multinomial distribution law expressed by Equation 5 . Whenever site-labeled compounds are analyzed special precautions or different relationships must be used ( 4 ) . The three methods represent increasingly more sophisticated and more generally-applicable techniques for calculating least squares fitted isotope abundances. They are also increasingly more complex and demanding. Consequently, the three methods are complementary and each
can find appropriate applications. The computer programs FIMS, MIMS, and FUMS have been filed with the American Society for Mass Spectrometry. A listing is available from the author.
RECEIVED for review April 26, 1974. Accepted August 9, 1974.
Pulse Voltammetry at Rotated Electrodes David J. Myers' and R. A. Osteryoung Department of Chemistry, Colorado State University, Fort Collins, Colo. 80523
Janet Osteryoung Department
of Microbiology and Civil Engineering, Colorado State University, Fort Collins, Colo. 80523
The dependence of pulse voltammetric currents at a rotated disk electrode on rotation rate from the region of pure diffusion control to the region of pure convection control is examined experimentally and theoretically. Data for normal pulse and differential pulse polarographic reduction of ferricyanide or oxidation of ferrocyanide agree well with a modification of an equation due to Bruckenstein and Prager. An approximate treatment of Levich is useful for defining the region of diffusion control.
The fact that normal pulse voltammetric currents are relatively insensitive to convective mass transport has been recognized for some time. However, the influence of convection on pulse voltammetry has been analyzed only to the extent of noting ( I , 2 ) that stirring has no effect as long as the Nernst layer thickness is small in comparison to the convective shear layer thickness. This assumption will not be true under all conditions, but the limits to convectionindependent behavior have not been identified and the nature of the current dependence, once these limits have been passed, has not been examined. Moreover, the effect of convection in the differential pulse mode has not been discussed. Since laminar flow at a rotated disk electrode is theoretically tractable and commonly used, it is this system which we consider. Understanding of instantaneous currents a t rotated electrodes has considerable analytical application, especially for problems in which it is desirable or necessary to use solid electrodes in turbulent solutions. Pulse voltammetry can be regarded as sampled chronoamperometry, so a solution to the problem of current transients produced by a potential step a t a rotated disk is required. This problem has been treated several times (39 ) , but none of the proposed solutions satisfactorily de-
'
Present address, Texas Instruments, P.O. Box 5012, M.S. 46, Dallas, Texas 75222. (1) K . B. Oldham and E. P. Parry, Anal. Chem., 38, 867 (1966). (2) E. P. Parry and R. A. Osteryoung, Anal. Chem., 36, 1366 (1964).
scribe the pulse voltammetry case accurately but simply. An equation derived by Bruckenstein and Prager (6) is successful if modified, and another equation by Levich (3) is useful even though it lacks generality.
EXPERIMENTAL The cell, including the reference electrode, was the same as described previously (10 j. The counter electrode was a 7 cm2 platinum foil. A Beckman Model 188501W rotating electrode assembly was used with Beckman No. 39086 (Pt), No. 390887 (Au), and No. 39084 (vitreous carbon) electrodes. These electrodes are interchangeable on the electrode assembly, and all have a geometric area of 0.283 cm2. Each electrode was polished to a mirror finish with 0.1-micron alumina on felt before use in a set of runs. Several 3.293-gram portions of K3Fe(CN)G (Mallinckrodt) were weighed out and stored in the dark in stoppered weighing bottles. Before each set of experiments, one of these portions was diluted to 100 ml with distilled water. Using a 100-pl Eppendorf micropipet, 500 p1 of stock solution was added to 50 ml of supporting electrolyte in the cell to give a 1.000 X 1OW3F ferricyanide solution. The same procedure was used for making ferrocyanide solutions by taking 4.224-gram portions of KdFe(CN)G. 3H20 (Baker Analyzed Reagent). The supporting electrolyte was 1M KCI, made from Fluka p.a. potassium chloride. Initial experiments were done on a Princeton Applied Research Model 174 Polarographic Analyzer. Although this instrument has only one pulse width (ea. 50 msec) available, it was felt that its behavior should be investigated because it is widely used. Most of the experiments, however, were done on a PDP-8/e computer system interfaced with a PAR 174 potentiostat. All instrumental experimental parameters were specified by user-interactive software. The system is similar to that described earlier (11 j .
Levich, "Physiochemical Hydrodynamics," Scripta Technica, Inc., Trans.. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. (4) V. Yu Fillnovskii and V . A. Kir'yanov, Dokl. Akad. Nauk., 156, 1412 (3) V. G.
