A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 39
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Physik. Chem., 202, 302 (1953)]found thatд log L00.03<Hog[Zn(II)]• = *0.41~± •д log in— = -0.28 ±0.02]д log L— — | - ^ ]— = 0.65 ± 0.03<Hog[NH3д log in*L°, = 0.57 ± 0.03tд log [Zn]136 • Chapter 3. Kinetics of Electrode Reactionswhere [Zn] refers to a concentration in the amalgam.(a) Give the equation for the overall reaction.(b) Assume that the process occurs by the following mechanism:HgZn(II) + e i^Zn(I) + i/1>NH NH 3 + ^ 1 O H - O HZn(I) + ^Zn(Hg) + i>2,NH NH 3 + F 2 , O H ~OH(fast pre-reactions)(rate-determining step)where Zn(I) stands for a zinc species of unknown composition in the + 1 oxidation state, andthe v's are stoichiometric coefficients.
Derive an expression for the exchange current analogousto (3.5.40), and find explicit relationships for the logarithmic derivatives given above.(b) Calculate a and all stoichiometric coefficients.(c) Identify Zn(I) and write chemical equations to give a mechanism consistent with the data.(d) Consider an alternative mechanism having the pattern above, but with the first step being ratedetermining. Is such a mechanism consistent with the observations?3.11 The following data were obtained for the reduction of species R to R~ in a stirred solution at a 0.1cm 2 electrode; the solution contained 0.01 M R and 0.01 M R~.7j(mV):i(jiA):-100-120-150-500-60045.962.6100965965Calculate: /0, k°, a, Rcb //, ra0, Rmt3.12 From results in Figure 3.4.5 for 10~2 M Mn(III) and 10~2 M Mn(IV), estimate j 0 and k°.
What is thepredicted j 0 for a solution 1 M in both Mn(III) and Mn(IV)?3.13 The magnitude of the solvent term (1/гор - l/ss) is about 0.5 for most solvents. Calculate the valueof Лo and the free energy of activation (in eV) due only to solvation for a molecule of radius 4.0 Aspaced 7 A from an electrode surface.3.14 Derive (3.6.30).3.15 Show from the equations for D O (E, Л) and D R (E, Л) that the equilibrium energy of a system, E e q , isrelated to the bulk concentrations, CQ and C R and E° by an expression resembling the Nernst equation.How does this expression differ from the Nernst equation written in terms of potentials, Ещ and £°?How do you account for the difference?3.16 Derive (3.6.36) by considering the reaction О + e ±± R at equilibrium in a system with bulk concentrations CQ and C R .CHAPTER4MASS TRANSFERBY MIGRATIONAND DIFFUSION4.1 DERIVATION OF A GENERAL MASSTRANSFER EQUATIONIn this section, we discuss the general partial differential equations governing mass transfer; these will be used frequently in subsequent chapters for the derivation of equationsappropriate to different electrochemical techniques.
As discussed in Section 1.4, masstransfer in solution occurs by diffusion, migration, and convection. Diffusion and migration result from a gradient in electrochemical potential, JL. Convection results from an imbalance of forces on the solution.Consider an infinitesimal element of solution (Figure 4.1.1) connecting two points inthe solution, r and s, where, for a certain species j , Jip) Ф ^(s). This difference of ^over a distance (a gradient of electrochemical potential) can arise because there is a difference of concentration (or activity) of species у (a concentration gradient), or because thereis a difference of ф (an electric field or potential gradient).
