would be added to an unknown of high concentration. By choosing a median value for the programmed, betweenpoints voltage range and a suitably small volume for the first addition, this system could be essentially run in an automated mode wherein samples of a wide range of concentrations could be analyzed without operator assistance. Further accuracy could possibly be obtained with a solution delivery system consisting of a large, precision-machined syringe delivery system driven by a computer-actuated stepping motor. Also the concentration range could possibly be extended and accuracy improved by developing and fitting an equation other than the Nernst equation t o the data; perhaps some exponential series which could accommodate changes in the concentration-electrode response function would be suitable. Computer programs used here, all of which were written in the DEC PAL I11 assembly language, are available from the authors upon request.
LITERATURE CITED (1) Paul E. Whitson, H. W. Vander Born, and Dennis H. Evans, Anal. Chem., 45, 1298 (1973). ( 2 ) John F. Holland, Richard E. Teets, and Andrew Timmick, Anal. Chem., 45, 145 (1973). (3) S. Ushioda, J. B. Valdez. W. H. Ward, and A. R. Evans, Rev. Sci. Instrum., 45, (4), 479 (1974). (4) John E. Wampler and Richard J. DeSa. Anal. Chem., 46, 563 (1974). (5) S.P. Perone. D. 0. Jones, and W. F. Gutknecht, Anal. Chem., 41, 1154 (1969). (6) M. J. D. Brand and G. A. Rechnitz, Anal. Chem., 42, 1172 (1970). (7) R. H. Moore and R. K. Zeigler, "The Solution of the General Least Squares Problem with Special Reference to High-speed Computers", LA-2367 (1959), U S . Government Printing Office, Washington, D.C. (8) John Mandel, "The Statistical Analysis of Experimental Data", Interscience, New York, 1964. (9) Henry Stone, J. Opt. SOC.Am., 52, 998 (1962). (10) "Logic Handbook", Digital Equipment Corporation, Maynard, Mass.. 1973-74. (11) "Introduction to Programming", Digital Equipment Corporation, Maynard, Mass., 1972.
ACKNOWLEDGMENT The authors acknowledge the efforts of Charles Lochmuller of the Department of Chemistry, Duke University, who assembled much of the computer interface used in this work.
RECEIVEDfor review August 7, 1974. Resubmitted July 15, 1975. Accepted November 6,1975.
Studies in Photoelectrochemistry: A Theoretical Model for induced Charging Currents in Potentiostatic Chronoamperometry S. S. Fratoni, Jr., and S. P. Perone" Purdue University, Department of Chemistry, Lafayette, Ind. 4 7907
It has been assumed previously that charging current is significant only for electrochemical experiments where the applied potential is varied (e.g., potential step, potential sweep). This paper deals with characterization of charging current contributions to chronoamperometric measurements under potentiostafic conditions, where the charging current is induced by the changing faradaic current. This case corresponds directly to the conditions for chronoamperometric measurements of flash photolytic processes, commonly called photopolarography. A theoretical model applicable to any electrolysis mechanism, and a general method of analysis, are presented to characterize this charging current. In addition, the model is extended to include potential-step chronoamperometry. Application of this model to these two specific cases demonstrates the significance of the charging current term relative to measured current. Moreover, the inability of blank measurements to adequately compensate for this background is illustrated. Thus, tabulated data are included which allow the extraction of pure faradaic signals from total measured currents, where the only parameter which must be known is the cell time constant.
Any capacitive element in a circuit draws current when the potential across it is changed. This phenomenon is well recognized in electrochemical measurements because of the capacitive double layer formed a t the working electrodesolution interface. This factor represents the major cause of background currents for controlled-potential electrochemical techniques, such as potential-step chronoamperometry or cyclic voltammetry. In all previously reported photopolarographic (1-28) and pulse radiolytic polarographic (29-34) studies, however, the tacit assumption has been made that chronoamperometric measurements under potentiostatic conditions (i.e., where the applied cell potential is held constant before, during, and after the flash) are not distorted significantly by double-layer charging currents, as charging currents are usually associated only with measurements involving a change in the applied potential. However, the work presented here shows that another source of charging current exists, even under applied potentiostatic conditions. This "induced" charging current is a direct result of the changing faradaic current. The contribution is particularly important for photoelectrochemical experiments a t constant
ANALYTICAL CHEMISTRY, VOL. 48, NO. 2, FEBRUARY 1976
287
CDL
'
Q
Figure 1. Simplified model of electrochemical cell
potential (potentiostatic), because it constitutes the primary source of background currents in these experiments. This paper considers the characterization of double-layer charging effects in chronoamperometric experiments by establishing a theoretical model and a generalized method of analysis. Two specific types of experiments have been characterized here. The first involves the instantaneous production of electroactive species in solution by flash photolysis, accompanied by continuous potentiostatic chronoamperometric measurements. T h e other case involves potential-step chronoamperometry, with or without simultaneous flash photolysis. In each case, it will be assumed that only diffusion-limited faradaic current exists on the time scale of the measurement. The induced charging current considered here is dependent on the specific nature of the electrolysis current. Thus, the nature of the induced charging current will vary with the magnitude of the faradaic current and type of electrolysis mechanism present. This interdependency thereby precludes the possibility of a n experimental "blank" and necessitates a theoretical approach. The theoretical approach assumes a n equivalent electrical representation of the potentiostat and electrochemical cell. Circuit analysis of this representation will then yield the respective contributions of faradaic and charging currents to chronoamperometric behavior. T h e model is also adaptable for analysis of any specific electrolysis mechanism.
