Frumkin Correction: Microscopic View

1023-1935/02/3802- $27.00 © 2002

åÄIä “Nauka

/Interperiodica”0132

Russian Journal of Electrochemistry, Vol. 38, No. 2, 2002, pp. 132–140. Translated from Elektrokhimiya, Vol. 38, No. 2, 2002, pp. 154–163.Original Russian Text Copyright © 2002 by Tsirlina, Petrii, Nazmutdinov, Glukhov.

INTRODUCTION

The slow-discharge theory of A.N. Frumkin, whosemost important positions had been formulated at thebeginning of the 1930s [1–3], phenomenologicallydescribes the kinetics of heterogeneous charge transferreactions and takes into account on a physical level theelectrostatic interaction of reactants and products witha charged interface. Hence, the brighter the manifesta-tion of the electrostatic effects, the higher the efficiencyof application of this theory to real processes. In partic-ular, a detalization of the theory became possible by thebeginning of the 1960s on the basis of an analysis of alarge accumulation of data concerning the reduction ofanions on negatively charged surfaces [4–8]. At thatperiod the weak sides of the theory, which manifestedthemselves in the absence of quantitative agreement withexperiment in the case of certain, albeit just a few, sys-tems or during the interpretation of results, were exposed.

Attempts to refine the slow-discharge theory, forexample, those undertaken in [9], in fact widely missedthe Frumkin approach proper; instead, they amountedto its combination with modeling assertions concerningthe structure of charged interfaces metal/solution. The“geometrical” corrections (allowance for the size andshape of the reactant) can be viewed as the simplest;these were introduced in the framework of the Gouy–Chapman model for a diffuse layer and were based onanalyses of phenomena of specific adsorption of support-ing-electrolyte ions and/or reactants. The intrinsicallycontradictory nature of these corrections (although someof them led to quite reasonable, on the face of it, results)is connected with three circumstances.

In the first place, when operating with the notion ofa psi-prime potential

1

as the potential at the point of

1

In what follows, we will specially dwell upon the problem of def-inition for a psi-prime potential. What is essential here is onlythat a psi-prime potential is a potential at a fixed point.

localization of a reactant or a transition state, in princi-ple, one must stay in the framework of a point-reactantapproximation. At the same time, the approximationaforementioned obviously breaks down in the case ofreal reactants whose size is commensurate with charac-teristic dimensions of the nonuniformity of the chargedistribution near an interface (Gouy–Chapman thick-ness of the diffuse layer, the dense layer thickness, andso on). Secondly, for a criterion of reasonableness of acorrelation, while remaining in the framework of theslow-discharge theory, one has to use the linearity ofcorrected Tafel plots (CTP). However, this criterion isin fact based on the phenomenological principle ofBroensted (the condition of constancy of the transfercoefficient). The broader the potential interval underanalysis, the poorer the reliability of this criterion.Finally, an analysis of experimental data is based on theassumption that a change in the potential (and, corre-spondingly, the charge) of the electrode makes noimpact on the magnitude of the heterogeneous rate con-stant of charge transfer in the Frumkin equation, i.e.there are no charge-dependent factors other than themagnitude of the psi-prime potential. The last two cir-cumstances are directly related to a phenomenologicalcharacter of the slow-discharge theory, the fact repeat-edly emphasized by Frumkin [7], and cannot be over-come without resorting to some physical models of anelementary act of charge transfer, which throw light onthe nature of the transfer coefficient and heterogeneousrate constant.

