Reaction/Diffusion at Electrode/Solution Interfaces: The EC 2 Reaction

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1 Int. J. Electrochem. Sci., 4(9) International Journal of ELECTROCHEMICAL SCIENCE Reaction/Diffusion at Electrode/Solution Interfaces: The EC Reaction Michael E G Lyons * Physical and Materials Electrochemistry Laboratory, School of Chemistry, Uniersity of Dublin, Trinity College, Dublin, Ireland. * melyons@tcd.ie Receied: 6 July 9 / Accepted: 15 August 9 / Published: 5 August 9 A closed form analytical expression is deried for the normalized current ersus potential response expected for a rotating disc electrode in which an initial fast electron transfer reaction is followed by a homogeneous electrochemical reaction which exhibits second order kinetics. The effect of the homogeneous reaction on the shape of the oltammetric response cure is calculated. Keywords: Reaction/diffusion equations ; Electrode kinetics ; EC reactions. 1. INTRODUCTION In this short paper we present a generalized theoretical analysis for second order EC processes at a rotating disc electrode. The analysis of diffusion processes with concurrent chemical reaction is well established in electrochemistry [1]. Howeer much of the published work has concentrated on the coupling between reactant diffusion and first order homogeneous chemical reaction and the mathematical analysis of the CE and EC reaction schemes [,] is well established. Howeer little work has been reported on reactant diffusion coupled with second order homogeneous reactant kinetics. More than years ago Ulstrup [4] examined second order kinetic processes at a rotating disc electrode and deried a simple expression for the limiting current for the case where an intermediate undergoes a fast disproportionation under steady state conditions. More recently Compton and co-workers [5] examined coupled diffusion and second order kinetics at a rotating disc electrode and obtained, ia numerical analysis, working cures for the expected current ersus potential response. Bartlett and Eastwick-Field [6] deried approximate analytical expressions for the limiting current at the rotating disc electrode for the ECE mechanism where the chemical step is irreersible. The situation corresponding to both pseudo-first order and second order kinetics was examined. Feldberg and co-workers [7] carried out a digital simulation of a second order ECE dimerization reaction at a rotating disc electrode.

2 Int. J. Electrochem. Sci., Vol. 4, In the present paper we describe a simple analysis of the EC reaction scheme in which an electro-generated intermediate reacts ia second order kinetics.. THEORETICAL MODEL.1 Model setup and general considerations We consider the following reaction scheme. m ne A B k B P (1) In this scheme A denotes a stable species in solution undergoing an oxidation or reduction reaction at a support electrode to an unstable intermediate species B which subsequently decomposes ia a chemical transformation within the diffusion layer to a product species P ia second order kinetics. The rate of the latter reaction is quantified by the bimolecular rate constant k (units: cm mol -1 s -1 ). If steady state behaiour is assumed then the pertinent reaction diffusion equations are gien by d a D = dx () d b D kb = dx In the latter expressions a and b denote the concentrations of species A and B at any distance x from the electrode surface and we hae assumed for simplicity that all species hae identical diffusion coefficients so that D A = D B = D. We further assume that diffusion is Ficksian and planar and that x represents the distance normal from the electrode surface. We hae also assumed that conection terms can be nglected in the transport equations. This assumption is well established [8]. We introduce the following scaling parameters which sere to make the reaction/diffusion equations dimensionless: a b x u = = χ = () a a δ where a denotes the bulk concentration of reactant species A and δ represents the Nernst Diffusion Layer thickness, which for a rotating disc electrode is gien by [9] : δ 1/ 6 1/ 1/ =.64ν D W (4) where ν denotes the kinematic iscosity of the solution and W is the electrode rotation speed (units: Hz). We finally introduce a reaction/diffusion parameter γ such that γ δ δ f D D δ f k a k a K = = = (5) D

3 Int. J. Electrochem. Sci., Vol. 4, This parameter represents the balance between homogeneous chemical reaction (quantified in terms of a kinetic flux f ) and diffusion (expressed in terms of a diffusie flux f ). K D form The system of reaction/diffusion equations therefore adopts the following adimensional d u d = γ = The boundary conditions defining the problem are gien by [ ξ ] χ = u = exp du d χ = + = dχ χ = 1 u = 1 = The first boundary condition pertains to the region immediately adjacent to the electrode surface and implies that the A/B redox transformation is kinetically facile and Nernstian. In this expression the parameter ξ denotes a normalised potential and is defined as F ( E E ξ = ) = θ θ (8) RT We also introduce a normalised current Ψ such that fσ fσ Ψ = = (9) f Da δ D In the latter expression f Σ denotes the flux arising at any applied potential E and f D represents the diffusion limited flux which is obsered at eleated potentials.note that the relationship between reaction flux f Σ and current i is gien by i f = Σ nfa (1) where n, F and A denote the number of electrons transferred, the Faraday constant and the electrode geometric area respectiely. Hence the normalised current is gien by du d Ψ = = (11) Note that the third statement presented in eqn.7 assumes that the concentration of the electrogenerated product species B at the edge of the diffusion layer when x = δ is zero. This assumption will be alid proided that B decomposes rapidly ia second order reaction and so the bimolecular rate constant k will be ery much larger than the rate of diffusion of B through the diffusion layer. Hence the boundary condition will be alid when the reaction/diffusion parameter γ is large. (6) (7)