In general, a flux of species jwill occur to alleviate any difference of /Zj. The flux, Jj (mol s ^ c m " 2 ) , is proportional tothe gradient of /xj:Jj oc grad^ujorJj oc V)itj(4.1.1)where grad or V is a vector operator. For linear (one-dimensional) mass transfer, V =i(d/dx)9 where i is the unit vector along the axis and x is distance. For mass transfer in athree-dimensional Cartesian space,V = i | - + j | - + k|dxPoint rJdyPoint s#(a)Figure 4.1.1(4.1.2)dz(b)A gradient of electrochemical potential.137138 • Chapter 4. Mass Transfer by Migration and DiffusionThe constant of proportionality in (4.1.1) turns out to be —CJO>^IRT\ thus,(4.1.3)For linear mass transfer, this is(4.1.4)The minus sign arises in these equations because the direction of the flux opposes the direction of increasing ~jlIf, in addition to this ~jx gradient, the solution is moving, so that an element of solution[with a concentration C}(s)\ shifts from s with a velocity v, then an additional term isadded to the flux equation:(4.1.5)For linear mass transfer,CD(4.1.6)IFTaking cij ~ Cj, we obtain the Nernst-Planck equations, which can be written as^-(RT In С д + ~-(z:F<£) +Cv(x)С/j(A)#W1 С(4.1.7)(4.1.8)or in general,(4.1.9)In this chapter, we are concerned with systems in which convection is absent.
Convective mass transfer will be treated in Chapter 9. Under quiescent conditions, that is, inan unstirred or stagnant solution with no density gradients, the solution velocity, v, iszero, and the general flux equation for species j , (4.1.9), becomes(4.1.10)For linear mass transfer, this isdCfx)(4.1.11)RT » Л dxwhere the terms on the right-hand side represent the contributions of diffusion and migration, respectively, to the total mass transfer.If species У is charged, then the flux, /j, is equivalent to a current density. Let us consider a linear system with a cross-sectional area, A, normal to the axis of mass flow.
Then,/j (mol s" 1 cm" 2 ) is equal to —Ц/ZJFA [C/s per (C mol" 1 cm 2 )], where ц is the currentdxD J C4.2 Migration139component at any value of x arising from a flow of species j . Equation 4.1.11 can then bewritten asJ¥A(4.1.12)¥Awith(4.1.13)(4.1.14)ЯГwhere /^j and / m j are diffusion and migration currents of species 7, respectively.At any location in solution during electrolysis, the total current, i, is made up of contributions from all species; that is,orF2A(4.1.16)where the current for each species at that location is made up of a migrational component(first term) and a diffusional component (second term).We will now discuss migration and diffusion in electrochemical systems in more detail.
The concepts and equations derived below date back to at least the work of Planck(1). Further details concerning the general problem of mass transfer in electrochemicalsystems can be found in a number of reviews (2-6).4.2 MIGRATIONIn the bulk solution (away from the electrode), concentration gradients are generallysmall, and the total current is carried mainly by migration. All charged species contribute.For species j in the bulk region of a linear mass-transfer system having a cross-sectionalarea А, ц = / m j orH = ~^fTx(4-2.1)The mobility of species y, defined in Section 2.3.3, is linked to the diffusion coefficient bythe Einstein-Smoluchowski equation:"iRT(4.2.2)hence ц can be reexpressed asij = |zj|FAwjCj^(4.2.3)For a linear electric field,d4z = ^ r(4.2.4)140Chapter 4.
Mass Transfer by Migration and Diffusionwhere Д£// is the gradient (V/cm) arising from the change in potential AE over distance /.Thus,\z;\FAuiCiAE1J/(4.2.5)and the total current in bulk solution is given byjJwhich is (4.1.16) expressed in particular for this situation. The conductance of the solution, L {ОГ1), which is the reciprocal of the resistance, R (£1), is given by Ohm's law,where к, the conductivity (П * cm l; Section 2.3.3) is given byк = / r 2l 2 jl M jCj(4.2.8)jEqually, one can write an equation for the solution resistance in terms of p, the resistivity(fl-cm), where p = 1/к:R =j(4.2.9)The fraction of the total current that a given ion7 carries is Ц, the transference numberof 7, given byti=J7(4.2.10)=See also equations 2.3.11 and 2.3.18.4.3 MIXED MIGRATION AND DIFFUSION NEAR ANACTIVE ELECTRODEThe relative contributions of diffusion and migration to the flux of a species (and of theflux of that species to the total current) differ at a given time for different locations insolution.