The Frumkin equation of the slow-discharge theorymay be derived in the framework of a Levich–Dogo-nadze–Kuznetsov quantum-chemical approach. In anarrower overvoltage interval, the equation may also bederived, in particular, in the framework of the widelyknown Marcus formula [10–12]. Modern approaches toanalysis of experimental data on the rate constants andthe slopes of polarization curves are based precisely on

Frumkin Correction: Microscopic View

G. A. Tsirlina, O. A. Petrii, R. R. Nazmutdinov*, and D. V. Glukhov*

Moscow State University, Vorob’evy gory, Moscow, 119899 Russia*Kazan State Technological University, ul. Karla Marksa 68, Kazan, Tatarstan, 420015 Russia

Received March 22, 2001

Abstract

—Approaches to perfecting analysis of psi-prime effects, based on modeling the reaction layer ofelectrochemical reactions and allowing for real molecular structure of reactants and products, are considered.Conditions of applicability of the traditional slow-discharge theory relations to systems with reactants of com-plicated structure are analyzed in a wide overvoltage interval. Special attention is paid to the problem ofaccounting for nonuniformity of charge distributions in particles of reactants and products. It is shown that itbecomes essential to account for the dielectricity of the effective cavity used for modeling the particles, pro-vided the particles are sufficiently large.

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FRUMKIN CORRECTION: MICROSCOPIC VIEW 133

such notions; however, they frequently fail to accountfor molecular specifics of systems under study: see, forexample, [13]. In the present work, on the basis of ageneralization of a series of previous papers [14–30],we briefly analyze the assumptions in the framework ofwhich one can apply simple equations of the slow-dis-charge theory to a treatment of experimental data.

THE STRUCTURE OF THE REACTION LAYER AND THE SELECTION OF MODEL

PARAMETERS OF THE THEORY

A transition from a phenomenological theory to aphysical one unavoidably calls for an adequate analysisof the structure of the reaction layer as a combination ofatoms, ions, and molecules that constitute the electrodesurface and the adjacent solution layer.

At this moment in time, it is difficult to offer anygeneral approach to this problem, which in the futurewill, no doubt, be solved in the framework of quantumchemistry [30]. However, at this junction it is of impor-tance to realistically evaluate the geometry of the reac-tion layer. Such an evaluation is most reliable in thecase of reactants sufficiently large compared with thesize of the supporting-electrolyte ions, in the absence ofspecific adsorption (penetration in the dense part ofEDL). In this case, the distance of the closest approachmay be determined to a first approximation from thecondition of contact between the reactant “edge” andthe outer Helmholtz plane (OHP). For the potential dis-tribution over the reaction layer, on the other hand, onecan use relations of the Gouy–Chapman theory. For theedge of a particle one can consider the perimeter of asphere or an ellipsoid, depending on the real form,which are used to approximate a particle of an electro-chemically active substance.

The selection of the size of a sphere (ellipsoid)depends on both the atomic structure of such a particleand the goal of the subsequent calculation. To deter-mine parameters that directly depend on distance andspatial configuration, it makes sense to construct enve-lopes. The finite size of atoms at the periphery of parti-cles introduces certain indefiniteness into this proce-dure. In connection with this, a procedure intended forthe estimation of the aforementioned size was proposedin [19]. The procedure accounted for effective charges

q

on the atoms and was based on the notion that theradius smoothly varies with increasing

q

from atomicradius

r

at

to ionic radius

r

ion

:

(1)

For the solvation calculations (in the first place, forcalculations of the reorganization energy of the sol-vent), with the reactant approximated by a sphere, themost satisfactory is probably the determination of aneffective Bornian radius,

r

Born

, from the magnitude ofthe solvation energy (for more details, the reader isreferred to [30]).

ref' 1 q–( )rat qrion.+=

Finally, for the selection of the electron transfer dis-tance in an analysis of the preexponential factor, of cru-cial importance is the overlap of molecular orbitals ofthe reactant and the product [30]. Only at a symmetricalmolecular-orbital structure of both redox forms is itpermissible to consider

r

ef

as a parameter of the transferdistance, of course with a correction for the distancefrom the particle edge to the electrode surface.