4 Int. J. Electrochem. Sci., Vol. 4, Solution of the reaction/diffusion equations We now sole the system of equations presented in eqn.6 subject to the boundary conditions outlined in eqn.7 to obtain an expression for the steady state oltammogram. The first expression presented in eqn.6 may be directly integrated to produce ( χ ) u = Aχ + B (1) The constants of integration A and B are readily shown ia application of the boundary conditions to du be A = = Ψ = 1 u, B = u. Hence the reactant concentration profile within the diffusion layer is linear and takes the form ( χ ) ( 1 ) u = u + u χ (1) A full integration of the -expression presented in eqn.6 can be accomplished ia the following idea. d d d ρ dρ d dρ We set ρ = and so = = = ρ and the second expression in eqn.6 reduces to dχ d d dρ ρ γ = (14) d Integrating the latter expression yields: p p = p = ( ) 1 1 (15) In performing the latter integration we specifically assume that at the edge of the diffusion layer at d χ = 1, ρ1 =, 1 =. This assumption will be alid for large alues of the 1 reaction/diffusion parameter. Noting that ρ = Ψ we immediately obtain the following useful result From the Nernst equation we note that 1/ = Ψ / [ ] ( ) [ ] (16) = u exp ξ = 1 Ψ exp ξ (17) Equating the expressions for outlined in eqn.16 and eqn.17 we obtain the following expression defining the current response / 1 Ψ = λψ (18) Where we hae set

5 Int. J. Electrochem. Sci., Vol. 4, 9 11 Alternatiely we write 1/ (, ) = exp[ ξ ] λ γ ξ 1/ 1/ ( ) λ ( ) (19) Ψ + Ψ 1 = () This is a cubic equation in Ψ 1/. We note immediately that as the potential is drien to extreme alues ξ, exp[ ξ ] and so λ and from eqn.18 we note that Ψ 1, f f and the diffusion controlled plateau region is reached as we would expect. Furthermore when the reaction/diffusion parameter γ is large signifying that the homogeneous reaction flux for the dimerization is much larger than the diffusie flux then again the reaction is under net diffusion control and again Ψ 1. Σ D.. Ealuation of the steady state oltammetric cure One general form [9] of a cubic equation is gien by the so called normal form: x + a x + a x + a = (1) 1 In the latter expression we hae set the coefficient a = 1 without loss of generality, and the coefficients a k, k = 1, define constant coefficients. If we define the following quantities: a a1 Q = 9 9a a 7a a R = { R } { R } S = + T = 1/ 1/ () We note that the discriminant is gien by: = Q + R () The cubic will admit the following real and complex roots proided that > : a1 x1 = S + T ( S + T ) a1 x = + ( S T ) i (4) ( S + T ) a1 x = ( S T ) i And we choose the single real root as being physically meaningful.

6 Int. J. Electrochem. Sci., Vol. 4, For the present problem we compare eqn. and eqn.1 and identify a1 = λ, a =, a = 1. Hence we note that Q λ λ λ λ Q =, R = = =. Furthermore we note that λ 6 6 λ 1 λ 1 λ λ 1 λ λ = = λ, R = = + = + λ and so the discriminant is gien by 1 λ 1 λ = Q + R = = >. This implies that our analysis will be alid proided / 7 that λ < = Using eqn.19 we note that the alue of λ depends both on the numerical 4 alues of the reaction/diffusion parameter γ and the normalized potentialξ. We can compute the minimum alue of normalized potential required to ensure the alidity of the condition λ < λ with λ = for a gien γ alue. For normalized potential alues more negatie than this the critical discriminant will not be positie. The results of the required computation is presented in table 1 below. This calculation in fact defines a lower bound for the potential which must be applied to ensure that the condition of rapid homogeneous kinetics is met. Table 1. Dependence of normalized potential on reaction/diffusion parameter γ ξ ( ) min λ < λ critical critical Note also that for large alues of λ, will be positie since as the normalized potential ξ increases the λ term will decrease and eentually approach zero. We can also show that 1 λ 1 λ S = solution to eqn. is: 1/ and also 1 λ 1 λ T = / 1/ 1/ 1/ 1 λ 1 λ 1 λ 1 λ λ Ψ = And the expression for the normalized current takes the form: and so the physically meaningful (5) 1/ 1/ 1 λ 1 λ 1 λ 1 λ λ Ψ = (6)

