Voltammetric Determination of Both Concentration and DiffusionCoefficient by Combinational Use of Regular and Microelectrodes

Hua Zhang, Koichi Aoki,* Jingyuan Chen, Toyohiko Nishiumi, Hirokazu Toda, Eiji Torita

Department of Applied Physics, University of Fukui, 3-9-1 Bunkyo, Fukui-shi, 910-8507 Japanphone/fax: +81-090-8095-1906, +81-776-27-8750*e-mail: kaoki@u-fukui.ac.jp

Received: December 13, 2009;&Accepted: April 6, 2010

AbstractTwo Pt disks with 1.6 and 0.1 mm diameter, whose sizes were accurately evaluated, were applied to voltammetricdiffusion-controlled currents to determine the concentrations of redox species. Since voltammetric peak currents ata regular electrode, Ip, and limiting currents at a microelectrode, IL, are proportional to D1/2 and D, respectively, theratio Ip

2/IL is independent of the diffusion coefficient but dependents on the concentration. We used this fact to de-termine the concentrations of ferrocenyl derivatives, hemin, hexacyanoferrate(II), dioxygen gas and hydrogen gas.New insights are not only the absolute determination of concentrations but also the sensitive detection of adsorp-tion of electrochemical products which blocked the voltammetric currents.

Keywords: Voltammetric concentration determination, Microelectrodes, Diffusion coefficients, Adsorption,Hygroscopic species

DOI: 10.1002/elan.200900603

1 Introduction

When solid species contain an unknown amount of wateror salt in preparation, the concentrations are obviouslyoverestimated. This type of error also occurs when a solu-tion is available only in aqueous form or as mixture,when a species is rapidly oxidized in air, is sensitive tomoisture, when a gas is dissolved through bubbling, andwhen a species is in micellar or ion-paired form. We hereintroduce a simple voltammetric technique for accurateconcentration determination which cannot be obtained atpreparation of solutions. Our technique consists of evalu-ating the ratio of the voltammetric peak currents at a reg-ular electrode and at a microelectrode.

Mass-transfer controlled voltammetric currents dependon the concentration, c, of an electroactive species, its dif-fusion coefficient, D, and the number of electrons trans-ferred, n, [1] in the form of their product, ncDx, [2] if nelectrons are transferred sequentially [3], where x (x�1)is proper to electrochemical techniques as well as themass transport mode. A value of n is often known or canreadily be estimated, whereas D-values range widely, de-pending on the viscosity of solutions and hydrodynamicradii of the diffusing species. If there are two electro-chemical data with different values of x, for example x=0.5 at a regular electrode and x=1 for the steady state ata microelectrode [4], the term of D can be eliminated bytaking the ratio of the two currents, retaining c. Thus, anabsolute value of the concentration can be determined.This concept is close to chronoamperometry of evaluating

n or D without knowing the value of either D or n, de-vised by Kakihana et al. [5], who took the ratio of slopeand intercept in Cottrell plots (current vs. inverse of thesquare-root of the electrolysis time) at a microelectrode.This method has been applied to accurate evaluation ofD [6–8] rather than nc [9]. The advantage has been justi-fied by other research groups [10–14], including apparentpotential dependence of n [15,16] and charge transportproperties in gels [17]. Since the determination of n, D orc depends not only on selecting a linear domain of theCottrell plot but also on applied potential and thenumber of experimental runs [8, 15], it is not practical foraccurate determination of c. The poor accuracy is as-cribed to interference of capacitive current and a delay ofa potentiostat.

The technique presented here is a replacement of chro-noamperometry with cyclic voltammetry [18]. It has ten-tatively been used to determine c and D in gels [19]. Anadvantage of cyclic voltammetry over chronoamperome-try is to provide extensive information about reacting spe-cies, reaction rates and rate-determining steps from vol-tammetric shape varying with scan rates. In order to ex-ploit the advantage of cyclic voltammetry, it is necessaryto solve the practical problem of performing voltammetryover three orders in magnitude of scan rates, which canreach [19, 20] from the steady-state to the proportionalityto v1/2. Fast scan voltammetry often distorts waves owingto reaction kinetics, capacitive current, a delay of a poten-tiostat, and solution resistance. The use of slow scan ratesis desirable to obtain both a steady-state voltammogram

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and a linear diffusion-voltammogram. A strategy of re-solving this problem is to use a microelectrode and a reg-ular electrode in one cell.

