A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 62
Текст из файла (страница 62)
No assumption related toelectrode kinetics or technique was made; hence (6.2.8) and (6.2.9) are general. Fromthese equations and the boundary condition for LSV, (6.2.3), we obtain(6.2.10)(6.2.11)where, as before, £ = (DQ/D^)1^2. The solution of this last integral equation would be thefunction i(t), embodying the desired current-time curve, or, since potential is linearly related to time, the current-potential equation. A closed-form solution of (6.2.11) cannot beobtained, and a numerical method must be employed.Before solving (6.2.11) numerically, it is convenient (a) to change from i(t) to i(E),since that is the way in which the data are usually considered, and (b) to put the equationin a dimensionless form so that a single numerical solution will give results that will beuseful under any experimental conditions.
This is accomplished by using the followingsubstitution:(6.2.12)\RTRTLet/(r) = g(crr). With z = or, so that r = z/cr, dr = dz/a, z= 0 at т = 0, and z = (7? atт = t, we obtainгfrat/(T)(?-rr1/2^r=•'O\ -1/2 i/«(z)r-^0\f(6.2.13)/so that (6.2.11) can be writtenratra1/21/2z)(o-f-z)- o-- rfz =1/21/2Q(7TD O)) ,or finally, dividing by CQ(7TDO1+(6.2.14)we obtain-z)l/2/.' (at-z)(6.2.15)230Chapter 6. Potential Sweep Methodswherei(at)g(z)cS(7rD o cr)1/2(6.2.16)nFAC%{TTD0(T)mNote that (6.2.15) is the desired equation in terms of the dimensionless variables x(z)>£, 0, S(at) and at.
Thus at any value of S(crt), which is a function of E, xiat) c a n be obtained by solution of (6.2.15) and, from it, the current can be obtained by rearrangementof (6.2.16):(6.2.17)s aAt any given point, х(°"0 * pure number, so that (6.2.17) gives the functional relationship between the current at any point on the LSV curve and the variables. Specifil/2cally, / is proportional to CQ and v . The solution of (6.2.15) has been carried outnumerically [Nicholson and Shain (3)], by a series solution [Sevcik (2), Reinmuth (4)],analytically in terms of an integral that must be evaluated numerically [Matsuda andAyabe (5), Gokhshtein (6)], and by related methods (7, 8). The general result of solving(6.2.15) is a set of values of xi&t) (see Table 6.2.1 and Figure 6.2.1) as a function of ator n(E - £ 1 / 2 ).
4TABLE 6.2.1 Current Functions for Reversible Charge Transfer (3)a>bn(E - El/2)RT/Fn(E - Em)mV at 25°C7Tl/2x(crt) <t>((Tt)n(E - El/2)RT/Fn(E - El/2)mV at 25°C7T4.673.893.112.341.951.751.561.361.170.970.780.580.390.190.001201008060504540353025201510500.0090.0200.0420.0840.1170.1380.1600.1850.2110.2400.2690.2980.3280.3550.380-0.19-0.39-0.58-0.78-0.97-1.109-1.17-1.36-1.56-1.95-2.34-3.11-3.89-4.67-5.84-5-10-15-20-25-28.50-30-35-40-50-60-80-100-120-1500.4000.4180.4320.4410.4450.44630.4460.4430.4380.4210.3990.3530.3120.2800.245a0.0080.0190.0410.0870.1240.1460.1730.2080.2360.2730.3140.3570.4030.4510.499X((Tt)ф(а00.5480.5960.6410.6850.7250.75160.7630.7960.8260.8750.9120.9570.9800.9910.997To calculate the current:1.i — /(plane) + /(spherical correction).2./ = nFAD^2CtalllTTll2X{(Tt)3.i = 602n3/2ADli2C$vy2{7rl/2x(crt)+ nFADoC%{UrQ)<l>((Tt).+ 0Л60[О^/2/(го«1/2У1/2)]ф(о-0} at 25°C with quantities in the2following units: /, amperes; A, cm ; Do, cm2/s; v, V/s; CQ, M\ r 0 , cm.bEm4l/2= E0' + (RT/nF) In(DR/D0)m.Note that In ^BS{crt) = nf(E - El/2), where Em= E0' + (RT/nF) In (DR/D0)1/26.2 Nemstian (Reversible) Systems < 231>0.4РJs о.з0.20.1Figure 6.2.1 Linear potentialsweep voltammogram in terms ofdimensionless current function.
