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ANALYTICAL CHEMISTRY, VOL. 51, NO. 13, NOVEMBER 1979
CORRESPONDENCE Diffusion Control in Linear Sweep Voltammetry Sir: In many linear sweep voltammetry experiments, even if the charge transfer is irreversible or there are coupled chemical reactions, the current will become purely diffusion controlled at a potential sufficiently past the peak (1). Then the current-voltage curve obeys effectively a simple Cottrell equation: current-time curve (2) with E = u t , where E = potential, u = sweep rate, and t = time. In some cases a further electrode process will interrupt the diffusion control (3) but in many situations one can use this part of the curve to obtain chronoamperometric (i-t) data, obviating the necessity for separate potential step experiments. The Cottrell equation is (2)
i = nFSCD112 .-1 ,112
t'/2
(1)
where i = current; n is the number of electrons involved in the process; F , the Faraday; S, the electrode area; C,the concentration of electroactive material; and D, its diffusion coefficient. In a potentiostatic experiment the initial time is obvious-it is the time at which the El to Ez pulse is applied (where El >E, + (0.16/n) V). It is indicated experimentally by a sharp rise in current as in the solid curve in Figure la. However, the corresponding time (to)in a linear sweep experiment, such that the diffusion part of the curve past the peak matches the equivalent potentiostatic current-time curve, is not known a priori, and therefore needs to be included in the equation. Thus we can write
Ginzburg gives calculated values for A and B for both reversible and irreversible charge transfer. However the basis for his calculations is not clear, especially as he expresses surprise that constant
A = or
A =
(%)"'
(reversible)
(s)112 (irreversible)
We have compared the theoretical values for the current linear sweep voltammetry with the diffusion controlled current-time values. In order to do this we needed to calculate additional values of r 1 / 2 ~ ( uwell t ) into the region of pure diffusion control. We have done this for the simple reversible electron transfer process following Nicholson and Shain's procedure (6) with a slight modification. From Equation 33 in ref. 6 normalized current function 1 1 +-I xbt) = (6) ,fi(i ye) 4 8
+
where
Let z = u t sin2 4, then dz/d4 = 2 u t sin 4 cos (6, so
I= Equation 2 becomes identical to Equation 1if one puts t o = 0. In linear sweep voltammetry t = E / u and therefore (3)
2 u t sin 4 cos 4 d 4
d-a t ( 1
- sin2 4) cosh2
(log
ye -2ut
sin2 4
where E = ut; Eo = ut, and G = u1IZK. (This is the form used by Ginzburg ( 4 ) with G = A and Eo = B). Re-arranging Equation 3 gives
_1 - E i2
G2
Eo G2
(4)
A plot of l/i2 vs. E should give a straight line for pure diffusion control with slope = 1 / G 2 and intercept at -Eo/G2. Hence Eo = -intercept/slope. Of major interest is G 2 from which n2D may be obtained and hence values of n and D if a rough value of D is known and n is small and integral. We have used this empirical approach in the study of the electrochemical reduction of carbon dioxide under various conditions ( 5 ) . to is a virtual zero time and can be more usefully expressed as a potential Eo with respect to the half-wave potential ( E l j z ) or the peak potential (E,). Ginzburg's ( 4 ) form of Equation 3 is
B , the "zero time" potential is expressed with respect to E, and the current is normalized so 5 = i/i,. 0003-2700/79/0351-2282$01 .OO/O
This integral was evaluated by Simpson's rule to six significant figures with log ye = 6.5 (ref. 6.) using a computer program. The data from these calculations, using at) for normalized dimensionless current at potentials given as ( E - E l / z )(as in ref. 6), was fitted by a linear least squares analysis computer program to Equation 4, where i = ~ ' / ~ x ( a Values t ) . of ( E Elj2)were taken from -0.2 to -1.0 V. The regression analysis was repeated using more cathodic initial potentials and then some more sets with cathodic potentials between -1.0 and -2.0 V. Over this whole range the correlation coefficient was between 0.99999 and 1.OOOOO. Standard deviations in G2were always better than *0.1% and in Eo better than f2%. The values of G2 slowly increased toward the theoretical value as the initial value of E was shifted cathodically. However from the practical point of view the lowest value of G2 was only 0.7% less than theory. At the same time the value of Eo shifted from $25 mV toward zero, i.e., the point a t which Eo = Ellz. Some data are shown in Table I. It will be noted that the values of G 2 and 0 1979 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 51, NO. 13, NOVEMBER 1979