The procedure that could be employed for reducingthe Marcus equations for the electron transfer rate to theequations of the slow-discharge theory was exhaus-tively considered in [21, 23]. Here recall only that rele-vant parameters are interrelated. In a general form, therelation for the current density

j

of a cathodic one-elec-tron reaction is

(2)

where [

c

] is the reactant concentration in the bulk solu-tion,

F

is Faraday’s number,

k

is a heterogeneous rateconstant, and

E

a

is an activation energy. In the slow-dis-charge theory,

(3)

where

α

is a transfer coefficient,

z

ox

is the charge of areactant particle, and

ψ

1

is the psi-prime potential. Inthe Marcus theory, the formula

(4)

includes work terms of the reactant (

W

ox

) and product(

W

red

) and parameter

λ

, which is the total reorganiza-tion energy. The meaning and the magnitude of con-stant

k

in (2) for the same reactions differ in the frame-work of the aforementioned theories [21].

The heterogeneous rate constant is a complicatedcombination of parameters. Specifically, it involves apenetration probability, which depends to one extent oranother on

r

ef

; the reorganization energy of the solvent,which depends on

r

Born

; the inner-sphere reorganizationenergy, which is insensitive to the reaction layer config-uration [15]; and, for adiabatic reactions, the character-istic times of the solvent relaxation. In the absence ofstrong electrostatic and other effects that depend on theelectrode charge, the transfer coefficient is definedlargely by the magnitude of the total reorganizationenergy. Experiment yields an observed transfer coeffi-cient whose magnitude is fundamentally affected byquantities

W

ox

and

W

red

. Finally, when considered in asimplest fashion in the framework of a point-reactantmodel, the psi-prime potential is connected with thework terms through expressions

(5)

A transition from a point-reactant model to a morerealistic “microscopic” consideration unavoidablyleads to a more complicated structure of the reactionlayer. In particular, even in the absence of various formsof the reactant in the bulk solution, for many processes

j c[ ] Fk Ea/RT–( ),exp=

Ea F α zox–( )ψ1– αFη ,–=

Ea Woxλ Fη– W red Wox–+( )+[ ] 2

4λ-----------------------------------------------------------------+=

Wox zoxFψ1, W red zox 1–( )Fψ1.= =

134

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et al

.

one must not ignore the nonuniformity of the reactionlayer associated with the orientation effects [14, 19].The orientation distributions as a factor affecting theshape of polarization curves was first considered on aqualitative level forty years ago [31]. By operating withvalues of work terms, it appears to be possible to detailthese notions and calculate shares of various orienta-tions at various distances from the electrode surface.

CORRECTING VALUES OF THE PSI-PRIME POTENTIAL

The most transparent, although rather complicatedproblem is a direct refinement of values of potentialsthat are used for introducing electrostatic corrections(in the English-language literature, the “Frumkin cor-rection”). The first attempts at such a refinement wereundertaken in [5–7], where various localizations of aneffective point charge (or a uniformly charged sphere)were compared. The position of a point (sphere) rela-tive to the electrode was varied in an arbitrary manner,thus trying to formally linearize CTP while using theGouy–Chapman dependences of the potential at thepoint of the reactant center localization on the distance

(6)

where

ψ

OHP

was the OHP potential, and the Debyeshielding length was determined for each value of theelectrolyte concentration

C

by the expression

(7)

On the other hand, if the center localization in thedense layer was considered, it was assumed that thepotential across it dropped linearly from

E

to

ψ

OHP

. Theapproach used to yield results that were difficult toexplain. For example, the linearity of CTP for

[Fe(CN)

6

]

3–

was reached with the reactant placed 1 nmand even farther away from OHP, i.e. at a distance fromthe electrode, which was almost twice the reactant size(sphere’s effective radius 0.43 nm [26]). Conversely,

the linearity of CTP for

S

2

was reached if the reac-tant was placed inside the dense part of EDL, i.e. at adistance from the surface, which was obviously smallerthan the anion size (effective ellipsoid of revolutionwith half-axes of about 0.35 and 0.65 nm [32]).