(1964). R. C. Bowers and A. M. Wilson, J. Amer. Chem. Soc., 80, 2968 (1958). S. Bruckensteinand S. Prager, Anal. Chem., 39, 1161 (1967). D. R. Olander, lnt. J. Heat Mass Transfer, 5, 826-36 (1962). V. S. Krylov and V. N . Babak, Sow. Nectrochem., 7 , 626-32 (1971); Elektrokhimiya, 7 , 649-54 (1971). (9) K. M. Nisancioglu and J. Newman, J. Hectroanal. Chem., 5 0 , 23 (1974). (10) D. J. Myers and J . Osteryoung. Anal. Chem., 45, 267 (1973). (11) H. E. Keller and R. A. Osteryoung, Anal. Chem., 43, 342 (1971). (5) (6) (7) (8)
ANALYTICAL CHEMISTRY, VOL. 46, NO. 14, DECEMBER 1974
2089
2
4
8
b
4
1 N. + r p s Figure 1. Normal pulse voltammetric oxidation of ferrocyanide on glassy carbon. Numbers on the curves are pulse widths ( t ) in msec
All experiments were carried out by stepping from rest potential to a final potential. In the normal pulse mode, rest potentials were +500 mV us. SCE for ferricyanide reduction and zero us. SCE for ferrocyanide oxidation. Pulse heights were large enough in each case to ensure that currents were from the potential-independent portion of the wave. All currents are corrected for background. The rest potential in the differential pulse mode was chosen so that the current measured was the maximum or peak current.
RESULTS AND DISCUSSION
data for N = 0, and for t = 500 msec, N > 3 (rpsjl12 showed that both the Cottrell and the Levich slopes, respectively, were somewhat low (4.1% and 3.0%, respectively) indicating a systematic error which probably was a combination of errors in concentration and electrode area. However, the relative error was small. The ratio of the slopes agrees within 1.0%. If the Cottrell current is assumed to be correct, the effective concentration in Equation 1 is 0.963mM (cf. the “true” value of 1.000mM). In order to compare theory and experiment, the concentration used to calculate theoretical curves was taken to be 0.963mM, the ratio of the calculated and experimental Cottrell slopes. T o generate the curves in Figure 1 from theory, one must solve the convective diffusion equation ( 3 )
ac/at +
nFACbD1l 2
4-2
where n is the number of electrons, F is the faraday, A is the electrode area, C b is the bulk concentration of electroactive material, D is the diffusion coefficient, and t is the time. At long pulse widths and rapid rotation rates, the current should be given by the Levich equation
i, =
’
(3)
fi
[g
-
arctan (-3----)]} 2R + 1
(4)
where 6, is the steady-state convective diffusion layer thickness as defined by their Nernst linear concentration profile model and is given by (6):
6, =
1. 8049 D 1 I 3 v i r 6
wi I2
(5)
Equation 4 is more concisely given by:
n F A C b 2 l 3w l 2 1. 6116v1/6
where w is the angular velocity in radians per second and Y is the kinematic viscosity. Experiments with ferricyanide reduction on platinum and gold gave the expected behavior in a qualitative fashion but there was a disagreement between the Cottrell slope (i vs. t-l’z), obtained by varying pulse width a t zero rotation rate, and the Levich slope (i us. w1I2j, obtained by varying rotation rate a t long times (pulse width 0.5 sec). The Levich current was essentially the same as predicted using the analytical concentration and the geometric electrode area in Equation 2 but the Cottrell current was high by 20% when the same values were used in Equation 1. Both lines were linear with zero intercept. The reason for this discrepancy is not understood. We do not believe it to be an experimental artifact; in a study of chronopotentiometry a t rotated electrodes, Buck and Keller (12j found that the transition times for ferricyanide reduction on platinum were longer than expected as w approached zero. Ferrocyanide oxidation on vitreous carbon proved to be a satisfactory model system. Figure 1 shows the family of curves obtained by varying rotation rate (given in rps and symbolized by N ) a t different pulse widths. Analyses of the
(6) In Equation 6, R is the ratio of steady-state to instantaneous currents:
R = iss/iinst= 6/6,
(7 1
Denoting the right-hand member of Equation 6 as f ( Rj for the sake of brevity and rearranging gives 6,
=
(8)
By definition
is, = n F A D C b / 6 , and iinst= n F A D C b 6
(9)
Combining Equations 7 , 8, and 9, and Equations 5 and 8, we obtain the dimensionless instantaneous current and the dimensionless rotation rate:
(12) R. P. Buck and H. E. Keller, Anal. Chern., 35, 400 (1963).
