1975. (6) J. H. Christie, Anal. Chem., in press. 17) R. S. Nicholson and M. L. Olmstead. “Comouters in Chemistrv and Instrumentation, Electrochemistry,” Vol. 3,J. S: Mattson. H. B. Mbrk, Jr., and H. C. MacDonald, Jr.. Ed., Marcel Dekker. New York, N.Y., 1972, p 119. (8) W. T. De Vries, J. Electroanal. Chem., 9, 448 (1965). (9) W. T. De Vries and E. Van D a h , J. Electroanal. Chem., 14, 315 (1967).
RECEIVEDfor review October 20, 1975. Accepted February 2, l976. This work was supported in part by the National Science Foundation under Grant No. MP575-00332 and by the Office of Naval Research under Contract N00014-67A-0299-0007.
Fundamental and Second Harmonic Alternating Current Cyclic Voltammetric Theory and Experimental Results for Simple Electrode Reactions Involving Solution-Soluble Redox Couples Alan M. Bond* and Roger J. O’Halloran Department of Inorganic Chemistry, University of Melbourne, Parkville, Victoria 3052, Australia
lvica Ruzic‘ and Donald E. Smith’ Department of Chemistry, Northwestern University, Evanston, 111. 6020 7
A quantitative theoretical and experimental study of fundamental and second harmonic ac cyclic voltammetry is presented for a simple electrode reactlon lnvolvlng solutlonsoluble redox forms, and rate control by diffusion and/or heterogeneous charge transfer. Rate laws are presented for the ac cycllc responses at stationary planar and spherical electrodes, and their predictions surveyed. Experimental data for several redox couples with widely varying ks values are found to confirm detailed predictlons of the theory. Particular attention is paid the sltuation where the dc process Is non-Nernstian, under which conditions varlatlon of the kinetic status of the dc process leads to some lnteresting and useful effects in the ac observables. Some novel bases for characterizing the heterogeneous charge transfer rate parameters are revealed for the cyclic ac experiment. It is concluded that fundamental and second harmonlc ac cyclic voltammograms both complement and, in many cases, provide more sensitive and convenient insights about the electrode reaction than wldely-used conventional (dc) cycllc voltammetry.
Conventional (dc) cyclic voltammetry probably is the most widely used electrochemical relaxation method for studying electrode processes. The use of a triangular voltage at stationary electrodes allows both the oxidation and reduction pathways to be studied conveniently from one experiment, and quantitative theory has been extensively developed in a readily implemented form (1-13). However, the predominant use of the method remains qualitative (e.g., see references 14-25), because of restrictions inherent in the readout format ( 1 4 ) .Correction for charging current, particularly a t high scan rates, and data analysis from an asymmetric peak-shaped curve present two of the more important difficulties discouraging more widespread quantitative use of the method. The extension from dc to ac polarography has been extremely fruitful in terms of quantitative studies of electrode processes a t a dropping mercury electrode (DME) On leave from the Center for Marine Research, Ruder Boskovic Institute, Zagreb, Yugoslavia, 1972-75. 872
ANALYTICAL CHEMISTRY, VOL. 48, NO. 6. MAY 1976
(26-28). It follows that the same type of extension with stationary electrodes should be similarly advantageous, notwithstanding the fact that stationary electrode ac voltammetry has received relatively little attention. Recently (29, 3 0 ) ,data were presented to demonstrate that superposition of a sinusoidal alternating potential onto the triangular wave “dc” voltage ramp, a technique referred to as ac cyclic voltammetry ( 3 0 ) ,did indeed appear to exhibit the expected merits, relative to dc studies undertaken at stationary electrodes. All the well-known, attractive features for rapidly obtaining qualitative data under dc conditions, such as qualitatively easily interpretable readout, are retained with the ac mode. In addition, the availability of the dual time domain (dc scan duration and ac period) (26, 28) plus additional measurement modes, such as phase-selective detection (26-30), second harmonic detection (26-30), and phase angle measurements (26, 28), appear to provide greater scope for obtaining high-quality quantitative data. In particular, the methodology allows for discrimination against charging current and results in well-defined, symmetrical voltammograms, which in both cases represent considerable improvement over the situation existing in