Approach [5–7] was developed in [14, 19, 25, 28] byaccounting for the real geometry and charge distribu-tions in electrochemically active species. The “micro-scopic psi-prime effect” under discussion is defined bythe magnitude of work term

W

, which is calculated as

ψ z( ) 4RTF

-----------FψOHP

4RT----------------

zκ---–

exptanh

,arctan=

κε0εRT

2CF2---------------.=

O82–

the sum of work terms for point charges on rigidlybound atoms

(8)

where ψ(zi) is a potential in a plane situated at distancezi from the electrode surface, and qi is the charge of theith atom. It should be noted that in the absolute majorityof cases the resultant quantity W is not too much sensi-tive towards the scheme selected for a quantum-chemi-cal calculation of charge distributions (after Mulliken,natural orbital analysis via scheme CHelpG [28]) evenif differences reach 0.1–0.2 elementary-charge units. Acalculation by scheme CHelpG is probably the mostcorrect. The charges in this scheme are selected self-consistently so as to induce at some point the samepotential as the entire continuous charge density of thereactant does.

Use of the obtained value of W in ordinary equationsof the slow-discharge theory is reached by expressingan effective value of the psi-prime potential through itin accordance with (5). For point charges, Wox – Wred =–Fψ1; after a linear expansion of the quadratic term in(4), this leads to ordinary formula (3). In this case, thepsi-prime potential has the meaning of a potential at thepoint of localization of the reactant, exactly as in theinitial version of the theory [1–3]. Simultaneously (andthis has never been discussed earlier), it uniquely corre-sponds to a potential at the point of localization of theproduct. Some workers of the Frumkin school used alater definition for the psi-prime potential, i.e. as apotential at the point of localization of an activatedcomplex. This definition qualitatively reflected a quitecorrect (probably) idea that the work term of the prod-uct affected the magnitude of the Franck–Condon bar-rier. It should be noted, however, that such “averaging”(in terms of an activated complex) is not enough if thepsi-prime potentials for effective charges that corre-spond to the reactant and product differ. Indeed, forcomplex-structure reactants, in the general case, (Wox –Wred) ≠ Wox/zox. Hence, when linearly expanding thequadratic term in (4), one cannot use the same value ofψ1 for both ψ1-dependent terms in (3). Correspondingmodifications of the equation of the slow-discharge the-ory are given in [23]. An example of a concrete analysisof numerical values of two psi-prime potentials withdifferent meaning (which must be used when construct-ing classic CTP by plotting corresponding quantities indifferent axes) is given in [26].

The main deficiency of the approach [14, 19, 25, 28]to a calculation of a microscopic psi-prime effect is thecrude assumption that the point charges are present inan environment with the same permittivity ε1 as insidea neat solvent. Below we for the first time attempt toestimate the effect of a change in the field of the diffusepart of EDL inside a modeling spherical cavity thatcontains atoms of the reactant. An external field insidea spherical cavity is screened in this case chiefly at the

W qiψ zi( ),i

∑=

RUSSIAN JOURNAL OF ELECTROCHEMISTRY Vol. 38 No. 2 2002


expense of vibrational modes; therefore, the expectedvalues of the permittivity in this region (ε2) lie in thelimits 2–5 [33].

The potential of the EDL field2 Ψ(zi) (at distance zifrom the electrode surface) perturbed by a dielectricsphere with ε < 78 may be represented as the sum

(9)

where Ψ0 = ψOHP is the potential of the diffuse part ofEDL, which is calculated in the framework of theGouy–Chapman theory (in the absence of a dielectricsphere); V is the perturbation caused by the reactant;and subscripts “1” and “2” refer to regions outside thereactant and inside it (inside an effective sphere).Potential Ψ satisfies the Laplace equation

(10)

and meets boundary conditions at the sphere surface

(11)