Near the electrode, an electroactive substance is, in general, transported byboth processes. The flux of an electroactive substance at the electrode surface controlsthe rate of reaction and, therefore, the faradaic current flowing in the external circuit(see Section 1.3.2). That current can be separated into diffusion and migration currentsreflecting the diffusive and migrational components to the flux of the electroactivespecies at the surface:i = id + *m(4.3.1)Note that /m and i$ may be in the same or opposite directions, depending on the direction ofthe electric field and the charge on the electroactive species. Examples of three reductions—of a positively charged, a negatively charged, and an uncharged substance—are shown inFigure 4.3.1.
The migrational component is always in the same direction as id for cationicspecies reacting at cathodes and for anionic species reacting at anodes. It opposes id whenanions are reduced at cathodes and when cations are oxidized at anodes.4.3 Mixed Migration and Diffusion Near an Active Electrode < 141Cu 2 + + 2e -> CuCu(CN)J~ + 2e -> Cu + 4CN'2CuCu(CN)2 + 2e -» Cu + 2CN~Cu(CN)J"00{a) i = id + \im\{b) i = id - \im\Cu(CN)20(c) i = idFigure 4.3.1 Examples of reduction processes with different contributions of the migrationcurrent: (a) positively charged reactant, (b) negatively charged reactant, (c) uncharged reactant.For many electrochemical systems, the mathematical treatments are simplified if themigrational component to the flux of the electroactive substance is made negligible.
Wediscuss in this section the conditions under which that approximation holds. The topic isdiscussed in greater depth in references 7-10.4.3.1Balance Sheets for Mass Transfer During ElectrolysisAlthough migration carries the current in the bulk solution during electrolysis, diffusionaltransport also occurs in the vicinity of the electrodes, because concentration gradients ofthe electroactive species arise there. Indeed, under some circumstances, the flux of electroactive species to the electrode is due almost completely to diffusion. To illustrate theseeffects, let us apply the "balance sheet" approach (11) to transport in several examples.Example 4.1Consider the electrolysis of a solution of hydrochloric acid at platinum electrodes (Figure 4.3.2a).
Since the equivalent ionic conductance of H + , Л+, and of Cl~, A_, relateas Л+ ~ 4A_, then from (4.2.10), t+ = 0.8 and t- = 0.2. Assume that a total currentequivalent to lOe per unit time is passed through the cell, producing five H2 molecules0.,.,©iЭ- Pt/H+, CI7Pt(a)©(p\(Cathode)(Anode)^0e10СГ-10е- >5CI2-10СГ+10H 8H +2СГdiffusiondiffusion+8СГ2СГ8H +2H(b)Figure 4.3.2 Balancesheet for electrolysis ofhydrochloric acid solution.(a) Cell schematic, (b)Various contributions tothe current when lOe arepassed in the externalcircuit per unit time.142Chapter 4. Mass Transfer by Migration and Diffusionat the cathode and five Cl 2 molecules at the anode.
(Actually, some O 2 could also beformed at the anode; for simplicity we neglect this reaction.) The total current is carried in the bulk solution by the movement of 8H + toward the cathode and 2C1~ towardthe anode (Figure 4.3.2b). To maintain a steady current, 10 H + must be supplied to thecathode per unit time, so an additional 2H + must diffuse to the electrode, bringingalong 2 C F to maintain electroneutrality. Similarly at the anode, to supply 10 Cl~ perunit time, 8C1~ must arrive by diffusion, along with 8H + . Thus, the different currents(in arbitrary e -units per unit time) are: for H + , i& — 2, /m = 8; for Cl~, i& = 8, /m = 2.The total current, /, is 10.
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