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Anal. Chern. 1985, 57,1503-1504
condition and the absorbance of the bis(Z,Z'-bipyridine)copper(1) species at 440 nm (t = 4800 M-l cm-l) (7)was monitored with time. The decrease in absorbance at 440 nm was less than 1%after 1 h.
ACKNOWLEDGMENT The authors express their appreciation to Ernest Yeager for fruitful discussions and suggestions, to Keith Schultz of CWRU for his advice and skillful machining during the course of the design and construction, and to Michael R. McDevitt and Suzanne Feke for their assistance in testing the cell.
LITERATURE CITED (1) Rhodes, R.; Kadish, K. M. Anal. Chem. 1981, 53, 1539. (2) Heineman, W. R.; Anderson, C. W.; Halsail, H. 6.;Hurst, M. M.; John-
son, J. M.; Kreishman, G. P.; Norris, B. J.; Simone, M. J.; Su, C.-H. I n "Electrochemical and Spectrochemical Studies of Biological Redox Components"; Kadish, K. M., Ed.; American Chemical Society: Washington, DC, 1982; ACS Advances in Chemistry Series No. 201, Chapter 1. (3) DeAngelis, T. P.; Heineman, W. R. J . Chem. Educ. 1976, 53, 594-597. (4) Ives, D. J. G.; Janz, G. J. "Reference Electrodes"; Academic Press: New York, 1961. (5) Haupt, G. W. J . Res. Natl. Bur. Stand. (US'.) 1952, 48, 414. (6) Koithoff, I. M.; Tomiscek, W. J. J . Phys. Chem. 1935, 39, 945-954. (7) Kitagawa, S.;Munakata, M. Inorg. Chem. 1981, 20,2261-2267.
RECEIVED for review September 20,1984.Accepted January 22,1985.Diamond Shamrock Corp. provided partial support for this work with a Vittorio de Nora Diamond Shamrock Postdoctoral Fellowship to D.S.
Simple Relationship for Calculating Backward to Forward Peak-Current Ratios in Cyclic Voltammetry Gin0 Bontempelli" and Franco Magno
Istituto di Chimica Analitica, Universitci di Padoua, via Marzolo 1, 35100 Padoua, Italy Salvatore Daniele
Dipartimento di Spettroscopia, Elettrochimica e Chimica Fisica, Universitd di Venezia, Dorsoduro 2137, 30123 Venezia, Italy The peak-current ratio (ipb/ipf; b, backward; f, forward) is probably the most significant quantity provided by cyclic voltammetry in that it is in general strongly affected by the nature of the process involved as well as by the values of the parameters characterizing this process. However, its measurement is made difficult by the need to individuate the appropriate base line for ipb(1). To overcome this difficulty, semiempirical relationships have been proposed for the significant but particular cases of reversible electrode processes followed by both first-order (2) and second-order (3) irreversible chemical reactions. In all other cases, the evaluation of the peak current relative to the backward process requires the troublesome estimation of the contribution given at Epb by the forward electrode process. This contribution can be experimentally determined by extending the forward scan beyond the switching potential ( I ) , but the use of this method is precluded when further electrode processes are able to interfere. Otherwise, some mathematical procedures have been proposed for extrapolating the descending branch of the forward wave to the back peak potential, all based on the consideration that the current becomes purely diffusion controlled a t a potential sufficiently past the peak. In particular, a semiempirical method has been presented by Polcyn and Shain ( 4 ) as a part of the multistep charge transfer analysis. I t is however restricted to totally reversible or irreversible electrode processes and, in addition, in the latter case its application requires the knowledge of the charge transfer coefficient a. Subsequently, rather different results have been obtained (5)for these two types of processes as well as for reversible charge transfers followed by irreversible chemical reactions, but the relevant equations have been criticized later (6). Furthermore, the increasing availability of digital circuits allows the extension of the forward voltammetric response to be performed through on-line data treatment by computers (7, 8). The need to study a rather complicated two-step electrode process for which the change of the peak-current ratio with
the scan rate appeared to be, within itself, a diagnostic criterion (9),prompted us to draw a simple and profitable operative equation which makes possible, for any type of electrochemical process involved, the easy and rapid calculation of the ratio ipb/ipffrom a single cyclic voltammogram, without the explicit extrapolation of the forward wave into the diffusion controlled region. The purpose of this paper is to report the procedure adopted to derive such equation. It is based on the same approach used also in ref 4 which takes advantage of the fact that the current exhibits typical diffusion controlled decay independent of potential, beginning from a definite potential sufficiently past the peak. This statement implies that when any further potential shift does not change the electrolysis conditions, the current-voltage curve obeys effectively the simple Cottrell equation
i ( t - t?'Iz = const
(1)
where t ' is the hypothetical origin for the current-time curve (in the linear sweep experiment scale) which matches the diffusion part of the stationary electrode voltammogram. In particular, with reference to Figure 1, this equation will be obeyed by the current iA,recorded at the switching potential E,, as well as by the current i, relative to the potential E , chosen in such a way that
E,
- EA =
EA - Epb
(2)
This last current represents the actual contribution of the forward process at E p b and hence it must be added to (ipb)' for a correct computation of the peak-current ratio ipb/ipf
= [(ipb)'
+ ixl/ipf
(3)
Thus, we can write
i x ( t x- t')l12= i,(t, - tq1Iz
(4)
Since experimental curves are recorded in current-voltage coordinates, it is more convenient to replace times with PO-
0003-2700/85/0357-1503$01.50/0@ 1985 American Chemical Society
1504
ANALYTICAL CHEMISTRY, VOL. 57, NO. 7, JUNE 1985
I
Figure 1. Base line evaluation for the reverse scan in a cyclic voltammetrlc experiment.
