.14 Better agreement with GCS Experimental Section theory resulted when this coefficient was evaluated from kiCr(OH2)50S03+ was separated from solutions of reagent netic measurements with 0.04 M La(C104)~as supporting grade chromic sulfate (J. T. Baker Chemical Co.) by cation electrolyte at potentials negative of -800 mV.14 For this reaexchange (BioRad AG 50W-X8 resin in its hydrogen ion form; son apparent transfer coefficients were also measured for the elution was with 0.15 M HC104).15,16Solutions of Cr(OH2),F2+ three Cr(II1) complexes in this supporting electrolyte (Table were prepared as described in ref 17. In some cases, the com11). plexes were eluted with acidified sodium perchlorate rather The values of aappin Table I were evaluated according to than pure perchloric acid to facilitate the preparation of test RT d log hap solutions of more widely varying pH. aapp= -2.3 F dE ' ) K Reduction rates were measured by means of chronocoulometry and dc polarography as previously described.14 ExcelThe apparent rate constant, hap', was evaluated from the lent agreement between the two techniques resulted when the measured ratio of the current density to the limiting current conventional Koutecky analysis of irreversible polarographic density according to the procedure of Koutecky.ls p is the ionic waved8 was employed without a correction for spherical difstrength of the supporting electrolyte and the other symbols have their customary significance. In replicate experiments, fusion.14 Rate constants a t the largest overpotentials were evaluated chronocoulometrically using relatively low (1-4 the values of aaPpcould be reproduced to better than f2%. mM) reactant concentrations; rate constants up to ca. 0.1 cm The values of aappin Tables I and I1 do exhibit a trend toward somewhat smaller values a t more negative potentials sY1 could be reproducibly evaluated with uncertainties below for two of the complexes. However, it is necessary to convert f 5 % . Rate constants a t the smallest overpotentials were evaluated polarographically using higher reactant concenthe values of aappto the corresponding intrinsic transfer trations (up to 50 mM) in order to obtain Faradaic currents coefficients,ll a ~in, order to compare the observed potential sufficiently larger than the residual current. This procedure dependences with the theoretical predictions. This conversion allowed the measurement of rate constants as small as ca. 4 can be made by means of X 10-7 cm s-1. The absence'of polarographic streaming maxima was confirmed by careful inspection of individual current-time curves and noting that the rate constants obtained were independent of the reactant concentration. where Z is the charge carried by the reactant. It is customary Maxima are encountered in supporting electrolytes of lower to obtain the necessary values of (d&/dE), from chargeionic strength or high pH3 but these were apparently suppotential data and GCS theory. This procedure was followed in calculating the values labeled a1GCS in Tables I and I1 (the pressed by the high ionic strength (1.0 M) utilized in these studies. charge-potential data were taken from ref 14). The concentrations of Cr(OH2)5F2+and c r ( O H 2 ) ~ were ~+ With the 0.040 M La(C104)3supporting electrolyte the re-
(
The Journal of Physical Chemistry, Voi. 80, No. 17, 1976
Potential Dependence of the Electrochemical Transfer Coefficient
1863
TABLE I: Apparent and Intrinsic Transfer Coefficients as a Function of Electrode Potential for the Reduction of Three Cr(II1) Complexes in 1 M NaCl0du Potential, -E, mV vs. SCE
Cr(OH2)63f aapp
b
Cr(OH2)5F2+
cyIGCSe
aapp
aIGCSe
Cr(OH2)50S03+ ff1g
cyIGCSe
aapp b
aIg
600 0.55' 0.55' 0.53 700 0.58' 0.55' 0.51 0.53 750 0.59' 0.45 0.546 800 0.585' 0.39 0.59' 0.47 0.545 0.556' 0.51 0.53 850 0.58'jd 0.41 900 0.58'jd 0.42 0.59' 0.50 0.54 0.555,' 0.54jd 0.52f 0.53f 1000 0.E17~ 0.44 0.585,' 0.5Sd 0.515 0.54f 0.55,' 0.545d 0.52f 0.52f 1100 0.57d 0.45 0.586,' 0.57bd1 0.515 0.5451 0.54bd 0.52 0.52 1150 0.54d 1200 0.57~~ 0.52 1250 0.575d 0.52 The pH of all solutions was adjusted to be between 1and 3. There was no effect of pH on the kinetics within this range. b Calculated from eq 1. Evaluated from polarographic data. Evaluated from chronocoulometric data. e Calculated from eq 2 with the values of (d@z/dE),from Gouy-Chapman-Stern theory (see ref 14). f Where the polarographic and chronocoulometric values of aappwere different, their average was taken before CUIvalues were calculated. g Calculated from eq 5 . See text.
