Comparison of Coulostatic Data Analysis Techniques J. M. Kudirka, P. H. Daum,’ a n d C. G . E n k e Michigan State Unioersity, East Lansing, Mich. 48823 The errors involved in the various methods of analyzing the data of the current impulse and coulostatic techniques when the relaxations are neither purely charge transfer nor purely diffusion controlled are discussed as a function of the ratio of the charge and diffusional time constants r,/rd and apparent rate constant k& The validity of the application of the simple charge transfer approximation is dependent on the time at which experimental measurements are obtained as well as r c / r d . The accuracies of a nomographic and a curve fitting technique of correcting the data for the influences of diffusion are discussed. The accuracy of the first is dependent on an accurate knowledge of the capacitance and short time measurements while the accuracy of the second is dependent upon the degree to which the observed portion of the decay is charge transfer controlled. Graphs are presented which show the experimental conditions or data analysis requirements for accurate exchange current measurements and which can provide good error correction estimates in the mixed rate region. Measurements of metal/metal ion systems and errors in CY measurement are also discussed.
IN THEIR ORIGINAL papers o n the coulostatic method, Delahay and Reinmuth ( I , 2 ) proposed several conditions which must be met for the application of either the charge transfer o r the diffusion limiting equations t o experimental relaxation data. Unfortunately, some of the electrochemical reactions which have been studied with this technique produce relaxations which are neither purely charge transfer nor purely diffusion controlled, but a combination of the two. Weir and Enke (3) and Daum and Enke ( 4 ) have said that satisfactory estimates of the charge transfer parameters can be obtained from decays of this type by obtaining the slope of the log(q) us. time curves at times sufficiently short t o apply the simple charge transfer assumption. However, this assumption has not been subjected to a rigorous mathematical analysis and correspondingly there has been some concern in the literature about the validity of some of the kinetic parameters which have been obtained by this method (5, 6). There have been several attempts t o correct relaxation data for mass transport phenomena (7, 8), but all of these techniques require a significant amount of tedious calculation, since the arguments of the general function which includes mass transport become complex in the region of greatest experimental interest. The purpose of the following discussion is threefold: to clarify some of the ambiguities involved in obtaining kinetic data from this measurement Present address, Department of Chemistry, Northern Illinois University, DeKalb. Ill. 601 15 ~~~~
~
(1) P. Delahay, J. Pliys. Cliem., 66, 2204 (1962). (2) W. H. Reinmuth, ANAL.CHEM.,34, 1272 (1962). (3) W. D. Weir and C. G . Enke, J . Pliys. Cliem., 71, 280 (1967). (4) P. H. Daum and C. G. Enke, ANAL.CHEM.,41,653 (1969). ( 5 ) F. C. Anson, in “Annual Review of Physical Chemistry,” Vol. 19, H. Eyring, Ed., Annual Review Inc., Palo Alto, Calif., 1968. (6) W. H. Reinrnuth, ANAL.CHEM., 40, 185R (1968). (7) R. R. Martin, Ph.D. Thesis, Louisiana State University. New Orleans, La., 1967. (8) D. J. Kooijman and J. H. Sluyters. Electrochim. Acto., 12, 1579 (1967).