7 Int. J. Electrochem. Sci., Vol. 4, 9 11 If we substitute the expression for λ outlined in eqn.19 into eqn.6 we obtain an explicit expression for the normalized current ersus potential cure as desired. 1/ λ Returning to the cubic equation illustrated in eqn. we can set Ψ = Ζ and hence the cubic transforms to Which simplifies to Where we hae set Proided that λ λ Ζ + λ Ζ 1 = (7) Ζ αζ β = (8) λ α = λ β = 1 7 7β > 4α then the cubic equation presented in eqn.8 will hae one real and two complex roots. Furthermore it can be shown that we can find ϑ such that And we can find the real root gien by We can readily show that where we hae set coshϑ = α / β Ζ = α cosh [ ϑ / ] λ = cosh = cosh [ ϑ ] κ [ ϑ ] 1 1 κ 1 κ 1 κ ϑ = cosh ln 1 = + κ κ κ {{ } 1/ [ ]} (9) () (1) () 1 κ = λ = exp ξ () Hence the normalized current response can be ealuated from { } ( κ ) κ ( cosh[ ϑ ] 1) Ψ = Ζ = (4) This is an alternatie expression for the normalized current response and can be used to define the oltammetric response when the explicit potential dependence of the κ parameter is inputted ia the expression outlined in eqn..

8 Int. J. Electrochem. Sci., Vol. 4, 9 11 We now ealuate the half wae potential. We note that the normalized potential is defined as: F ( E E ξ = ) = θ θ, and when 1/ RT θ = θ, Ψ = 1/. Hence from eqn. we obtain that 1/ 1/ / 1 exp ( θ1/ θ ) 1 + = 1.This expression is readily shown to simplify to: 1/ / 1 1 exp ( θ1/ θ ) =, and so we obtain on further rearrangement the following expression for the half wae potential: 1 1 θ1/ = θ + ln + ln (5) We note that since the reaction/diffusion parameter γ is always positie then the term ln{ } < dθ1/ 1 and so for an oxidation process, θ1/ < θ and d ln γ =. Hence we predict that the half wae potential should be shifted to more negatie alues with increasing alues of the reaction/diffusion parameter γ. This ariation, computed from eqn.5 is presented in figure 1 and in semi-logarithmic format in figure. These results are in agreement with data preiously reported by Compton and coworkers [5] who attacked the problem using numerical methods θ=θ 1/ θ e+4 4e+4 6e+4 8e+4 1e+5 γ Figure 1. Variation of θ = θ1/ θ with reaction/diffusion parameter γ for an EC process computed from eqn.5. Returning to eqn.14 we can integrate across the width of the diffusion layer to obtain:

9 Int. J. Electrochem. Sci., Vol. 4, We recall that p and so : p pdp = γ { } p = + p p d d = = d = dχ / (6) (7) Separating the ariables and integrating produces the following expression for the concentration profile of the electro-generated reactant species B in the diffusion layer: 1 1 γ = + χ (8) θ=θ 1/ θ e+ 1e+1 1e+ 1e+ 1e+4 1e+5 γ Figure. Presentation of eqn.5 in semi-logarithmic format. Furthermore since = χ = 1 then = and so the concentration profile is gien by γ From eqn.15 we also note that 1 γ = { χ 1} (9) p d du = = = dχ dχ / (4)

10 Int. J. Electrochem. Sci., Vol. 4, Also we note that du = 1 u = 1 exp dχ From eqn. 4 and eqn.41 we immediately obtain [ ξ ] (41) The latter may be written in the following form / exp[ ] 1 + ξ = (4) ( ) [ ξ ]( ) + exp = (4) The latter expression defines a cubic in which may also be soled ia procedures similar to those outlined earlier in the paper. This approach was adopted by Compton and co-workers [5]..4 Extension of analysis to a general mth order homogeneous reaction: the EC m process. We now briefly discuss how the analysis deeloped in the present paper may be extended to a more general reaction sequence in which one has an initial rapid interfacial heterogeneous electron transfer reaction followed by a homogeneous chemical reaction which follows m th order kinetics with a rate constant k. We focus specific attention on the homogeneous reaction step occurring within the diffusion layer of thickness δ inoling the electro-generated reactant in which the reaction stoichiometry is: k mb Products Again the reaction/diffusion equation takes the form: d m mγ dχ = (44) Where we hae defined the reaction/diffusion parameter in this case as: kδ m 1 γ = ( a ) (45) D Proceeding as before we get d d m d = mγ (46) dχ dχ dχ d d d d d mγ Now again we note that = 1+ m m d dχ and we also note that = mγ d 1+ m hence from eqn.46 we obtain: d d mγ 1+ m d = dχ (47) ( 1+ ) m Integrating both sides produces