TABLE 11: Apparent and Intrinsic Transfer Coefficients as a Function of Electrode Potential for the Reduction of Three Cr(II1) Complexes in 0.04 M La(C104)3" Potential -E. mV
Cr(OH2)63+
750
0.595'
800 850
0.585'
900 1000 1100
0.58'jd 0.56'~~
0.54d 0.52ijd
0.48 0.485 0.495 0.49 0.485 0.48
Cr(OH&F2+
0.57' 0.57' 0.555'
0.54 0.54 0.536
Cr(OH2)50S03+
-
0.55'
0.53
0.545'
0.53 0.52
0.53d
0.52
The pH of all solutions was adjusted to be between 1and 3. There was no effect of pH on the kinetics within this range. Calculated from eq 1. Evaluated from polarographic data. Evaluated from chronocoulometric data. e Calculated from eq 2 with the values of (d@z/dE), from Gouy-Chapman-Stern theory (see ref 14). sulting values of a1GCS are essentially independent of electrode potential for all three complexes (Table 11). With the 1 M NaC104 supporting electrolyte, the values of a~~~~for the singly charged complex (Cr(OH2)50S03+)are virtually constant, while the more highly charged complexes show appreciable increases in q G C S a t more negative potentials. This trend, which is opposite to the theoretical prediction? probably arises from the inaccurate estimates of (d&/dE), that result when GCS theory is applied to this supporting electrolyte.14 The magnitude of the correction for diffuse layer effects is quite small for the singly charged sulfato complex so that the calculated values of cqGCS are not very sensitive to uncertainties in the estimated values of (d+2/dE),. Nevertheless, it seemed desirable to try to obtain values of a1 which did not depend on theoretical estimates of (d&/dE), and would therefore not be influenced by shortcomings in conventional GCS theory. The following procedure was developed with this objective in mind: Inasmuch as the three complexes under study have very similar standard rate constantslZb (and, therefore, similar reorganizational energy barriers to reaction) they are predicted to exhibit very similar potential dependences of their intrinsic transfer coefficients.s However, since the three complexes bear different charges they are expected to show quite dissimilar potential dependences of cyapp which can be used to estimate values of (d+z/dE), without direct
recourse to GCS theory. Thus, if we consider two complexes with charges 2' and 2" and intrinsic transfer coefficients a; and a;(which exhibit apparent transfer coefficients aippand aippat a potential E , the changes in the apparent transfer coefficient, Aaapp, resulting from a change in electrode potential, AE,will be given by
For complexes with equal, or nearly equal, standard rate constants Aa; = AaY, and it may be assumed that a; = a;. Applying these two conditions and substracting eq 4 from eq 3 yields
The values of Aaapp are obtained from the experimental rate data and 2" and 2' are known so that changes in (d+z/dE), with potential can be calculated from eq 5 without resort to GCS theory. It is still necessary to know one value of (d+z/dE), in order to use eq 2 to calculate cy1 from aaPpWe employed the value of this coefficient calculated from GCS theory at -900 mV in 0.040 M La(C104)3 for this purpose because our previThe Journal of Physical Chemistry, Vol. 80, No. 17, 1976
1864
Michael J. Weaver and Fred C. Anson