The present work is devoted to the absolute and accu-rate determination of concentrations of redox species byslow cyclic voltammetry at two kinds of electrodes, ca.1.6 mm and 0.1 mm in diameter. The concentrations willbe evaluated by taking the ratio of the voltammetric peakcurrents at the two electrodes to eliminate the diffusioncoefficients. The determination requires accurate evalua-tion of the diameters of the electrodes because the preci-sion of the concentration depends on the forth powers ofthe diameter of the regular electrode. The redox speciesare a ferrocenyl derivative as a prototypical example,hemin as a hygroscopic species, hexacyanoferrate(II) as awater-containing species, and dioxygen gas and hydrogengas as much requirement of their accurate concentrations.

2 Experimental

2.1 Chemicals

Ferrocene was purified by sublimation. (Ferrocenylme-thyl)trimethylammonium hexafluorophosphate (FcTMA)was synthesized by mixing (ferrocenylmethyl)trimethyl-ammonium iodide (Wako) with ammonium hexafluoro-phosphate [21], and was purified by re-crystallization inwater. Hemin (Wako) with 97 % purity was stored in a re-frigerator. It was weighed after attaining room tempera-ture in order to avoid mixing with water. Potassium hexa-cyanoferrate(II) was newly purchased from Wako. Theother chemicals were used as received.

Sulfonated ferrocene (FcSO3) was synthesized bymixing ferrocene with chlorosulfonic acid in acetic anhy-dride, was then pH-neutralized, and at least recrystallized[22].

2.2 Working Electrodes

Two kinds of Pt working disk electrodes were employed,ca. 1.6 mm (BAS, Tokyo) and ca. 0.1 mm (home-made) indiameter. The former electrode was sealed with PTFE,whereas the latter was sealed with glass. These electrodeswere polished on a wet polishing pad with aluminapowder. Photographs of their surfaces were takenthrough an optical microscope, VH-Z450 (Keyence,Osaka), together with a graticule of tick-marks 10 mmeach. In order to evaluate an average radius of the ex-posed disk, we read the x–y coordinates at N points onthe circumference of the disk each by approximately 2p/N angle from a magnified photograph, where N>10. Theradius, a0, was evaluated by the following least squaremethod. Let the read coordinates be (xi, yi) for i�N, andthe coordinates of the real center of the circle be (X,Y).Then the equation for the circle is given by (xi�X)2 +(yi�Y)2 =ao

2. It is rewritten as 2 xiX+2yiY+Z=xi2 +yi

2,where Z=ao

2�X2�Y2. We determined X, Y, and Z fromN values of (xi, yi) by means of the least square method.

Values of X, Y, and Z provided the ao-value. The varianceof the radius was defined by

(1/N)S [((xi�X)2 + (yi-Y)2)1/2�ao]2.

Electrodes 2a0<0.05 mm were fabricated by coating athin Pt wire with anodic electrophoretic paint [23].

Diameters of the two Pt disk electrodes were deter-mined to be 2a1 =1.643�0.003 mm and 2a2 =0.0964�0.0011 mm as three times averages, where the errorsstand for the standard deviations.

2.3 Instrumentation

An electrochemical cell for gas-measurements was madeair-tightly so that the inner pressure of the cell was1.05 atm. The reference and the counter electrodes wereAg/AgCl and platinum wire, respectively. The potentio-stat was HECS 972 (Huso, Kawasaki), which was con-trolled with a home-made software.

Thermogravimetry was made by Thermo-Plus(TG8120, Rigaku) from the room temperature to 200 8Cin nitrogen gas atmosphere. A UV-spectrometer was V-570 (JASCO).