11 Int. J. Electrochem. Sci., Vol. 4, d mγ 1+ m K = + ( 1+ m) The integration constant K is obtained by appropriate use of the boundary conditions. When d d χ = 1 = 1 = = dχ and so d K =. If we can make the further assumption that 1 1 d then K = and we obtain from eqn.48 that 1 Specifically at χ = the current is gien by We recall that u = 1 Ψ hence d mγ dχ ( 1+ m) electrode surface we note [ ξ ] From eqn.5 we obtain that: u + m ( ) (1 ) 1+ m d mγ Ψ = = ( 1+ m) m ( 1 m) = 1 γ + = exp.hence we obtain that: [ ξ ] ( + m) 1 (48) (49) (5). Furthermore from the Nernst equation at the 1 Ψ = exp (51) 1 ( 1+ m) 1+ m ( 1+ m) = Ψ (5) mγ Hence from eqn,51 and eqn.5 we obtain the following expression for the current ersus potential cure for a m th order reaction: 1 ( 1+ m) ( 1+ m) 1+ m 1 Ψ = Ψ mγ exp[ ξ ] (5) This expression may be written as: 1+ m Ψ + Ψ exp ξ * 1 = (54) ( ) [ ] Where we hae introduced 1 mγ ξ* = ξ + ln (55) 1+ m 1+ m This expression is in agreement with a result first proposed by Amatore [1]. Specifically for m = we note that eqn.5 reduces to eqn.16 and eqn.54 transforms to eqn. as indeed it should.. CONCLUSIONS In the present paper an analysis of an EC reaction sequence inoling a Nernstian interfacial electron transfer reaction followed by a homogeneous dimerization reaction occurring in the diffusion layer is presented. The pertinent reaction-diffusion equations hae been formulated and soled under steady state conditions subject to defined boundary conditions. The analysis has enabled an algebraic expression for either the normalized current or the surface concentration of electro-generated reactant

12 Int. J. Electrochem. Sci., Vol. 4, to be formulated in terms of a cubic equation. Two closed form analytical solutions to the cubic hae been proposed using standard methodologies. The ariation in half wae potential with a characteristic reaction/diffusion parameter has been proposed. Finally the extension of the analysis to an EC P type process has been demonstrated. Acknowledgements This work was supported by Enterprise Ireland (Grant number SC//49), IRCSET (Grant number SC//169) and the HEA-PRTLI Programme. References 1. C.P. Andrieux and J.M. Saeant in Inestigation of rates and mechanism of reactions. Part II. (C.F. Bernasconi, Ed.,), 4 th Edition, Techniques of Chemistry Series, Vol. VI, Wiley, New York, 1986, Chapter 7, pp A.J. Bard, L.R. Faulkner, Electrochemical Methods: Fundamentals and Applications, Wiley, New York, 198, Chapter 11, pp (a) J. Koutecky and R. Brdicka, Collect. Czech Chem. Commun., 1 (1947) 7 ; (b) P. Delehay, S. Ora, J. Am. Chem. Soc., 8 (196) 9 ; L.H.J. Tong, K. Liang and W.R. Ruby, J. Electroanal. Chem., 1 (1967) 45 ; Z. Galus, R.N. Adams, J. Electroanal. Chem., 4 (196) J. Ulstrup, Electrochim Acta., 1 (1968) (a) R.G. Compton, D. Mason and P.R. Unwin, J. Chem. Soc. Faraday Trans.I. 84 (1988) 47 ; (b) R.G. Compton, P.R. Unwin, J. Chem. Soc. Faraday Trans.I. 85 (1989) P.N. Bartlett, V. Eastwick-Field, J. Chem. Soc. Faraday Trans.I. 89 (199) L.S. Marcoux, R.N. Adams and S.W. Feldberg, J. Phys. Chem., 7 (1969) W.J. Albery, Electrode Kinetics, Clarendon Press, Oxford, 1975, pp W.J. Albery and M.L. Hitchman, Ring-Disc Electrodes, Clarendon Press, Oxford, 1971, p H.T. Dais, Introduction to nonlinear differential and integral equations, Doer, New York, 196, pp M.R. Spiegel, Mathematical Handbook, McGraw Hill, New York, 1968, pp C. Amatore, in Organic Electrochemistry, (H. Lund, M.M. Baizer, Eds.), rd Edition, Marcel Dekker, New York, 1991, Chapter, pp by ESG (

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