A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 50
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The only difference is that the steady-state wave is displaced at everypoint along the potential axis by (RT/2nF)]n(DR/Do) from the wave based on early transients. Unless the two diffusion coefficients differ markedly, this displacement is not experimentally significant.184Chapter 5. Basic Potential Step Methods(d) Concentration ProfilesBecause the transformed concentration profiles, (5.4.45) and (5.4.46), contain y, which isa function of s, they are not readily inverted to produce a equations covering all timeregimes. However, we can obtain limiting cases for the early transient and steady-stateregimes simply by recognizing limiting forms, just as we did earlier.When ro(s/D)x/2 »1, so that the diffusion layer is thin compared to r 0 , then у —> Щand r/r0 ~ 1 throughout the diffusion layer.
Thus the transformed profiles for the early transient regime are as given in (5.4.14) and (5.4.15), with x recognized as r — r 0 . Inversiongives the concentration profiles in (5.4.23) and (5.4.24).In the steady-state regime, ro(s/D)l/2 «1, and у approaches unity. Thus the transformed profiles becomeCo(r, s) = —- -(5.4.57)rsCK(r, s) =(5.4.58)rswhich can be inverted to give the desired results(5.4.59)0\l— erfc'Г7-\2(DRt)l>(5.4.60)The surface concentrations are then(5.4.61)(5.4.62)Since (5.4.54) identifies 1/(1 + £20) as i/id,(5.4.63)(5.4.64)(e) Steady-State Voltammetry at a Disk UMEThe results in this section have been derived for spherical geometry; thus they applyrigorously only for spherical and hemispherical electrodes. Because disk UMEs are important in practical applications, it is of interest to determine how well the results forspherical systems can be extended to them.
As we noted in Section 5.3, the diffusionproblem at the disk is considerably more complicated, because it is two-dimensional.We will not work through the details here. However, the literature contains solutionsfor steady state at the disk showing that the key equations, (5.4.55), (5.4.56), (5.4.63),and (5.4.64), apply for reversible systems (26, 27). The limiting current is given by(5.3.11).1855.4 Sampled-Current Voltammetry for Reversible Electrode Reactions5.4.3Simplified Current-Concentration RelationshipsOur treatments of sampling in both the early transient regime and the steady-state regimeproduced simple linkages between the surface concentrations and the current.
For theearly transient regime, the relationships (5.4.31) and (5.4.32) can be rearranged and reexpressed by recognizing /d as the Cottrell relation:2/ w=HFADHi/2 m*o ~ c o ( 0 , 01[ C(5.4.65)7T ' t 'W =nFAD^* CR(0, t)m(5.4.66)Since these relations hold at any time along the current decay, for sampled voltammetrywe can replace t by the sampling time т. Likewise, for the steady-state regime at a sphere,equations (5.4.63) and (5.4.64) can be rearranged and reexpressed asnFADo±i = — ^ [ Cg - C o (0, t)](5.4.67)nFADR/ = -~CR(0,0(5.4.68)where the distance variable r has been converted to r — r 0 in the interest of comparabilitywith (5.4.65) and (5.4.66) and related equations elsewhere in the book.For either sampling regime, we arrive with rigor at a set of simple relations of precisely the same form as those assumed in the naive approach to mass transport used inSection 1.4.
When early transients are sampled, one need only replace MQ withDol/2/7rl/2tl/2 and mR with DRl/2/7rl/2tl/2 to translate the relationships exactly. For sampled-current voltammetry under steady-state conditions at a sphere or hemisphere, one instead identifies m0 with Do/r0 and mR with DR/r0. Similarly, mo and mR for steady stateat a disk UME are (4/тг)Оо/г0 and (4/7r)DR/r0, respectively (Table 5.3.1). The two approaches to deriving the i-E curve can be compared as follows:Naive ApproachNernstian behaviorand i = nFAmo[Co - C o (0, t)]i = nFAmR[CR(Q, t) - CR]were assumedSimple>"m a t hi-EcurveRigorous ApproachNernstian behavior,diffusion equations,and boundary conditionswere assumedMore complex,i-E curveas before and/ = nFAmo[ CQ — C o (0, t)]/ = nFAmR[CR(0, t) — CR] alsoas beforeThe rigorous treatment has therefore justified the i-C linkages used before, and it increases confidence in the simpler approach as a means for treating other systems.The essential reason for the general applicability of these equations is that, in reversible systems, the potential controls the surface concentrations directly and maintainsuniformity in these concentrations everywhere on the face of the working electrode.
Thusthe geometry of the diffusion field, either at steady state or as long as semi-infinite lineardiffusion holds, does not depend on potential, and the gradient of that field is simply proportional to the difference between the surface and bulk concentrations.186 • Chapter 5. Basic Potential Step Methods5.4.4Applications of Reversible UE Curves(a) Information from the Wave HeightThe plateau current of a simple reversible wave is controlled by mass transfer and can beused to determine any single system parameter that affects the limiting flux of electroreactant at the electrode surface.
