A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 98
Текст из файла (страница 98)
Further analysis of Rcb such as to obtain the rate constant of the RDS, requiresknowledge of the i-E characteristic for the mechanism, which can, however, be difficult todevelop (see Section 3.5.4).10.4 ELECTROCHEMICAL IMPEDANCE SPECTROSCOPYIn Section 10.3, we concentrated on the components of the faradaic impedance, Rs and C s .We assumed that they can be extracted readily from direct measurements of the total impedance, which also includes the solution resistance, /fo, and the double-layer capacitance, C d . In this section we will consider measurements of the total cell or electrodeimpedance as a function of oo and methods of extracting the faradaic impedance, /fo, andQ from the results.At a given frequency, the equivalent circuit of the cell can be taken as in Figure10.1.14, but we measure its impedance as a resistance value J?B and the capacitance valueCB in series [or equivalently as ZR C = R% and Z\m = l/(oC%].
One approach to obtainingthe faradaic impedance from these values is to measure the cell impedance in a separateexperiment under identical conditions, but in the absence of the electroactive couple. Thismeasurement should yield the values of ifo and Qj (assuming they are not changed by thepresence of the electroactive species), since the faradaic path is inactive. One can thensubtract these graphically or analytically from the R% and CB values. This approach is384 s Chapter 10. Techniques Based on Concepts of Impedanceoften used when impedance bridge measurements are made and was discussed in somedetail in the first edition.8 A more direct approach involves a study of the way in whichthe total impedance Z = RB — j/(coCB) = Z R e - jZim varies with frequency. From thisvariation, one can extract /fo, C d , Rs, and C s directly.
This method circumvents the needfor separate measurements without the electroactive species, and it eliminates the need toassume that the electroactive species has no effect on the nonfaradaic impedance.10.4.1Variation of Total Impedance (4,16)The electrochemical impedance spectroscopic approach, which is largely based on similarmethods used to analyze circuits in electrical engineering practice, was developed bySluyters and coworkers (4) and later extended by others (8-12). It deals with the variationof total impedance in the complex plane [as represented in Nyquist plots (Section10.1.2)]. Let us consider this approach for our standard system.The measured total impedance of the cell, Z, is expressed as the series combinationof /?B and C B .
These two elements provide the real and imaginary components of Z, thatis, Z R e = RB and Z I m = VcoCB The electrochemical system is described theoretically interms of an equivalent circuit such as that in Figure 10.1.14. Its impedance is readily written down according to the methods of Section 10.1.2. The real part, which must equal themeasured Z R e , iswhere A = (Cd/Cs) + 1 and 5 = a>RsCd. Similarly,B2/a)Cd + A/coC,< 1 0 A 2 )Substitution for Rs and C s by (10.2.25) and (10.2.26) providesRct1)2 +_ (oCd(Rcim+ ао)~У2)22ma(o~1/2(col/2Cda+21/2+ I ) + ^ C;j(# c t + crco~ r(Cdaa)Chemical information can be extracted by plotting Z ^ vs. Z R e for different со.
For simplicity let us first consider the limiting behavior at high and low со.(a) Low-Frequency LimitAs со —> 0, the functions (10.4.3) and (10.4.4) approach the limiting forms:Z R e - Ru + Rct + ao)~l/2mZ ^ - aco~+ 2(?Cd(10.4.5)(10.4.6)Elimination of со between these two givesZZIm - Re " Ru^ct+2a" Q(10.4.7)Thus, the plot of Z I m vs. Z R e should be linear and have unit slope, as shown in Figure210.4.1. The extrapolated line intersects the real axis at Ru + Rct - 2a Cd. One can see8First edition, pp. 347-349.10.4 Electrochemical Impedance Spectroscopy <i 385zReFigure 10.4.1 Impedance plane plot for lowfrequencies.from (10.4.5) and (10.4.6) that the frequency dependence in this regime comes only fromWarburg impedance terms; thus the linear correlation of Z R e and Z I m is characteristic of adiffusion-controlled electrode process.
As the frequency rises, the charge-transfer resistance, Rcb and the double-layer capacitance become more important elements, and we canexpect a departure from (10.4.7).(b) High-Frequency LimitAt very high frequencies, the Warburg impedance becomes unimportant in relation to Rcband the equivalent circuit converges to that of Figure 10.4.2. The impedance is(10.4.8)= *а-лwhich has the componentsZ(10.4.9)R e "~2ZIm=2coCd1R2ct+ со C%(10.4.10)a) CdRctElimination of со from this pair of equations yields(10.4.11)Hence Z I m vs. Z R e should give a circular plot centered at Z R e = /fo + Rct/2 and Z I m = 0and having a radius of Rct/2.
