A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 42
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However when the electroactive component is present at a high concentration,large changes in solution properties, such as the local viscosity, can occur during electrolysis. For such systems, (4.4.12) is no longer appropriate, and more complicated treatments are necessary (24, 25). Under these conditions, migrational effects can also becomeimportant.We will have many occasions in future chapters to solve (4.4.12) under a variety ofboundary conditions. Solutions of this equation yield concentration profiles, CQ(X, t).The general formulation of Fick's second law for any geometry isdt2=DOV2CC(4.4.17)2where V is the Laplacian operator.
Forms of V for different geometries are given inTable 4.4.2. Thus, for problems involving a planar electrode (Figure 4.4.5a), the lineardiffusion equation, (4.4.12), is appropriate. For problems involving a spherical electrodeTABLE 4.4.2Forms of the Laplacian Operator for Different Geometries'*TypeLinearSphericalCylindrical (axial)DiskBandaVariablesxrгr, zx, zV22Example2Shielded disk electrodeд /дх22Hanging drop electroded /dr ^- (2/r)(d/dr)Wire electrodeд2/дг2 Нh (l/r)(d/dr)22- d /dz Inlaid disk ultramicroelectrode^д2/дг2 Н- (l/r)(d/dr) H622Inlaid band electrodeд2/дх2 -f a /azSee also J.
Crank, "The Mathematics of Diffusion," Clarendon, Oxford, 1976.r = radial distance measured from the center of the disk; z = distance normal to the disk surface.cx — distance in the plane of the band; z = distance normal to the band surface.4.4 DiffusionIXТ/iii\\\\151А \\\\\Figure 4.4.5 Types of diffusionoccurring at different electrodes.(a) Linear diffusion to a planarelectrode, (b) Spherical diffusionto a hanging drop electrode.\ifl)(Figure 4.4.5b), such as the hanging mercury drop electrode (HMDE), the spherical formof the diffusion equation must be employed:(4.4.18)The difference between the linear and spherical equations arises because spherical diffusion takes place through an increasing area as r increases.Consider the situation where О is an electroactive species transported purely by diffusion to an electrode, where it undergoes the electrode reactionО + ne <± R(4.4.19)If no other electrode reactions occur, then the current is related to the flux of О at the electrode surface (x = 0), /Q(0, t), by the equation(4.4.20)because the total number of electrons transferred at the electrode in a unit time must beproportional to the quantity of О reaching the electrode in that time period.
This is an extremely important relationship in electrochemistry, because it is the link between theevolving concentration profile near the electrode and the current flowing in an electrochemical experiment. We will draw upon it many times in subsequent chapters.If several electroactive species exist in the solution, the current is related to the sumof their fluxes at the electrode surface. Thus, for q reducible species,iFA4.4.3q2k=lqk=l(4.4.21)Boundary Conditions in Electrochemical ProblemsIn solving the mass-transfer part of an electrochemical problem, a diffusion equation (or, ingeneral, a mass-transfer equation) is written for each dissolved species (O, R , .
. . ) . The solution of these equations, that is, the discovery of an equation expressing CQ,CR, . . . as functions of л: and t, requires that an initial condition (the concentration profile atChapter 4. Mass Transfer by Migration and Diffusiont = 0) and two boundary conditions (functions applicable at certain values of x) be givenfor each diffusing species. Typical initial and boundary conditions include the following.(a) Initial ConditionsThese are usually of the form(4.4.22)Co(x,0)=f(x)For example, if О is uniformly distributed throughout the solution at a bulk concentrationCQ at the start of the experiment, the initial condition isCo(Jt, 0) = Cg(for all x)(4.4.23)If R is initially absent from the solution, thenCR(JC, 0) = 0(for all x)(4.4.24)(b) Semi-infinite Boundary ConditionsThe electrolysis cell is usually large compared to the length of diffusion; hence the solution at the walls of the cell is not altered by the process at the electrode (see Section5.2.1).
