A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 57
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If tT and t{ are the times at which the current measurements aremade, then for the purely diffusion-limited case described by (5.7.15),K10First edition, pp. 178-180.tr-r210Chapter5. Basic Potential Step MethodsFigure 5.7.3 Currentresponse in double-stepchronoamperometry.1.0Figure 5.7.4 Working curve for-Шг)Н№)0.01.02.03.0for tT = t{ + т. Thesystem is О + ne <=^ R, with bothО and R being stable on the timescale of observation. Responses inboth phases are diffusion-limited.If tY and tf values are selected in pairs so that tT — т = t$ always, then(5.7.17)When one calculates these ratios for several different values of tT, they ought to fallon the working curve shown in Figure 5.7.4.
A convenient quick reference for a stablesystem is that —ir(2r)/if(r) = 0.293. Deviations from the working curve indicate kineticcomplications in the electrode reaction. For example, if species R decays to an electroinactive species, then |/r| is smaller than predicted by (5.7.15) and the current ratio -/ r // f deviates negatively from that given in Figure 5.7.4. Chapter 12 covers in more detail theways in which these experiments can be used to diagnose and quantify complex electrodeprocesses.5.8 CHRONOCOULOMETRYTo this point, this chapter has concerned either current-time transients stimulated by potential steps or voltammograms constructed by sampling those curves.
An alternative, andvery useful, mode for recording the electrochemical response is to integrate the current, sothat one obtains the charge passed as a function of time, Q{t). This chronocoulometricmode was popularized by Anson (41) and co-workers and is widely employed in place ofchronoamperometry because it offers important experimental advantages: (a) The measured signal often grows with time; hence the later parts of the transient, which are mostaccessible experimentally and are least distorted by nonideal potential rise, offer better sig-5.8 Chronocoulometry211nal-to-noise ratios than the early time results.
The opposite is true for chronoamperometry.(b) The act of integration smooths random noise on the current transients; hence thechronocoulometric records are inherently cleaner, (c) Contributions to Q(t) from doublelayer charging and from electrode reactions of adsorbed species can be distinguished fromthose due to diffusing electroreactants. An analogous separation of the components of acurrent transient is not generally feasible. This latter advantage of chronocoulometry is especially valuable for the study of surface processes.Large-Amplitude Potential StepThe simplest chronocoulometric experiment is the Cottrell case discussed in Section5.2.1. One begins with a quiescent, homogeneous solution of species O, in which a planarworking electrode is held at some potential, E^ where insignificant electrolysis takesplace.
At t = 0, the potential is shifted to Ef, which is sufficiently negative to enforce adiffusion-limited current. The Cottrell equation, (5.2.11), describes the chronoamperometric response, and its integral from t = 0 gives the cumulative charge passed in reducing the diffusing reactant:(5.8.1)1/27ГAs shown in Figure 5.8.1, Q^ rises with time, and a plot of its value vs. tl/2 is linear. Theslope of this plot is useful for evaluating any one of the variables n, A, Do, or CQ, givenknowledge of the others.Equation 5.8.1 shows that the diffusional component to the charge is zero at t = 0,yet a plot of the total charge Q vs. t^2 generally does not pass through the origin, becauseadditional components of Q arise from double-layer charging and from the electroreduction of any О molecules that might be adsorbed at E-v The charges devoted to theseprocesses are passed very quickly compared to the slow accumulation of the diffusionalcomponent; hence they may be included by adding two time-independent terms:Q =0.10.22nFADxgC%txn1/2ТГ0.3Qdl + nFATo0.4(5.8.2)0.5Figure 5.8.1 Linear plot of chronocoulometric response at a planar platinum disk.
System is 0.95mM 1,4-dicyanobenzene (DCB) in benzonitrile containing 0.1 M tetra-w-butylammoniumfluoborate. Initial potential: 0.0 V vs. Pt QRE. Step potential: -1.892 V vs. Pt QRE. T = 25°C,A = 0.018 cm2. Eo> for DCB + e ?± DCB7 is -1.63 V vs. QRE. The actual chronocoulometrictrace is the part of Figure 5.8.2 corresponding to t < 250 ms. [Data courtesy of R. S. Glass.]212Chapter 5. Basic Potential Step Methodswhere Qdh is the capacitive charge and nFAT0 quantifies the faradaic component given tothe reduction of the surface excess, Г о (mol/cm2), of adsorbed O.The intercept of Q vs.
t112 is therefore Q d l + nFAT0. A common application ofchronocoulometry is to evaluate surface excesses of electroactive species; hence it is ofinterest to separate these two interfacial components. However, doing so reliably usuallyrequires other experiments, such as those described in the next section.
An approximatevalue of HFATQ can be had by comparing the intercept of the Q-t112 plot obtained for a solution containing O, with the "instantaneous" charge passed in the same experiment performed with supporting electrolyte only. The latter quantity is Qd\ for the backgroundsolution, and it may approximate Qdi for the complete system. Note, however, that thesetwo capacitive components will not be identical if О is adsorbed, because adsorption influences the interfacial capacitance (see Chapter 13).5.8,2Reversal Experiments Under Diffusion ControlChronocoulometric reversal experiments are nearly always designed with step magnitudesthat are large enough to ensure that any electroreactant diffuses to the electrode at its maximum rate.
