电化学

Theory of the reversible electrode process in case of cyclicvoltammetry at finite thickness film electrodeB.F.Nazarov,rionova *,A.G.Stromberg ,I.S.AntipenkoTomsk Polytechnic University,NPP Technoanalit,Lenin Avenue,30,634050Tomsk,Russian FederationReceived 2February 2007;received in revised form 21April 2007;accepted 23April 2007Available online 30April 2007AbstractIn this report the theory of the reversible process of amalgam formation and dissolution reaction at a mercury film electrode of dif-ferent thickness is developed using an additional boundary condition approach for asymmetricaldiffusion.The profile of the entire vol-tammetric peak is simulated using a wide range of values of the parameter H ¼l ffiffiffir p =ffiffiffiffiD p (where l is the film thickness,r =nFW /RT ,D is diffusion coefficient,n is number of electrons,W is scan rate,and F and R are the Faraday and Universal gas constants,respectively)from 20to 0.001.In addition the influence of the initial potential E i and the switching potential E k is included in the calculations.The dependence of the anodic and cathodic peak currents,peak potentials and half-height-full-widths on the initial potential,E i ,and the value of ln H ,are described using approximate equations;the dependence of the ratio of peak currents and peak-to-peak separation on the switching potential E k and value of ln H ,using cyclic voltammetry are also described.Ó2007Elsevier B.V.All rights reserved.Keywords:Stripping voltammetry;Cyclic voltammetry;Simulation;Reversible electrode process;Film electrode1.IntroductionThe study of reversible electrochemical processes at film electrodes has been the subject of much attention during the last thirty years.This matter has fundamental signifi-cance for the theory of stripping and cyclic voltammetry,because the electrochemically reversible process is the lim-iting case for heterogeneous redox processes coupled to preceding or subsequent homogeneous chemical reactions (e.g.CE or EC mechanisms),irreversible and quasi-revers-ible processes at electrodes made of different film and sub-strate materials (Hg,Bi,Hg–Au,Hg–Ag and others)and different geometries of electrodes [1].To obtain exact solutions of the diffusion problem over a finite area in the case of linear sweep voltammetry it is required to use the powerful mathematical tool.Initialattempts to take into account the asymmetrical area of dif-fusion were made for the case of mercury hanging drop electrodes [2–4]and mercury film electrodes [5–8].De Vries and Van Dalen’s works [9–11]are one of the most well known and powerful approaches to mathemati-cally model the thin film system.The exact integral equa-tion was obtained by Laplace transformation method.In order to calculate the full profile of the linear sweep cur-rent–potential curves obtained during oxidation or reduc-tion,the numerical integration method of Huber is required.Numerical simulations are widely applied to solve a variety of voltammetric problems.Numerical methods rely on approximating the mass transport field by dividing it up into discrete elements,which are handled mathematically [12].The system of Fick’s equation has directly been solved by finite difference method [13–15].The time dependent backward implicit method was used to simulate anodic stripping voltammetry at mercury thin film electrode under hydrodynamic conditions for electrochemically reversible1388-2481/$-see front matter Ó2007Elsevier B.V.All rights reserved.doi:10.1016/j.elecom.2007.04.014*Corresponding author.E-mail address:evs@anchem.chtd.tpu.ru (rionova).Deceased./locate/elecomElectrochemistry Communications 9(2007)1936–1944[16–19],quasi-reversible and irreversible processes[20,21]. It should be noted that the advantages of numerical meth-ods in comparison with semi-analytical methods include