A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 6
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Whether the charge on the metal is negative orpositive with respect to the solution depends on the potential across the interface and thecomposition of the solution. At all times, however, qM — -qs. (In an actual experimentalarrangement, two metal electrodes, and thus two interfaces, would have to be considered;we concentrate our attention here on one of these and ignore what happens at the other.)The charge on the metal, qM, represents an excess or deficiency of electrons and resides ina very thin layer (<0.1 A) on the metal surface.
The charge in solution, qs, is made up ofan excess of either cations or anions in the vicinity of the electrode surface. The chargesqM and qs are often divided by the electrode area and expressed as charge densities, suchas, ( j M = qM/A, usually given in /лС/ст2. The whole array of charged species and oriented dipoles existing at the metal-solution interface is called the electrical double layer(although its structure only very loosely resembles two charged layers, as we will see inSection 1.2.3).
At a given potential, the electrode- solution interface is characterized by adouble-layer capacitance, C<j, typically in the range of 10 to 40 /^F/cm2. However, unlikereal capacitors, whose capacitances are independent of the voltage across them, Q isoften a function of potential.41.2.3Brief Description of the Electrical Double LayerThe solution side of the double layer is thought to be made up of several "layers." Thatclosest to the electrode, the inner layer, contains solvent molecules and sometimes otherspecies (ions or molecules) that are said to be specifically adsorbed (Figure 1.2.3).
Thisinner layer is also called the compact, Helmholtz, or Stern layer. The locus of the electri4In various equations in the literature and in this book, Cj may express the capacitance per unit area and begiven in fxF/cm2, or it may express the capacitance of a whole interface and be given in JJLF. The usage for agiven situation is always apparent from the context or from a dimensional analysis.1.2 Nonfaradaic Processes and the Nature of the Electrode-Solution Interface:13IHP OHPф1ф2Diffuse layerSolvated cationMetalSpecifically adsorbed anion= Solvent moleculeFigure 1.2.3 Proposed model of thedouble-layer region under conditionswhere anions are specifically adsorbed.cal centers of the specifically adsorbed ions is called the inner Helmholtz plane (IHP),which is at a distance x\. The total charge density from specifically adsorbed ions in thisinner layer is а1 (/лС/ст2).
Solvated ions can approach the metal only to a distance x2; thelocus of centers of these nearest solvated ions is called the outer Helmholtz plane (OHP).The interaction of the solvated ions with the charged metal involves only long-range electrostatic forces, so that their interaction is essentially independent of the chemical properties of the ions. These ions are said to be nonspecifically adsorbed. Because of thermalagitation in the solution, the nonspecifically adsorbed ions are distributed in a threedimensional region called the dijfuse layer, which extends from the OHP into the bulk ofthe solution.
The excess charge density in the diffuse layer is <7d, hence the total excesscharge density on the solution side of the double layer, cr s , is given by=_ом(1.2.5)The thickness of the diffuse layer depends on the total ionic concentration in the solution;for concentrations greater than 10~2 M, the thickness is less than ~100 A. The potentialprofile across the double-layer region is shown in Figure 1.2.4.The structure of the double layer can affect the rates of electrode processes. Consideran electroactive species that is not specifically adsorbed.
This species can approach theelectrode only to the OHP, and the total potential it experiences is less than the potentialbetween the electrode and the solution by an amount ф2 — </>s, which is the potential dropacross the diffuse layer. For example, in 0.1 M NaF, ф2 — <£s is —0.021 V at E = -0.55V vs. SCE, but it has somewhat larger magnitudes at more negative and more positive potentials. Sometimes one can neglect double-layer effects in considering electrode reactionkinetics. At other times they must be taken into account.
The importance of adsorptionand double-layer structure is considered in greater detail in Chapter 13.One usually cannot neglect the existence of the double-layer capacitance or the presence of a charging current in electrochemical experiments. Indeed, during electrode reactions involving very low concentrations of electroactive species, the charging current canbe much larger than the faradaic current for the reduction or oxidation reaction.
For thisreason, we will briefly examine the nature of the charging current at an IPE for severaltypes of electrochemical experiments.14Chapter 1. Introduction and Overview of Electrode Processes- Metal — > ЦSolutionN'I i0!©)\j>(+) Solvated cationч~' "Ghost" of anion repelledfrom electrode surfaceф2Figure 1.2.4 Potential profile across thedouble-layer region in the absence of specificadsorption of ions. The variable ф, called theinner potential, is discussed in detail inSection 2.2.