2283
Note Ginzburg's value for A2 = RT/nF = 0.02568, as he has omitted the all2term from the Cottrell equation ( 4 ) . Apart from that, his data are clearly accurate to f l %as indicated in the condition for diffusion control that it >0.991. The dimensionless current ratio it/ip used by Ginzburg does not however give the result he quotes, despite the i, = i, (max) = nFSCD'12 r1I2x(at) max (ref. 6.) nFu 'I2 = nFSCD112 0.4463 at E =
(E)
E, (ref. 6.) therefore
therefore
RT 1 n F ~ ( 0 . 4 4 6 3 ) ~(E- Eo) Thus A2 should be = RT/(nF~(0.4463)~). Ginzburg gives one value for B = +0.0445 V (ref. 4). This is with respect to E,. As Ell2is 0.0285 anodic of E , this value is equivalent to Eo = +0.016-which is within the range of values we obtained. Polcyn and Shain ( 3 ) give an empirical factor 0 (= 0.96 - 0.99) to adjust the theoretical G2 factors to the computed values for some arbitrary ranges of potential given the calculated values of their equivalent of Eo, which is E' - Eo which again fall within the range of values we obtained. However a more systematic approach is needed extending farther out into the diffusion controlled region. The latter authors (3)confine themselves within the range 50 to 830 mV (rev) and 36 to 845 mV (irrev.) Cyclic voltammetry of benzoquinone in acetonitrile yields two reversible one-electron waves (7). Analysis of tails of the first cathodic peaks for four scans at different scan rates and concentrations of quinone gave data from which values of the diffusion coefficient of benzoquinone were calculated using Equation 4. The mean value was 2.9 f 0.3 cm2 s-' which is comparable with the value of 2.4 cm2 s-' obtain in ref. 7 using the Randles-Sevcik equation and with the value of 2.9 cm2 s-' obtained by potential step chronoamperometry (8). LITERATURE CITED i2
t
to
Figure 1. Comparison of potentiostatic i l t curve with linear sweep vottammetric ilEcurve. (a) Potentiistatic i l t curve (-) compared with linear sweep vottammetric ilEcurve (---) such that E = vt. (b) Potential step determining the initiation of the i l t curve
Table I. The Shift of E , Values from + 2 5 mV toward 0 range of EImV
G 2x 10
-200 to-1000 -300 to-1000 -400 to-1000 -600 to-1000 -800 to-1000 -920 to-1000 - 1000 to - 2000 -1940 to -2000
8.1135 8.1323 8.1418 8.1519 8.1570 8.1609 8.1676 8.1710 8.1740
theory
i computed % error E,/mV in E,
25.0 19.6 16.7 13.5 11.7 10.3 2.3 1.6 0.0
1.6 0.9 0.6 0.2 0.09 0.03
Eo differ from Ginzburg's values of A and B ( 4 ) . The theoretical value of G2can be found from the point at which the linear sweep voltammetric current (i,) becomes equal to the potential step chronoamperometric current (it). Now (7) and
i, = nFSCD1l2 all2 a1I2x ( u t ) a = -nFu
RT
When it = i,
(8)
=
.-
(1) A d a m , Ralph N. "Electrochemistry at Soli Electrodes"; Marcel Dekker: New York, 1969; p 125. (2) Ref. 1, p 50. (3) Polcyn, Daniel S.; Shain, Irving Anal. Chem. 1966, 38, 370-5. (4) Ginzburg, Gregory. Anal. Ch8m. 1970, 5 0 , 375-6. (5) Eggins, Brian R.;McNeill, Joanne, Unpublished results, 1977. (6) Nicholson, Richard S.; Shain, Irving. Anal. Chem. 1964, 36, 706-23. (7) Eggins, Brian R. Chem. Commun. 1969, 1267-8. (8) Eggins, Brian R.; Chambers, James Q. J. Electrochem. Soc.1970, 117, 186-92.
B r i a n R. Eggins* School of Physical Science Ulster Polytechnic Shore Road Newtownabbey Co. Antrim, BT37 OQB, N. Ireland Norman H. S m i t h
Now u ( t - to) can be replaced by E - Eo and therefore
Comparing Equations 3 and 10
RT G2 = - = 0.008174 at 298 K anF
School of Mathematics Ulster Polytechnic Shore Road Newtownabbey Co. Antrim, BT37 OQB, N. Ireland RECEIVED for review November 22,1978. Accepted July 19, 1979. The financial support of The Chemical Society is gratefully acknowledged.