THEORY A simplified model of the electrochemical cell is assumed. In Figure 1 CDL represents the double layer a t the working electrode-solution interface and is assumed to be equivalent to a constant capacitance (independent of potential) of value C farads. F , represents the electrical properties of the electrolysis process, which are dependent on the specific electrolysis mechanism (e.g., diffusion-limited, competing chemical reaction, etc.). R , represents the uncompensated solution resistance between the reference electrode and the double layer, and is assumed to act as a pure resistance of value R ohms. The potentiostat is assumed t o have infinite current and voltage capabilities, such that it appears to be a n ideal constant potential source. (The use of a potentiostat with positive feedback (35) is also consistent with further discussions, except that the percent compensation must be specified to determine the effective R,. A detailed theoretical treatment of this case will be included in a later publication ( 3 6 ) . ) Interpretation of the model presented here is based on the recognition that a changing faradaic current, even when a constant potential is applied to the cell by the potentiostat, must produce charging current. Any electrolysis cur-
+ Flgure 2. Laplace domain equivalent circuit representation of electrochemical cell (arrows indicate direction of anodic current)
rent ( i ~ )will result in a potential drop across R,, thus causing the potential across the double layer (CDL)to be differe n t then that applied to the cell. Any change in electrolysis current ( i ~ )will cause a different potential drop, a corresponding change in potential across CDL, and subsequent charging current (ic). Analysis of the equivalent circuit (Figure 1)will be based on standard electrical circuit analysis ( 3 7 ) .This consists of first transforming each component into its equivalent frequency domain term by taking the Laplace transform. After analysis of the circuit in the Laplace domain based on fundamental electrical laws (e.g., Ohm's, Kirchhoff's), the inverse transform can be taken to find the desired relationship in the time domain. For the following discussion, three points must be kept in mind. First, it is assumed that the experiment starts ( i ~ begins to flow) instantaneously a t t = 0. This is approximately true for the specific cases of flash photolysis-, pulse radiolysis-, or potential-step-initiated measurements. Second, it is assumed that the double layer will be charged t o a n initial potential ( E , ) before the experiment is started ( t < 0). (The equivalent representation in the Laplace domain of a charged capacitor is a capacitor (C) in series with a potential source ( E , )( 3 7 ) . )The last point is that the electrolysis process, F,, will be represented as a current source rather than an equivalent impedance, as is the normal convention. Transforming the circuit given in Figure 1 into the Laplace domain gives the representation shown in Figure 2. T h e current that is normally measured, by either a difference reading across a resistor in the feedback loop or a current-to-voltage converter after the working electrode, cor, total current supplied by the potentioresponds to i ~ the stat. This term is the net sum of the faradaic (or electrolysis) current ( i ~ and ) any charging current (ic).It is evident a t this point that, although i~ is measured experimentally, we are really interested in extracting the faradaic current ( i ~ )from measured values of i ~ . A simplified analysis of the model for the potentiostatic case will yield diT dt
i ~ = - iR C~ -
where the last term represents the induced charging current. Application of this relationship will give a faradaic current corrected for induced charging current. This expression is very useful because it is applicable even if the nature of the faradaic process is unknown. (Specific applications will be discussed elsewhere ( 3 6 ) . ) The concern of this work is to develop rigorous mathematical expressions for the total currents. This requires some assumptions concerning the faradaic process, but allows one to predict the nature and magnitude of induced charging current and its dependence on experimental pa-
200 * ANALYTICAL CHEMISTRY, VOL. 48, NO. 2, FEBRUARY 1976
From this expression, the respective current terms can be found. T h e total current is the current that flows through t h e uncompensated resistance ( R in Figure 2), and can be found by dividing the potential difference across the resistance by the value of R .
rameters. T h e explicit relationship between the total current, i ~ and , t h e faradaic current, i ~ can , be derived if the time-dependence of i F is predictable and the Laplace transform can be obtained. This is illustrated in the following examples. (Note that, for simplicity, R , and CDLare represented as R and C in all mathematical expressions and in Figure 2.) Flash Photocurrents, Diffusion-Limited, Potentiostatic Conditions. For potentiostatic measurements of flash-initiated electrolysis currents, a typical example of the electrode process might be:
This can be simplified by letting
hv
S.
O*RH*+ protic
solvent
+ +
RH. 0 eH+ (2b) If RH. is a stable species, the flash is infinitesimaly short, and photolysis occurs homogeneously around a planar working electrode, the Cottrell equation should describe faradaic currents. +
and
m
Then Equation 13 reduces to
i ~ T( ) = K'F ( T )
(3)
+ iF
(17)
T h e faradaic current (as given in Equation 3) can be rewritten as:
where K = nFAD1I2CRo,n = number of electrons, F = Faraday, A = electrode area, D = diffusion coefficient, and C R O = initial concentration of RH. formed by the flash. I t is also assumed that any variations in the double layer voltage d o not exceed the limits of the diffusion plateau. Referring to Figure 2, iT = ic
K
K'=-
T h e charging current is the difference between these terms
(4)
.
I('
=
.
