(7) H. W. Lakin and D. F. Davidson, U.S., Geol. Surv. Prof. Pap., 820, 573 (1973). ( 8 ) Y. Hashimoto and J. W. Winchester, Environ. Sci. Technol., 1, 338 (1967). (9)E. Sawicki, Anal. Chem., 29, 1377 (1957). (10)J. H. Watkinson, Anal. Chem., 32, 981 (1980). (11)P. F. Lott, P. Cukor, G. Moriber, and J. Solger, Anal. Chem., 35, 1159 (1963). (12)J. W. Young and G. D.Christian, Anal. Chim. Acta, 65, 127 (1973). (13)E. Orvini, T, E. Gills, and P. D. LaFleur, Anal. Chem., 46, 1294 (1974). (14) D. C. Manning, At. Absorp. Newsl., 10, 123 (1971). (15)E. N. Pollock and S. J. West, At. Absorp. Newsl., 12, 6 (1973). (16)K. C. Thompson and D. R. Thomerson, Analyst (London), 9g, 595 (1974). (17)A. E. Smith, Analyst(London), 100, 300 (1975). (18)J. J. Lingane and L. W. Niedrach, J. Am. Chem. Soc., 70, 41 15 (1948). (19) S.Araki, S. Suzuki, and T. Kuwabara, Jpn Anal., 22, 700 (1973). (20)F. Vadja, Acta Chim. Acad. Sci. Hung., 63, 257 (1970). (21)R. W. Andrews and D. C. Johnson, Anal. Chem., 47, 294 (1975). (22)G. Colovos, G. S. Wilson, and J. L. Moyers, Anal. Chem., 46, 1051 (1974). (23)G. Colovos, G. S. Wilson, and J. L. Moyers, Anal. Chem., 46, 1045 (1974). (24) A. D. Shenrikar and P. W. West, Anal. Chim. Acta, 74, 189 (1975). (25)J. H. Christie, J. Osteryoung, and R. A. Osteryoung, Anal. Chem., 45,210 (1973).
(26)J. A. Gardiner and J. W. Collat, J. Am. Chem. Soc.,87, 1692 (1965). (27)L. E. Ranweiler and J. L. Moyers, Environ. Sci. Technol., 8, 152 (1974). (28)F. J. Fernandez, At. Absorp. Newsl., 12, 93 (1973). (29)F. J. Schmidt and J. L. Royer, Anal. Lett., 6, 17 (1973). (30)L. E. Ranweiler, unpublished results, 1975. (31)R. H. Wood, J. Am. Chem. Soc., 80, 1559 (1958). (32)E. P. Parry and R. A. Osteryoung, Anal. Chem., 37, 1634 (1965). (33)J. B. Flato, Anal. Chem., 44 (ll),75A (1972). (34)L. I. Kleshchinskii, A. A. Makeev, and P. V. Sharavskii, Flz. Tekh. Poluprov., 2, 1002 (1968);Chem. Abstr., 69, 111013a (1968). (35) 0.G. Mikolaichuk, Y. I. Dutchak, and D. M. Freik, Ukr. Fiz. Zh. (Ukr. Ed.), 13, 154 (1968);Chem. Abstr., 69, 1108751 (1968).
RECEIVEDfor review April 26,1976. Accepted June 7,1976. This work was supported in part by grants from the Arizona Mining Association and the Electric Power Research Institute (RP-438-1).Presented in part at the 169th National Meeting, American Chemical Society, Philadelphia, Pa., April 1975, Symposium on “Recent Advances in Analytical Voltammetry”.
General Equation for Voltammetry with Step-Functional Potential Changes Applied to Differential Pulse Voltammetry Steven C. Rifkin and Dennis H. Evans* Department of Chemistry, University of Wisconsin, Madison, Wis. 53706
A general equation is presented which gives the current for a reversible electrode reaction as a function of time throughout a series of any number of step-functional potential changes of arbitrary direction, magnitude, and duration. The general expression was used to predict the dependence of the differential pulse voltammetric peak current on the time between pulses and the pulse duration. The oxidation of 9,lO-diphenylanthracene in acetonitrile was found to agree with the predicted response.