3 Methods

The diffusion-controlled voltammetric peak current foran n-electron oxidation at a regular disk electrode isknown to be expressed by Ip =0.446(pa1

2)Fn3/2c (FDv/RT)1/2, where a1 is the radius of the disk electrode. How-ever, for n�2 this equation is valid only for concomitantn-electron transfer reactions rather than successive n-electron transfer reactions [3]. There are very few exam-ples of concomitant n-electron transfer reactions [24],whereas successive transfer reactions have often beenfound. The peak current for the latter is expressed by Ip =0.446(pa1

2)Fnc(FDv/RT)1/2. The difference between n3/2

and n is ascribed to coupling of the Nernst equation withthe rate of the potential-varying diffusion. Then, theslope of the proportionality between Ip and v1/2 is givenby

slope ¼ Ip v�1=2 ¼ 0:446 p a12FncðFD=RTÞ1=2 ð1Þ

In contrast, the steady-state limiting current at the mi-crodisk electrode with radius a2 is represented by

IL ¼ 4FncDa2 ð2Þ

Eliminating D from the two equations yields

nc ¼ 2:037 ðRT=F2Þ ða2Ip2=vÞ=a1

4IL ð3Þ

or

nc=M ¼ 0:542ðIp2=vILÞða2=a1

4Þ ð4Þ

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at 25 8C, where the unit of the currents is A, that ofa2 and a1 is mm, that of v is V/s and that of the concentra-tion is M (=mol dm�3). On the other hand, elimination ofc from Equation 1 and 2 yields

D ¼ 0:123 ðF=RTÞða14IL

2Þða22 Ip

2=v�1

¼ 0:0474 ða14IL

2Þða22 Ip

2=vÞ�1ð5Þ

where the unit of D is cm2 s�1, and units of a1, a2, Ip, IL

and v are the same as in Equation 4.In order to search experimental conditions under which

c can be determined with an error below 3 %, we estimateerrors involved possibly in Equation 4. Variations of Ip

with v1/2 at slow scans may be deviated from a proportion-al relation owing to the edge effect. According to thetheory of the voltammetric peak current [18],

IP ¼ IL ð0:34 e�0:66p þ 0:66� 0:13 e�11=p þ 0:352 pÞ for

p ¼ aðvF=RTDÞ1=2ð6Þ

the edge effect can be neglected within 3 % errors of Ip

for v>0.1 V s�1 at D=10�5 cm2 s�1, 2a1 =1.6 mm at 25 8C.This domain of v is too fast to neglect capacitive currents.We use the slope of the plot of Ip vs. v1/2, being identicalwith Equation 1, without regard to the intercept. Thelargest errors may be caused by the accuracy of the diam-eter evaluation of the regular electrode because it ap-pears with forth order in Equation 4. Therefore, we haveto determine the diameter of the regular electrode withan error �0.75 %.

In contrast, a microelectrode provides the steady-statecurrent within 3 % errors when p<0.21, which is satisfiedfor v<0.45, 5.0, and 45.0 mV s�1, respectively at 2a=100,30 and 10 mm. Since scan rates greater than 10 mV s�1 canprovide practically stable voltammograms, electrodes for2a<30 mm are desirable. However, these electrodes oftenexhibit not only nonideal geometry but also poor accura-cy of the diameter by a microscope. This contradictioncan be solved by extrapolating the quasi steady-state cur-rent, IL’, at 2a=100 mm to v=0 in the plot of the limitingcurrent against v1/2, according to the Tailor expansion ofEquation 6, e.g.,

IL0 � IL ½1þ 0:129 a ðvF=RTDÞ1=2� ð7Þ

4 Results and Discussion

4.1 Ferrocenyl Compounds

Voltammograms of 0.110 mM ferrocene in 0.5 M tetrethy-lammonium (TBA) perchlorate of acetonitrile solutionwere obtained for v<0.2 Vs�1 at the regular electrodeand for v<0.05 V s�1 at the microelectrode. Voltammo-grams at the microelectrode were almost under thesteady-state. Figure 1 shows plot of the anodic peak cur-

rents at the regular electrode and the anodic limiting cur-rent, IL’, at the microelectrode against v1/2. Both plots fellon each line without ambiguity. Insertion of both theslope of the regression line for Ip and the intercept for IL

into Equation 4 yields values of c and D, respectively,which are listed in Table 1. The concentration agreedwith the prepared concentration within 3 % errors, sup-porting the present technique. The hydrodynamic radius,r, of ferrocene in the medium with the viscosity h wasevaluated from the diffusion coefficient through theStokes–Einstein equation [25]

D ¼ kBT=6phr

to be 0.815 nm.We obtained voltammograms of crude ferrocene, which

was not purified by sublimation. The voltammetric shapeas well as the plot of Ip or IL vs. v1/2 were actually thesame as those of purified ferrocene. Only the differencelied in values of the peak current, which was smaller thanthat for purified ferrocene. The concentration obtainedfrom Equation 4 was (in Table 1) smaller than that for pu-rified ferrocene by 10 %, whereas the diffusion coefficientwas identical. Therefore the crude ferrocene included im-purity by 10 % molar weight.