17'
- I.F
where all currents follow the normal polarographic sign for cathodic currents. In conventions, - for anodic and the Laplace domain,
(19)
+
~T(s= )
i c ( s )+ iF(s)
(5)
Also, from Figure 2,
E1 + sc For the potentiostatic case, E ; = Ef = E . Thus Equation 7' becomes:
T h e Laplace transform of the faradaic current term (Equation 3) yields
1 &(s
In order t o evaluate the characteristics of these relationships, the value of the integral given in Equation 15 ( F ( T ) ) must first be found. Because simple integration of this expression is not possible, a numerical method of integration based on Simpson's one-third rule (38) was used for evaluation. This method was chosen over a series approximation because it is more suitable for computer evaluation. T h e number of segments used in the integration was selected in order t o use a sufficient number for accurate integration and yet use minimum computer time. At least 200 segments were used for integration. As is evident in Equations 17, 18, and 20, a factor of K / m , or K', is common to all current terms. This indicates that, when T is used as the time variable, the ratios between these currents are independent of experimental conditions ( e . g . ,R,, CI)L,K ) . Therefore, the currents were tabulated (Tables I and 11). normalized by this factor. In order to find the actual magnitude of the currents, it is, therefore, necessary to multiply each value in the tables by K'. Normal polarographic sign conventions were used for these data, assuming a cathodic faradaic process.
DISCUSSION
Substituting Equation 9 into 8 gives:
E K VC(S)= - - - * s C
(20)
+ l/RC)
(10)
T o find the time dependence of this potential, the inverse transform is taken.
A plot of the total current vs. T (based on Equation 13) is given in Figure 3, showing that I ' T ( T ) increases to a maximum a t T = 0.85, and then decreases. Although similar behavior has been observed before experimentally in flash photocurrent studies, the deviation from pure faradaic current (i = K / G ) has been attributed to experimental limitations, including non-instantaneous flash and finite response time of the electronics (19, 2 4 ) . Also, some workers have suggested that the distorted currents are due to adsorption effects (32).Figure 4 is a log-log plot of 1'747) VS. T , ANALYTICAL CHEMISTRY, VOL. 48, NO. 2, FEBRUARY 1976
289
Y \
.-I-
1
OO
2
4
8
6
IO
Figure 4. Time dependence of
TIME ( t / R C ) Flgure 3. Theoretical total current fusion limited case ( T = 0.0-10.0)
(blK') vs.
7
In ( t / R C ) for potentiostatic-diffusion limited
case for potentiostatic-dif-
Table 11. Cirrent Terms for Diffusion Limited-Potentiostatic Casea Table I. Current Terms for Diffusion Limited-Potentiostatic Casea T
I( T ) ~ 'K
0.05 0.43260 0.10 0.59193 0.15 0.70159 0.20 0.78419 0.25 0.84887 0.30 0.90054 0.35 0.94220 0.40 0.97591 0.45 1.00314 0.50 1.02499 0.55 1.04232 0.60 1.05580 0.65 1.06600 0.70 1.07338 0.75 1.07831 0.80 1.08112 0.85 1.08208 0.90 1.08145 0.95 1.07941 1.00 1.07616 1.10 1.06661 1.20 1.05384 1.30 1.03865 1.40 1.02170 1.50 1.00349 1.60 0.98443 1.70 0.96482 1.80 0.94493 1.90 0.92496 2.00 0.90508 0 7 = 0.05-2.0.
7
I(c)iK'
I(F)iK'
-4.03953 -2.57035 -1.88040 -1.45188 -1.1 51 13 -0.9 2521 -0.74811 -0.605 22 -0.487 57 -0.38922 -0.30608 -0.23519 -0.17434 -0.12185 -0.07639 -0.03692 -0.00257 0.02735 0.05343 0.07616 0.11315 0.1.4 096 0.16160 0.17655 0.18700 0.19386 0.19785 0.19957 0.19949 0.19797
4.47214 3.16228 2.58199 2.23607 2.00000 1.82574 1.69031 1.58114 1.49071 1.41421 1.34840 1.29099 1.24035 1.19523 1.15470 1.11803 1.08465 1.05409 1.02598 1.00000 0.95346 0.91287 0.87706 0.84515 0.81650 0.79057 0.76696 0.74536 0.72548 0.70711
in which the slope indicates the approximate exponential dependence. From 7 = 2.4to 8.2,the slope is nearly constant a t -0.66,and from 7 = 18 to >50 the slope is -0.51. This indicates that the time dependence of experimentally measured currents might appear to satisfy the Cottrell equation (i 0: t-'I2) for 7 > 2.4, depending on the time domain covered. The predicted induced charging current (Figure 5) starts as a large negative value, then crosses zero a t 7 = 0.85,and forms a broad positive peak, slowly decreasing back to zero. Because the potential across the double-layer is the same a t t = 0 and t = -, the net charging coulombs should be zero. 290
I(T)iK'
2.00 0.90508 2.50 0.81047 3.00 0.72819 3.50 0.65940 4.00 0.60268 4.50 0.55602 0.51745 5.00 0.48528 5.50 6.00 0.45817 0.43505 6.50 0.41511 7.00 0.39772 7.50 8.00 0.38240 0.36877 8.50 9.00 0.35654 0.34549 9.50 0.33544 10.00 10.58 0.32624 0.31777 11.00 11.50 0.30995 0.30268 12.00 0.29592 12.50 0.28959 13.00 13.50 0.28366 0.27809 14.00 0.27283 14.50 16.00 0.26786 0.25870 16.00 0.25042 17.00 0.24289 18.00 0.23601 19.00 0.22968 20.00 21.00 0.22384 22.00 0.21842 23.00 0.21338 0.20868 24.00 0.20427 25.00 a 7 = 2.0-25.0
I(C)/K'
I(F)/K'
