A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 18
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Consider Аф at the interface Zn/Zn 2 + ,Cl~. Shown in Figure 2.2.4a is the simplest approach one could make to Аф using a potentiometric instrument with copper contacts. The measurable potential difference between the copper phases clearly includes interfacial potential differences at the Zn/Cuinterface and the Cu/electrolyte interface in addition to Аф. We might simplify mattersby constructing a voltmeter wholly from zinc but, as shown in Figure 2.2.4b, the measurable voltage would still contain contributions from two separate interfacial potentialdifferences.By now we realize that a measured cell potential is a sum of several interfacial differences, none of which we can evaluate independently.
For example, one could sketch thepotential profile through the cellCu/Zn/Zn2+,Cr/AgCl/Ag/Cu;(2.2.5)13according to Vetter's representation (24) in the manner of Figure 2.2.5.Even with these complications, it is still possible to focus on a single interfacial potential difference, such as that between zinc and the electrolyte in (2.2.5). If we can maintain constant interfacial potentials at all of the other junctions in the cell, then any changein E must be wholly attributed to a change in Аф at the zinc/electrolyte boundary. Keeping the other junctions at a constant potential difference is not so difficult, for the metalZn f^\ZnFigure 2.2.4 Two devices for measuringthe potential of a cell containing the Zn/Zn2interface.12Sometimes it is useful to break the inner potential into two components called the outer (or Volta) potential,ф, and the surface potential, x- Thus, ф = ф + х- There is a large, detailed literature on the establishment, themeaning, and the measurement of interfacial potential differences and their components.
See references 23—26.13Although silver chloride is a separate phase, it does not contribute to the cell potential, because it does notphysically separate silver from the electrolyte. In fact, it need not even be present; one merely requires asolution saturated in silver chloride to measure the same cell potential.60Chapter 2.
Potentials and Thermodynamics of CellsCu'ElectrolyteZnJ= £CuFigure 2.2.5 Potential profileacross a whole cell atequilibrium.Distance across cellmetal junctions always remain constant (at constant temperature) without attention, andthe silver/electrolyte junction can be fixed if the activities of the participants in its half-reaction remain fixed. When this idea is realized, the whole rationale behind half-cell potentials and the choice of reference electrodes becomes much clearer.2.2.4 Electrochemical PotentialsLet us consider again the interface Zn/Zn 2+ , Cl~ (aqueous) and focus on zinc ions inmetallic zinc and in solution.
In the metal, Zn 2+ is fixed in a lattice of positive zinc ions,with free electrons permeating the structure. In solution, zinc ion is hydrated and may interact with Cl~. The energy state of Zn 2 + in any location clearly depends on the chemical environment, which manifests itself through short-range forces that are mostlyelectrical in nature. In addition, there is the energy required simply to bring the +2charge, disregarding the chemical effects, to the location in question.
This second energyis clearly proportional to the potential ф at the location; hence it depends on the electrical properties of an environment very much larger than the ion itself. Although one cannot experimentally separate these two components for a single species, the differences inthe scales of the two environments responsible for them makes it plausible to separatethem mathematically (23-26). Butler (27) and Guggenheim (28) developed the conceptual separation and introduced the electrochemical potential, Jlf, for species / withcharge zx in phase a:z? = tf +(2.2.6)The term fxf is the familiar chemical potential(2.2.7)where щ is the number of moles of / in phase a.
Thus, the electrochemical potential wouldbe=я (ж)(22- *8)where the electrochemical free energy, G, differs from the chemical free energy, G, by theinclusion of effects from the large-scale electrical environment.2.2 A More Detailed View of Interfacial Potential Differences -> 61(a) Properties of the Electrochemical Potential1.For an uncharged species: Jif = fjbf.2.For any substance: [xf = ix®a + RT In af, where /if" is the standard chemicalpotential, and a? is the activity of species / in phase a.3.For a pure phase at unit activity (e.g., solid Zn, AgCl, Ag, or H 2 at unit fugacity):aHf = rf 4.5.For electrons in a metal (z = — 1): ~jx% = /л®а - ¥фа.
Activity effects can be disregarded because the electron concentration never changes appreciably.For equilibrium of species / between phases a and /3: JLf = jitf.(b) Reactions in a Single PhaseWithin a single conducting phase, ф is constant everywhere and exerts no effect on achemical equilibrium. The ф terms drop out of relations involving electrochemical potentials, and only chemical potentials will remain. Consider, for example, the acid-baseequilibrium:HO Ac <=± H + + OAc~(2.2.9)This requires thatДн + + ДолеМн+ + рФ + Доле- - РФА*Н+ + МОАс-(2.2.10)(2.2.11)(2.2.12)(c) Reactions Involving Two Phases Without Charge TransferLet us now examine the solubility equilibriumAgCl (crystal, c) +± Ag + + С Г (solution, 5),(2.2.13)which can be treated in two ways.
