A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 97
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However, we are interested now in derivatives of surface concentrations, and they will be dominated bythe higher-frequency ac signal. Relations (10.2.21) and (10.2.22) will still apply to a very high approximation.This is a mathematical manifestation of the way in which the ac part of the experiment can usually beuncoupled from the dc part.380Chapter 10. Techniques Based on Concepts of Impedancewhere1( Po _(10.2.24)It is easy to identify Rs and C s by comparison with (10.2.4):Rc = Rnt + a/co 1/2(10.2.25)(10.2.26)Q =The complete evaluation of Rs and C s depends on finding relations for Rcb /3Q, and /3R.We will see below that Rct is primarily determined by the heterogeneous chargetransfer kinetics, and we have already observed above that the terms cr/a)l/2 and llaoo>mcome from mass-transfer effects. Recognition of this situation has led to a division of thefaradaic impedance into the charge-transfer resistance, Rcb and the Warburg impedance,Z w , as shown in Figure 10.1.14b.
Equations 10.2.25 and 10.2.26 demonstrate that this latter impedance can be regarded as a frequency-dependent resistance, /?w = a/o)1/2, in series with the pseudocapacitance, Cw = Cs = l/acom. Thus the total faradaic impedance,Z f , can be written= Rct + [cr(o~m - j(aa)~l/2)](10.2.27)with the term in brackets representing the Warburg impedance.10.3 KINETIC PARAMETERS FROM IMPEDANCEMEASUREMENTS (1, 4, 6, 8-14, 16)From the description of the faradaic impedance experiment given in Section 10.1, it isclear that measurements are made with the working electrode's mean potential at equilibrium. Since the amplitude of the sinusoidal perturbation is small, we can use the linearized i-rj characteristic to describe the electrical response to the departure fromequilibrium.
For a one-step, one-electron process, О + e <=^ R, the linearized relationship is (3.4.30), which can be rewritten in terms of the electronic current convention as77= RThence4CO(Q,0c R (0,C*C*RTFi0t](10.3.1)(10.3.2)RT(10.3.3)(10.3.4)Now we see thatRTFi0(10.3.5)10.3 Kinetic Parameters from Impedance Measurements «t 381so that the exchange current, and therefore k°, can be evaluated easily when Rs and C s areknown. The bridge method allows a precise definition of these electrical equivalents; thusit can yield kinetic data of very high quality.Equation 10.3.5 shows that one can, in principle, evaluate /0 from data taken at a single frequency.
However, doing so is not really wise, because one has no experimental assurance that the equivalent circuit actually mirrors the performance of the system. Thebest way to check for agreement is to examine the frequency dependence of the impedance. For example, (10.2.25) and (10.2.26) predict that Rs and l/coCs should both be linearwith a)~m and should have a common slope, <7, which is quantitatively predictable fromthe constants of the experiment; that is,о- =RT(-(10.3.6)Figure 10.3.1 is a display of these relationships.The plot of Rs should have an intercept, Rct, from which /0 can be evaluated.
Extrapolation to the intercept is equivalent to estimating the system's performance atinfinite frequency. The Warburg impedance drops out at high frequencies, becausethe time scale is so short that diffusion cannot manifest itself as a factor influencingthe current. Since the surface concentrations never change significantly from themean values [see (10.2.19) and (10.2.20)], charge-transfer kinetics alone dictate thecurrent.If the linear behavior typified in Figure 10.3.1 is not observed, then the electrodeprocess is not as simple as we assume here, and a more complex situation must beconsidered. The availability of this kind of check for internal consistency is an extremely important asset of the impedance technique. See Section 10.4 for more details.The conclusions of the foregoing discussion are also applicable to a quasireversiblemultistep mechanism, for which Rct is defined asSee Section 3.5.4(d) for more about the interpretation of /0 in such a system.Let us now consider the general impedance properties of a reversible system, which isan important limiting case.