0.19797 0.17801 0.15084 0.12487 0.10268 0.08462 0.07024 0.05888 0.04992 0.04282 0.03715 0.03257 0.02885 0.02577 0.02321 0.02105 0.01921 0.01763 0.01626 0.01506 0.01401 0.01308 0.01224 0.01150 0.01082 0.01021 0.00966 0.00870 0.00788 0.00719 0.00659 0.00607 0.00562 0.00522 0.00487 0.00455 0.00427
0.70711 0.63246 0.57735 0.53452 0.50000 0.47140 0.44721 0.42640 0.40825 0.39223 0.37796 0.36515 0.35355 0.34300 0.33333 0.32444 0.31623 0.30861 0.30151 0.29488 0.28868 0.28284 0.27735 0.27217 0.26726 0.26261 0.25820 0.25000 0.24254 0.23570 0.22942 0.22361 0.21822 0.21320 0.20851 0.20412 0.20000
The direction of the initial current is always opposite to that of the faradaic current, and the broad peak is the same direction as the faradaic current. T o illustrate this, Figure 6 shows a composite of all the current functions. Qualitatively, the results seem reasonable. I t appears that the charged double-layer (CDL,Figure 1) first supplies some of the current necessary for the cathodic faradaic process, resulting in the negative ic (negative being electron
ANALYTICAL CHEMISTRY, VOL. 48, NO. 2, FEBRUARY 1976
2'6iA
I r
tI 0
-Y
I .6
A
iF/K'
B
iT/K'
C
ic/K'
-I
\
.- 0
CHARGING C U R R E N T
t/c
-2
-0
2
4
6
8
-o'8 -20;
IO
TIME ( t / R C ) Figure 5. Theoretical charging current (iclK') vs. tic-diffusion limited case ( T = 0.0-10.0)
T
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 110 1.20 1.30
10.33779 5.34230 3.68019 2.85141 2.35606 2.02739 1.79400 1.62016 1.48604 1.37973 1.29365 1.22276 1.16355 1.11352 1.07085 1.03415 1.00237 0.97471 0.95050 0.92923 0.89392 0.86624 0.84442 1.40 0.82720 1.50 0.81365 1.60 0.80308 1.70 0.79493 1.80 0.78880 1.90 0.78433 2.00 0.78126 0 7 = 0.05-2.0.
,
I
,
1
J
2
4
6
8
IO
TIME ( t / R C ) Figure 6. Composite plot of predicted k / K ' , ic/K', and iF/K' vs. potentiostatic-diffusion limited case ( T = 0.0-1 0.0)
Table 111. Ratios of Currents for Diffusioli Limited-Potentiostatic Casea w ) / I ( n
'
for potentiosta-
flow in the direction of the arrow in Figure 2). Then, the potentiostat begins to supply more of the current, until it is supplying all the faradaic current and additional current to recharge the double-layer (positive charging current). Quantitatively, the theoretical model predicts a significant contribution to the measured current from the charging current for the case presented. Tables I11 and IV present the values of iF/iT, iC/iT, and iC/iF for various values of T . I t is obvious from Tables 111 and IV that the measured signal is seriously distorted for T < -4, and t h a t the charging current is still significant a t long times compared to the cell time constant. (The charging current is greater than 5% of the measured currents for T = 11, and is still more than 2% a t T = 25.)
T
t
I ( C ) / I (T )
I(C)/I(F)
-9.33779 -4.34230 -2.68019 -1,85144 -1.35606 -1.02739 -0.79400 -0.62016 -0.48604 -0.3797 3 -0.29365 -0.22276 -0.16 355 -0.11 352 -0.07085 -0.034 15 -0.00237 0.02529 0.04950 0.07077 0.10608 0.13376 0.15558 0.17280 0.18635 0.19692 0.20507 0.21120 0.21567 0.21874
-0.903 27 -0.81281 -0.7 2827 -0.64930 -0.57556 -0.50676 -0.44259 -0.382 78 -0.32707 -0.275 22 -0.22700 -0.18218 -0.14056 -0.10195 -0.0661 6 -0.03302 -0.00237 0.02595 0.05208 0.07616 0.11867 0.15442 0.18425 0.20890 0.22902 0.24521 0.25797 0.26776 0.27497 0.27998
T
for
The applicability of this model to experimental work is straightforward. First, the cell time constant ( R C ) should be determined under conditions as identical to those of the experiment as possible. Then, based on the time (in terms of t / R C ) ,the value of iF/iT can be determined (see Tables 111 and IV). The current measured a t that point can then be multiplied by this correction term to obtain the correTable IV. Ratios of Currents for Diffusion Limited-Potentiostatic Cased I(F)Il(T)
I(c)/I(T)
I(C)II(F)
2.00 0.78126 2.50 0.78036 3.00 0.79285 3.50 0.81063 4.00 0.82963 4.50 0.84782 5.00 0.86427 5.50 0.87867 6.00 0.89104 6.50 0.90157 7.00 0.91051 7.50 0.91810 8.00 0.92457 8.50 0.93012 9.00 0.93491 9.50 0.93907 10.00 0.94273 10.50 0.94596 11.00 0.94882 11.50 0.95140 12.00 0.95371 12.50 0.95581 13.00 0.95772 13.50 0,95947 14.00 0.96108 14.50 0.96256 15.00 0.96393 16.00 0.96638 17.00 0.96852 18.00 0.97040 19.00 0.97207 20.00 0.97355 21.00 0.97489 22.00 0.97609 23.00 0.97719 24.00 0.97819 25.00 0.97910 ' 7 = 2.0-25.0
0.21874 0.21964 0.20715 0.18937 0.17037 0.15218 0.13573 0.12133 0.10896 0.09843 0,08949 0.08190 0.07543 0.06988 0.06509 0.06093 0.05727 0.05404 0.05118 0.04860 0.04629 0.04419 0.04228 0.04053 0.03892 0.03744 0.03607 0.03362 0.03148 0.02960 0.02793 0.02645 0.02511 0.02391 0.02281 0.02181 0.02090
0.27998 0.28146 0.26127 0.23362 0.20536 0.17950 0.15705 0.13809 0.12228 0.10917 0.09829 0.08921 0.08159 0.07513 0.06963 0.06488 0.06075 0.05713 0.05394 0.05109 0.0485 3 0.04623 0.04414 0.04224 0.04050 0.03890 0.03742 0.03478 0.03250 0.03050 0.02874 0.02717 0.02576 0.02450 0.02334 0.02230 0.02134
I4NALYTICAL CHEMISTRY, VOL. 48, NO. 2, FEBRUARY 1976