A more quantitativerepresentation of this profile is shown inFigure 12.3.6.x21.2.4Double-Layer Capacitance and ChargingCurrent in Electrochemical MeasurementsConsider a cell consisting of an IPE and an ideal reversible electrode. We can approximate such a system with a mercury electrode in a potassium chloride solution that is alsoin contact with an SCE. This cell, represented by Hg/K+, CF/SCE, can be approximatedby an electrical circuit with a resistor, Rs, representing the solution resistance and a capacitor, C(j, representing the double layer at the Hg/K+,C1~ interface (Figure 1.2.5).5 SinceндdropelectrodeHISCEWv II-AMоFigure 1.2.5 Left: Two-electrode cell with an ideal polarized mercury drop electrode and an SCE.Right: Representation of the cell in terms of linear circuit elements.Actually, the capacitance of the SCE, С$СЕ, should also be included.
However, the series capacitance of C d andCSCE is C T = CdCSCEJ[Cd + CSCEL and normally C S C E » Q> so that C T « C d . Thus, C S C E can be neglectedin the circuit.1.2 Nonfaradaic Processes and the Nature of the Electrode-Solution Interface «I 15Cd is generally a function of potential, the proposed model in terms of circuit elements isstrictly accurate only for experiments where the overall cell potential does not changevery much.
Where it does, approximate results can be obtained using an "average" Cdover the potential range.Information about an electrochemical system is often gained by applying an electricalperturbation to the system and observing the resulting changes in the characteristics of thesystem. In later sections of this chapter and later chapters of this book, we will encountersuch experiments over and over. It is worthwhile now to consider the response of the IPEsystem, represented by the circuit elements Rs and Q in series, to several common electrical perturbations.(a) Voltage (or Potential) StepThe result of a potential step to the IPE is the familiar RC circuit problem (Figure1.2.6). The behavior of the current, /, with time, t, when applying a potential step ofmagnitude E, is(1.2.6)RThis equation is derived from the general equation for the charge, q, on a capacitor asa function of the voltage across it, EQ\(1.2.7)q = CdEcAt any time the sum of the voltages, £ R and EQ, across the resistor and the capacitor, respectively, must equal the applied voltage; henceE=Er = iR* + 4-(1.2.8)Noting that / = dq/dt and rearranging yieldsdqdt-qRsCd(1.2.9)RsIf we assume that the capacitor is initially uncharged (q = 0 at t = 0), then the solution of(1.2.9) isq = ECd[l-e~t/RsCd](1.2.10)By differentiating (1.2.10), one obtains (1.2.6).
Hence, for a potential step input, there isan exponentially decaying current having a time constant, т = RsCd (Figure 1.2.7). Thecurrent for charging the double-layer capacitance drops to 37% of its initial value at t = т,and to 5% of its initial value at t = 3r. For example, if Rs = 1 ft and Cd = 20 fxF, thenт = 20 /JLS and double-layer charging is 95% complete in 60 /xs.Figure 1.2.6circuit.Potential step experiment for an RC16 P Chapter 1.
Introduction and Overview of Electrode ProcessesResultant (/)• Applied(E)Figure 1.2.7 Currenttransient (/ vs. t) resulting froma potential step experiment.(b) Current StepWhen the RsCd circuit is charged by a constant current (Figure 1.2.8), then equation 1.2.8again applies. Since q = Jidt, and / is a constant,E = iRK + 4r\dt(1.2.11)orE = i(Rs + t/Cd)(1.2.12)Hence, the potential increases linearly with time for a current step (Figure 1.2.9).(c) Voltage Ramp (or Potential Sweep)A voltage ramp or linear potential sweep is a potential that increases linearly with timestarting at some initial value (here assumed to be zero) at a sweep rate и (in V s" 1 ) (seeFigure 1.2.10a).E = vtConstant current source(1.2.13)Figure 1.2.8 Current step experiment for an RCcircuit.1.2 Nonfaradaic Processes and the Nature of the Electrode-Solution Interface-Slope = i -17Resultants• Applied(0_tFigure 1.2.9 E-t behavior resultingfrom a current step experiment.If such a ramp is applied to the RSC^ circuit, equation 1.2.8 still applies; hencevt = Rs(dq/dt) + q/Cd(1.2.14)lfq = OaU = 0,(1.2.15)The current rises from zero as the scan starts and attains a steady-state value, vCd (Figure1.2.10b).
This steady-state current can then be used to estimate Cd. If the time constant,Applied E(t)(a)Resultant i(b)Figure 1.2.10 Current-timebehavior resulting from a linearpotential sweep applied to an RCcircuit.18 • Chapter 1. Introduction and Overview of Electrode ProcessesApplied ESlope = - иResultant [/ =f{t)]Resultant [/=/(£)]vCAFigure 1.2.11 Current-time and current-potentialplots resulting from a cyclic linear potential sweep (ortriangular wave) applied to an RC circuit.RsCd, is small compared to v, the instantaneous current can be used to measure C«j as afunction of E.If one instead applies a triangular wave (i.e., a ramp whose sweep rate switches fromv to —v at some potential, £ A ), then the steady-state current changes from vC& during theforward (increasing E) scan to — y Q during the reverse (decreasing E) scan.
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