Theory of Square Wave Voltammetry Louis Ramaley and Matthew S . Krause, Jr.1 Department of Chemistry, University of Arizona, Tucson, Ariz. 85721
When the sum of a synchronized square wave and staircase potential i s applied to a stationary electrode, the double layer charging current may be made negligible by measurement at a suitable time after the pulses occur. The staircase allows the electrode potential to be swept over the useful potential range. The theory for a reversible system predicts that the resulting current-time behavior should be symmetrical and bellshaped with a peak at El/*. The peak current is a linear function of concentration and the square root of the square wave frequency. The peak current is a more complicated function of square wave amplitude, measurement time, and staircase amplitude. The similarity between this technique and square wave polarography is discussed.
THEUSEFULNESS of square wave polarography as developed by Barker ( I ) is well established in trace analysis. The ability to present the current output in the form of a symmetrical peak, rather than the waveforms obtained in dc polarography or linear sweep voltammetry, and the ability to measure Faradaic current at a time when the double layer charging current is negligible, are primarily responsible for its success. The sensitivity is limited by capillary noise (2) and the necessary resolution is obtained only at sweep rates of 1 V in 10 to 60 minutes. The use of stationary electrodes should eliminate capillary noise and allow rapid scan rates while maintaining high resolution. Mann (3, 4 ) has applied a staircase potential waveform to stationary electrodes and significantly reduced the effect of double layer charging current. The current waveforms obtained were similar to those of linear sweep voltammetry. The superimposition of a square wave on the staircase potential should result in higher sensitivity than in the case of the staircase alone, and a “derivative” or peak current waveform similar to that of square wave polarography should be observed. The waveforms for this technique, which the authors propose be called square wave voltammetry (SWV), are shown, somewhat exaggerated, in Figure 1. Since the potential sweep is discontinuous rather than linear, the tops of the pulses remain flat, even at rapid sweep rates. Since the double layer charging current is proportional to e-IIRC,where t is time, R is the solution resistance, and C is the double layer capacitance, and the Faradaic current is approximately proportional to t - l / * , the charging current decays much more rapidly than the Faradaic current, allowing measurements to be made at a time when the charging current can be considered negligible. The final current waveform is the differential sum of the current flowing at an instant, selected to reduce the effect of the charging current, along the cathodic half cycle of the square wave and that flowing at the same instant along the preceeding or following anodic half cycle. 1 Present address, E. I. du Pont de Nemours & Co., Inc., Marshall Research Laboratory, 3500 Grays Ferry Ave., Philadelphia, Pa. 19146
(1) G. C. Barker and I. L. Jenkins, Analyst, 77,685 (1952). ( 2 ) G. C. Barker, Anal. Chim. Acta, 18, 118 (1958). HL EM . . ,1484(1961). ~~, (3) C. K. M ~ ~ ~ , A NCA (4) C. K. Mann, ibid.,37, 326 (1965). 1362
ANALYTICAL CHEMISTRY
MEASURED CURRENT
DIFFERENTIAL CURRENT El/*
n TOTAL DIFFERENTIAL CURRENT
E-
[-)
Figure 1. Waveforms employed and obtained in square wave voltammetry
THEORY
Barker et al. (9,Matsuda (6), and Kambara (7) have discussed the theory of square wave polarography. Christie and Lingane (8) have derived an equation for staircase voltammetry. The theory presented below for square wave voltammetry is approached in a manner similar to that of Barker et al. and Kambara. The current resulting from the application of the potential waveform of Figure 1 to an electrode at which the reversible reaction Ox
+ ne-
+ Red
can occur may be easily determined if semi-infinite linear diffusion to the electrode is assumed. It has been demonstrated (5, 7) that for a planar electrode and a reversible redox couple, the concentrations at the electrode surface are functions only of the electrode potential and the bulk concentrations and not of any previous potential established at the electrode. Barker, R. L. Faircloth, and A. W. Gardner, Atomic Energy Research Establ. (Gt. Brit.) C/R-1786. (6) H. Matsuda, 2.Elektrochem., 60,489 (1957). (7) T. Kambara, Bull. Chem. SOC.Japan, 27 527 (1954). (8) J. H. Christie and P. J. Lingane, J. Electroanal. Chem., 10, ( 5 ) G. C .
176 (1965).
where DO is the diffusion coefficient of the oxidized species. The current may be obtained in the usual manner by differentiating Equation 2 with respect to .r and setting x = 0 in the resulting expression. This produces
The C’s may be obtained from the following expressions
c, = /
C“
(1 -exp (E,( E -
exp
.) nFiRT ~-_ El _ _ - El .) nF RT
=
C*Q,
(4)
/
In the above expressions, .4 is the electrode area, Dn the diffusion coeffiiiznt of the rzduced species, and the other symbols have their usual meaning. If it is further specified that the current will only be measured at a certain time during each half period of the square wave, then
AE
t = ( j - 1)r T 87 0 0, Cox
0