Since recrystallized FcTMA may contain salt andwater, the present method was applied to a FcTMA solu-tion. Voltammograms of nominal 0.675 mM FcTMA+0.5 M KCl aqueous solution showed a waveform similarto ferrocene without any potential shift for v<0.2 V s�1.Variations of Ip vs. v1/2 and IL vs. v1/2 were very similar toferrocene. Inserting the slope and the intercept intoEquation 4 yielded c=0.62 mM. The difference in theconcentration may be ascribed to mixing of salts, proba-bly iodide compounds.

Since FcSO3 is an anion, it was dissolved well in water.It was difficult to purify sodium salt of FcSO3 because ofother sodium salts. The aqueous solution of FcSO3 at theregular electrode had anodic and cathodic peaks at 0.47and 0.40 V, respectively. The variations of Ip and IL’ with v1/

2 were similar to those in Figure 1. From the slope and theintercept, we determined the concentration, as listed inTable 1. The actual concentration was able to be obtained.

This technique was applied to dichloromethane solu-tion in which FcSO3 was sparingly soluble. Voltammo-grams in FcSO3-saturated dichloromethane solution had awaveform similar to those in the aqueous solution, show-ing much smaller currents. The slope of Ip vs. v1/2 and theintercept provided the value of the sparingly solubleFcSO3 as listed in Table 1. The values of D were almostindependent of the prepared concentration, and were rea-sonable in dichloromethane. Therefore the dissolved con-centration ought to range from 0.01 to 0.015 mM.

4.2 Fe(CN)64�

Aqueous solution of 0.500 mM Fe(CN)64� including 0.5 M

KCl showed well-defined diffusion-controlled voltammo-

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Determination of Concentration and Diffusion Coefficient

grams at the two kinds of electrodes. Poor reproducibilityof waves of Fe(CN)6

4�, which was often found at an elec-trode with a2<5 mm and has already been reported [26],was not detected at the microelectrode. The slope and theintercept yielded c=0.514 mM. The overestimation of theconcentration may be ascribed to a loss of crystal waterrather than comingling of impurity. Weight-loss by TGAwas 11.7% around 100 8C, which corresponds to 2.71water molecules rather than nominal three. Thus theactual molecular weight is 417 rather than the nominalvalue of 422.

4.3 FeCl3

Figure 2 shows voltammograms of FeCl3 in 1 M HCl solu-tion at the (a) regular and the (b) microelectrode. Thepeak potentials of voltammograms (a) varied with thescan rates only by 15 mV for 0.01<v<0.1 V s�1, and thepeak currents were proportional to v1/2. The peak differ-ence, DEp, ranged from 120 to 150 mV. In contrast, noplateau of the limiting currents at the microelectrode wasobtained. The halfwave potential was shifted by 110 mVfrom the middle of the peak potentials. Therefore, thevoltammograms involve more complications than diffu-

sion or electron transfer kinetics. From the slope and theintercept, we obtained unreasonably large values of c andsmall values of D (see Table 1). This trend was more re-markable for larger concentrations at smaller electrodes.Higher current density at the microelectrode seems tocause complications, as is similar to the reduction of ben-zoquinone in acetonitrile [23], which formed film on theelectrode. Indeed, film was found on the 0.1 mm elec-trode after voltammetry at a slow scans. The film mayblock diffusion of Fe3+ . The block is stronger at the mi-croelectrode than at the regular electrode because ofhigher current density. Since IL is more underestimatedthat Ip in Equation 3, the value of c was overestimated.