sponding faradaic current free from induced charging effects. Potential-Step Chronoamperometry, Diffusion-Limited Faradaic Currents. The case described above considers diffusion-limited currents initiated by flash photolysis or pulse radiolysis under potentiostatic conditions. In this section, faradaic diffusion-limited currents will be considered, but under potential-step conditions. This involves initiation of the electrolysis reaction and the potential step simultaneously at t = 0. In addition to the charging current due to the step, a charging current also will arise from the faradaic current similar to that described in the preceding section. The true faradaic current can only be found after correcting for the contributions from both of these charging current sources. (A brief mention of this combined effect was reported by Booman (39), but his concern was with the error it caused in the controlled potential, rather than the resulting induced charging current.) It is sometimes desirable to monitor photolytic products by stepping the potential from one point on the diffusion plateau to another potential on the same plateau, simultaneously with the flash, in order to avoid electrode photoemission problems ( 4 0 ) . This type of experiment, assuming an “instantaneous flash”, corresponds rigorously to the theory developed below. The electrolysis experiment can also be initiated by a step in the absence of a flash. A necessary assumption here would be that the potential across the double layer reaches the diffusion plateau a t t = 0. Experimentally this might correspond to a potential step from the foot of the plateau to a potential well out on the plateau, a significant distance beyond El/z. This is necessary to ensure that the potential across the double layer reaches the plateau region in “negligible” time (even if it takes several time constants to reach the final potential), thus initiating the diffusion current “instantaneously”. This case would also include potentialstep experiments conducted with flash photolysis studies used to monitor the unphotolyzed photolyte after a flash (23). (For the experimental situation where double-layer charging prevents “instantaneous” attainment of diffusionplateau potentials, as studied by Berntsen (41), the expression for the faradaic current (Equation 3) could be modified to include the potential-dependent faradaic current in addition to the diffusion limited current.) Previous workers have minimized charging current contributions in potential-step chronoamperometry by restricting measurements to T > -4 ( 4 2 ) , or by employing potentiostats with positive feedback (35). The first approach throws away information a t short times and uses data at longer times which are not completely accurate. Also, as pointed out in the first case above, blank measurements cannot correct the data properly. (This fact has also been pointed out recently by Rodgers ( 4 3 ) . ) The positive feedback approach is the best, theoretically, assuming 100% compensation and appropriate electronics to provide “instantaneous” charging of the double layer, as only faradaic currents would be measured a t observable times. However, this ideal is not generally approached in practice. Thus, the theoretical approach described below is presented here in terms of a conventional 3-electrode potentiostat design, and this treatment should lay the foundation for further considerations incorporating the effects of alternative potentiostat characteristics (36). The theoretical approach for this case is the same as above. Referring again to Figure 2, Equations 3 to 7 still hold. For this case, however, the potential across the double layer before the experiment (E,) is not equal to the potential (Ef)after the start of the experiment (Le., after the
292
potential step). Continuing from Equation 7 :
Again considering only diffusion-limited current described by the Cottrell equation (iF = K l f i ) , then I F ( s )= K/&. Therefore,
Transforming back into the time domain
+ [EieVr]-
V c ( t )= [Ef-
T o find the total current, as in Equation 12, use the expression,
Thus, ,
1T =
(Ef-Ei)e-’
R
-1
+ m2Ke-1
o
d7 ex2dX
(25)
This can be simplified based on Equations 15 and 16, and by defining a parameter, y, where
Equation 25 then reduces to iT(r)
= K’[ye-r
+ F(T)]
(27)
The faradaic current is given in Equation 18, with the difference again equal to the charging current ye-T
‘I
+ F ( T )- 4
Before proceeding, the sign conventions should be clarified. According to Figures 1 and 2, for a cathodic step, AE will be positive, giving a positive y, while an anodic step gives a negative y. Also, the plus sign in Equations 27 and 28 is for a cathodic faradaic current, while an anodic reaction would necessitate a negative sign. The expression for faradaic current (Equation 18) would also be positive for cathodic reactions and negative for anodic reactions.