A typical experiment begins exactly like the one described just above. At t = 0,the potential is shifted from E{ to Ef, where О is reduced under diffusion-limited conditions. That potential is enforced for a fixed period r then the electrode is returned to Ebwhere R is reconverted to O, again at the limiting rate. This sequence is a special case ofthe general reversal experiment considered in Section 5.7, and we have already found thechronoamperometric response for t > т in (5.7.15), which isBefore r, the experiment is clearly the same as that treated just above; hence the cumulative charge devoted to the diffusional component after т isQi(t > т) =^+itdt(5.8.4){t1/2т)or2«™^cS[CdOr)—^—Y1J(5 8 5)--This function declines with increasing t, because the second step actually withdrawscharge injected in the forward step.
The overall experimental record would resemble thecurve of Figure 5.8.2, and one could expect a linear plot of Q(t > r) vs. [t112 - (t - r ) 1 / 2 ] .Note that there is no net capacitive component in the total charge after time r, because thenet potential change is zero. Although Qd\ was injected with the rise of the forward step, itwas withdrawn upon reversal.Now consider the quantity of charge removed in the reversal, QT(t > r) which experimentally is the difference Q(r) - Q(t > r), as depicted in Figure 5.8.2.Л(5.8.6)where the bracketed factor is frequently denoted as в.
For simplicity, we consider the casein which R is not adsorbed. A plot of Qv vs. в should be linear and possess the same slopemagnitude seen in the other chronocoulometric plots. Its intercept is Qd\.5.8 Chronocoulometry100200t, msec300400500213Figure 5.8.2Chronocoulometricresponse for a double-stepexperiment performed onthe system of Figure 5.8.1.The reversal step wasmade to 0.0 V vs.
QRE.[Data courtesy of R. S.Glass.]The pair of graphs depicting Q(t < r) vs. t112 and QT(t > r) vs. в in the manner of Figure 5.8.3 (often called an Anson plot) is extremely useful for quantifying electrode reactions of adsorbed species. In the case we have considered, where О is adsorbed and R isnot, the difference between the intercepts is simply nFATo. This difference cancels Q d land leaves only a net faradaic charge devoted to adsorbate, which in general is nFA(To —FR). For details of interpretation concerning the various possible situations, the originalliterature should be consulted (41-43).
(See also Section 14.3.6.)Note that (5.8.3), (5.8.5), and (5.8.6) are all based on the assumption that the concentration profiles at the start of the second step are exactly those that would be produced byan uncomplicated Cottrell experiment. In other words, we have regarded those profiles asbeing unperturbed by the additions or subtractions of diffusing material that are impliedby adsorption and desorption.
This assumption obviously cannot hold strictly. Christie etal. avoided it in their rigorous treatments, and they showed how conventional chronocoulometric data can be corrected for such effects (42).Q(t < T)0.20.30.4VS.0.5Figure 5.83 Linearchronocoulometric plots for datafrom the trace shown in Figure5.8.2. For Q(t<T)vs.tm,the slope is 9.89 /JLC/S112 andthe intercept is 0.79 JJLC. ForQr(t > T) VS. 0, the slope is9.45 /JLC/S 1 / 2 and the intercept is0.66 JJLC. [Data courtesy of R. S.Glass.]214Chapter 5.
Basic Potential Step MethodsReversal chronocoulometry is also useful for characterizing the homogeneous chemistry of О and R. The diffusive faradaic component Qd(t) is especially sensitive to solution-phase reactions (44,43), and it can be conveniently separated from the overall chargeQ(t) as described above.If both О and R are stable, and are not adsorbed, then Qd(t) is fully described by (5.8.1)and (5.8.5). Let us consider the result of dividing Qd(t) by the Cottrell charge passed in theforward step, that is, Qd(r).
This charge ratio takes a particularly simple form:(5.8.7)(5.8.8)which is independent of the specific experimental parameters n, CQ, DQ, and A. For agiven value of t/т, the charge ratio is even independent of r. Equations 5.8.7 and 5.8.8clearly describe the essential shape of the chronocoulometric response for a stable system.If the experimental results for any real system do not adhere to this shape function, thenchemical complications are indicated. For a quick examination of chemical stability, onecan conveniently evaluate the charge ratio Qd{2r)IQd{r) or, alternatively, the ratio[Qd(T) ~ Qd(2T)]/Qd(r). Equation 5.8.8 shows that these ratios for a stable system are0.414 and 0.586, respectively.In contrast, consider the nernstian O/R couple in which R rapidly decays in solutionto electroinactive X.
In the forward step О is reduced at the diffusion-controlled rate and(5.8.7) is obeyed. However, (5.8.8) is not followed, because species R cannot be fully reoxidized. The ratio Qd(t > r)IQd{r) falls less rapidly than for a stable system, and in thelimit of completely effective conversion of R to X, no reoxidation is seen at all. ThenQdit > r)/Qd(r) = 1 for all t > r.Various other kinds of departure from (5.8.7) and (5.8.8) can be observed. See Chapter 12 for a discussion concerning the diagnosis of prominent homogeneous reactionmechanisms.
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