dc cyclic voltammetry. Our interest in ac cyclic voltammetry, therefore, is stimulated to some extent because we think the above-cited advantages should make it preferable to the dc approach in many instances. Also important is the simple consideration that ac cyclic experiments represent a natural approach any time ac data are desired a t a stationary electrode. Finally, an important incentive arises because the dc and ac readout formats often will be complementary, providing more information than either individual data format ( 2 8 ) . In this regard, it is of significance to recognize that modern instrumentation makes possible simultaneous measurement of the dc and ac (fundamental and second harmonic) responses (28, 31-33). Thus, access to both dc and ac data can be considered the natural result of a single experimental run. The concept of what we define as ac cyclic voltammetry appears to have originated with Okamoto (34-38),who developed instrumentation (with Saito) (34) and theory (35, 36) for the case where a small amplitude square-wave alternating potential is superimposed on a triangular wave dc
potential scan. The technique was referred to as “Oscillographic Square Wave Polarography”. The Okamoto theory addressed simple, single-step Nernstian (reversible) and non-Nernstian (quasi-reversible and irreversible) electrode processes, using an approximate solution to the dc component of the boundary value problem (35, 36). Significant influences of the charge transfer coefficient and heterogeneous rate constant on predicted ac wave characteristics were reported. Satisfactory agreement between theory and experimental results were obtained only for the initial, forward (cathodic) potential sweep. Characteristics of the ac response on the reverse (anodic) sweep deviated significantly from predictions, particularly with amalgam-forming systems (38). A significant influence of the triangular wave’s switching potential on the ac response was indicated and discussed. The possibility of “interaction” between the ac signal and the dc triangular wave scan was mentioned briefly. Later, a theory and experimental results were developed by Shain and Underkofler (39) for Nernstian electrode reactions with a small amplitude sinusoidal potential superimposed on a linear sweep dc potential (i.e., forward scan only). This work more clearly defined restrictions arising from possible interactions between the dc potential sweep and the applied ac perturbation. Treatment of the non-Nernstian case, or other more complicated mechanisms has not been given for ac cyclic voltammetry with a sinusoidal ac potential. However, such rate laws are readily derived using guidelines developed for conventional ac polarography (constant dc potential) and are conveniently adapted to cases where dc potential programs are characterized by linear or triangular wave temporal variations, as will be shown below. We present here theoretical rate laws and their verification for fundamental and second harmonic ac cyclic measurements with the simple electrode reaction O+ne*R
(1)
where both redox forms are soluble in the solution phase. Heterogeneous charge transfer kinetic behavior ranging from the purely Nernstian case through the quasi-reversible and irreversible domains of non-Nernstian behavior is encompassed by the rate laws presented. We employ the framework of stationary plane and sphere electrode models, the “slow scan limit” (26, 39, 40), and the usual assumptions invoked in conventional ac polarographic theory (small amplitude perturbations, negligible migration effects, etc.) (26,27). We exclude from consideration here the closely-related amalgam-formation case because it is attended by such profound electrode curvature effects that even qualitative response characteristics differ substantially from those exhibited by solution-soluble redox couples. The amalgam formation case will be addressed in a separate communication. Chemical systems and instrumental conditions selected to match the conditions assumed in the theory (solution soluble redox couples, no coupled chemical reactions) were used in the experimental work. Data treatment strategies which are specific to ac cyclic voltammetry were emphasized somewhat, relative to the more conventional modes of extracting heterogeneous rate parameters from ac polarographic data (cot4 - w ~ profiles, / ~ etc.) (26-28, 31). We are able to report excellent consistency between experimental observations and predictions of the theoretical rate laws.