DISCUSSION A t this point the difference between the potentiostatic and potential-step cases becomes apparent. The last term of Equation 27 is the same as in Equation 17, representing the faradaic current present and its corresponding induced charging current. The first term on the right side of Equation 27 represents the additional charging current arising from the potential step. Thus, the total current measured from the potential-step experiment represents the sum of three contributions: the faradaic current, its related induced charging current, and the charging current due to the potential step. A review of the derivation indicates that this will always be the case, independent of the specific electrolysis mechanism chosen. T h a t is, the total current for a potential-step chronoamperometric measurement will be the sum of the total current for the corresponding po-
ANALYTICAL CHEMISTRY, VOL. 48, NO. 2, FEBRUARY 1976
CATHODIC S T E P / A N O D I C REACTION GAMMA VALUES A 10.0 B 5.0 c 1.0 D 0.1
CATHODIC S T E P / CATHODIC REAC'I ION GAMMA VALUES A 10.0 B 5.0 c 1.0 D 0.1 Y \
.-I-
-1.2,L
'
I
I
I
I
1
I
2
3
4
5
TIME (t/RC) Figure 7. Total current vs. T with y = 10.0, 5.0, 1.0, 0.1 for potential step-diffusion limited case (anodic reaction/cathodicstep), (y = 1.o, T = 0.0-5.0)
tentiostatic experiment and the charging current due to the step. The value of y in the generalized solution (Equation 2 7 ) , is indicative of the ratio between the current contribution from the step ( A E I R ) and the electrode reaction ( K / a ) .A large y indicates the major current factor is the step, while a small y indicates t h a t the faradaic process is more significant. T h e charging current due to the step may therefore be negligible a t T < 4 or significant a t 7 > 4, depending on the value of y. Obviously, one limiting case for Equation 21, when A E 0, is Equation 17, the potentiostatic case. As was indicated above in regard to sign convention, there are four possible combinations of anodic and cathodic reactions and potential steps. T h e total current vs. T for two of these combinations, for various values of 7,are given in Figures 7 and 8. Because the plot of the anodic reaction/ anodic step combination is the mirror image of the cathodic reaction/cathodic step combination, (same behavior but with the opposite sign), only the latter figure is presented. The cathodic reaction/anodic step combination exhibits mirror image behavior with respect to the anodic reaction/ cathodic step, and is not shown here either. Tables V and
-
l
I
I
I
2
3
4
I
5
TIME ( t / R C ) Figure 8. Total current vs. T with y = 10.0, 5.0, 1.0, 0.1 for potential step-diffusion limited case (cathodic reaction/cathodic step), ( 7 = 0.0-5.O)
VI give the specific current values and ratios for the cathodic reaction/cathodic step and the anodic reactionlcathodic step combinations, respectively, with their inverse combinations being of the same magnitudes but of opposite signs. Figures 9 and 10 give combined plots of the three current terms for y = 1.0. The application of this case to experimental work is accomplished in a similar manner as mentioned above. T h e only difference is t h a t the value of y must be known before the iF/iT ratio can be determined. In most cases, y can be estimated from the experimental data by comparison with the theoretical family of curves represented in Figures 7 and 8. For example, referring to Figure 7 , the 7 value a t which the total current crosses zero is indicative of a specific y. Equation 27 indicates that this value of y is equal t o F(r)-e', for the cathodic reaction/anodic step or its inverse case, for the value of 7 a t the point where the measured current equals zero. Specific application of this model to experimental cases will be described elsewhere ( 3 6 , 4 4 ) .
CONCLUSIONS Chronoamperometric experiments necessarily involve
Table V. Current Terms for Potential Step-Diffusion Limited Case, Cathodic Reaction/Cathodic Step y = 0.1 T
0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00
y = l
-
-
y=
10
I(F)/K'
I ( T ) /K '
I(C)IK'
I(T)/K'
I(C)/K'
I(T)IK'
I(C)IK'
1.11803 0.79057 0.64550 0.55902 0.50000 0.45644 0.42258 0.39528 0.37268 0.35355 0.28868 0.25000 0.22361 0.20412 0.18898 0.17678 0.16667 0.15811
0.47397 0.55499 0.58278 0.58549 0.57487 0.55704 0.53551 0.51240 0.48899 0.46607 0.36908 0.30317 0.25940 0.22933 0.20765 0.19123 0.17828 0.1677 2
-0.64 40 7 -0.23558 -0.06271 0.02647 0.07487 0.10060 0.11293 0.11712 0.11632 0.11252 0.08040 0.05317 0.03579 0.02521 0.01867 0.01446 0.01162 0.00961
1.21083 1.15828 1.07671 0.98989 0.90596 0.82811 0.75745 0.69411 0.63776 0.58788 0.41388 0.31966 0.26546 0.23156 0.20847 0.19153 0.17839 0.16776
0.09279 0.36771 0.43122 0.43087 0.40596 0.37168 0.33487 0.29882 0.26509 0.23432 0.12521 0.06966 0.04186 0.02744 0.01949 0.01476 0.01173 0.00965
8.57940 7.19116 6.01602 5.03385 4.21687 3.53886 2.97682 2.51118 2.12546 1,80589 0.86197 0.48450 0.32610 0.25387 0.21668 0.19455 0.17951 0.1 6817
7.46137 6.40059 5.37052 4.47483 3.71687 3.08242 2.55425 2.11589 1.75278 1.45234 0.67329 0.23450 0.10250 0.04975 0.02769 0.017 78 0.01284 0.01006
ANALYTICAL CHEMISTRY, VOL. 48, NO. 2, FEBRUARY 1976
293
-k
2.5r 1.5
i.
