A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 66
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However, with the convolutive technique, this assumption is not essential, forthe rate law can be written in the general form (27),i(0 = FAkf(E)[Co(P91) - 0CR(O, 01(6.7.18)where kf(E) is the potential-dependent rate constant of the forward reaction and в is alsorecognized as kb/k{. With (6.7.6) and (6.7.7),р'-'у1}(6.7.19)Analysis of experimental linear potential sweep experiments according to (6.7.19) or theequivalent expression for a totally irreversible one-step, one-electron reduction (£6 « 1),In kf (E) = In Dxg - In [//j(°](6.7.20)yields In kf(E) as a function of £ at different v.
If classic Butler-Volmer kinetics apply, then aplot of In kf(E) vs. E should be linear with a slope —aF/RT. In an analysis of experimentaldata for the electroreduction of te^nitrobutane in aprotic solvents, Saveant and Tessier (27)noted significant deviations from linearity (after necessary corrections for double-layer effects were carried out), demonstrating that the a value was potential dependent, as is indeedpredicted by the microscopic theories of electron-transfer reactions (Section 3.6).6.7 Convolutive or Semi-Integral Techniques251ApplicationsThe convolution technique offers a number of advantages in the treatment of linear sweepdata (and perhaps also in other electrochemical techniques).
For a reversible reaction in acyclic voltammetric experiment, the curves of I(f) vs. E for the forward and backwardscans superimpose, with I(t) returning to zero at sufficiently positive potentials [whereCR(0, t) = 0]. This behavior has been verified experimentally (20, 25, 28) (Figure 6.7.3a).For a quasireversible reaction, however, the forward and backward I(t) curves do notcoincide (Figure 6.13b). One can regard this effect as a consequence of the shifting of theEpc and £ p a values from their reversible values. The procedure used to obtain the "reversible" Ец2 value for such a system is shown in Figure 6.1.3b (27).-1.0-1.1-1.2-1.3E vs. SCE, volts-1.4(a)5хЮ"5- БхЮ"-I-0.9II-1.3ENI6I-1.7(b)Figure 6.7.3 Experimental cyclic voltammogram and convolution of (a) 1.84 mMp-nitrotoluene in acetonitrile containing 0.2 M TEAP at HMDE, v = 50 V/s.
[Reprinted withpermission from P. E. Whitson, H. W. Vanden Born, and D. H. Evans, Anal. С hem., 45, 1298(1973). Copyright 1975, American Chemical Society.] (b) 0.5 mM tert-nitrobutane in DMFcontaining 0.1 M TBAI, v = 17.9 V/s. Ещ determined for quasireversible system from £1/2 =E{i = 0) - (RT/F) In [(// - // =o)/// = 0 ] [From J.-M. Saveant and D. Tessier, / . Electroanal. Chem.,65, 57 (1975), with permission.]252Chapter 6. Potential Sweep MethodsCorrection for the uncompensated resistance, Ru, is also much more straightforwardfor I{t)-E curves than for the i(t)-E curve, since Ru affects the linearity of the potentialsweep at the electrode (see Section 6.2.4).
For the I(t)-E curves, correction is accomplished simply by replacing the applied potential E by E' = E + iRu (20, 25, 28).A procedure for correcting the /(0 curves for charging current has also been presented (10, 28). The charging current for a blank experiment in the absence of electroactive compound, ib, is(6 7 21)'"5—sr---where Q is the potential-dependent capacitance (fiF) and E' is the potential corrected for Ru:E' =E + ibcRu = E{-vt+ihcRu(6.7.22)Thus, at any given value of E\Cd =v -l^—rRu(dibc/dt)(6.7.23)If Ru is known, the measured values of /£, and dibjdt allow the determination of Q as afunction of £".
When substance О is introduced into solution, the total current, /t, is(6-7.24)it = if + kwhere /c may differ from i\ because of the presence of O. However, it is still true thatE' = E - vt + itRu(6.7.26)and finally(§)(6-7.27)where Сд is assumed to be the same function of potential as determined in the blank experiment. Correction of it for Q can be accomplished by (6.7.27) to give /f before theconvolution is performed.Convolution methods simplify to some extent the treatment of data for electrodeprocesses with coupled chemical reactions (20) and may be useful in analytical applications (22).Although data acquisition is carried out digitally in modern electroanalytical instrumentation, and convolution can be readily applied in computer-based instruments, thisapproach has not been widely used. Most practitioners appear to prefer comparing conventional voltammograms to those obtained via digital simulation.6.8 CYCLIC VOLTAMMETRY OF THELIQUID-LIQUID INTERFACEIn Section 2.3.6 we considered ion transfer at the interface between two immiscible electrolyte solutions (ITIES), where we found that a potential difference can arise because ofdifferential transfer of ions.