ous measurements indicated that GCS theory was reasonably satisfactory in this electr01yte.l~Equation 5 was then applied by comparing aappvalues for Cr(OH~)50S03+and Cr(OHz)5F2+with those for Cr(OH2)e3+to calculate A(d&/dE), and, subsequently, (@z/dE), a t each potential. Values of a1 for the sulfato and fluoro complexes were then calculated from eq 2 and are listed in Table I. Note that the small potential dependence of aIGCSfor the latter complex is now eliminated; the values of a1 for both complexes show virtually no dependence on electrode potential. Thus, it seems unlikely that the observed lack of potential dependence can be attributed to inaccuracies in the corrections for diffuse-layer effects. Note in particular that the values of aappand a1 for the sulfato complex are very similar because even rather large changes in the value of (d&ldE), do not produce large changes in a1 for this singly charged reactant. In order to compare the values of a1 and qGCS in Tables I and I1 with those predicted from theory, the magnitude of the predicted potential dependence of CUIis needed. This can be calculated from Marcus theorys according to eq 620(see also the Appendix)
F ( E - Eo) QI = 0.5 (6) 2x where Eo is the standard potential, and 1 is the Marcus intrinsic reorganization terms8 is related to k,, the rate constant at the standard potential, by
+
4
-+/
-60
1
I
I
(7) where 2, is the diffusion limited frequency of molecular collisions with the electrode [ Z , = ( k T / P ~ m ) lwhere /~ m is the mass of the reactant, 2, is usually estimatedg as lo4 cm s-l, and values near this figure are obtained for the present reactants using this formula]. Apparent rate constants at the standard potentials for the three complexes arel2b7.5 X 10-5, 4X and 6 X cm s-l for the aquo, fluoro, and sulfato complexes, respectively. Correction of these values for the diffuse layer acceleration of the reduction rates using the Frumkin formulalg yields12bk , = 2.5 X 10-6,2.1 X and 3.6 X cm s-l. These values are close enough to each other that an average value, 2.7 X cm s-l, was used to obtain from eq 7 a value of X/F of 2.3 V which, with eq 6, yields a1 = 0.5
+ 0.22(E - Eo)
(8)
Thus, a1 is predicted by Marcus theorys to decrease by ca. 0.022 for each 100 mV of overvoltage for all three of the complexes studied. (Note that the magnitude of the coefficient of ( E - Eo) in eq 8 is only weakly dependent on the absolute values of the (somewhat uncertain) complex formation constants used in evaluating the E o values for the fluoro and sulfato complexes.12b) Data were obtained for the sulfato complex over a potential range of 550 mV within which eq 8 predicts that a1 should decrease by 0.13 (i-e.,over 25%). In fact, cy1 is observed to decrease by only 0.01 for this complex (Tables I and 11) which is within the experimental reproducibility. The fluoro and aquo complexes also do not exhibit the changes in the values of a1 or cylGCS that are predicted by eq 8. The extent of the discrepancies between the predictions of eq 8 and the experimental rate-potential data is also illustrated in Figure 1for the sulfato complex. The experimental data are the plotted points. Curve 1was obtained by applying the Frumkin diffuse layer correction to the data. Curve 2 is the log kapp vs. potential curve calculated from assuming (arbitrarily) that the slopes of the theoretical and FrumkinThe Journal of Physical Chemistry, Vol. BO, No. 17, 1976
corrected experimental curves could be matched a t -600 mV. Even larger discrepancies between curves 1and 2 would result if curve 2 were forced to have the predicted slope corresponding to a1 = 0.5 a t E = E o (ca. -700 mV) as predicted by the Marcus models because the slope of curve at this potential corresponds to a = 0.53.