tentials by using the well-known relationship (1)linking these two quantities in linear sweep voltammetry E = Ei - U t (5) where E, is the initial potential and u is the scan rate. (This relationship holds only for cathodic forward scans: the change of the sign required by anodic forward scans does not modify however the final equation.) By substituting in such a way t x , t,, and t’, eq 4 becomes
i,,(E’-
= ix(E’- EX)li2
(6)
which can be rearranged, after combination with eq 2 for eliminating E,, so that i, is obtained as
i, = iA(
E’- E , E‘ Epb- 2Eh
(7)
+
Its insertion in eq 3 affords the following relationship for the computation of the peak-current ratio
It includes the term E ’which is not known a priori. This parameter can be however evaluated by employing two potential values E, and Ez taken in such a way that the corresponding currents il and i2 obey in their turn eq 1
il(tl - t’)lI2 = i2(t2- t?’/’
(9)
(note that E, or E, may coincide with EA). By adopting once again eq 5 for replacing times with potentials and rearranging, E’ can be in fact obtained in the following simple form:
E’ =
i12E,- iz2Ez i12- i2’
= E2
+
ii2(E, - E,) i12
- .2
(10)
which represents an effective and rigorous tool for the calculation of the ratio ipb/ipfby quantities easily drawn from a single experimental voltammogram, provided that the potentiostatic condition (1) applies to the currents relative to the potential values El and EA. Therefore, it is of major importance to individuate where this condition is satisfied, that is, where the voltammogram is coincident with the chronoamperometric curve (see Figure 1). In order to locate the “convergence point”, several voltammograms concerning both uncomplicated and complicated electrochemical reactions were considered. With this purpose, the voltammetric curves relative to different mechanisms already available in the literature (including EE, EC, ECE, proportionation, and disproportionation processes, all characterized by different reversibility degrees for the charge transfer steps) were simulated (10). The relevant programs, available on request, were written in FORTRAN IV and the calculations were carried out on a MINC-11/23 minicomputer (Digital Equipment Corp.). Each theoretical voltammogram was then compared with the appropriate current-time curve whose origin E’was calculated by eq 10. With this aim, values of E , and E2well into the region of pure diffusion control (at least 1.0 V from Epf)were inserted. This comparison points out that this coincidence is accurate within 1%from a potential of at least 300 mV past the forward peak in the worst cases (i.e., when an irreversible electrode step is involved or when a two-step charge transfer characterized by a sufficiently small &E”, so that the two waves merge and become distorted, is operative). This distance between the “convergence point” and E p flowers to 230 mV if an accuracy within 2% is agreed and, moreover, it decreases rapidly to the inferior limit of 120 mV (for an accuracy within 1%) as a reversible process is approached. Consequently, these limits and these accuracies must be A which borne in mind in the choice of the potentials E, and E make possible the calculation of the peak-current ratio by the use of eq 11. Note that although the “convergence points” are referred to Epf,the potentials which must be inserted in the operative relationship (11)may be quoted with respect to the desired reference potential (Epf,Elj2,independent reference, etc.) provided that they are all consistent.
LITERATURE CITED ( 1 ) Nicholson, R. S.; Shah, I . Anal. Chem. 1964, 3 6 , 706-723. (2) Nicholson, R. S Anal. Cbem 1966, 3 8 , 1406 (3) Olmstead, M. L.;Hamilton, R G ; Nicholson, R. S Anal. Cbem 1969, A. I. , 7fin-3fi7 - .. Polcyn, D. S.; Shah, I . Anal. Chem. 1966, 3 8 , 370-375. Ginzburg, G. Anal. Chem. 1978, 5 0 , 375-376, Eggins, B. R.; Smith, N. H. Anal. Chem. 1979, 5 1 , 2282-2283. Perone, S. P.; Frazer, J. W. Anal. Chem. 1971, 43, 1485-1490. Whitson, P. E.; Van den Born. H W.: Evans. D. H Anal. Chem. 1973, 45. 1298-1306 G.; Daniele, S.;Fiorani, M.Ann. Cbim. (Rome) 1985, 75, (9) Bontempelli, 19-31.
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(10) Magno, F.; Bontempelii, G.; AndreuzzCSedea, M. Anal. Chim. Acta 1982, 140, 65-76.
12
The insertion of this equation, written by choosing E 2 = E A ,in relationship 8 allows one to obtain
RECEIVED for review January 28, 1985. Accepted March 11, 1985.