theirflexibility in applying them to different voltammetric conditions and kinetic regimes,and their relative mathe-matical simplicity.However,in our opinion the important advantages of semi-analytical methods are that the solu-tions in the analytical form are obtained.Among the advantages of semianalytical methods,a high degree of accuracy and ready applicability to different forms of the applied voltage should also be emphasized.Keller and Reinmuth[22]have developed a new approach to solvefinite and infinite diffusion problems in potential scan voltammetry using equations in the form of infinite series.Investigation of the electrochemically reversible process at afilm electrode by applying the Laplace transformation method with additional boundary conditions is also an alternative and fruitful mathematical approach[3,23–27]. This approach,in which additional boundary conditions holds true for semi-infinite diffusion,wasfirst used by Reinmuth[3].For successful application of this method tofinite diffusion problem,it is necessary to choose the cor-rect boundary condition for the solution/electrode inter-face.The additional boundary condition reported by Nemov and coworkers[23–27]defines the concentration of oxidized and reduced forms on the solution–electrode interface in case of asymmetrical diffusion.The use of cyclic voltammetry atfilm electrodes can pro-vide more information about a given system compared to stripping voltammetry alone.Cyclic voltammetry at mer-cury electrodes of various geometries has been applied to the study of the electrochemical properties of oxidation–reduction couples and the mechanism of these electrode processes,such as amalgam formation,etc.[1].Cyclic vol-tammetry is used as a tool to study the electrochemical characteristics of mercury microelectrodes[28–32].The theory of cyclic voltammetry involving amalgam formation at a hanging mercury drop electrode was described generally elsewhere[33–38].Theoretical treat-ment of cyclic voltammetry for hanging mercury drop elec-trode,taking into consideration the diffusion coefficient of the metal ion,sphericity andfinal volume of the mercury drop and other factors,has been developed by authors of the work[36].The direct solution of differential equa-tions system was obtained by difference explicit method [37,38].Authors of works[39,40]further developed the theory of cyclic voltammetry to include a stage of amalgam forma-tion with metal initially present in the mercuryfilm and ions of the metal in the solution.To calculate current–potential curves,the approximation of De Vries and Van Dalen’s equation for thin-film systems was used,though De Vries and Van Dalen’s equation describes only one-direction scan processes.In our recently published papers[26,27]we have obtained simple,yet exact explicit expressions describing linear scan stripping voltammetry at a thinfilm mercury electrode with the help of Nemov’s additional boundary condition.The purpose of the present work is theoretically to investigate current–potential curves for linear sweep cyc-lic voltammetry at afilm electrode to test the application of Nemov’s additional boundary condition.2.TheoryLet us consider a simple electrochemical reaction of reversible anodic and cathodic electrode processes at afilm electrode in the case of linear sweep voltammetry:M nþþne$MðHgÞ;ð1ÞE¼E iþWt;E¼E kÀWt;ð2Þwhere E is the potential,E i is the initial potential,E k is the switching potential,W is the scan rate.Fig.1shows the form of change of the potential of the working electrode in case of practical realization of cyclic process(1).As can be seen from Fig.1,cyclic process con-sists of several parts:A1B1–the amalgam formation at constant potential during the time of electrolysis t,B1C1–the dissolution step at linear sweep potential,C1A2–the amalgam