Discussion The present data for Cr(OHz)s3+ are in good accord with those presented previously3 in which no potential dependence of a1 was found. The fact that a similar potential independence is also exhibited over a much wider potential range by c r ( O H ~ ) 6 ~Cr(OH2)5F2+, +, and especially Cr(OH&OSOZ+, for which diffuse layer corrections are much smaller, clearly indicates that these three complexes do not behave in the way predicted by the Marcus theory despite the apparent simplicity of their electrode reactions and their demonstrated adherence to outer-sphere pathways.12 A number of factors may be responsible for this circumstance: The free energies of activation for the three electrode reactions varied from 6 to 15 kcal mol-l over the potential range investigated. These are considerably larger free energies of activation than those for the homogeneous electron transfer reactions which have provided the most convincing experimental evidence in support of a quadratic term in the Marcus t h e o r y . 5 1 ~Unfortunately, ~,~~ presently available electrochemical kinetic techniques are unable to cope with reaction
Potential Dependence of the Electrochemical Transfer Coefficient
1865
rates as large as those that result when systems with signifiUncertainties in double-layer corrections were also faced cantly smaller activation energies are employed. by Bindra et al. in their kinetic studies of the HgZ2+1HgreThe constancy of a1 for the Cr(III)/Cr(II) couple implies action? These authors obtained more or less parabolic-shaped that the free energy-reaction coordinate profile for these reTafel plots for this reaction but, in order to account for the actions is linear rather than parabolic, a t least within the acunusual kinetic behavior they observed, extremely large, cessible potential range. The predicted quadratic dependence positive, and potential-independent values of 42 were reof the free energy of activation upon the free energy of reaction quired. Such values of 4 2 are of the opposite sign than those arises from the assumption that the changes in system free to which previously measured values of q52 in perchlorate soenergy with changes in the reaction coordinate (arising from l u t i o n ~would ~ ~ extrapolate, and very much larger than our solvent polarization and metal-ligand bond stretching and kinetic data on the oxidation O f C1(OH2)e2+and Eu2+would compression) are harmonic. This assumption, however, is suggest.26Moreover, there is e v i d e n ~ eto~ support ~ , ~ ~ the claim probably more appealing for its convenience than for its acthat electrode processes which involve the deposition of recuracy.23 For instance, if the important bond stretching modes action products on the electrode surface will usually involve were distinctly anharmonic, as has been suggested, for exactivated states which lie closer to the electrode surface than ample, for the ligand-metal “breathing” vibration^?^ it is the outer Helmholtz plane so that the effects of adsorbed conceivable that a more linear activation energy-reaction anions on electron-transfer rates would be expected to be coordinate profile could result. Thus, the present results need much greater than is allowed for in the conventional Frumkin not be regarded as a t variance with the essential notions of correction.lg Thus, the behavior observed by Bindra et ala6 Marcus theory8 (and the other similar appro ache^^,^^) alcould conceivably be the result of nothing more than a dethough neither do they offer strong support (except that the creasing adsorption of perchlorate anion at more negative measured intrinsic transfer coefficients a t the standard popotentials. tential are very close to the value of 0.50 predicted by Marcus In a very recent report7 Saveant and Tessier described kitheory and most other models). netic measurements on the reduction rates of tert-nitrobutane Inasmuch as it is our conclusion that a1 for the Cr(II1) in nonaqueous solvents in which curved plots of log rate vs. complexes studied does not vary with potential as predicted electrode potential were obtained. The nonlinearity was arby Marcus theory it seems appropriate to comment on pregued not to be the result of a diffuse layer effect on the basis viously published experimental results which have been inof double layer data which showed (d&/dE), to be approxiterpreted as providing support for this prediction. Mohilner2b mately independent of E for the electrolyte and potential pointed to the kinetic data of Randles and W h i t e h ~ u s efor ~ ~ range involved (cf. eq 2). The magnitude of the curvature the V3+ (V2+ couple in concentrated sodium perchlorate observed, while comparable to the experimental uncertainties, electrolytes as support for the predictions of Marcus theory. was nevertheless in fair agreement with the predictions of It is true that