2090
D(~~c/E$)
where y is the distance normal to the plane of the electrode surface and u is the solution velocity in that direction. An analytical solution to this modified form of Fick’s second law has been found by Olander for the heat transfer case (7). Other solutions have been obtained for the voltammetric case by Krylov and Babak (8) and by Nigancioglu (9). These solutions appear to describe the potential pulse (concentration step) case accurately, but the solutions are sufficiently complicated that it is difficult for the non-expert to use them. An approximate solution has been reported by Bruckenstein and Prager (6):
From the simple and qualitative Nernst layer/shear layer argument, it can be predicted that a t short pulse widths and low rotation rates, the current will obey the Cottrell equation
i, =
c,(ac/ay) =
ANALYTICAL CHEMISTRY, VOL. 46, NO. 14, DECEMBER 1 9 7 4
v-’ ’ w
As Bruckenstein and Prager point out, this is only an approximate solution. However, it is very nearly an exact solution, as the following argument will show. As R approaches unity, Equations 8 and 11 should reduce to the Levich equation (Equation 2). It is seen by inspection of Equation 9 that this can be accomplished by equating 6, to 6 ~which , is given by
6, =
”
-7
3OL
2 8. 2 624b
1. 6116 D’’3u”6 w1/2
and multiplying the rhs of Equation 6 by (1.8049/1.6116)2, This transformation includes changing Equation 7 to read
i
0 4 b
02!
O
which now makes the definition internally consistent, because 61, is the value of the steady state diffusion layer thickness which would be calculated from the measured steady state current. This change also affects Equation 11; the factor 1.8049 should be changed to 1.6116:
Now consider the Cottrell end of the proaches zero, we should find
wt
h L L 0 02 06
U
I 10
NORMALIZED
I , 14
1
IS
,
22
,
1 I 26
30
JROTATION RATE
Figure 2. Analysis of experimental values using Equations 10 and 12 as modified by Equation 14 for ferrocyanide oxidation on glassy car-
bon an2/ n = 1, f = 96,500, A = 0.2827 cm2, C = 0 9626rnM. D = 8 X 0 00893 stokes The data are those of Figure 1 The solid line IS sed f ( R ) / R vs while the points are ft”2/nfAD”2Cb vs 1 61 1 6 ( ~ / D ) ” ~
range. As R ap400
so that Equation 10 reduces to the Cottrell equation (Equation 1). In fact, applying L’Hospital’s rule to Equation 6, it is found that
t.ms
200
Notice that the factor &/2 = 0.88623 differs from the factor (1.6116/1.8049) = 0.89290 by only 0.8%. In other words, the equations are incorrect by almost the same factor a t both extremes. In fact, it can be shown that any solution of the form of Equation 6 fits a t the Levich end if 6 , is chosen , that a more general derivation gives the soluto be 6 ~ and tion of Equations 10 and 1 2 provided that f ( R ) is given by (13);
f(R)=
N.rps
Figure 3. Values of parameters for a given discrepancy between the
Cottrell current and the true current The discrepancy is less than 1% or 10% for choices of (t, N) within the fields bounded by the curves
The predictions of Equations 10 and 12 as modified by Equation 14 are tested by using the data set displayed in Figure 1. The results are shown in Figure 2. Agreement with theory is within experimental error. I t is interesting that the range over which the transition from Cottrell t o Levich control takes place is rather narrow. The current exceeds the Cottrell current by no more than 10%up to about f ( R ) = 0.57, while the current exceeds the Levich current by no more than 10% down to about f ( R ) = 0.64. This corresponds to the range 0.85 5 R 5 0.90. Another approach to the current transient problem is given by Levich ( 3 ) . In this treatment, convective mass transport is considered as a perturbation on the Cottrell current and an equation is derived which expresses this perturbation. Because of the assumptions made in its derivation, the equation is valid only if the convective contribution is relatively small. However, his result: (13) S.Bruckenstein and J. Osteryoung. J. Nectroanal. Chern., submitted for publication.