THEORY A key assumption invoked in the theoretical rate law presented here is that the dc scan rate is sufficiently slow, relative to the applied ac potential fluctuations, that the
time derivative of the dc sweep (scan rate, u ) is negligible relative to the corresponding characteristic for the ac potential. That is, one requires that WAE
>> U
(1)
This condition was discussed and invoked by Shain and Underkofler (39) and was implicit in the derivation of Okamot0 (35, 36). It will be referred to as the “slow scan limit”. It amounts to a requirement that there is insignificant overlap of the Fourier spectra associated with the “dc” and “ac” waveforms, with regard to either the input potentials, or the current responses. Such conditions are conveniently achieved experimentally and greatly simplify the theoretical problem by enabling separation of the dc and ac parts of the boundary value problem in the manner described for conventional ac polarography where a constant dc potential (zero scan rate limit in theory presented here) is assumed (26, 28, 41, 42). A theoretical development which is not confined by the slow scan rate restriction has been achieved in our laboratories to allow mapping the limits of validity of the slow scan rate approximation more quantitatively than provided by Equation 1. This work will be communicated when the investigation is completed. By invoking the slow scan rate restriction, along with other assumptions normally associated with ac polarographic theory, one may progress through the solution of the boundary value problem for Mechanism I following well-documented procedures (26, 41, 42) without making explicit assumptions regarding the applied dc potential program. The mathematical procedures differ trivially from those discussed recently in the literature, where quantities related to the dc polarization process are expressed in terms of depolarizer surface concentrations (41, 42). These exercises will not be repeated here. Results one obtains for the fundamental and second harmonic responses are as follows.
Z ( w t ) = Z(w),,,F(t)G(w)
sin (at
+ 4)
Z ( 2 w t ) = Z ( B w ) , , , F ( t ) G ( 2 w ) W ( w ) sin ( 2 w t Z(L)rev
=
+ 42)
n 2F2AC&(1W D0)1/2AE
4RT cosh2
(i)
1 = 1,2
4
-2
) - - - ( 2=L
= cot-1 (1
(ale;
+
F)
);e:
(10)
ae-J - fi 1 e-J
+
ANALYTICAL CHEMISTRY, VOL. 48, NO. 6, MAY 1976
873
[
L
42
= cot-1
(y+
1)
2u1/2
L- (x+l)P
+P
]
(13)
(14)
Notation definitions used above are conventional (26-28, 41, 42) and are tabulated at the end of this report. These expressions are made applicable to a specific mode of control of the dc polarization process by computing the appropriate relationships for the dc surface concentration com, the type of dc potential proponents, and C R * = ~for gram employed (constant, linear sweep, single and multicycle triangular wave, etc.) and the operative electrode geometry. For present purposes, we have computed and C R , , ~by digital simulation procedures (7, 8 ) for a symmetrical triangular wave dc potential “scan” with stationary plane and stationary sphere electrodes. Simulation algorithms employed for the stationary plane are those of Feldberg (7). The stationary sphere case was treated with the aid of the algorithm,
C(t
+ At,Z) = C(t,Z) + DD. TR(Z)[C(t,Z+ 1 ) -
C(t,Z)]- DD. T L ( Z ) [ C ( t , Z) C(t,Z - l ) ] (20)
for Z > 1. The simulation constant DD was chosen as 0.32 to ensure a stable solution. Here Z represents the number of space increments from the electrode surface, and the quantities T L ( I ) and T R ( I )are:
TL(Z)= 3 / { R S U M [ 1 / ( 1- 1 / R S U M ) 2 - 11
+ 11
T R ( Z ) = 3 / ( R S U M [ 1- ( 1 - 1 / R S U M ) 3 ] ) RSUM = R
+Z
(21) (22) (23)
where R is the radius of the electrode measured in units of the space increment. In order to obtain sufficiently accurate results, the potential increment for each time loop was set to 0.2 mV, corresponding to a value of approximately 0.008 for the Feldberg dimensionless parameter, n F A E / R T . The surface boundary value problem was treated in the manner proposed by Feldberg ( 7 ) without additional change. The simulation program yielded results for dc voltammograms at a stationary sphere which were in agreement with the Shain-Nicholson results (1). The FORTRAN programs used in this work are available from the authors on request.