formation at linear sweep potential.It is to be noted that equilibrium is rapidly reached at point C1. In the second cycle the section A2B2corresponds to the reduction of the metal ion during time of electrolysis t. The time periods A1B1,A2B2etc.serve to establish the equilibrium condition determined by Nernst equation. Such form of the working electrode potential change gives us the possibility of carrying out multicycle processes.According to works[26,27]in order to obtain expression for c sRand c sOxin the case of a limited region of diffusion Fick’s equations:o c Rðx;tÞ¼DRo2c Rðx;tÞ2o c Oxðx;tÞo t¼D Ox o2c Oxðx;tÞo x29=;ð3Þshould be solved with the following initial and boundary conditions:t¼0:c Rðx;0Þ¼c0R¼const;ð06x6lÞð3aÞc Oxðx;0Þ¼c0Ox¼const;ðx>lÞð3bÞc0Ox¼h0c0Rð3cÞB.F.Nazarov et al./Electrochemistry Communications9(2007)1936–19441937t >0:o c R ðx ;t Þo xx ¼0¼0;ð3d Þlim x !1ðc Ox ðx ;t ÞÞ¼c 0Ox ;ð3e ÞD Ro c R ðx ;t Þo x x ¼l¼D Ox o c Ox ðx ;t Þo xx ¼l ¼i nFS ;ð3f Þc s Ox ¼h c sR :ð3g ÞThis system of Fick’s Eq.(3)with initial and boundary conditions (3a)–(3g)was solved using the Laplace transfor-mation method to obtain equations for c s R and c sOx [26].Next the additional boundary condition for asymmetric diffusion [26]was obtained as:c s R ¼c 0R 1þh 1þh 0À2ðh Àh 0Þ1þh X 1k ¼11Àh 1þh k À1erfc kHðr t Þ1=2"#()ð4Þwhere t is time,x is the distance from electrode substrate,x =l is the film thickness,D R ,D Ox are the diffusion coeffi-cients of reduced (M(Hg))and oxidized form (M n +),c 0R and c 0Ox are the initial concentrations of reduced (M(Hg))andoxidized form (M n +),c s R and c sOx are concentrations of re-duced (M(Hg))and oxidized form (M n +)at the electrodesurface.For the anodic process h ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD Ox =D R p h 0exp ðr t Þ,h 0¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD Ox =D R p exp nF RT E i ÀE 0ÀÁÀÁ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD Ox =D R p exp ðw i Þ,r t ¼nFW t ¼nF ðE ÀE i Þ,where w i is dimensionless initial potential;E 0is standard potential of the electrode couple(arbitrarily,let the standard potential be zero),whilstfor the cathodic process:h ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD Ox =D R p h 0exp ðÀr t Þ,h 0¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD Ox =D R p exp nF ðE k ÀE 0ÞÀÁ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD Ox =D R p exp ðw k Þ,r t ¼nFW t ¼nF E k ÀE ðÞ,where w k is dimensionless switching potential;H ¼l ffiffiffir p =ffiffiffiffiffiffiffiD R p ,n is number of elec-trons,F and R are the Faraday and universal gas constants,respectively.It is obvious that as H !1expression (4)be-comes the additional boundary condition for symmetrical diffusion [26].The equation for the anodic current i a (in amperes)was developed previously,using the superposition principle [26]:i a¼n 2F 2Sc 0R lW RTv aðr t Þ;ð5Þwhere S is the electrode area,v a (r t )is the dimensionless anodic current function:v a ðr t Þ¼2H Z t 0o c 0ðs Þo s X 1k ¼1exp Àð2k À1Þ2p 24r ðt Às ÞH d s ;ð6Þwhere c 0¼c s R =c 0R is the dimensionless concentration and is determined from the additional boundary condition (4).More detailed mathematical description and analysis of Eqs.(4)and (5)were presented in our previous papers [26,27].There is disadvantage at the application of equitation (6).It is connected with constraint of use of direct comput-ing facilities in case of H !0.Let us consider the limiting case of equitation (6)for H !0.It is known that [41]Z 1d ðx Àx 0Þf ðx Þd x ¼f ðx 0Þ:ð7ÞIf d (x Àx 0)is (Dirac)delta function,then it has the fol-lowing properties:(i)d (x Àx 0)=0provided that x ¼x 0;(ii)d (x Àx 0)=1provided that x =x 0and R 1d ðx Þd x ¼paring expression (6)to expression (7)it can be shown that:q ðr t Þ¼2H 2X 1k ¼1exp Àð2k À1Þ2p 24r ðt Às ÞH 2 ð8Þis a delta function at H !0.It is evident thatlim H !0q (r t )=0provided that t ¼0and the followingexpressions lim H !0q (r t )=1and 2R 10P 1k ¼1exp Àð2k À1Þ2p 24r ðt Às ÞH 2h i d r tH 2¼1are true provided that t =0.Therefore the limiting equation for current–potential curves at vanishing values of H <0.01can be written as lim H !0v aðr t Þ¼d c 0ðt Þd t:ð9ÞFirstly the Eq.