One can normally assume that at large distances from the electrode (x —> °°) theconcentration reaches a constant value, typically the initial concentration, so that, forexample,lim Co(x9 t) =C%(at all 0(4.4.25)lim CR(JC, t) = 0(at all t)(4.4.26)For thin-layer electrochemical cells (Section 11.7), where the cell wall is at a distance, /,of the order of the diffusion length, one must use boundary conditions at x = I instead ofthose for лс—> °°.(c) Electrode Surface Boundary ConditionsAdditional boundary conditions usually relate to concentrations or concentration gradients at the electrode surface. For example, if the potential is controlled in an experiment,one might haveCo(0,t)=f(E)(4.4.27)where f(E) is some function of the electrode potential derived from the general currentpotential characteristic or one of its special cases (e.g., the Nernst equation).If the current is the controlled quantity, the boundary condition is expressed in termsof the flux at x = 0; for example,FHL<4A29)The conservation of matter in an electrode reaction is also important.
For example,when О is converted to R at the electrode and both О and R are soluble in the solutionphase, then for each О that undergoes electron transfer at the electrode, an R must be produced. Consequently, /o(0, t) = - / R ( 0 , t), andr<?C R (jt O l4.5 References i 1534.4.4Solution of Diffusion EquationsIn the chapters that follow, we will examine the solution of the diffusion equations undera variety of conditions. The analytical mathematical methods for attacking these problemsare discussed briefly in Appendix A.
Numerical methods, including digital simulations(Appendix B), are also frequently employed.Sometimes one is interested only in the steady-state solution (e.g., with rotating diskelectrodes or ultramicroelectrodes). Since dC0/dt = 0 in such a situation, the diffusionequation simply becomesV2Co = 0(4.4.31)Occasionally, solutions can be found by searching the literature concerning analogous problems. For example, the conduction of heat involves equations of the same formas the diffusion equation (26, 27);дТ/dt = a^2T(4.4.32)where T is the temperature, and ax = к/ps (к = thermal conductivity, p = density, and s= specific heat). If one can find the solution of a problem of interest in terms of the temperature distribution, such as, T(x, t), or heat flux, one can easily transpose the results togive concentration profiles and currents.Electrical analogies also exist.
For example, the steady-state diffusion equation,(4.4.31), is of the same form as that for the potential distribution in a region of space notoccupied by electrically charged bodies (Laplace's equation),V2<£ = 0(4.4.33)If one can solve an electrical problem in terms of the current density, j , where-j = KV<J>(4.4.34)(where к is the conductivity), one can write the solution to an analogous diffusion problem (as the function CQ) and find the flux from equation 4.4.20 or from the more generalform,- / = D O VC O(4.4.35)This approach has been employed, for example, in determining the steady-state uncompensated resistance at an ultramicroelectrode (28) and the solution resistance between anion-selective electrode tip and a surface in a scanning electrochemical microscope (29,30).
It also is sometimes possible to model the mass transport and kinetics in an electrochemical system by a network of electrical components (31, 32). Since there are a numberof computer programs (e.g., SPICE) for the analysis of electric circuits, this approach canbe convenient for certain electrochemical problems.• 4.5 REFERENCES1. M.
Planck, Ann. Physik, 39, 161; 40, 561 (1890).2. J. Newman, Electroanal. Chem., 6, 187 (1973).3. J. Newman, Adv. Electrochem. Electrochem.Engr., 5, 87 (1967).4. С W. Tobias, M. Eisenberg, and С R. Wilke, /.Electrochem. Soc, 99, 359C (1952).5. W. Vielstich, Z. Elektrochem., 57, 646 (1953).6. N. Ibl, Chem. Ing. Tech., 35, 353 (1963).7. G.
Chariot, J. Badoz-Lambling, and B. Tremillion, "Electrochemical Reactions," Elsevier,Amsterdam, 1962, pp. 18-21, 27-28.g. I. M. Kolthoff and J. J. Lingane, "Polarography," 2nd ed., Interscience, New York, 1952,Vol. 1, Chap. 7.154 • Chapter 4. Mass Transfer by Migration and Diffusion9. K. Vetter, "Electrochemical Kinetics," Academic, New York, 1967.22.
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Bard, Anal. Chem., 65, 3605 (1993).18. W. Jost, "Diffusion in Solids, Liquids, andGases," Academic, New York, 1960.30. С Wei, A. J. Bard, G. Nagy, and K. Toth, Anal.Chem., 67, 1346 (1995).19. S. Chandrasekhar, Rev. Mod. Phys., 15, 1 (1943).20. L. B. Anderson and C. N. Reilley, /. Chem.Educ, 44, 9 (1967).31. J. Homo, M. T. Garcia-Hernandez, and С F.Gonzalez-Fernandez, /.
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