J. Chem. SOC. Faraday Trans. I 1986,82 1099-1104 Convolution of Voltammograms as a Method of Chemical Analysis Keith B. Oldham Trent University Peterborough Ontario Canada K9J 7B8 Voltammetry is extensively employed for chemical analysis but in all existing methods the analyte concentration is calculated from some feature (usually a peak or plateau) of the voltammogram so that most of the experimental data remain unused. Here a concept is proposed that permits measurement of the bulk analyte concentration from every point on the voltammogram. The principle exploited is to apply a suitable convolution procedure to the Faradaic current thereby generating the concentration excursions of the electroactive reactant and product at the electrode surface. These excursions are then combined with the Nernst equation to measure the analyte concentration.The concept is tested experimentally and is shown to have promise as a means of enhancing analytical precision and rejecting interferences. Voltammetric techniques occupy a secure position in the repertoire of the analytical chemist they constitute the preferred method for the assay of a large number of organic and inorganic species in solution. By a ' voltammetric technique ' is meant one of the many methods1 that share the following characteristics. In voltammetry an electroactive species reaches a stationary electrode by diffusion alone. A programmed potential waveform is imposed on the electrode previously at equilibrium as a result of which a Faradaic current flows.Often the readout of the voltammetric technique is a graph a so-called voltammogram displaying the current as a function of potential. Alternatively the current may be processed in some way (differentiated differenced semi-integrated etc.) prior to display versus potential. The large number of voltammetric techniques arises from the diversity of potential programmes from differences in current processing and from the variety of possible cell geometries. Except in the experimental example that concludes this article no particular voltammetric technique will be assumed here so that the discussion will apply generally to all such techniques. When voltammetry is used as an electroanalytical method it is of course the bulk concentration of the electroactive analyte that is sought.The standard methods of determining this concentration are based on some feature of the voltammogram usually the height of a voltammetric peak or wave. Thus these analytical procedures use the current data only from the immediate vicinity of the feature whereas at least in principle information about the bulk analyte concentration is present in the current values at all potentials. Greater analytical precision is to be expected from a procedure that uses all the current data. This is the philosophy that motivates the present article. Concentration Excursions If 0 represents the analyte then c$ will denote the sought bulk concentration. Let us assume that the voltammetric reaction is a reduction O(so1n) + ne + R(so1n) (1) 1099 Convolution of Voltammograms c& = cg exp [nF(E- P ) / R T ] (2) must apply where E is the instantaneous electrode potential and Ee is the standard electrode potential F R and T being the Faraday constant the gas constant and temperature respectively.Let t = 0 be the instant at which the voltammetric experiment commences; then the concentration of 0 at the electrode surface is c& prior to this instant. At any time during the experiment the difference c&-c& represents the change that has occurred in the concentration of 0 at the electrode surface. This difference plays a salient role in what follows and we shall call it the ‘concentration excursion’ of 0. The corresponding quantity for the reduction product R would generally be ck-cg but in experimental practice the bulk product concentration cg is usually zero and it is convenient to assume that this is always the case.Hence the concentration excursion of R is simply ck. Notice that the signs in the definitions of the concentration excursions have been selected such that each will normally be positive. (3) (4) 1100 to a product species R. The symbols c6 and cg will be used to denote the concentrations of species 0 and R at the surface of the working electrode (more exactly at the outer Helmholtz plane2 adjacent to the working electrode). These surface concentrations will each change in value as the voltammetric experiment proceeds but thermodynamic constraints prevent their varying independently. If reaction (1) is reversible the Nernst relationship and Convolution Procedures In the last two decades a number of procedures have been discovered that provide a direct method of calculating the concentration excursions ck-c& and cg (or cg-cg if the solution is not initially devoid of R) from measurements of the Faradaic current i.These procedures a catalogue of which has been p~blished,~ have the following features in common (a) all are independent of the way in which potential varies with time; (b) all are independent of the rate and mechanism of the electron-transfer process provided that no significant accumulation of intermediates occurs ; (c) knowledge of the electrode area A and of the diffusion coefficient Do or D of the appropriate species is required; ( d ) in order to generate surface excursion data