(9)was suggested by Baev [42]but the detailed study of applicability of the expression (9)was not carried out by this author.We estimated that the errors D v a ðr t Þ¼ðv a ðr t ÞÀlim H !0ðv a ðr t ÞÞ100%=v a ðr t Þcalculated by the Eqs.(6)and (9)rapidly decreases with decreasing H and those become less then 1%at H =0.1.The dimensionless cathodic current v c (r t )can be calcu-lated by Eqs.(6)and (9)with use of the parameters of cathodic process.These are the negative value of W ,the switching potential E k and the initial concentration of oxi-dized form c 0Ox determined by the Nernst Eq.(3c).The function v (r t )allows one to calculate the theoretical cyclic curves under the condition where the initial quantity of metal (Q R =const)is constant.This is achieved by increasing the concentration of M n +with a corresponding decrease in film thickness.Let us now introduce a new function v 0(r t ):v 0ðr t Þ¼H v ðr t Þ;ð10Þcorresponding to the condition where the initial concentra-tion of M n +is constant,that is equivalent to constant ini-tial concentration of M ðHg Þðc 0R ¼const Þ.Thus,the major parameters determining the shape of v (r t )is H ,w i and w k .To calculate theoretical curves v (r t )Eq.(6)should be used and Eq.(9)at vanishing values of H <0.1should be applied.1938 B.F.Nazarov et al./Electrochemistry Communications 9(2007)1936–19443.Results and discussion3.1.Anodic processThe full contour of anodic peak(Fig.2)in the dimen-sionless coordinates v a—r t for following values of H: 0.001;0.01;0.05;0.2;0.5;1.11;2.5;5;10are calculated by Eqs.(6)and(9).Deformation of peak form depended on the value of the dimensionless initial potential w i(À8;À5;À2;1;4)or E i,mV(À205.43;À128.39;À51.36;25.68;102.71),that corresponds to the following relations of concentration c Ox/c R0.0003;0.0067;0.1353; 2.7183;54.5982,for the parameter H given above is investigated. The calculations were performed under the condition that Q R=const.The basic anodic peak parameters(height of peak v amax ,peak potential e ap ¼ðnF=RTÞðEÀE iÞand d a=(nF/RT)E1/2,where E1/2–width at half peak height/V)as a function of ln H at different values of w i are shown in Fig.3.The basic peak parameters which depend on ln H and w i can be pre-sented by the following approximation model:P1¼a1þb1ln Hþd1w iþe1ðln HÞ2þf1w2iþg1ðln HÞw ið11Þfor1ÆHæ0.001andÀ8Æw iæÀ2,where P1is v amax ,e ap,d a.Coefficients of the approximation formula(11)are found by a method of nonlinear optimization and also coefficients of correlation r are estimated(Table1).It is apparent from Fig.3,that the values of H and w i significantly influence the form of the anodic peak.As can be seen from Fig.3,the value of v amaxdepends on thevalues of w i for semi-infinite linear diffusion and increases to the limiting value,1,for H!0and for all values ofw i.It is to be noted that on the v amaxðln HÞ–curve at w i=À8plateau of value0.298is observed as it was founded by De Vries and Van Dalen.The width at half height E1/2changes from203.4mV to17.7mV at all valuesof w i.The peak potential e apbecomes more negative and forH!0the limiting value of e apis individual for every value of w i.It is to be emphasized that in the practical area of strip-ping voltammetry(l<10l m,w i=À8;À5)the basic peak parameters strongly depend on the value of w i.This fact is necessary to take into account during analytical data processing and interpretation.3.2.Cathodic processThe full contour of the cathodic peak(Fig.4)in the dimensionless coordinates v0c–r t for following values of H:0.001;0.01;0.05;0.2;0.5;1.11;2.5;5;10is calcu-lated by Eqs.