4.4 Hemin (Iron-Coordinated Carboxyl Porphyrin)

Figure 3 shows voltammograms (a, b) of hemin in 0.5 Mtetrabutylammonium perchlorate (TBAClO4)-included di-methylsulfoxide (DMSO) solution. The cathodic currentsincreased with bubbling air, being due to catalytic reduc-tion of oxygen. The value of DEp of the deaerated heminsolution varied from 0.08 to 0.14 V for 0.01<v<0.1 V s�1.The peak current at the regular electrode was proportion-al to v1/2 for v<0.1 V s�1. The peak and steady-state cur-rent can be regarded as the diffusion-controlled. The in-

Fig. 1. Variations of Ip and IL’ with v1/2 for anodic voltammo-grams of ca. 0.110 mM Fc in 0.5 M TBAClO4 included acetoni-trile, where Ip and IL’ were obtained at Pt disk electrodes with1.643 mm and 0.096 mm in diameter, respectively.

Table 1. Concentrations and diffusion coefficient evaluated from Equations 4 and 5.

Species Prepared c (mM) Determined c (mM) D� 10�5 (cm2 s�1) 2r (nm) Salt+ solvent

Ferrocene 0.110 0.113 1.55 0.815 0.5 M TBAClO4 in acetonitrileCrude ferrocene 1.00 0.90 1.48 – 0.5 M TBAClO4 in waterFcTMA 0.675 0.62 0.63 0.78 0.5 M KCl in waterFcSO3 1.00 0.90 0.63 0.78 0.5 M KCl in water

0.1 0.015 1.1 0.9 0.1 M TBAClO4 in dichloromethane0.05 0.010 1.0 0.9

K4Fe(CN)6 0.500 0.514 0.66 0.51 0.5 M KCl in waterFeCl3 3.4 (8.3) (0.14) – 1 M HCl in water

0.17 (0.23) 0.59 –Hemin 1.0 0.496 0.24 0.91 0.5 M TBAClO4 in DMSOHydrogen 0.78 0.75 3.1 0.16 0.5 M KCl in waterDioxygen 1.27(sat) (3.5 at n=2) (0.69) – 0.5 M KCl in water

0.263(air) (0.10 at n=2) (0.96)

Fig. 2. Voltammograms of nominal 3.4 mM FeCl3 in 1 M HClaqueous solution at (a) the regular electrode for v=0.1 V s�1 and(b) the microelectrode for v=0.01 Vs�1.

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sertion of the slope and the intercept into Equation 4yielded half the value of the prepared concentration, aslisted in Table 1. Since the molecular diameter, 0.91 nm,is only slightly smaller than the maximum diameter evalu-ated from the molecular image, the value of the diffusioncoefficient may be reasonable. The commercially avail-able hemin turns out to contain a large amount of impuri-ty or electroinactive moiety. Since TGA demonstrated0.5 % weight less at 120 8C, the impurity should be salt ororganics. The absorption coefficient at 405 nm was 9.8 �105.

4.5 Hydrogen Gas

Figure 4 shows voltammograms of hydrogen gas-saturatedKCl solution. The one at the regular electrode had acouple of redox peaks at any scan rate as well as thesecond oxidation wave at ca. �0.2 V at high scan rates.The peak potential varied with v only by 5 mV for 0.01<v<0.2 V s�1. The value of DEp was 60–70 mV, suggestingthe successive two step reactions each being a one-elec-tron transfer. In contrast, the microelectrode showed onlyone wave which was approximately below the steadystate. The log-plot of the steady-state voltammograms atthe microelectrode showed a slope of 43 mV, indicatingthat the oxidation proceeds via a combination of one-and two-electron transfer. The current density at the mi-croelectrode higher than that at the regular electrodemay make the first step oxidation delayed.

The plot of Ip vs. v1/2 was linear, exhibiting a small posi-tive value of the intercept, probably owing to the secondwave. From the slope and IL, we evaluated c and D forn=2. The value of c was close to the saturated concentra-tion of hydrogen gas (0.78 mM). The values of the diffu-sion coefficient and the molecular diameter are reasona-ble.

4.6 Dioxygen Gas

Voltammograms in dioxygen-saturated KCl solution wereobtained and are shown in Figure 5. A cathodic wave at

the regular electrode was shifted in the negative directionwith an increasing scan rates, whereas no anodic wavewas found. The cathodic wave decreased with the numberof scans because of irreversibility effects. In contrast, thevoltammograms at the microelectrode were almost atsteady-state. The half-wave potential was shifted in thenegative direction from the peak at the regular electrodeby 40 mV. The shift indicates consumption of the reducedspecies (O�C) by follow-up chemical reactions during thevoltammetric period. The vague limiting currents suggestgradual blocking of diffusion of dioxygen.

