A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 10
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This equation is derived and discussed in more detail in Chapter 4. The three terms on the right-hand side represent thecontributions of diffusion, migration, and convection, respectively, to the flux.While we will be concerned with particular solutions of this equation in later chapters, a rigorous solution is generally not very easy when all three forms of mass transferare in effect; hence electrochemical systems are frequently designed so that one or moreof the contributions to mass transfer are negligible. For example, the migrational component can be reduced to negligible levels by addition of an inert electrolyte (a supporting electrolyte) at a concentration much larger than that of the electroactive species (seeSection 4.3.2).
Convection can be avoided by preventing stirring and vibrations in theelectrochemical cell. In this chapter, we present an approximate treatment of steadystate mass transfer, which will provide a useful guide for these processes in later chapters and will give insight into electrochemical reactions without encumbrance bymathematical details.1.4.2Semiempirical Treatment of Steady-State Mass TransferConsider the reduction of a species О at a cathode: О + ne *± R. In an actual case, the oxidized form, O, might be Fe(CN)£~ and R might be Fe(CN)6~, with only Fe(CN)^" initially present at the millimolar level in a solution of 0.1 M K2SO4.
We envision athree-electrode cell having a platinum cathode, platinum anode, and SCE reference electrode. In addition, we furnish provision for agitation of the solution, such as by a stirrer. Aparticularly reproducible way to realize these conditions is to make the cathode in theform of a disk embedded in an insulator and to rotate the assembly at a known rate; this iscalled the rotating disk electrode (RDE) and is discussed in Section 9.3.Once electrolysis of species О begins, its concentration at the electrode surface,CQ(X = 0) becomes smaller than the value, CQ, in the bulk solution (far from the electrode).
We assume here that stirring is ineffective at the electrode surface, so the solutionvelocity term need not be considered at x = 0. This simplified treatment is based on theidea that a stagnant layer of thickness 8O exists at the electrode surface (Nernst diffusionlayer), with stirring maintaining the concentration of О at CQ beyond x = 8O (Figure1.4.1). Since we also assume that there is an excess of supporting electrolyte, migration isnot important, and the rate of mass transfer is proportional to the concentration gradient atthe electrode surface, as given by the first (diffusive) term in equation 1.4.2:vmt <* (dCo/dx)x=0= Do(dCo/dx)x=0(1.4.3)If one further assumes a linear concentration gradient within the diffusion layer, then,from equation 1.4.3*>mt = ^otCcS - Co(x = 0)]/8 o(1.4.4)30 • Chapter 1.
Introduction and Overview of Electrode ProcessesCo(x =Figure 1.4.1 Concentration profiles (solid lines) and diffusion layer approximation (dashedlines), x = 0 corresponds to the electrode surface and 80 is the diffusion layer thickness.Concentration profiles are shown at two different electrode potentials: (7) where C0(x = 0)is about CQ/2, (2) where C0(x = 0) « 0 and i = ihSince 8Q is often unknown, it is convenient to combine it with the diffusion coefficient toproduce a single constant, YHQ — DQ/8Q, and to write equation 1.4.4 as~ Co(x = 0)](1.4.5)The proportionality constant, m o , called the mass-transfer coefficient, has units ofcm/s (which are those of a rate constant of a first-order heterogeneous reaction; seeChapter 3). These units follow from those of v and CQ, but can also be thought of asvolume flow/s per unit area (cm 3 s" 1 c m " 2 ) . 1 1 Thus, from equations 1.4.1 and 1.4.5and taking a reduction current as positive [i.e., / is positive when CQ > CQ(X = 0)], weobtainnFA= mo[C% - Co(x = 0)](1.4.6)Under the conditions of a net cathodic reaction, R is produced at the electrode surface, sothat CR(x = 0) > CR (where CR is the bulk concentration of R).
Therefore,= mR[CR(x = 0) - C*](1A7)11While m0 is treated here as a phenomenological parameter, in more exact treatments the value of m0 cansometimes be specified in terms of measurable quantities. For example, for the rotating disk electrode,m 0 = 0.62Do/3(ol/2v~l/e, where со is the angular velocity of the disk (i.e., 2тг/, with/as the frequency inrevolutions per second) and v is the kinematic viscosity (i.e., viscosity/density, with units of cm2/s) (seeSection 9.3.2). Steady-state currents can also be obtained with a very small electrode (such as a Pt diskwith a radius, r 0 , in the \xva range), called an ultramicroelectrode (UME, Section 5.3).
At a disk UME,m0 = 4Do/7rr0.311.4 Introduction to Mass-Transfer-Controlled Reactionsor for the particular case when CR = 0 (no R in the bulk solution),= 0)(1.4.8)The values of Co(x = 0) and CR(x = 0) are functions of electrode potential, E. Thelargest rate of mass transfer of О occurs when CQ(X — 0) = 0 (or more precisely, whenC o (x = 0) < < Co, so that CQ - Co(x = 0) « CQ).
The value of the current under theseconditions is called the limiting current, //, where(1.4.9)When the limiting current flows, the electrode process is occurring at the maximumrate possible for a given set of mass-transfer conditions, because О is being reduced asfast as it can be brought to the electrode surface. Equations 1.4.6 and 1.4.9 can be used toobtain expressions for Co(x = 0):(1.4.10)Co(x = 0) =nFAmo(1.4.11)Thus, the concentration of species О at the electrode surface is linearly related to the current and varies from CQ, when / = 0, to a negligible value, when / = //.If the kinetics of electron transfer are rapid, the concentrations of О and R at the electrode surface can be assumed to be at equilibrium with the electrode potential, as governed by the Nernst equation for the half-reaction 12= 0)(1.4.12)Such a process is called a nernstian reaction.