CATHODIC STEP/CATHODIC REACTION A
iF/K'
B
iT/K' ic/K'
C
0
iF/K'
A
B iT/K' C QK'
Y 1.5
I /A
-0.51
-1.5
CATHODiC STEP/ANODiC REACTION
I
2
3
-"'p
5
4
-3 L
TIME ( t / R C )
0
,
I
I
I
I
I
2
3
4
5
TIME ( t / R C )
Figure 9. Composite plot of k/K', ic/K' and iF/K' for potential stepdiffusion limited case (cathodic reaction/cathodic step), (y = 1.O, T = 0.0-5.0)
Figure 10. Composite plot of hlK', ic/K', and iF/K' for potential step-diffusionlimited case (anodic reaction/cathodic step), (y = 1.O, T = 0.0-5.0)
measurements of induced charging currents in addition to the faradaic currents even under "potentiostatic" conditions. The theoretical model described here provides the general mathematical relationships between the desired faradaic current and the experimentally measured total current. These have been calculated here specifically for a diffusion-controlled electrode process. Application of correction factors obtained from the theoretical relationships allows for more accurate determination of faradaic currents from potentiostatic and potential-step chronoamperometric experiments, so long as the mechanism of the electrode process and the related i - t expression are known. Theoretically, these corrections can be applied from time zero regardless of cell time constant. For cases where the form of the faradaic i - t behavior is not known in advance, the general correction approach implicit in Equation 1 is applicable, and more recent studies evaluate this alternative (36, 45). In this work, however, a general model for the prediction of total currents and relative contributions of faradaic and charging currents was desired. The relationships derived here indicate t h a t the magnitude of the charging current contribution, even under potentiostatic conditions, is significant at times up to twenty times the cell time constant for diffusion-limited faradaic currents. This observation has far-reaching consequences for the interpretation of chronoamperometric measure-
ments in photoelectrochemical experiments, and suggests that a re-evaluation of previous photopolarographic observations may be in order. I t is also appropriate to contrast our work with that of Delahay, Pilla, and co-workers (46-49). Although Delahay's work invovles the same concept of inseparability of faradaic and charging currents, his analysis, based on solution/electrode interface phenomena, results in a complex expression for the faradaic impedance where up to nine double-layer parameters must be known for evaluation. Our work is based on a simplified model and is directly applicable to most chronoamperometric experiments with the prior knowledge of only a minimum number of variables (usually only the cell time constant). Pilla's work ( 4 9 ) has included frequency domain analysis of potentiostatic measurements. One objective was to account for contributions of non-ideal potentiostat characteristics to overall potentiostat-cell response. His approach involves the definition of an equivalent circuit similar to that described in this work except that actual potentiostat characteristics are included. Moreover, all circuit components (including the faradaic elements) are described in terms of equivalent impedances. Data analysis according to Pilla's model allows evaluation of the faradaic impedance. Thus,
Table VI. Current Terms for Potential Step-Diffusion Limited Case, Anodic Reaction/Cathodic Step y = 1
y = 0.1 T
0.20 0.40 0.60 0.80 1.00
1.20 1.40 1.60 1.80 2.00 3.00 4.00 5.00 6.00 7.00 8.00
9.00 10.00
294
Y =
W)iK'
I(T)lK'
I(C)iK'
I(T)IK'
I(c)iK'
I(T)/K'
-1.11803 -0.79057 -0.64550 -0.5 5902 -0.50000 -0.4 5 644 -0.42258 -0.39528 -0.37268 -0.35355 -0.28868 -0.2 5 000 -0.22361 -0.2041 2 -0.18898 -0.17678 -0.16667 -0.15811
-0.3 1022 -0.4 2093 -0.47 30 2 -0.4956 3 -0.501 29 -0.49 6 8 0 -0.48619 -0.47202 -0.45 594 -0.4 390 1 -0.359 1 2 -0.29951 -0.25805 -0.2 2 88 4 -0.2074 7 -0.191 1 7 -0.17 826 -0.167 7 1
0.80781 0.36964 0.17248 0.06339 -0.00129 -0.04036 -0.0 6 3 6 2 -0.07 6 74 -0.08326 -0.0 8545 -0.07044 -0.04951 -0.03444 -0.0 247 1 -0.01848 -0.0 1439 -0.0 1159 -0.00960
0.42664 0.18236 0.02091 -0.09 123 -0.17 0 20 -0.22 5 7 2 -0.26426 -0.290 3 2 - 0.30717 -0.3 17 2 0 -0.3 1 4 31 -0.28302 -0.25 199 -0.2 2 66 1 -0.20664 -0.19086 -0.17815 -0.16767
1.54467 0.97293 0.66641 0.46779 0.32980 0.23071 0.15832 0.10497 0.06551 0.03635 -0.02564 -0.03302 -0.028 38 -0 .O2 24 8 -0.0 1766 -0.0 1409 --0 .O1148 -0.00 9 5 6
7.79521 6.21524 4.96021 3.95273 3.14071 2.48502 1.95512 1.52675 1.18052 0.90081 0.13377 -0.1 1818 -0.19 135 -0.20430 -0.19844 -0.187 84 -0.17704 -0.16 7 27
ANALYTICAL CHEMISTRY, VOL. 48, NO. 2, FEBRUARY 1976
10
I(C)iK'
8.91325 7.00681 j . 6 0 57 1 4.51175 3.64071 2.94146 2.37769 1.92204 1.55320 1.25437 0.42245 0.13 182 0.03226 -0,000 17
-0.0094 6 -0.01107 -0.01 0 3 7 -0.0 0 9 15
information equivalent t o t h a t provided by the analysis presented in this paper is obtained. However, the model presented here is directly applicable to time-domain data and is more appropriate for the kinds of experimental measurements normally made in flash photopolarography and other controlled-potential chronoamperometric techniques. To demonstrate this, experimental studies reported elsewhere ( 4 4 ) provide verification of the theoretical discussions presented here, and illustrate the application of this theoretical model in removing charging current contributions from experimentally measured currents. In addition, further work is in progress ( 3 6 ) ,to provide theoretical expressions for potentiostatic photocurrents with various electrolysis mechanisms (e.g., coupled first- and secondorder chemical reactions), and to provide experimental verification of the predicted relationships ( 4 5 ) .
ACKNOWLEDGMENT The authors would like to express their indebtedness to Jackson Harrar and Charles Pomernacki of Lawrence Livermore Laboratory for their helpful suggestions in the development of the theoretical model presented here. The authors would also like to acknowledge the helpful discussions with s. W. Feldberg and A. A. Pilla.