Apparently, in view of cylindrical symmetry, poten-tials V1 and V2 , when written in a spherical frame of ref-erence bound to the cavity center, depend solely on thedistance r and angle θ. In the present work, the wellknown solution of the Laplace equation for the case ofa uniform field is generalized to a system where thefield is described by a nonlinear Gouy–Chapmanpotential (equation (6)). It can be shown that

(12)

where z = rcosθ, a is the radius of the sphere, and (z)are expressed through adjoint Legendre polynomialsPn(z):

(13)

where

(14)

Integrals (14) were calculated numerically.In order to estimate the influence the effect under

study exerts on the work term of the reactant, we car-ried out a series of modeling calculations for hypothet-ical reactants with different arrangement of two point

2 In what follows we denote the potential considered in the frame-work of a dielectric-sphere model by symbol Ψ, as opposed tosymbol ψ adopted in the framework of an ordinary consideration.

Ψ zi( )

= Ψ1 Ψ0 V1—outside the reactant cavity+=

Ψ2 Ψ0 V2—inside the reactant cavity,+=

∆Ψ 0=

Ψ1 Ψ2 and ε1

Ψ1∂n∂

--------- ε2

Ψ1∂n∂

---------.= =

V1ar---

n 1+

Pn z( ) and V2

n 0=

∞

∑ ra---

n

Pn z( ),n 0=

∞

∑= =

Pn

Pn cn z( )Pn z( )ε1 ε2–( )a

ε1 n 1+( ) ε2n+------------------------------------,=

cn z( ) 2n 1+2

--------------- Ψ0 z( )∂z∂

-----------------r a=

Pn z( ) θ.cosd

1–

1

∫=

charges in the sphere. For simplicity, we considered anasymmetric reactant with an overall charge of –2. Fig-ure 1 shows the arrangement of charges 1–3 of identicalmagnitude. The distance between the charges was fixedat 0.2 nm.

Maximum deviations from values of W obtainedwithout allowance for the effect of “dielectricity” werediscovered for the reactant in which point charges lienear OHP (Fig 1, configuration 1). It is precisely thisreactant, for which we varied the sphere radius in sub-sequent calculations (Fig. 1, configurations 4–6). Thedielectricity effect increased with the radius (Table 1).Finally, we analyzed the dependence of the effect understudy on the supporting-electrolyte concentration C(Table 1). We see in the table that the magnitude of cor-rection substantially rises with increasing C, i.e. withdecreasing Debye shielding length.

Similar calculations were performed for two realreactants with a clearly pronounced nonuniformness ofcharge distributions, namely, heteropolyanion[CoMo6O24H6]3– and persulfate anion [S2O8]2–. Quan-tum-chemical calculations of the geometry and chargedistributions of these two species, carried out by theHartree–Fock–Roothaan ab initio approximation in theframework of the density functional theory (hybridscheme B3LYP), are discussed in detail in, respectively,[35] and [32]. The calculations (Figs. 2, 3) show thatthe dielectricity effect does increase with increasingreactant size.

We conclude that a calculation of microscopic workterms in the framework of a simpler scheme [14, 19, 25,28] is satisfactory enough for low-molecular-weightreactants even if charge distributions are rather nonuni-form and that such a calculation is most accurate at lowconcentrations of the supporting electrolyte.

1

2

3

4

5

6

Fig. 1. Hypothetical reactants: unit point charges in adielectric sphere (ε2 = 5); vertical line represents OHP.

136


TSIRLINA et al.

TRANSFER COEFFICIENT AND ITS DEPENDENCE ON THE POTENTIAL

The nonlinearity of classic CTP is easily explainedin the framework of the Marcus theory, which gives thetransfer coefficient as

(15)

The first attempt to experimentally verify the Mar-cus theory for electrochemical reactions was probablyundertaken by Randles [36]. Quite a few subsequentworks of different arrangement [37–49] threw no lighton the question of applicability of the theory to thedescription of the overvoltage dependence of the trans-fer coefficient and yielded seemingly contradictoryresults. Problems associated with arranging and run-ning informative experiment for a verification of thiskind are quite a few and may be classified in the follow-ing manner.