First, one can consider separate equilibria involvingAg + and Cl~ in solution and in the solid. Thus1^crC1g(2-2.14)= Ma-(2- 2 - 1 5 )Recognizing that^f(2-2.16)^ r(2-2.17)one has from the sum of (2.2.14) and (2.2.15),MAgl?' = M V+Expanding, we obtainA^cf 1 = MAg+ + RTln aA g + + F<£s + M c r +RTln acr~ рФ8(2.2.18)and rearrangement givesЙ(2.2.19)where ^ s p is the solubility product.
A quicker route to this well-known result is to writedown (2.2.17) directly from the chemical equation, (2.2.13).Note that the </>s terms canceled in (2.2.18), and that an implicit cancellation of </>AgC1terms occurred in (2.2.16). Since the final result depends only on chemical potentials, the62Chapter 2. Potentials and Thermodynamics of Cellsequilibrium is unaffected by the potential difference across the interface. This is a generalfeature of interphase reactions without transfer of charge (either ionic or electronic).When charge transfer does occur, the ф terms will not cancel and the interfacial potentialdifference strongly affects the chemical process.
We can use that potential difference either to probe or to alter the equilibrium position.(d) Formulation of a Cell PotentialConsider now the cell (2.2.5), for which the cell reaction can be writtenZn + 2AgCl + 2e(Cu') <=* Z n 2 + + 2Ag + 2СГ + 2e(Cu)(2.2.20)At equilibrium,M+ 2/#fg + 2Д Си ' = Jfo* + 2j#§ + 2м&+ 2Д Си(2.2.21)But,2(ДеСи' - /ZeCu) = - 2F(^ C u ' - фСа) = - 2FE(2.2.23)Expanding (2.2.22), we have-2FE = /4n 2+ + RTln«zn2+ + 2рФ* + 2^Kg+ 2 Мсг(2.2.24)-2FE = AG° + ЯГ In a|n2+ (asa-)2(2.2.25)whereд y--tOZAw"OsLH,r7~/lJt_|_ ^Os~\ ZjjJLr^"\~I оOAg\ Ai LJL A *OZnr^Txx^OAgClr^ A2CI^тгтг®(^ о О/%\yjL.Z*,Z*K))Thus, we arrive atE = £ ° -Щ; ln(asZn2+)(ascr)2,(2.2.27)which is the Nernst equation for the cell.
This corroboration of an earlier result displaysthe general utility of electrochemical potentials for treating interfacial reactions withcharge transfer. They are powerful tools. For example, they are easily used to considerwhether the two cellsCu/Pt/Fe2+, F e 3 + , СГ/AgCl/Ag/Cu'Cu/Au/Fe2+, F e 3 + , СГ/AgCl/Ag/Cu'(2.2.28)(2.2.29)would have the same cell potential. This point is left to the reader as Problem 2.8.2.2.5Fermi Level and Absolute PotentialThe electrochemical potential of electrons in a phase a, JIQ , is called the Fermi level orFermi energy and corresponds to an electron energy (not an electrical potential) Ep.
TheFermi level represents the average energy of available electrons in phase a and is relatedto the chemical potential of electrons in that phase, /x£, and the inner potential of a.14 TheFermi level of a metal or semiconductor depends on the work function of the material (seeSection 18.2.2). For a solution phase, it is a function of the electrochemical potentials of14More exactly, it is the energy where the occupation probability is 0.5 in the distribution of electrons among thevarious energy levels (the Fermi—Dirac distribution). See Sections 3.6.3 and 18.2.2 for more discussion of Ep.2.3 Liquid Junction Potentials л: 63the dissolved oxidized and reduced species.
For example, for a solution containing F e 3 +and F e 2 +Де = Й е 2 + - Й е 3 +(2.2.30)For an inert metal in contact with a solution, the condition for electrical (or electronic) equilibrium is that the Fermi levels of the two phases be equal, that is,E | = E^(2.2.31)This condition is equivalent to saying that the electrochemical potentials of electrons inboth phases are equal, or that the average energies of available (i.e., transferable) electrons are the same in both phases. When an initially uncharged metal is brought into contact with an initially uncharged solution, the Fermi levels will not usually be equal.
Asdiscussed in Section 2.2.2, equality is attained by the transfer of electrons between thephases, with electrons flowing from the phase with the higher Fermi level (higher /Ze ormore energetic electrons) to the phase with the lower Fermi level. This electron flowcauses the potential difference between the phases (the electrode potential) to shift.For most purposes in electrochemistry, it is sufficient to reference the potentials ofelectrodes (and half-cell emfs) arbitrarily to the NHE, but it is sometimes of interest tohave an estimate of the absolute or single electrode potential (i.e., vs.
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