Figure 10.4.3 depicts the result.The general features of the plot are readily grasped intuitively. The imaginary component to the impedance in the circuit of Figure 10.4.2 comes solely from C d . Its contribution falls to zero at high frequencies, because it offers no impedance. All of the current ischarging current, and the only impedance it sees is the ohmic resistance. As the frequencydrops, the finite impedance of Cd manifests itself as a significant Z I m . At very low frequencies, the capacitance C d offers a high impedance; hence current flow passes mostlyRctFigure 10.4.2 Equivalent circuit for a system inwhich the Warburg impedance is unimportant.386Chapter 10. Techniques Based on Concepts of Impedancen=VRctCd/У-1со1+RctFigure 10.4.3 Impedance plane plot forthe equivalent circuit of Figure 10.4.2.through Rct and RQ.
Thus the imaginary impedance component falls off again. In general, we can expect to see a departure from this plot in this lower-frequency regime, because the Warburg impedance will become important.(c) Application to Real SystemsAn actual plot of impedance in the complex plane will combine the features of our twolimiting cases as in Figure 10.4.4. However, both regions may not be well defined for anygiven system. The determining feature is the charge-transfer resistance, Rcb and its relation to the Warburg impedance, which is controlled by a. If the chemical system is kinetically sluggish, it will show a large Rcb and may display only a very limitedfrequency region where mass transfer is a significant factor. This case is shown in Figure10.4.5a.
At the other extreme, Rct might be inconsequentially small by comparison to theohmic resistance and the Warburg impedance over nearly the whole available range of a.Then the system is so kinetically facile that mass transfer always plays a role, and thesemicircular region is not well defined. An example is shown in Figure 10.4.5b.10.4.2Limits to Measurable k° by the FaradaicImpedance Method (1-6,9-12)The foregoing paragraphs highlight the limitations in interpreting impedance data, and0they lead naturally to the idea that A: must fall in some fairly well-defined range inKineticcontrolRCl + RctFigure 10.4.4 Impedance plotfor an electrochemical system.Regions of mass-transfer andkinetic control are found at lowand high frequencies, respectively.10.4 Electrochemical Impedance Spectroscopy387Figure 10.4.5 Impedance plane plots for actual chemical systems.
Numbers by points arefrequencies in kHz. (a) For the electrode reaction Z n 2 + + 2e ^± Zn(Hg). C*n2+ = Czn(Hg) =8 X 1СГ3 M. Electrolyte was 1 M NaClO4 plus 1(Г 3 М HC1O4. (b) For the electrode reactionUgl+ + 2e ±± Hg in 1 M HC1O4. C H g 2+ = 2 X 10~3 M. [From J. H. Sluyters and J. J. С Oomen,Rec. Trav. Chim. Pays-Bas, 79, 1101 (1960), with permission.]order to be reliably measured by an impedance method. We can define the range semiquantitatively.(a) Upper LimitThe parameter Rct must make a significant contribution to Rs hence Rcttuting from (10.3.2), (10.3.6), and (3.4.7) and assuming Doa/col/2. Substi-= DR and C$ = C | , we ob-tain the condition that k° ^ (ZW2) 1 / 2 .
The highest practical value of со is determined bythe cell time constant, /? U Q, which must remain much smaller than the cycle period ofthe applied ac stimulus. With a UME, one can do useful work at several MHz, so thatсо < 107 s~\ and with D ~ 10~5 cm2/s, we have k° ^ 7 cm/s.9 In addition, there are requirements that C s ^ C d and Rs > R^.(b) Lower LimitFor very large Rcb the Warburg impedance is negligible, and the equivalent circuit of Figure 10.4.2 can be applied. One problem here is that Rct cannot be so large that all the current takes the path through C d .
That is, Rct < l/(oCd or k° > RTCdco/F2C*A. If we choosethe most favorable conditions of C* = 10~2 M and со = 2тт X 1 Hz, 1 0 then at T = 298 Кand with Cd/A = 20 /^F/crn2, we obtain ^ ° > 3 X 1 0 " 6 cm/s.9The reductions and oxidations of aromatic species to anion and cation radicals in aprotic solvents are generallyamong the fastest known heterogeneous charge-transfer reactions.
The values of k° can exceed 1 cm/s. Seereferences 17 and 18 for measurements of such systems by impedance methods.10It is also essential that the period of the ac stimulus not be so long that convection becomes a factor within afew cycles. The lower frequency limit was set here at 1 Hz because convection would become a problem in therange of several seconds in most liquid systems with water-like viscosity. Current equipment for EIS can operateat much lower frequencies (as low as 10 ^Hz) and can be usefully applied in the low-frequency (long-time)regime when the processes being examined are not controlled by convection.
Examples include transport orreaction at a solid-solid interfaces or diffusion and reaction in extremely viscous media, such as glasses orpolymers.388Chapter 10. Techniques Based on Concepts of Impedance10.4.3Other Applications of ElectrochemicalImpedance SpectroscopyThe approach discussed above for EIS of a simple heterogeneous electron-transfer reaction of solution components can be applied to more complicated electrochemical systems,like those with coupled homogeneous reactions or with adsorbed intermediates.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