corresponding to the time instant t current values i(z) are needed over the time interval 0 < z < t from the beginning of the experiment or z) up to time s,’ i(z)g(t t ; (e) - all dz involve equivalently integrals of the s form i(t - z) g(z) dz where g is an appropriate function.Mathematically the operation in eqn (3) is known as a ‘convolution’ and therefore the procedures of deriving concentration excursions from current values are termed convolution procedures. The different varieties of convolution procedure arise from differences in electrode geometry and from whether or not the species 0 or R are involved in homogeneous reaction with components of the electrolyte solution.The archetypal convolution procedure is semi-integration; this is the appropriate procedure when the electrode is planar the diffusion field is semi-infinite and there are no homogeneous reactions. The convolution function g(z) is then simply (nz)-f and the expressions c&- co-- s - nAFD$ m - nAF(nD,); 1 [ ( t - z ) f i(z) dz The convolution function g(z) in this case has (m)+ exp (- kz) replacing the (m)+ term that is adequate for semi-integration. Whereas simple semi-integration yields the concentration excursions at the surface of a planar electrode more complicated convolution functions are needed when the electrode is spherical. These have been discussed recently by Myland et aL8 These authors define functions fs and fa that represent the appropriate convolution functions when 0 is the cation of an amalgamable metal and the voltammetric reaction is (6) occurring at a mercury sphere of radius r.They demonstrated that the convolution procedures (7) K. B. Oldham hold m (sometimes denoted I ) being the Faradaic semi-integral. Under the names ' semi-integral electroanalysis ' and ' neopolarography ' this particular convolution procedure as well as related derivative techniques has been used extensively for electr~analysis,~ while under the name 'convolution voltammetry' it has been employed by Saveant and coworkers5 as a tool in mechanistic studies. More recently Bond et a1.6 have used semi-integration innovatively to measure electrode kinetic parameters. Another convolution procedure known as ' kinetic convolution ',7 is appropriate when the product of an electron-transfer process at a planar electrode is involved in a subsequent homogeneous reaction O+ne + R + P.k O(so1n) + ne -+ R(ama1) nAFr and cb,-cf) = - i(t - z) fs(Do z / r 2 ) dz ck = - nAFr 1 1 - I, P t i(t - z) fa(D z / r 2 ) dz which they term ' spherical convolutions ' generate the concentration excursions in these circumstances. Yet other convolution procedures are appropriate in other circumstance^,^ but the above three examples suffice to demonstrate the generality of the concept. [See ref. (3) for information on how in practice one may implement the convolution procedures.] Several applications of convolution procedures may be envisaged not all of which have yet been exploited.Thus these procedures may be applied to studies of electrode reversibility to measure standard electrode potentials to determine the kinetic parameters of electron-transfer processes to elucidate the mechanisms of homogeneous reactions and measure their rate constants to determine diffusion coefficients to estimate the thickness of films on modified electrodes and to perform chemical analysis both by classical feature-reliant methods and the ' featureless' concept that is the subject of this article. Implementation of the New Concept Once the two concentration excursions cb,-c& and cg are determined it is trivially simple for a reversible electrode reaction to combine their values with the Nernst term [see eqn (2)] so as to give the bulk analyte concentration (c8 - c&) + ck exp [nF(E- P ) / R T ] = c;.Such a calculation can be carried out for each instant during the experiment so that an array of bulk analyte concentrations becomes available. Such values should in theory all be equal. In practice errors will contaminate each datum especially those cb values corresponding to the more positive potentials (which are especially prone to errors arising (10) 1101 (8) (9) 1102 I t Table 1. Format of the experimental data and quantities derived therefrom I1 IV I11 I cg-c; I Convolution of Voltammograms V cg -0.50 I 0 t = A t = 2A t = 3A tj = j A -0.40 20 10 I 2 0 .................. .J “ I .-10 - - -20 2 0 from small discrepancies in the E - Ee term). A weighted mean will represent the ‘ best’ bulk concentration. Nowadays electrochemical instrumentation is mostly digital so that current data are usually available as a set of i values at equally spaced time instants. The potential programme provides E values at the same instants so that the results of a voltammetric experiment take the form of the first three columns in table 1. Column IV the concentration excursion of species 0 is computed by an appropriate convolution algorithm [that given in ref. (3) for example] that uses all the preceding current data; i.e. (cB-cS~)~ is based upon i,, i, i, ... as well as ij. Similarly the appropriate convolution procedure is used to generate values for column V of table 1.The final step in the calculation of (~5)~ simply applies eqn (10) to values drawn from the j row of columns 11 IV and V. Constants # Do and D are required in order to perform the convolution procedures as