(1) Selecting a process whose slow stage is theelectron transfer proper. The two principal groups ofreactants that satisfy, to a first approximation, this con-dition are inorganic complexes of transition metals andorganic molecules that form radical anions in thecourse of a one-electron reduction. It should be notedthat selecting the electrode material is of no lesserimportance than selecting a reactant. All other condi-tions being the same, noncomplicated outer-spheremechanisms on adsorption-active metals (for example,Pt and Au) are less probable than on Hg and sp metals.

(2) Allowing for the dependence of a on h causedby the reactant–EDL interaction. Kinetic equationsthat describe the ψ1 effects in the framework of theMarcus theory were specially analyzed in [50]. Themost convincing example of their application to realsystems is probably [43]. The ψ1 effects were mini-mized largely by two techniques, specifically, by takingmeasurements at high concentrations of the supportingelectrolyte and by using low-charge reactants. In theformer case, however, it was impossible to avoid addi-tional complications caused by ionic association and,correspondingly, by transition to bridging transfermechanisms. In the latter case, the selection of possibleobjects for studying was extremely limited (at the very

α 0.5Fη– W red Wox–+( )

2λ------------------------------------------------.+=

least, in an aqueous environment), mainly to different-ligand complexes with small stability constants and arelatively high lability.

Introducing corrections for the EDL structureinvolves acute problems connected with an indepen-dent determination of psi-prime potentials in the casewhere reactants are localized nonuniquely and/or repre-sented by point charges. In connection with this, thedata obtained at negative charges of the electrode sur-face are, indisputably, more reliable. Moreover, thevalidity of the selected localization can be verified fromthe dependence of the current on the supporting-electro-lyte concentration. The systems, where specifically-adsorbed components and effects of discreteness of theEDL structure are absent, are obviously more preferable.

(3) Determining kinetic currents in broad poten-tial intervals. Correct corrections for a concentrationpolarization when using polarography, cyclic voltam-metry, rotating disk electrode, or pulsed chronoamper-ometry and various modifications of these methodsappear to be possible in intervals of about a few tenthsof a volt only for exceedingly slow processes. More-over, the accuracy of corrections at the interval edges,where the expected effect is most pronounced, is as arule poorer than in the middle.

(4) Determining the model parameter of the the-ory (reorganization energy). The most commonexample is a calculation of λ from the rate constantdetermined experimentally at a zero overvoltage. Thetechnique is in principle correct only for adiabatic reac-tions, but even for them it ensures no quantitativeresults, as the preexponential factors are known towithin orders of magnitude. An independent calcula-tion of λ from the experimentally determined value ofthe activation energy (heat) is employed less frequently.The accuracy of this calculation is also wanting, due tothe problems associated with the determination of tem-perature dependences. Theoretical calculations of con-stituents of the reorganization energy have enjoyedconsiderable development only in the last decade [11];however, the systems for which reliable results havebeen obtained are small in number, and examples ofapplication of results of such calculations for in-depth

Table 1. Work terms (kJ mol–1), calculated for hypothetical reactants (Fig. 1) in 1,1-electrolyte of concentration C without(A) and with (B) allowance for changing EDL field inside dielectric cavity (C is the difference between A and B, %)

Radius, nm

C, M

0.01 0.1 1

A B C A B C A B C

0.3 24.5 26.0 5.8 15.3 17.0 10.9 5.1 6.5 29.0

0.4 23.6 25.4 7.7 14.2 16.5 16.3 4.3 6.3 46.4

0.5 22.8 25.1 10.1 13.3 16.1 21.7 3.6 6.0 65.6

0.6 22.0 24.8 12.4 12.4 15.8 27.3 3.1 5.7 87.3



quantitative analysis of electrode kinetics are ratherscarce.