(8)and(11).Deformation of peak form depends on the value of the dimensionless switching potential w k(8;5;2;À1;À4)or E k,mV(205.43; 128.39;51.36;À25.68;À102.71),that corresponds to the following relations of concentration c Ox/c R(2980; 148;7;0.37;0.018),for the parameter H given above is investigated.The calculations were carried out pro-vided that the initial amount of metal at electrode is con-stant(c0R¼constÞ.B.F.Nazarov et al./Electrochemistry Communications9(2007)1936–19441939Fig.5shows basic cathodic peak parameters (height ofpeak v 0cmax ,peak potential e c p ¼ðnF =RT ÞðE k ÀE Þ,and d c=(nF /RT )E 1/2,where E 1/2is width at half peak height)as a function of ln H at different values of w k .The following properties of v 0(Àr t )from Fig.5are apparent:(i)As can be seen from Fig.3,the value of v 0cmaxdepends on the values of w k for semi-infinite linear diffusion (H !0)and increase to the limiting value,1.243,for H !0and for all values of w k .Withdecreasing H the value of v 0cmax increases so with decreasing film thickness and increasing the value of scan rate the maximum of cathodic currents decreases.In the cathodic process,during the initial time of electrolysis,the currents decrease with decreasing film thickness.In the following,the rate of this process grows accordingly exponentially,and the descending branch of the cathodic peak will only be defined by the rate of transport of fresh material.Thereby the descending branches of catho-dic current–potential curves have root character.This fact proves that at r t !1the value v 0ffiffiffiffiffir t p is equal to 1=ffiffiffip p .(ii)The width at half peak height E 1/2changes from203.4mV to 78.5mV at all values of w k .(iii)The peak potential e c p becomes more negative.ForH !0,the limiting value of e c p is individual for each value of w k .(iv)All dependences in the area of small H and positivevalue w k have linear character.For negative value w k the dependence of the basic cathodic peak param-eters on ln H is more complicated.In that case it is observed that the condition applied by Berzins and Delahay [43]at which concentration of M(Hg)is con-stant applies.We have shown that the same occurs at more positive values of w i for the anodic peak.The width at half peak height,d c ,which depends on ln H and w k ,can be presented by the following approximation model:P 2¼a 2þb 2ln H þd 2w k þe 2ðln H Þ2þf 2w 2k þg 2ðln H Þw kð12Þfor 1ÆH æ0.001and 8Æw k æ2,where P 2is d c .The approxi-mate equation for the dependence of v 0c max and e c p on ln H and w k can be written as:P 3¼a 3þb 3ln H þd 3w kð13Þfor 1ÆH æ0.001and 8Æw k æ2,where P 3is v 0cmax and e c p .Thecoefficients of the approximate expressions (12)and (13)are found using the method of nonlinear optimization and coefficients of correlation r are also estimated (Table 1).As it can be seen from Table 1,correlation between peak parameters and approximation model in case of anodic process is not so good than in case of cathodic processes.1940 B.F.Nazarov et al./Electrochemistry Communications 9(2007)1936–1944In case of anodic process with change of parameter H and initial potential,the effect of asymmetry of diffusion pro-cess is much stronger than in case of cathodic one.For this reason the dependences for anodic peak parameters from H and E i are more complex than for cathodic one in the given variation range of H and E i(1ÆHæ0.001and À8Æw iæÀ2).However,accuracy of calculations of anodic peak parameters by approximation equations,resulted in Table1,is quite relevant to practical application.3.3.Cyclic voltammetryThe effect of parameter H and switching potential w k on the form of cyclic voltammogram is illustrated by