In recent years, several electroanalytical techniques have been described in which a series of step-functional potential changes is applied to a stationary working electrode. These include staircase voltammetry (1-3), square wave voltammetry ( 4 , 5 )and differential pulse voltammetry (6). The potential-time waveforms are shown in Figure l. Similar waveforms are used for pulse polarography with the dropping mercury electrode, but only one pulse is applied during each drop life. By contrast, when a stationary electrode is employed, the entire sequence of potential changes is applied to one and the same electrode surface. Consequently, the current after any given potential change is affected in greater or lesser degree by all preceding steps, and theory for these techniques must explicitly take into account the effects of all potential steps employed in the waveform. For a reversible electrode reaction, this objective is readily achieved by solving the appropriate boundary value problem using the method of the Laplace transform often in conjunction with the superposition theorem (7). Specific equations have been derived for staircase voltammetry (I),square wave voltammetry ( 4 ) , and a limiting case of differential pulse voltammetry (6) at stationary electrodes. In connection with our studies of analytical applications of differential pulse voltammetry a t solid electrodes, a general equation was required. The existence of a completely general equation giving the current resulting from an arbitrary series of step-func1616
ANALYTICAL CHEMISTRY, VOL.
tional potential changes is widely recognized by workers in the field. Nevertheless, this equation seems not to have been explicitly reported elsewhere so we present it here along with an example of its application.
EXPERIMENTAL A digital pulse voltammetry system was constructed around a Raytheon 706 (Raytheon Data Systems, Norwood, Mass.) minicomputer. The potential waveform was generated by software utilizing a programmable interval timer and a 12-bit DAC (Data Technology Corp., Santa Ana, Calif.). The output from the DAC was fed to a Princeton Applied Research (Princeton,N.J.) Model 173potentiostat. A second programmable interval timer was used to control the acquisition of data from a Princeton Applied Research Model 176 current-to-voltage converter. Details of the computer hardwarehoftware, which are similar to those of Keller and Osteryoung (61, will be presented at a future date. Acetonitrile was Burdick and Jackson Spectroquality,tetraethylammonium perchlorate was prepared by the method of Kolthoff and Coetzee (8) and the 9,lO-diphenylanthracene was supplied by Aldrich Chemical Co., Milwaukee. The circular disk platinum electrode (0.15-cmdiameter) was sealed in soft glass. Construction details are presented elsewhere (9). The three-electrode cell was equivalent to the Type 1 polarographic cell available from the Princeton Applied Research Corporation. Solutions contained 0.10 M tetraethylammonium perchlorate and they were not deaerated. THEORY An arbitrary potential-time waveform is shown in Figure 1. Step-functional changes in potential of arbitrary magnitude and direction occur a t arbitrary switching times with the potential being changed to E , at time 7, ( T is ~ defined as time zero, Eo = initial potential). The electrode reaction 0 + n e = R is assumed to be reversible. The solution may initially contain any desired mixture of 0 and R so long as the initial potential is the equilibrium potential. Assuming semiinfinite linear diffusion, the diffusion equations with the relevant initial and boundary conditions may be written:
48, NO. 1 1 , SEPTEMBER 1976
Square Wave
S to i rcase
t 1.4
1.3
1.2
1.1
1.0
VOLTS VS.
SCE
0.9
Figure 2. Differentialpulse voltammogram of 1.8 X phenylanthracene
M 9,lO-di-
td = 50 ms, tr = 100 ms. E, = 100 mV, and E, = 10 mV, x = experimental, curve is theoretical (Equation 11) using & = 1.7 X IO+ cm2/s Differential Pulse Figure 1. Potential-time
0