In view of the above, the available set of experimen-tal data obtained specially with the aim of verifying theMarcus formulas (2), (4), and (15) (Table 2) is appar-ently insufficient. None of the systems studied satisfiesto the full all the enumerated requirements. The mostsystematic and convincing are probably the data fororganic reactants reported in [37, 38, 44, 45], althoughtraditional “double-layer” tests (varying the support-ing-electrolyte concentration, estimating psi-primeeffects for different localizations) have not been done inthe works cited, which makes the conclusions ambigu-ous in this case.

The maximum absolute values were substantiallybelow 1 V even if the measurements were taken in suf-ficiently broad overvoltage intervals (Table 2). With theλ values estimated for these systems taken into account,

deviations of α' from 0.5 were expected (and proved tobe) not too great (up to ±0.15, see the table). Thedecrease in α' to 0.2, which was recorded for anodicprocesses in [43], was, according to conclusions drawnby the authors, “supermarcusian” and stemmed proba-bly from adsorption complications.

Performing a similar verification for obviouslyouter-sphere reactions in the region of higher overvolt-ages, where deviations of α' from 0.5 could be moresignificant and distinguishable against the backgroundof the EDL effects, was indisputably of fundamentalvalue. Only in the region of even higher overvoltagesone could hope to verify a quantum-mechanical theory,which predicts that the transfer coefficient smoothlyapproaches zero in the vicinity of the activationlessregion and in the inverted marcusian region, i.e. underconditions where equation (15) predicts negative values

25

20

15

100.6 0.7 0.8 1.0

z, nm

Wi, kJ mol–1

1

2

3

4

5

(A)

(B)

Fig. 2. Work terms for heteropolyanion [CoMo6O24]3– (Wi), calculated as a function of distance between the center of an effectivesphere and OHP (z); considered are orientations A and B, (1, 3) with and (2, 4) without allowance for the dielectricity effect; (5) isfor point charge in the reactant center; in orientations A and B, reactant is positioned normally and horizontally to the electrodesurface, respectively.

138


TSIRLINA et al.

of α' that are not realized in the case of ordinary elec-trode reactions.

Reactions of reduction of anions are classic systemsfor which it proved possible to realize detailed investi-

gation of the electron transfer kinetics [5–8]. In partic-

ular, for reactions of reduction of S2 , Fe( ,

and Pt [5], which were thoroughly examined in thepotential region –1.5 to 0 V (in a reduced scale, i.e. withpotentials referred to the point of zero charge), the equi-librium potentials of corresponding oxred systems werepositive; therefore, extremely high overvoltages of 1 to3 V were realized in experiment. At the same time, het-erogeneous rate constants happen to be so small that itbecomes possible to reliably measure kinetic currentsof reduction in the η intervals about 1 V wide bymethod of classic polarography, at least in the region ofnot too large (by absolute values) charges and concen-trations of the supporting electrolyte, in conditions ofsharp electrostatic hampering on a negatively chargedelectrode surface. Of maximum interest is the one-elec-

tron reduction of Fe( not complicated by thecleavage of bonds and (in some conditions) the forma-tion of anion–cation pairs.

To analyze processes in potential intervals that wide,a procedure for constructing so-called marcusian CTP,i.e. for the linearization of experimental polarizationcurves in coordinates that correspond to the equation

(16)

was proposed in [21]. The obvious superiority of thisprocedure over an ordinary one is the existence of an

O82– CN )6

3–

Cl42–

CN )63–

jlog χlog–( )RT zoxFψ1+[ ]–{ } 0.5

= λ /2F

2 λ---------- η ψ1+( )+

20

17

0.5 0.7 0.9z, nm

Wi, kJ mol–1

12345

(A) (B)

Fig. 3. Same as in Fig. 2, for [S2O8]2–.