Fig.6.Parameters i cp =i ap(i apand i cp–maximum of anodic andcathodic peak,accordingly,in amps)and D e p¼e ap Àe cpare presented in Fig.7at different values of switching potentials w k.For semi-infinite linear diffusion,the param-eter i cp =i apequals1,for H!0the parameter i cp=i apdependson w k.For semi-infinite linear diffusion for8>w k<2the parameter D e p has the well known value2.22RT/nF[1]. For H!0,the limiting value of D e p depends on the value of w k.From the data presented in Fig.7and experimentally measured values of D e p,it is possible to determine the ‘‘equilibrium’’value of H for the reversible electrode pro-cess,which can differ from that calculated using experimen-tal data,for example,because of roughness of the interface, Ag/Hg,dissolution of the mercury in silver,etc.The dependence of the parameters i cp=i apand D e p on ln H and w k can be presented by the following approximate model:P4¼a4þb4ln Hþd4w kþe4ðln HÞ2þf4w2kþg4ðln HÞw k;ð14Þwhere P4is i cp=i apand D e p.The coefficients of the approxi-mate expression(14)are found by a method of nonlinear optimization and also coefficients of correlation r are esti-mated(Table1).In case of practical realization multicycle processes for periods A2B2etc.the process of an equilibrium achievement is enough complex.The time of electrolysis depends on the initial potential,switching potential and parameter of H. We can estimate the time of electrolysis when concentrationTable1Value of coefficients of expressions(11)–(14),standard errors S r and coefficients of correlation r Coefficients of model Anodic processv a max e apd aCoefficient estimation S r Coefficient estimation S r Coefficient estimation S ra10.510.060.650.20 1.70.4 b10.0730.025À0.460.08À0.320.14 c1À0.1060.0180.480.060.240.08 d10.00530.00240.0470.008–ae1À0.01150.0019À0.1060.006À0.0130.007 f1–a0.0620.007–ar0.94900.98700.9275Cathodic processv0cmax e cpd cCoefficient estimation S r Coefficient estimation S r Coefficient estimation S ra2,30.5560.007À1.0790.022 5.260.09 b2,3À0.04830.0012 1.0190.004À0.2450.036 c2,30.04800.0011À0.02780.00330.3170.026 d20.01100.0035 e2À0.01680.0028 f20.01950.0031 r0.99740.99990.9860Cyclic processi c p =i apD e pCoefficient estimation S r Coefficient estimation S ra4À2.9 1.80.740.06 b40.30.40.1550.014 c4À2.20.7À0.370.03 d40.120.07À0.00800.0024 e4À0.80.1À0.0570.003 f4 2.10.40.0180.004 r0.93190.9933a Coefficient estimation is not significant.B.F.Nazarov et al./Electrochemistry Communications9(2007)1936–194419411942 B.F.Nazarov et al./Electrochemistry Communications9(2007)1936–1944c s Ox calculated by the solution of system of Eq.(3)achievesconcentration c0Ox calculated by Nernst equation.It is esti-mated that for H<0.1the time of electrolysis will exceed the time of periods B1C1,B2C2etc in tens times.For example,in thefield of potentials from the switch-ing potential to a potential of peak maximum the time of electrolysis is negligible quantity.On the descending branch at stopping potential on the level of3/4heights of cathodic peak if the time of electrolyses is equal10periods of B2C2than the deviation of concentration from equilib-rium does not exceed20%.It corresponds to the error of equilibrium achievement which is equal to5mV or less. At stopping cathodic process at1/4height of peak the time of electrolysis increases on two orders.Stirring of solution during electrolysis considerably increases the time of equi-librium achievement.4.ConclusionThus,the additional boundary condition approach for calculation of cyclic voltammograms in case offinitefilm thickness electrode is realized.The full profile of current–potential curves obtained during oxidizing and reducing linear potential sweep at differentfilm thickness(from0 to1)is calculated.It is to be noted that application of Eq.