Valueson the potential axis are for 25°C6.2.2Peak Current and PotentialThe function TT^xicrt), and hence the current, reaches a maximum where 7T^2x(crt) =0.4463. From (6.2.17) the peak current, /p, isn3/2ip =9 ^(6.2.18)KJh= (2.69 X 10 5 )n 3 /1/2(6.2.19)The peak potential, E]P ,is found from Table 6.2.1 to be= E 1 / 2 - 1 . 1 0 9 ^ | = 28.5/л mV at 25'nb(6.2.20)Because the peak is somewhat broad, so that the peak potential may be difficult to determine, it is sometimes convenient to report the potential at /p/2, called the half-peak potential, £рд, which is£ p /2 =+ 1-09 ^= El/2 + 28.0/и mV at 25°C(6.2.21)Note that Ey2 is located just about midway between £ p and Ev/2, and that a convenient diagnostic for a nernstian wave is\EV - Ev/2\ - 2.20 Щ= - 56.5/л mV at 25°CnF(6.2.22)Thus for a reversible wave, E p is independent of scan rate, and /p (as well as the current at any other point on the wave) is proportional to vl/2.
The latter property indicatesdiffusion control and is analogous to the variation of /d with t~l/2 in chronoamperometry.A convenient constant in LSV is ivlvll2C% (sometimes called the current function),which depends on пъ12 and Z)Q/2- This constant can be used to estimate n for an electrode232Chapter 6. Potential Sweep Methodsreaction, if a value of DQ can be estimated, for example, from the LSV of a compound ofsimilar size or structure that undergoes an electrode reaction with a known n value.6.2.3Spherical Electrodes and UMEsFor LSV with a spherical electrode (e.g., a hanging mercury drop), a similar treatment canbe presented (4); the resulting current isi = /(plane) +nFADoCo<l>(ort)(6.2.23)where r 0 is the radius of the electrode and ф(сгГ) is a tabulated function (see Table 6.2.1).For large values of v and with electrodes of conventional size the / (plane) term is muchlarger than the spherical correction term, and the electrode can be considered planar underthese conditions.Basically, the same considerations apply to hemispherical and ultramicroelectrodes atfast scan rates.
However, for a UME, where r 0 is small, the second term will dominate atsufficiently small scan rates. One can show from (6.2.23) that this is true when(6.2.24)v«RTDInFrl5so that the voltammogram will be a steady-state response independent of v. For r 0 = 5 /xm,D = 10" 5 cm2/s, and T = 298 K, the right side of (6.2.24) has a value of 1000 mV/s; thus ascan made at —100 mV/s or slower should permit the accurate recording of steady-state currents.
The limit depends on the square of the radius, so it is generally impractical to recordsteady-state voltammograms with electrodes much larger than those normally considered tobe UMEs. Conversely, with very small UMEs, one requires a high sweep rate to see any behavior other than the steady state. For example, at an electrode of 0.5-/Ш1 radius and with Dand T as given above, steady-state behavior would hold up to 10 V/s.-4.00.30.20.10-0.1Potential (V)-0.2-0.3Figure 6.2.2 Effect of scan rate on cyclic voltammograms for an ultramicroelectrode(hemispherical diffusion) with 10 /Ш1 radius.
Simulations for a nernstian reaction with n = 1,Eo> = 0.0 V, D o = DR = 1 X 10" 5 cm2/s, Cg = 1.0 mM, and T = 25°C. At 1 V/s, the responsebegins to show the peak expected for linear diffusion, but the height of the current at the switchingpotential and the small peak current ratio show that the steady-state component is still dominant.Relationship 6.2.24 involves a comparison of a diffusion length to the radius of the electrode in the mannerdiscussed in Section 5.2.2. The diffusion length is [D0/(nfv)]l/2, which corresponds to the time \/(nfv). This isthe period required for the scan to cover an energy 6T along the potential axis (25.7'In mV at 25°C).
It is oftenregarded as the characteristic time of an LSV or CV experiment (Chapter 12).6.2 Nernstian (Reversible) Systems233The transition from typical peak-shaped voltammograms at fast sweep rates in thelinear diffusion region to steady-state voltammograms at small v is shown for cyclicvoltammetry in Figure 6.2.2. In the steady-state region, the voltammograms areS-shaped and follow the treatment in Sections 1.4.2 and 5.4.2. Ultramicroelectrodes arealmost always employed in the limiting regions: the linear region when v^2/r$ is large andthe steady-state region when v1/2/r0 is small. There is nothing additional to be gained fromworking in the mathematically more complicated intermediate region.6,2.4Effect of Double-Layer Capacitanceand Uncompensated ResistanceFor a potential step experiment at a stationary, constant-area electrode, the charging current dies away after a time equivalent to a few time constants (RuCd) (see Section 1.2.4).Since the potential is continuously changing in a potential sweep experiment, a chargingcurrent, /c, always flows (see equation 1.2.15):|/c| = ACdv(6.2.25)and the faradaic current must always be measured from a baseline of charging current(Figure 6.2.3).
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