Although the behavior is different from that of the dif-fusion-controlled voltammograms, the peak current waslinear with v1/2 with a small intercept for v<0.2 Vs�1. Weattempted to evaluate c and D by inserting values of theslope of Ip vs. v1/2 and the intercept into Equation 4 result-ing in nc=7.1 mM and D=0.69 �10�5 cm2 s�1. The formeris much larger than the saturated value (c=1.27 mM)even for n=4, whereas the latter is too small for a di-molecule. We found that this tendency appeared morestrongly at smaller electrodes (2a=25 and 10 mm). Wecan explain this by a higher current density at smaller

Fig. 3. Voltammograms of nominal 1 mM hemin in 0.5 MTBAClO4 included DMSO at (a) the regular electrode for v=0.1 Vs�1 and (b) the microelectrode for v=0.01 Vs�1.

Fig. 4. Voltammograms of hydrogen gas-saturated water includ-ing 0.5 M KCl at (a) the regular electrode for v=0.1 Vs�1 and(b) the microelectrode for v=0.01 Vs�1.

Fig. 5. Voltammograms of dioxygen gas-saturated water includ-ing 0.5 M KCl at (a) the regular electrode for v=0.1 Vs�1 and(b) the microelectrode for v=0.01 Vs�1. 1, 2 and 3 denote the or-dinal scan number.

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Determination of Concentration and Diffusion Coefficient

electrodes, which may accumulate the products and blockthe reduction of dioxygen, as has been found for reduc-tion of benzoquinone [23].

The KCl aqueous solution in ambient air showed a ten-dency similar to the above except for poor reproducibili-ty. Values of c and D are half the predicted values. There-fore the present technique is not suitable for determina-tion of dioxygen.

5 Conclusions

The advantage of the presented method is the determina-tion of effective concentrations without knowing D underthe voltammetric criteria for waveforms, potential shifts,and complications by side reactions. Concentrations anddiffusion coefficients were evaluated independently fromtaking the ratio of the Ipv

�1/2 to IL by use of Equation 4and 5, respectively, within 3 % errors, where Ip is the vol-tammetric peak current at the 1.6 mm disk electrode andIL is the steady-state current at the 0.1 mm disk electrode.Any size of a regular electrode can be used if the area isaccurately evaluated and if planar diffusion is ensured. Incontrast, a microelectrode should be a disk with a knownaccurate diameter, exhibiting the steady-state current. Itis the diameter of the microelectrode that is apt to losesignificant digits in Equation 4. The practical diametermay range from 0.05 to 0.12 mm in accuracy from meas-urements by an optical microscope. Since very small mi-croelectrodes increase the current density to make differ-ence in reaction mechanisms at the regular electrode,they are not suitable for determination of concentrations.

The present method succeeded not only in Fc andFcTMA exhibiting typically diffusion-controlled currentsbut also in weakly adsorbed species such as Fe(CN)6,FcSO3, hydrogen gas and hemin. It also succeeded in de-termination of sparingly soluble species. These species ex-hibit nearly diffusion-like current. The present techniqueis useful for evaluating net concentrations of redox spe-cies although they may be moist or include crystal water,salt or impurities.

Strong adsorption of electrochemical products some-times blocked the voltammetric currents to yield underes-timated diffusion coefficients and overestimated concen-trations, as exemplified by ferric chloride and dioxygengas. This is observed more remarkably at smaller electro-des employed, because the higher current density suppliesproducts faster than their dissolution rate. The diffusioncoefficients should be checked by comparison with valuesfrom the Stokes–Einstein equation. In other word, thepresent technique can access an adsorption level even if

no deposit can be optically recognized on electrode surfa-ces.

The shape of a voltammogram, represented by a log–plot or Ep–E1/2, was not discussed here because it includesmore detailed analysis than the determination of c and Dfrom peak currents. Kinetic effects in electron transfer re-actions were not examined, here too, because typical ex-amples without adsorption or chemical complicationswere not available.

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