We can derive the steady-state i-E curvesfor nernstian reactions under several different conditions.(a) R Initially AbsentWhen C | = 0, CR(x = 0) can be obtained from (1.4.8):CR(x = 0) = i/nFAmR(1.4.13)Then, combining equations 1.4.11 to 1.4.13, we obtainnF(1.4.14)An i-E plot is shown in Figure 1 A.2a. Note that when / = ///2,nF(1.4.15)12Equation 1.4.12 is written in terms of E° , called the formal potential, rather than the standard potential E°.The formal potential is an adjusted form of the standard potential, manifesting activity coefficients and somechemical effects of the medium.
In Section 2.1.6, it will be introduced in more detail. For the present it is notnecessary to distinguish between E° and E°.32 * Chapter 1. Introduction and Overview of Electrode ProcessesCathodicH-( - ) •AnodicFigure 1.4.2 (a) Current-potential curve for a nernstian reaction involving two soluble specieswith only oxidant present initially, (b) log[(// - /)//] vs.
E for this system.where Ещ is independent of the substrate concentration and is therefore characteristic ofthe O/R system. Thus,(1.4.16)When a system conforms to this equation, a plot of E vs. log[(// — /)//] is a straight linewith a slope of 23RT/nF (or 59.1/л mV at 25°C). Alternatively (Figure 1.4.2b), log[(// i)li\ vs. E is linear with a slope of nF/23RT (or л/59.1 mV" 1 at 25°C) and has an ^-intercept of Ещ. When mo and mR have similar values, Ещ ~ E°'.(b) Both О and R Initially PresentWhen both members of the redox couple exist in the bulk, we must distinguish between acathodic limiting current, //c, when CQ(X = 0) ~ 0, and an anodic limiting current, z/a,when CR(x = 0) « 0.
We still have CQ(X = 0) given by (1.4.11), but with // now specifiedas // c . The limiting anodic current naturally reflects the maximum rate at which R can bebrought to the electrode surface for conversion to O. It is obtained from (1.4.7):(1.4.17)(The negative sign arises because of our convention that cathodic currents are taken aspositive and anodic ones as negative.) Thus CR(X = 0) is given by(1.4.18)CR(x = 0) = nFAmRCR(x = 0)_CR- 1—l(1.4.19)The i-E curve is then0>RT,~n~FlnЩ+ fjln(1-W(1.4.20)A plot of this equation is shown in Figure 1.4.3.
When / = 0, E = Eeq and the system is atequilibrium. Surface concentrations are then equal to the bulk values. When current flows,1.4 Introduction to Mass-Transfer-Controlled Reactions33Figure 1.4.3 Current-potential curve for a nernstian systeminvolving two soluble species with both forms initiallypresent.the potential deviates from £ e q , and the extent of this deviation is the concentration overpotential.
(An equilibrium potential cannot be defined when CR = 0, of course.)(c) R InsolubleSuppose species R is a metal and can be considered to be at essentially unit activity as theelectrode reaction takes place on bulk R.13 When aR = 1, the Nernst equation is(1.4.21)or, using the value of CQ(X = 0) from equation 1.4.11,(1.4.22)III111\11IWhen i = О, Е = Ещ = Е0' + (RT/nF) In CQ (Figure 1.4.4). If we define the concentration overpotential, rjconc (or the mass-transfer overpotential, 7]mt), asn'/cone= F —F^^eq(\ 4 1 \ \yLs-r.^o)then(1.4.24)When / = //, i7 c o n c —> oo. Since 17 is a measure of polarization, this condition is sometimescalled complete concentration polarization.^EFigure 1.4.4 Current-potential curve for a nernstian systemwhere the reduced form is insoluble.13This will not be the case for R plated onto an inert substrate in amounts less than a monolayer (e.g., thesubstrate electrode being Pt and R being Cu).
Under those conditions, aR may be considerably less than unity(see Section 11.2.1).34 • Chapter 1. Introduction and Overview of Electrode ProcessesEquation 1.4.24 can be written in exponential form:(1.4.25)The exponential can be expanded as a power series, and the higher-order terms can bedropped if the argument is kept small; that is,- ! + . * (when* is small)e*=l+jt + y(1.4.26)Thus, under conditions of small deviations of potential from Eeq, the /-т?Сопс characteristicis linear:-RTi(1.4.27)Since -r]/i has dimensions of resistance (ohms), we can define a "small signal" masstransfer resistance, Rmb as=i?m tRT(1.4.28)nF\it\Here we see that the mass-transfer-limited electrode reaction resembles an actual resistance element only at small overpotentials.1.4.3Semiempirical Treatment of the Transient ResponseThe treatment in Section 1.4.2 can also be employed in an approximate way to timedependent (transient) phenomena, for example, the buildup of the diffusion layer, either ina stirred solution (before steady state is attained) or in an unstirred solution where the diffusion layer continues to grow with time.
Equation 1.4.4 still applies, but in this case weconsider the diffusion layer thickness to be a time-dependent quantity, so thatUnFA = vmt = D o [ Cg - Co(x = 0)]/8 o (0(1.4.29)Consider what happens when a potential step of magnitude E is applied to an electrodeimmersed in a solution containing a species O. If the reaction is nernstian, the concentrations of О and R at x = 0 instantaneously adjust to the values governed by theNernst equation, (1.4.12). The thickness of the approximately linear diffusion layer,> grows with time (Figure 1.4.5).
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