(8)for vanishing values of H leads to an increase in the volume of calcula-tions.However,using the limiting Eq.(11)for vanishing values of H allows us to extend the range of variation of H(H<0.05).It is relevant because voltammetry at solid modified electrodes and microelectrodes has become increasingly popular.The anodic and cathodic curves are given at values of H(from10to0.001)and different values of initial and switching potential.It is established that the dependence basic parameters of anodic peak on H in thefield of their small values is more complicated than has been shown in previous works[9–11].It is also been shown that the results of De Vries and Van Dalen for cathodic processes are valid only for thickfilms,which are practi-cally unrealizable in voltammetry using mercuryfilm electrodes.B.F.Nazarov et al./Electrochemistry Communications9(2007)1936–19441943Significant effect of initial and switching potentials on the form of anodic and cathodic curves is shown.The approximate equations for the key parameters characteriz-ing the anodic,cathodic and cycle curves are given.Acknowledgementrionova thanks Tomsk Polytechnic University for a young scientist grant.References[1]Z.Galus,Fundamentals of Electrochemical Analysis,Polish scientificpublishers PWN,Warsaw,1994.[2]W.H.Reinmuth,Anal.Chem.33(1961)185.[3]W.H.Reinmuth,Anal.Chem.34(1962)1446.[4]I.Shain,J.Lewinson,Anal.Chem.33(1961)187.[5]M.M.Nicholson,J.Am.Chem.Soc.79(1957)7.[6]V.A.Igolinsky,Ph.D.Thesis,Tomsk Polytechnic University,Tomsk,1963.[7]D.K.Roe,J.E.K.Toni,Anal.Chem.37(1965)1503.[8]E.M.Roizenblat,I.M.Reznik,Russian J.Anal.Chem.36(1981)48.[9]W.T.De Vries,E.Van Dalen,J.Electroanal.Chem.10(1965)183.[10]W.T.De Vries,E.Van Dalen,J.Electroanal.Chem.12(1966)9.[11]W.T.De Vries,E.Van Dalen,J.Electroanal.Chem.14(1967)315.[12]P.J.Mahon,J.C.Myland,K.B.Oldham,J.Electroanal.Chem.537(2002)1.[13]M.Penczek,Z.Stojek,J.Electroanal.Chem.170(1984)99.[14]M.Penczek,Z.Stojek,J.Electroanal.Chem.181(1984)83.[15]M.Penczek,Z.Stojek,J.Electroanal.Chem.191(1985)91.[16]J.C.Ball,pton,Electroanal.9(1997)765.[17]J.C.Ball,J.A.Cooper,pton,J.Electroanal.Chem.435(1997)229.[18]J.C.Ball,pton,J.Phys.Chem.B102(1998)3967.[19]J.C.Ball,pton,C.M.A.Brett,J.Phys.Chem.B102(1998)162.[20]J.C.Ball,pton,Electroanal.9(1997)1305.[21]C.Agra-Gutierrez,J.C.Ball,pton,J.Phys.Chem.B102(1998)7028.[22]H.E.Keller,W.H.Reinmuth,Anal.Chem.44(1972)434.[23]V.A.Nemov,Ph.D.Thesis,Tomsk Polytechnic University,Tomsk,1972.[24]B.F.Nazarov,V.A.Nemov,in:Proceedings of Tomsk PolytechnicUniversity.Tomsk,250,1975,p.111.[25]B.F.Nazarov,V.A.Nemov,in:Proceedings of Tomsk PolytechnicUniversity.Tomsk,250,1975,p.115.[26]B.F.Nazarov, A.G.Stromberg,Russ.J.Electrochem.41(2005)49.[27]B.F.Nazarov,A.G.Stromberg,rionova,Russ.J.Electro-chem.41(2005)63.[28]J.Golas,Z.Galus,J.Osteryoung,Electrochim.Acta.32(1987)66.[29]M.A.Baldo,S.Daniele,G.A.Mazzocchin,Electrochim.Acta.41(1996)811.[30]J.Golas,Z.Galus,J.Osteryoung,Anal.Chem.59(1987)389.[31]A.M.Bond,T.L.E.Henderson,W.Thormann,J.Phys.Chem.90(1986)2911.[32]M.Ciszkowska,Z.Stojek,J.Electroanal.Chem.191(1985)101.[33]J.Golas,Z.Kowalski,Anal.Chim.Acta.221(1989)305.[34]F.H.Beyerlein,R.S.Nicholson,Anal.Chem.44(1972)1647.[35]C.Guminski,Z.Galus,Roczniki.Chem.43(1969)2147.[36]K.Tokuda,N.Enomoto,H.Matsuda,N.Koizumi,J.Electroanal.Chem.159(1983)23.[37]J.E.Spell,R.H.Philp,Anal.Chem.51(1979)2287.[38]sia,J.Electroanal.Chem.191(1985)185.[39]M.Donten,Z.Stojek,Z.Kublik,J.Electroanal.Chem.163(1984)11.[40]M.Donten,Z.Kublik,J.Electroanal.Chem.195(1985)251.[41]V.G.Bagrov,V.V.Belov,V.N.Zadorozhniy,A.Y.TrifonovMethodsof mathematics physics,vol.1,Technical-scientific literature publish-ers,Tomsk,2002.[42]B.C.Baev,Ph.D.Thesis,Institute of organic catalysis and electro-chemistry,Alma-Ata,1977.[43]T.Berzins,P.Delahay,J.Am.Chem.Soc.75(1953)555.1944 B.F.Nazarov et al./Electrochemistry Communications9(2007)1936–1944。