For waves based on either the sampling of early transients orsteady-state currents, the accessible parameters are the «-value of the electrode reaction,the area of the electrode, and the diffusion coefficient and bulk concentration of the electroactive species. Certainly the most common application is to employ wave heights todetermine concentrations, typically either by calibration or standard addition. The analytical application of sampled-current voltammetry is discussed more fully in Sections 7.1.3and 7.3.6.The plateau currents of steady-state voltammograms can also provide the critical dimension of the electrode (e.g., TQ for a sphere or disk).
When a new UME is constructed,its critical dimension is often not known; however, it can be easily determined from a single voltammogram recorded for a solution of a species with a known concentration anddiffusion coefficient, such as Ru(NH 3 )^ + [D = 5.3 X 10~6 cm2/s in 0.09 M phosphatebuffer, pH 7.4 (8)].(b) Information from the Wave ShapeWith respect to the heterogeneous electron-transfer process, reversible (nernstian) systems are always at equilibrium. The kinetics are so facile that the interface is governedsolely by thermodynamic aspects.
Not surprisingly, then, the shapes and positions of reversible waves, which reflect the energy dependence of the electrode reaction, can be exploited to provide thermodynamic properties, such as standard potentials, free energies ofreaction, and various equilibrium constants, just as potentiometric measurements can be.On the other hand, reversible systems can offer no kinetic information, because the kinetics are, in effect, transparent.The wave shape is most easily analyzed in terms of the "wave slope," which is expected to be 2303RT/nF (i.e., 59A/n mV at 25°C) for a reversible system.
Larger slopesare generally found for systems that do not have both nernstian heterogeneous kineticsand overall chemical reversibility [Section 5.5.4(b)]; thus the slope can be used to diagnose reversibility. If the system is known to be reversible, the wave slope can be used alternatively to suggest the value of n. Often one finds the idea that a wave slope near 60mV can be taken as an indicator of both reversibility and an л-value of 1. If the electrodereaction is simple and does not implicate, for example, adsorbed species (Chapter 14), onecan accurately draw both conclusions from the wave slope.
However, electrode reactionsare often subtly complex, and it is safer to determine reversibility by a technique that canview the reaction in both directions, such as cyclic voltammetry (Chapter 6). One can thentest the conclusion against the observed wave slope in sampled-current voltammetry,which can also suggest the value of n.(c) Information from the Wave PositionBecause the half-wave potential for a reversible wave is very close to E°, sampledcurrent voltammetry is readily employed to estimate the formal potentials for chemicalsystems that have not been previously characterized. It is essential to verify reversibility,because Ещ can otherwise be quite some distance from E0' (see Sections 1.5.2 and 5.5and Chapter 12).By definition, a formal potential describes the potential of a couple at equilibrium in asystem where the oxidized and reduced forms are present at unit formal concentration,even though О and R may be distributed over multiple chemical forms (e.g., as both5.4 Sampled-Current Voltammetry for Reversible Electrode Reactions;:187members of a conjugate acid-base pair).
Formal potentials always manifest activity coefficients. Frequently they also reflect chemical effects, such as complexation or participationin acid-base equilibria. Thus the formal potential can shift systematically as the mediumchanges. In sampled-current voltammetry the half-wave potential of a recorded wavewould shift correspondingly. This phenomenon provides a highly profitable route tochemical information and has been exploited elaborately.As a first example, let us consider the kinds of information that are contained in thesampled-current voltammogram for the reversible reduction of a complex ion, such asZn(NH 3 )4 + in an aqueous ammonia buffer at a mercury drop electrode,8Zn(NH 3 )| + + 2e + Hg «± Zn(Hg) + 4NH 3(5.4.69)To treat this problem, we derive the i-E curve using the simplified approach, as justified inthe preceding sections. For generality, the process is represented asMX p + ne + Hg «± M(Hg) + pX(5.4.70)where the charges on the metal, M, and the ligands, X, are omitted for simplicity.
ForM + ne + Hg <± M(Hg),EE+~^ln7П(g)U> 4(5471)and for M + pX *± MX p*c = | T £(5.4.72)The presumption of reversibility implies that both of these processes are simultaneouslyat equilibrium. Substituting (5.4.72) into (5.4.71), we obtain,5.4,3,Let us now add the assumptions (a) that initially C]vi(Hg) = 0, C MX = Cj^x » m&Cx = Cx and (b) that Cx » Смх • For the specific example involving the zinc amminecomplex, the latter condition would be assured by the strength of the buffer, in which ammonia would typically be present at 100 mM to 1 M, very much above the concentrationof the complex, which would normally be at 1 mM or even lower.
Even though reductionliberates ammonia and oxidation consumes it, the electrode process cannot have an appreciable effect on the value of Cx at the surface, and Cx(0, i) ~ C X . Then the following relations apply:i(r) = nFAmc[ C^x p - Смхр(0, 0](5.4.74)i(0 = nFAmACM{Ug)(0,(5.4.75)0i d (0 = nFAmcC^Xv(5.4.76)or,^^(5A77)( 5 A 7 8 )8Zinc deposits in the mercury during the potential steps; thus a question arises about how the initial conditionsare restored after each cycle in a sampled-current voltammetric experiment.
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