the apparent (cathodic) transfer coefficient Marcus theory whether or not the small diffuse layer correcdecreased significantly at increasingly negative potentials in tions were applied. Although the standard rate constant is this study but this is to be expected from the effect of the somewhat larger for the reduction of tert-nitrobutane (7 X diffuse layer on the kinetics in the perchlorate electrolytes cm s-l in dimethylcm s-1 in acetonitrile, 5 X employed. Mohilner2b used the limited double-layer data formamide) than that for the reduction of Cr(II1) (2 X given in ref 2a to argue that the observed variation in aapp was cm s-l), they are not sufficiently disparate to encourage a too great to be attributable only to a diffuse layer effect. search for an explanation of the contrasting behavior in terms However, in recent experiments,26we have measured the oxof significantly different activation energies. Indeed, the idation rates of Cr( in perchlorate electrolytes over Cr(II1) reaction shows no indication of curved plots even when a potential range similar to those involved in ref 25 and obits rate is measured a t potentials which yield rates as large as tained values of (1 - aapp)which exhibited an appreciable those a t which the plots for tert-nitrobutane are reported to potential dependence despite the absence of a comparable be quite curved.7 It remains to be seen why a potential depotential dependence of aappfor the reduction of C I ( O H ~ ) ~ ~ +pendence . of the transfer coefficient is evidently present with We conclude that the observed variation in (1- aapP)results this simple, one-electron organic electrode reaction in nonentirely from the effect of the diffuse layer on the kinetics and aqueous media but is not observed with simple one-electron this conclusion would apply to the V3+ I V2+ couple as well. inorganic electrode reactions in aqueous media. This assertion is also supported by measurements of the kinetics of Eu2+ oxidation in perchlorate electrolytes.26 The Acknowledgments. This work was supported by the Nadiffuse layer potentials required to account for the kinetic tional Science Foundation and the US.Army Research Office behavior are in qualitative disagreement with those calculated (Triangle Park). from conventional electrocapillary data and diffuse layer theory, i.e., the procedure used to obtain the potentials used Appendix in Mohilner’s analysis.” The message appears to be that kinetic data obtained in the presence of significant specific Two procedures, involving two distinct definitions of the adsorption of components of the supporting electrolytes are electrochemical transfer coefficient, have been utilized in the poor choices for tests of the Marcus theory because of the previous literature to compare the predictions of Marcus considerably greater uncertainties in applying the necessary theory with experiment. Although the two procedures are in diffuse-layer corrections. fact equivalent, it seems desirable t o point out their superficial The recent attribution4of measured curvature in Tafel plots differences in order to avoid misunderstandings-we were for the Fe3+I Fe2+ reaction at platinum electrodes in perinitially confused by the apparent differences in the two chlorate electrolytes to a potential dependence of a1 suffers procedures. not only from the difficulty just cited but also from the lack In the first experimental paper on this topic, Parsons and of any reliable data on the extent of perchlorate specific adPasseron2a analyzed their kinetic data in the conventional sorption on platinum. way30 by defining a transfer coefficient, a l , in terms of the The Journal of Physical Chemistry, Vol. 80, No, 17, 1976
1866
Michael J. Weaver and Fred C. Anson
slopes of plots of the logarithm of the rate constant vs. electrode potential, Le., "Tafel slopes". Thus, for a one-electron reduction these authors wrote (in effect): -RT d In k a1=--- 2.3(Tafel slope)-l (AI) F dE (For convenience, any diffuse-layer corrections are assumed to have already been applied.) The observed potential dependence of the resulting transfer coefficients were then compared with the predictions of Marcus theory as expressed in8 X F ( E - Eo) [F(E- E0)]2 -RT In = (A21 2 4x
(t,
+
+
Differentiating eq A2 with respect to potential and combining the result with eq A1 leads to eq A3 which was employed by Parsons and Passeronza = 0.5
r +( E - EO) 2x
(A3)
Marcus also gives an expression very similar to eq A3 (eq 87d, ref 8a). The work terms arising from the effect of the diffuse layer which were explicitly included in the Marcus equation are essentially eliminatedz0from equations such as eq A1 by employing rate constan'ts which have been corrected for the diffuse-layer effects. The second approach, which has been employed in some studies,2b,7 begins with the classical expression for the potential dependence of the heterogeneous rate constant, for a one-electron reduction k = k,exp [-;:(E -