is heuristically useful. For example, it allows a simple estimation of the degree to which normal pulse currents will be independent of rotation. We write the inequality
where f is the fraction by which the current is increased due to rotation. For any given pulse width, one can easily calculate the rotation rate which will give, say, a 1%current increase. Figure 3 shows the result of some of these calculations. I t is seen that extremely short pulse widths are needed if one is to obtain currents that are independent of rotation rate over the entire range of rotation rates available with most commercial equipment. For currents to be within about 10% of the Cottrell current, the N t product (rps X sec) must be less than about 1.4. For the parameters of the ferrocyanide oxidation reported here, this corresponds to f ( R )= 0.57 or R 0.85. Equation 15 also is useful in explaining one rather re-
-
ANALYTICAL C H E M I S T R Y , VOL. 46, NO. 14, D E C E M B E R 1974 * 2091
where x is the distance from the electrode surface. Assuming a linear concentration gradient, this equation can be formulated in terms of a diffusion layer thickness 6:
This gives
.
z =
40
20
60
80
100
J
~ F A Cb~ 6
1
(19)
where P = exp { ( - n F / R T ) ( E - E 1 / 2 ) )and E112 is the half-wave potential. Parry and Osteryoung ( 1 4 ) have shown that when the current difference is maximum, the bracketed expression in Equation 22 can be formulated in terms of pulse height AE:
N,rps
Figure 4. Comparison of normal pulse and differential pulse currents for the reduction of ferricyanide on Pt Curve A: NPP; Curve B: DPP; (. experiment: (-) 0.750 times Curve A. DPP: A € = 100 mV. u = 7.001,( u 1 ) / ( u 1) = 0.750 e)
-
+
markable-and initially baffling-result which was obtained during the course of this study. As the rotation rate is increased and a convective contribution appears, an w3IL dependence is predicted by Equation 15. At higher rotation depenrates, the ordinary Levich equation predicts an d2 dence. One might suspect that there exists a particular pulse width wheie the currents observed would fall exactly between a 3/2 and 1h dependence-ie., where a linear dependence would be found. Indeed, this is the case. As shown by curve A in Figure 4, this pulse width happens to be 50 msec, the fixed pulse width of the PAR 174. The data in Figure 4 are for the reduction of ferricyanide on Pt, and as explained above deviate absolutely from predicted values. However. the agreement of the data with theory in the mixed control region of rotation rates is good, in the sense that the functional dependence on rotation rate is theoretical. If the currents a t 50 msec from Figure 1 are plotted us. rotation rate, the plot is also linear in the range 20-100 rps. The same is true for currents calculated from the modified Bruckenstein-Prager equation. Normal pulse results for ferricyanide reduction are shown here so that they can be compared with experiments done on the same system in the differential pulse mode. The following simple argument can be used to predict the dependence of differential pulse currents for reversible reactions on rotation rate. In a differential pulse experiment, the current is sampled a t a potential El just prior t o application of a pulse of height AE. Before the potential is returned to E 1, the current is sampled again a t potential E2 = E 1 LE. The current difference is
+
i =
i,
-i, 2
= 1
where u = exp (AE/2) ( n F / R T ) .Since the limiting current for normal pulse can be expressed as
we see that the differential pulse current is given by the normal pulse current times a constant which depends only on pulse height. This derivation gives the same result as found previously (14 ), but it shows that differential pulse currents can be formulated in terms of the diffusion layer thickness 6. Such a formulation is very general: in quiet solution 6 = -\/.lrDt, under steady-state conditions a t a rotated electrode 6 = 6 ~ , and for cases between these extremes 6 can be calculated from Equations 10 and 12. According to this simple treatment, differential pulse should depend on rotation rate exactly as does normal pulse for a given pulse width; the current will be lower than normal pulse currents by (u - l)/(a 1). Figure 4 shows the currents given by normal and differential pulse voltammetry of the same solution. A pulse height of -100 mV was used in the differential experiments, corresponding (for n = 1 electron) to a (u - l)/(0 1)ratio of 0.7500. The normal pulse current multiplied by this factor agrees well with the experimental differential pulse currents.
+
+
ACKNOWLEDGMENT Discussions with S. Bruckenstein clarified our interpretation. The help of J. Turner, R. Abel, and B. Vassos in programming and interfacing the computer system is gratefully acknowledged.
RECEIVEDfor review May 6, 1974. Accepted August 5, 1974. This work was supported in part by NSF Grant GP31491X and by a Biological Sciences Support Grant from Colorado State University. (14) E. P. Parry and R. A. Osteryoung, Anal. Chem., 37,1634 (1965)
2092
ANALYTICAL CHEMISTRY, VOL. 46, NO. 14, DECEMBER 1 9 7 4