LITERATURE CITED (1) H. Berg and H. Schweiss, Monatsber. Dtsch. Akad. Wiss. Berlin, 2, 546 (1960). (2) H. Berg and H. Schweiss, Naturwiss., 47, 513 (1960). (3) H. Berg, 2.Electrochem., 65, 710 (1961). (4) H. Berg and H. Kapulia, 2.Elektrochem., 64, 44 (1960). (5) H. Berg and H. Schweiss, Nature, (London), 191, 1270 (1961). (6) H. Berg, 2.Chem., 2, 237 (1962). (71 . . K. E. Reinert and H. Berg, Monatsber. Dtsch. Akad. Wiss. Berlin, 4, 26 (1962). (8) H. Berg, Rev. Polarogr., 11, 29 (1963). (9) H. Berg, Abh. Dtsch. Akad. Wiss. Berlin, Kl. Med., 213 (1964) (10) H. Berg, Abh: Dtsch. Akad. Wiss. Berlin, Kl. Chem., Geol. Biol., 128 (1964). (11) H. Berg and K. Kramarczyk, Ber. Bunsenges. Phys. Chem., 68, 296 (1964). (12) H. Berg and H. Schweiss, Electrochim. Acta, 9, 425 (1964). (13) H. Berg, H. Schweiss, and D. Tresselt, f x p . Tech. Phys., 12, 116 (1964). (14) F. A. Gollmick and H. Berg, Ber. Bunsenges. Phys. Chem., 69, 196 (1965).
(15) H. Berg, 2.Anal. Chem., 216, 165 (1966). (16) H. Berg, Nectrochim. Acta, 13, 1249 (1968). (17) H. Berg, H. Schweiss, E. Stutter, and K. Weller, J. Nectroanal. Chem., 15, 415 (1967). (18) S. P. Perone and J. R. Birk, Anal. Chem., 38, 1589 (1966). (19) J. R. Birk and S. P. Perone. Anal. Chem., 40, 496 (1968). (20) H. E. Stapelfeldt and S. P. Perone, Anal. Chem., 40, 815 (1968). (21) H. E. Stapelfeldt and S. P. Perone, Anal. Chern., 41, 623 (1969). (22) H. E. Stapelfedlt and S. P. Perone, Anal. Chem., 41, 628 (1969). (23) S. P. Perone and H. D. Drew, in "Analytical Photochemistry and Photochemical Analysis: Solids, Solutions, and Polymers", J. Fitzgerald. Ed., Marcel Dekker. New York, N.Y., 1971. (24) G. L. Kirschner and S. P. Perone, Anal. Chem., 44, 443 (1972). (25) R . A. Jamieson and S.P. Perone, J. Phys. Chem., 76, 830 (1972). (26) J. I. H. Patterson and S. P. Perone. Anal. Chem., 44, 1978 (1972). (27) J. I. H. Patterson and S. P. Perone, J. Phys. Chem., 77, 2437 (1973). (28) E. Stutter, J. Nectroanal. Chem., 50, 315 (1974). (29) M. Gratzel and H. Henglein, Ber. Bunsenges Phys., Chem.. 77, 2 (1973). (30) M. Gratzel, A. Hengiein, and K. M. Bansal. Ber. Bunsenges Phys. Chem.. 77, 6 (1973). (31) /bid., p 11 (32) A. Henglein and M. Gratzel, ibid., p 17. (33) K. M. Bansal, M. Schoneshofer. and M. Gratzel, Z.Naturforsch., Teil B, 28, 528 (1973). (34) K. M. Bansal and A. Hengiein, J. Phys. Chem., 78, 160 (1974). (35) R . R . Schroeder and I. Shain, Chem. lnstrum.. 1, 233 (1969). (36) S. S. Fratoni, Jr.. and S. P. Perone, in preparation. (37) M. E. Van Valkenburg, "Network Analysis", Prentice-Hall. Englewood Cliffs, N.J., 1964. (38) F. B. Hiidebrand, "Advanced Calculus for Applications", Prentice-Hall, Englewood Cliffs, N.J., 1962. (39) G. L. Booman and W. 8 . Hoibrook. Anal. Chem., 37, 795 (1965). (40) R. Baldwin and S. P. Perone, unpublished results. (41) J/ H. Berntsen, Ph.D. Thesis, University of Wisconsin, Madison, Wis., 1974. (42) J. E. Mumby and S. P. Perone, Chem. lnstrum., 3, 191 (1971). (43) R. S. Rodgers, Anal. Chem., 47, 281 (1975). (44) K. F. Dahnke, S.S. Fratoni, Jr., and S. P. Perone, Anal. Chem., 48, 296 (1976). (45) K. F. Dahnke and S. P. Perone, in preparation. (46) P. Delahay, J. Phys. Chem., 70, 2373 (1966). (47) P. Delahay and G. Susbielles, J. Phys. Chem., 70, 3150 (1966). (48) K. Holub, G. Tessari, and P. Delahay, J. Phys. Chem., 71, 2612 (1967). (49) A . A. Pilia, in "Electrochemistry, Calculations, Simulation, and Instrumentation'', Mattson, Mark, and MacDonald. Ed., Marcel Dekker, New York. N.Y., 1972.
RECEIVEDfor review May 5 , 1975. Accepted September 22, 1975. This work was supported by Public Health Service Grant No. CA-07773 from the National Cancer Institute and by the Office of Naval Research, Contract N00014-75'2-0874. One of the authors, S.S.F., also received the Procter & Gamble Fellowship in the Department of Chemistry a t Purdue University for the 1974-75 academic year.
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