When charge-transfer kinetics are very facile, IQ —> °°; henceRct —> 0. Thus Rs —» a/col/2. The corresponding impedance plot is shown in Figure 10.3.2a.•8Slope = оFigure 10.3.1frequency.Dependence of R s and l/o)Cs on382Chapter 10. Techniques Based on Concepts of Impedance= G/(0 1 / 236= 45°ф = 45°\X уXIIоFigure 10.3.2 (a) Vector diagramshowing the components of thefaradaic impedance for a reversiblesystem, (b) Phase relationshipbetween the ac current and theac component of potential.г-\Since the resistance and the capacitive reactance are exactly equal, the magnitude of thefaradaic impedance ismwhich is the magnitude of the Warburg impedance alone.Since this is a mass-transfer impedance that applies to any electrode reaction, it is aminimum impedance.
If kinetics are observable, another factor, Rcb contributes, and Zfmust be greater, as Figure 10.3.3a depicts. Thus the amplitude of the sinusoidal currentflowing in response to a given excitation signal, Ё а с , is maximal for a reversible systemand decreases correspondingly for more sluggish kinetics.
If the heterogeneous redoxprocess is very immobile, Rct and Z f are so large that there is only a very small ac component to the current, and the limit of detection sets the lower bound on rate constants thatcan be measured by this approach. See Section 10.4.2 for more details about quantitativeworking ranges.It is interesting also to examine the effect of concentration, which is manifestedthrough a. In general, higher concentrations reduce the mass-transfer impedance, as wewould expect intuitively.
Of greater interest, however, is the effect of the concentrationratio CQ/CR. One can change it experimentally in order to vary the equilibrium potentialfor a series of impedance measurements. Both large and small ratios imply that one of theconcentrations is small; hence a and Zf must be large. The current response to Eac is notvery great, because the supply of one reagent is insufficient to permit a high reaction ratefor the cyclic, reversible electrode process that causes the ac current.
Large rates can beachieved only when both electroreactants are present at comparable concentrations; hencewe expect Z f to be minimal near E°'. Impedance measurements are most easily made inthat potential region, and they gradually become more difficult as one departs from it ei-о/ш 1 ' 2XXXt 'ф < 45°X\z/>\zw\(a)(b)Figure 10.3.3 (a) Vectordiagram showing the effectof Rct on the impedance,(b) Phase relationship between/ and E for a system withsignificant Rci.10.4 Electrochemical Impedance Spectroscopy383ther positively or negatively.
This effect presages the shape of the ac voltammetric response, which will be derived in Section 10.5.A final point of interest is the phase angle between the current phasor, / ac , and the potential, Eac. Since / a c lies along Rs and Eac lies along Zf, the phase angle isreadily calculated asф = tan" 1 —±— = tan" 1 — — — —w/4csRct + a/couz(10.3.9)For the reversible case, Rct — 0; hence ф — тг/4 or 45°. A quasireversible system showsRct > 0; hence ф < тг/4. However, ф must always be greater than zero, unless Rct —> °°;but then the reaction would be so sluggish that little alternating current would flow anyway in a conventional impedance measurement. This sensitivity of ф to kinetics suggeststhat Rct might be extracted from the phase angle.
It can be, and often it is, by ac voltammetric experiments. Before we proceed to a discussion of them, let us note that since0 < ф < 45°, there is always a component of /ac that is in-phase (0°) with £ ac , and it canbe measured with a phase-sensitive detector (e.g., a lock-in amplifier) referenced to £ ac .This feature is extremely useful as a basis for discriminating against charging current inac voltammetry.Even though this section has been developed with the assumption that the electrodereaction is a one-step, one-electron process, many of the conclusions apply generally forchemically reversible multi-electron mechanisms.
The nernstian limit is still described by(10.3.8) and Figure 10.3.2, but with a given byRT(12 л А2I l / 2 *2г24nrП Г A\1 \JJConИV\Hr)l/2 *M? Cr R(10.3.10)When charge-transfer kinetics manifest themselves in a chemically reversible n-electronsystem, they do so in the manner discussed in relation to Figure 10.3.3. The kinetic effectscan be expressed in terms of a charge-transfer resistance Rct defined operationally as in(10.2.8).
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