Table 2. Basic information about available experimental works on the verification of the Marcus formula

Redox process* Supporting electrolyte ElectrodeOvervol-

tage interval width, V

Maximumovervoltage,

V

Intervalof α beforecorrection

Refer-ence

tert-Nitrobutane (ox) R4NClO4, 0.5 M in AN Hg 0.8 0.37–0.55 [42]Ditto Bu4NI, 0.5 M in AH and L Hg 1.0 0.5 0.3–0.6 [37]

Nitrodurene (ox) Ditto Hg 0.6 0.35 0.3–0.5 [37]Nitromesitylene (ox) Ditto Hg 0.4 0.2 0.35–0.5 [37]Fe(H2O)6]

3+ (ox); [Fe(CN)6]3– (ox) K2SO4, 0.5 M in H2O Pt 0.4 0.2 0.35–0.5 [47]

[Cr(H2O)6]3+ (ox) NaClO4 + HClO4, 0.5 Min H2O

Hg 0.5 0.25 0.34–0.5 [48]

Ditto Various salts,0.01–1 M in H2O

Hg 0.5 0.25 0.43–0.73 [39]

(ox) HClO4, KNO3 + HNO3, 0.01–1 M in H2O Hg 0.5 0.5 0.17–0.37 [42]

[Cr(H2O)5X]3 + n,

Xn = , F–, H2O (ox)

Hg [49]

[M(H2O)6]m+, M = Cr2+, Eu2+, V3+

(red)KPF6 + HPF6, NaClO4 + HClO4, 0.5 M in H2O

Hg 0.5–1.0 0.5–0.7 0.17–0.38 [43]

Fe(H2O)x(SO4)y Na2SO4 + H2SO4 in H2O Pt 0.4 0.25 [24]

* Brackets indicate the reactant form (ox/red). In the table, AN and DMF stand for acetonitrile and dimethylformamide.

Hg22+

SO42–



unambiguous criterion of the construction correctness,which is based on a verification of the ratio between theslope and the intercept in the ordinate. Both these quan-tities depend solely on one model parameter, specifi-cally, the total reorganization energy λ.

In [26], on the basis of calculations of two constitu-ents of the reorganization energy and microscopic psi-prime effects, it proved possible to show that the truetransfer coefficient, calculated from an experiment

involving the Fe( reduction on mercury, quanti-tatively corresponded to relations of a quantum-mechanical theory of an elementary act. In [32], thisapproach was extended to a more complicated process

with a bond cleavage, i.e. to the reduction of S2 .

CONCLUSION

In this compilation we disregarded many importantaspects of the slow-discharge theory in its modern ver-sion, such as the introduction of electrostatic correc-tions when measuring rate constants of charge transferat the semiconductor/solution in [51], the direct mea-surements of temperature dependences of kineticparameters [52], the improvement of models of the dif-fuse layer [53], and the psi-prime effects in multistageelectrode reactions [54, 55] and during the occurrenceof parallel electrochemical conversions involving dif-ferent forms of the reactant that form in the bulk solu-tion [22, 27]. A microscopic description of the reactionlayer in the case of the reactant penetration into thedense part of EDL constitutes an individual importantproblem [30]. In the near future, research in these direc-tions may easily accumulate microscopic approachessummarized in this communication.

Of crucial importance for in-depth experimentalinvestigations of an elementary act is the developmentof experimental techniques [56] and fundamentallynew procedural approaches based on the realization ofprocesses of charge transfer in a scanning tunnelingmicroscopy configuration [57] and also the invoking ofnew modeling systems with a rigidly fixed charge trans-fer distance [17, 24].

Finally, analysis of the already available reliableexperimental data, which were earlier considered onlyin the framework of the initial version of the slow-dis-charge theory is a vast and fruitful sphere of action.

ACKNOWLEDGMENTS

This work was supported by the International Asso-ciation for Furthering Cooperation between Scientistsof Independent States of the Former Soviet Union(INTAS), project no. 99-21093, and the EducationMinistry of the Russian Federation.

CN )63–

O82–

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Frumkin Correction: Microscopic View