well as information about the electrode geometry. Literature values of the constants may be employed or better they may be obtained from the analysis of similar experiments carried out on solutions of known composition. Fig. 1. Experimental cyclic voltammogram. E i ’ . .. VI 43 -0.41 10 EO El E E Ej -0.50 I 1 EIV us. Ag I AgCl -0.60 I -0.60 I I . . . . I 8 4 6 ti S 1103 10 8 8 2 4 I K. B. Oldham EIV us. Agl AgCl 6 I tl s Fig. 2. Concentration excursions (a) cg - c& (b) cg and (c) the calculated bulk analyte concentration cg.It cannot be emphasized too strongly that any potential-time programme may be employed with this concept. Moreover the potentials encountered during the experiment need not span the entire range of the voltammetric wave a valuable consideration when chemical interferences are present. The theory of the concept is insensitive to whether the potential is changed continuously (linear or non-linear ramps triangles) or discon- tinuously (steps pulses) but the discontinuous current that results from the latter perturbations is more difficult to convolve accurately. An Example Fig. 1 shows the cyclic voltammogram obtained when a triangular potential waveform was imposed on a hanging mercury drop immersed in a solution containing cadmium ions so that (1 1) Cd2+(aq) + 2e -+ Cd(ama1) was the electrode reaction.Other experimental conditions were as described by Myland.8 The convolution procedures indicated by eqn (4) and (5) were applied to the current data shown in fig. 1 and produced the concentration excursions shown as the two lower curves in fig. 2. The shapes of these curves are interesting. Note that during the backward potential scan the concentration excursion c t - c6 actually becomes negative. This reflects the fact that the surface Cd2+ concentration then exceeds its bulk value as a result of the concentrating of Cd that occurred in the mercury drop during the time when its potential was negative. This build-up of Cd within the drop is evidenced by the steady climb of cg apparent in fig.2 during the period of negative polarization. The topmost curve in fig. 2 was calculated via eqn (10) and should be a horizontal line corresponding to the actual analyte concentration of 5.00 mmol dm-3. At all times in the 1.7 < t / s < 6.7 range the measured cb concentration lies within the range 4.9-5.2 mmol dm-3 acceptably close to the true value. The potentials to which these measurements correspond range from -0.486 V to the reversal potential of -0.633 V and back to -0.536 V (ie. 60 -+ -87 -+ 10 mV with respect to P). Outside this range c becomes inconstant much too large and is evidently erroneous. Perhaps the most remarkable aspect of these experimental results is that accurate 1104 Convolution of Voltammograms analyte concentrations are calculable as early as 2.0 s into the experiment.A glance at fig. 1 shows that at this time the cyclic voltammogram had barely departed from the base line! Thus Cd2+ could have been accurately assayed even in the presence of other electroactive species provided these do not reduce at potentials more positive than -0.50 V versus AgIAgCl. Summary Each corresponding to a specific geometric and kinetic circumstance there exist a variety of convolution procedures that may be applied to voltammetric currents. These procedures which are independent of the details of the potential programme and of the electron-transfer process generate the concentration excursions of the electroactive reactant and product.The two concentration excursions may be combined with Nernst’s equation to produce a measurement of the analyte concentration. Such a measurement is possible at each and every point in the voltammogram and not merely at its features. This new concept offers the possibility of increased analytical precision as well as an opportunity to alleviate interference. The generous financial support of the National Science and Engineering Research Council of Canada is acknowledged with gratitude. Portions of this work were described at the 68th Canadian Chemical Congress (Kingston June 1985) and at the 30th Congress of the International Union of Pure and Applied Chemistry (Manchester September 1985). References 1 A. J. Bard and L. R. Faulkner Electrochemical Methods (John Wiley Chichester 1980) chap. 5 and 6. J. Chem. SOC. Dalton Trans. 1985 1213. 8 J. C. Myland K. B. Oldham and C. G. Zoski J. Electroanal. Chem. 1985 193 3. 2 Laboratory Techniques in Electroanalytical Chemistry ed. P. T. Kissinger and W. R. Heineman (Marcel Dekker New York 1984) chap. 1. 3 K. B. Oldham Anal. Chem. submitted. 4 M. Grenness and K. B. Oldham Anal. Chem. 1972 44 1121; F. C. Soong and J. T. Maloy J. Electroanal. Chem. 1983 153 29; M. Goto and D. Ishii J. Electroanal. Chem. 1975 61 361; G. Zhu and E. Wang Acta Chim. Sinica 1982 40 897. 5 J. C. Imbeaux and J. M. Saveant J. Electroanal. Chem. 1973 44 169; L. Nadjo J. M. Saveant and D. Tessier J. Electroanal. Chem. 1974 52 403; J. M. Saveant and D. Tessier J. Electroanal. Chem. 1975 61 251; 1975 65 57. 6 A. M. Bond T. L. E. Henderson and K. B. Oldham J. Electroanal. Chem. 1985 191 75. 7 F. E. Woodward R. D. Goodin and P. J. Kinlen Anal. Chem. 1984 56 1920; A. Blagg S . W. Carr G. R. Cooper I. D. Dobson J. B. Gill D. C. Goodall B. L. Shaw N. Taylor and T. Boddington Paper 511883; Received 10th October 1985