technique by discussion of some of the determinate errors which are present in the various methods of analyzing the relaxation data as a function of the ratio of the charge transfer and diffusional time constants; t o suggest techniques which experimenters can use t o minimize their errors and/or reduce the complexity of data analyses or, alternatively, t o determine the simplest analytical technique that can be applied t o a given set of expeIimenta1 data; and t o clarify the limits of rate constants measurable within given experimental conditions. It will also be shown that the value of experimentally measurable rate constants is very different for the metallmetal ion case and the case where both species of the redox couple are in solution. This discussion of determinate errors is based o n a n analysis of the current impulse technique. The relaxation curves produced by this technique ars of the same mathematical form as those produced by the coulostatic technique. However, the form of the perturbing impulse is a square wave and this allows a separate measurement of the capacitance. This measurement will be shown not t o add significantly to the accuracy of the determination of the capacitive or kinetic parameters of the system. DISCUSSION
The basic equation, including mass transport processes which the decay follows ( 2 ) is: q = qt,o(p+
- &)-l v+exp(p-zt)erfc(p-t1’2) P-exp(P+ ‘t)erfc(p+ t l l 91 ( 1 )
where
p*
=
7d’/2/2TC
f
l/rc’~yrd/4rc-1)1~~
(2)
and the charge transfer time constant is re = RTCd/nFIo
(3)
and the diffusional time constant is rd
=
[RTC,/n’F’(l/CooDo’’’
+ 1/CRoDR1”)]2
(4)
If 7 c >> Td, Equation 1 reduces t o the simple charge transfer equation q = qI,oexp(-t/7,) (which is the result which could be obtained if diffusion processes were ignored in the derivation). If 7 d >> T ? , Equation 1 reduces to q = qt=Oexp(t/Td) erfc(t/Td)1/2which is the diffusion limiting equation. When neither of the above inequalities is satisfied, the decay is charge transfer ccntrolled at short times and diffusion controlled at long times in the decay. Kinetic information can be obtained from relaxations of this type in a variety of ways. The simplest and most direct method of estimating the reaction rate is t o assume the simple charge transfer limiting equation. The application of this assumption t o data of this type obviously presents some difficulties, but these difficulties can be minimized by paying proper attention to the way in which the measurements are made. DETERMINATION OF DOUBLE LAYER CAPACITANCE
Tied inextricably t o the accuracy of these kinetic measurements is the accuracy of the value of the double layer capacitance. There are a variety of methods available for the ex-
ANALYTICAL CHEMISTRY, VOL. 44, NO. 2, FEBRUARY 1972
*
309
Example 2
.'i
7
I
2.5
21)
-11)
Log(rc/rd)
Time, psec
Figure 1. Theoretical decay curves with Co = CR = lO-jrn/ cma, Do = DR = 10-j cm2/sec, T = 300 OK, n = 1,and Cd = 2.0 X lO-bF/cm2 Example 1 is for r C / 7 d = 2.5 and shows how the measured exchange current and capacitance can vary with extrapolation from various time regions. Example 2 is for T J T ~ = 25 and is essentially linear over the time range shown
perimental determination of the capacitance under nonreactive conditions-Le., when the electrode is ideally polarized. Many investigators have assumed that the capacitance for a n electrode at a certain potential and in a specified supporting electrolyte should be the same regardless of whether the electroactive species is present or nct. There is no a priori reason to believe this is true in all cases. For example, if specific absorption of the electroactive species occurs, then the capacitance may differ widely from its value with n o electroactive species. Or, if solutions of extremely high concentrations of electroactive species are studied, the capacitance may change because of the addition of these ions to the double layer. Furthermore, in the case of solid electrodes, the value of the capacitance is dependent on small additions of oxide or other films t o the electrode surface and is particularly sensitive to the adsorption of any foreign organic species in the solution. It is important, therefore, t o make the measurement of the double layer capacitance in the same solution and at the same time that the relaxation measurement is made. The value of the capacitance can be estimated from coulostatic data by extrapolation of the log (7) us. time curve to zero time and calculating for the relationship Cd = Aq/vI=,,. The current impulse method provides an additional estimate from the slope of the charging curve, from the relationship Cd = iL/(dq/dt) where it is the magnitude of the applied current. Both of these measurements can involve some very large determinate errors depending on the time at which the measurements are made, the definition of zero time, and the ratio 7 4 r d of the charge transfer and diffusional time constants. m/cm3, Do = DR = For example with Co = CR = 10-5 cmZ/sec,T = 300 OK, and Cd = 2.0 X 1O-j faradsjcmz rd is 0.1149 psec. If we assume an exchange current density of 1.8 A/cm2 corresponding t o a rate constant of 1.86 cmjsec assuming cr = 0.5 then 7