-Eo)]
in which a2 is regarded as constant. An equivalent form of eq A4 is a2
1
=-
Now, if a2 is allowed to be potential dependent, the combination of eq A5 and A2 yields2$'
The coefficients of the ( E - Eo) term in eq A3 and A6 are clearly different but this apparent difference arises from the alternative definition of a1 and az: 012 corresponds to the slope of a chord drawn between the points for In k and In k, on the In k-potential curve, while cy1 corresponds to the slope of the tangent to the curve a t In k . The relationship between these two definitions of the transfer coefficient is given by 011
= 2az
- 0.5
(-47)
Tests of the adherence of experimental kinetic data to the predictions of Marcus theory by means of eq A3 or A6 will
The Journal of Physical Chemlstry, Vol. BO, No. 17, 1976
clearly be equivalent so long as the difference between the definitions of a1 and 012 is appreciated. Saveant and Tessier evaluated a2 and its potential dependence in their recent study.' Our preference has been to follow Parsons and PasseronZain evaluating a1 because this parameter is directly related to the differential change in the free energy of activation with the change in the free energy of reaction at each potential,sa and, therefore, can be identified with the intrinsic transfer coefficient.11
References and Notes (1) Contribution No. 5293. (2) (a) R . Parsonsand E. Passeron, J. Electroanal. Chem., 12, 525 (1966): (b) D. M. Mohilner, J. Phys. Chem., 73, 2652 (1969). (3) F.C. Anson, N. Rathjen, and R. D. Frisbee, J. Electrochem. Soc., 117,477 (19701. - -, (4) T. Dickinson and D. H. Angell, J. Electroanal. Chem., 35, 55 (1972). (5) K. Suga, J. Mizota. Y. Kanzaki, and S. Aoyagi, J. Electroanal. Chem., 41, 313 (1973). (6) P. Bindra, A. P. Brown, M. Fleishmann, and D. Pletcher, J. Electroanal. Chem., 58, 39 (1975). (7) J. M. Saveant and D. Tessier, J. Electroanal. Chem., 65, 57 (1975). (8) (a) R. A. Marcus, J. Chem. Phys., 43, 679 (1965); (b) Electrochim. Acta, 13, 995 (1968). (9) N. S. Hush, J. Chem. Phys., 28, 962 (1958). (10) V. G. Levich in "Advances in Electrochemistry and Electrochemical Engineering", Voi. 4, P. Delahay and C. Tobias, Ed., interscience, New York, N.Y., 1966. (11) R. Parsons, Croat. Chem. Acta, 42, 281 (1970); M. J. Weaver and F. C. Anson, J. Electroanal. Chem., 58, 81 (1975). (12) (a) M. J. Weaver and F. C. Anson, J. Am. Chem. Soc., 97,4403 (1975); (b) lnorg. Chem., in press. (13) Suga et have argued that outer-sphere electron exchange between complexes of Cr(lil) and Cr(1l) is unlikely because of the substitutional lability of the latter. Our experience is that these complexes may follow either inner-sphere or outer-sphere mechanisms in their electrode reactions depending on the chemical (e.g., coordinating) properties of the ligands attached to the metal cation.12 (14) M. J. Weaver and F. C. Anson, J. Electroanal. Chem., 65, 711 (1975). (15) N. Fogel, J. M. J. Tal, and J. Yarborough, J. Am. Chem. SOC., 84, 1145 (1962). (16) That the OS032- group is monodentate in the complex is demonstrated in J. E. Finhott, R. W. Anderson, J. A. Fyfe, and K. G. Cauton, lnorg. Chem., 4, 43 (1965). (17) T. W. Swaddle and E. L. King, lnorg. Chem., 4, 532 (1965). (18) J. Koutecky, Collect. Czech. Chem. Commun., 18, 597 (1953); J. Weber and J. Koutecky, ibid., 20, 980 (1955). (19) P. Deiahay, "Double Layer and Electrode Kinetics", Interscience, New York, N.Y., 1965, Chapter 9. (20) Rigorously, ( E - E?) should be replaced by [ E - 4 2 - E) 4-@pol where @2 and are the potentials at the reaction site when the measured electrode potential is E and E), respectively. However, the small values of (d@,/de) in the electrolytes employed in the present study14 Justifythe simplifying approximation embodied in eq 6 and A3. (21) D. Rehm and A. Weller, Ber. Bunsenges. Phys. Chem., 73,934 (1969). (22) Note that the free energy of activation for an electrochemical reaction at its standard potential is predicted to be only half as large as that for the corresponding homogeneous self-exchange reaction. Cf. R. A. Marcus, J. Phys. Chem., 67, 853 (1963). (23) R. A. Marcus, Discuss. Faraday Soc., 29, 21 (1960); 45, 7 (1968). (24) W. L. Reynolds and R. A. Lumry, "Mechanisms of Electron Transfer", Ronald Press, New York, N.Y., 1966, p 125 ff. (25) J. E. B. Randles and D. R. Whitehouse, Trans. Faraday Soc. 64, 1376 (1968). (26) M. J. Weaver and F. C. Anson, unpublished experiments. (27) R. Payne, J. Phys. Chem., 70,204 (1966); R. Parsons and R. Payne, Z.PhYS. Chem. (Frankfurt am Main), 98, 9 (1975). (28) R. Parsons, J. Electroanal. Chem., 21, 35 (1969). (29) W. R. Fawcett and S. Levine, J. Electroanal. Chem.. 43, 37 (1973). (30) R. Parsons, Trans. Faraday Soc., 47, 1332 (1951); ref 19, Chapter 7 . \