Ion movement across the interface can also be driven by the application of an external potential, and the rate of ion transfer can be detected as a currentflow. This response allows one to examine the ITIES via voltammetric methods in the sameway that electron transfer can be monitored at electrode surfaces (29-32).6.8 Cyclic Voltammetry of the Liquid-Liquid Interface -4 253As shown in (2.3.47), the junction potential between phases a and /3 is given byфР - фа = Д0ф = ( - l / z i F ) [ A G ^ 4 i+RT\n(aP/a?)](6.8.1)If one defines the standard free energy required to transfer species /, with charge zi? between the two phases, A G ^ T ^ , in terms of a standard potential difference, Д^ф? (thestandard Galvani potential of ion transfer of species / from a to /3) that is,(6.8.2)then (6.8.1) can be written in the form of a Nernst equation:g0 = bfcfi + (RT/z{F) In (af/a?)(6.8.3)As shown in Section 2.3.6, AG^^s^i, a n d hence Д^ф°, can be obtained from thermochemical, solubility, or potentiometric measurements (with an extrathermodynamic assumption).Thus by sweeping the potential drop across the ITIES, the activities of species in the twophases can be varied, with a resulting flow of current as ions cross the interface.Consider the system shown in Figure 6.8.1, which can be represented asRef j8/(nitrobenzene) TBuATPB/o-/(H2O) LiCl/Ref a(6.8.4)+where TBuA is tetra-n-butylammonium, TPB is tetraphenylborate, Ref a and Ref P are reference electrodes, and a represents the interface.
The values of the ionic transfer free energiesand the standard potential differences for the species in this system are given in Table 6.8.1(29). The distributions of the four ions in this system, calculated from the data in this table and(6.8.3), are shown in Figure 6.8.2a. Note that the interface in this case is unpoised, since nocommon ion exists in both phases at open circuit. The salt LiCl is very hydrophilic and, withno potential applied across the interface, resides almost totally in the aqueous phase, whileTBuATPB is hydrophobic and remains in the nitrobenzene. As shown in Figure 6.8.2, this remains true for —250 mV < A^</>° < 150 mV.
The situation is equivalent to having a platinum electrode immersed in a solution that does not contain both halves of a redox couple.If the potential across the interface is varied by applying a voltage between the working electrodes as shown in Figure 6.8.1, ions will tend to move across the interface. Apositive potential applied to the electrode in the aqueous phase (a negative А^ф) will tendto transport Li + into the nitrobenzene and TPB~ into the aqueous phase.
A negative potential in the aqueous phase (producing a more positive Д^ф) will drive C\~ into the+nitrobenzene or TBuA into the aqueous phase. At any potential across the interface (in-WkocRef af0.01 M Li + СГ0.01MTBuA + TPB"NitrobenzeneWk(3Refp0Figure 6.8.1 Schematic diagram for theapparatus for cyclic voltammetry at the ITIESbetween water and nitrobenzene.
Ref a and Ref /3are reference electrodes, Wk a and Wk /3 aremetal working electrodes.254Chapter 6. Potential Sweep MethodsTABLE 6.8.1 Gibbs Energy of Transfer and StandardInterfacial Potential Differences for Ion Transfer BetweenaWater and Nitrobenzene (29)SpeciesAG? r ^iLiСГ+TBuATPB"+TMA38.243.9-24.7-35.94.0-396455256-372-41aNote that a positive Gibbs energy implies that work must beperformed to move the species from the aqueous phase to thenitrobenzene.
For cations, a negative А^Ф? goes with a greatertendency to remain in the aqueous phase, while for anions a positiveA ^ f shows greater tendency to remain in the aqueous phase.dicated by that measured between the two reference electrodes), the relative activities ofthe ions at the interface can be calculated with (6.8.3), assuming equilibrium is attained(see Problem 6.11). Within the potential range of about —0.2 to 0.15 V no appreciabletransfer occurs in this system, because the ratios (af/af) for the different ions are suchthat negligible amounts of Li + and Cl~ are in the nitrobenzene and negligible amounts ofTBuA + and TPB~ are in the water. When the potential is swept beyond these limits, iontransfer can occur, with TBuA + moving into the aqueous phase and Cl~ moving into thenitrobenzene at the positive extreme of А%ф, and TPB~ moving into the water and Li +moving into the nitrobenzene at the negative extreme.
Since these ionic movements represent passage of a net charge across the interface, they result in a current flow in the external circuit. A current-potential curve for this system is shown in Figure 6.8.ЗА. The limitsshown are controlled by the commencement of movement of TBuA + into the water at aА%ф of about 100 mV and the movement of TBP~ into the water at Д^ф of about -200mV.
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