EXPERIMENTAL Chemicals. Metal complexes and reagents used in this work were prepared and purified by standard literature methods. Concentrations of electroactive species were millimolar and solutions were degassed with argon. Instrumentation. Either a Princeton Applied Research Corp. (PAR) Model 170 Electrochemistry System, or the Model 174 Po874
ANALYTICAL CHEMISTRY, VOL. 48, NO. 6, MAY 1976
larographic Analyzer was used to record the cyclic voltammograms reported in this work. A PAR Model 129A Two Phasenector Lock-In Amplifier was used in conjunction with the Model 174 Polarographic Analyzer for ac measurements. With the 170 system, the second harmonic response was obtained by multiplication of the reference signal as described elsewhere ( 4 3 ) . Circuitry modifications used to record the ac cyclic voltammograms are the same as before (29, 30), except that the time constants present in the commercially available instrumentation were removed to eliminate distortion which otherwise occurs at the faster scan rates. Mechanical damping provided by the X-Y recorder and/or an external low pass filter eliminated high frequency noise components. Positive feedback circuitry was used in all measurements to minimize iR drop effects. The amplitude of the alternating potential used in this work was 10 mV peak-to-peak. Frequencies employed are stated in the text and tables. Other experimental parameters are indicated below as necessary. Electrochemical Cell. All work was performed using acetone0.1 M tetraethylammonium perchlorate (TEAP). Ag/AgCl in 0.1 M LiC1-acetone was used as the reference electrode, and platinum wire provided the auxiliary electrode. The working electrodes were either platinum, constructed as described by Adams (14) to ensure linear diffusion, or a slowly growing dropping mercury electrode (natural drop life greater than 10 s). The synchronization of the dc potential sweep and mercury drop life are described in the literature (30).
RESULTS AND DISCUSSION Theoretical Predictions. Pure Nernstian Case. When the electrode reaction is sufficiently rapid that Nernstian (reversible or diffusion-controlled) conditions exist on the dc and ac time scales, for planar diffusion one has G(w) = G(2o) = F ( t ) = cot4 = -Cot42 = 1
W(w)= sinh
(i)
/cosh
(24)
(f,
The consequence is that Equations 2-13 reduce to the wellknown expressions for the so-called “reversible” fundamental and second harmonic ac polarographic waves, which were originally derived using the constant dc potential assumption. That is, as Shain and Underkofler (39) and Okamot0 (35, 36) have previously shown for the fundamental harmonic case, the rate law for these conditions is independent of the nature of the applied dc potential program, as long as the slow scan rate limit is valid. The ac responses depend on the dc potential’s value, but not on how the latter is reached. Thus, the ac response vs. potential profiles (current amplitude and phase angle polarograms) will be independent of whether the scan direction is positive or negative, how many triangular wave cycles have ensued previously, the choice of initial and switching potentials characterizing the triangular wave dc scan, and the scan rate. This behavior contrasts sharply to properties of the Nernstian dc cyclic voltammetric response which is sensitive to all of the foregoing experimental parameters. One concludes that, with Nernstian systems, planar diffusion and slow scan rates (Equation l ) ,well-known data analysis procedures advanced for conventional (constant E d c ) fundamental and second harmonic ac polarography (26, 27) are applicable without modification to the corresponding modes of ac cyclic voltammetry. Curvature associated with spherical diffusion alters the expression for F ( t ) under Nernstian conditions (Equation 24), while other relevant quantities such as G(w), G(2w), W(w), cot4, and cot42 are unchanged for electrode radii and alternating potential frequencies normally employed (44-46). For a general dc potential program with Nernstian behavior and diffusion to a stationary sphere, one obtains for F ( t ) ( 4 4 )
where &c(t) is the applicable dc current function
Thus, strictly speaking, some of the foregoing conclusions might be altered a t spherical electrodes. However, casual inspection of Equation 27 leads to the conclusion that, as with conventional ac polarography, the sphericity correction disappears when DO = D R , and is small for normally encountered differences in the oxidized and reduced form diffusion coefficients. Even with a factor of ten difference in diffusion coefficients ( D o = 10 D R = 1.0 X cm2 s-l), we compute that there will be a difference of only 0.6% between the fundamental harmonic ac peak currents on forward and reverse scans, introduced by the sphericity effect. Such negligibility relative to experimental error characterizes the spherical diffusion perturbation on other properties, such as peak potentials, scan rate and switching potential dependences, etc. One concludes that, for the case in question, sphericity effects may be ignored. We emphasize the importance of the existence of solution phase-soluble redox couples in combination with Nernstian conditions in reaching this conclusion. Dramatically different conclusions are reached with amalgam formation and/or with non-Nernstian conditions. DC Nernstian Case. We now consider the situation where the electrode process is sufficiently rapid to achieve Nernstian behavior in the dc sense, but the response to the ac perturbation is non-Nernstian (quasi-reversible). Thus, C(w), G(2w), W ( o ) ,cot$, and cot42 no longer assume the simplified forms defined by Equations 24 and 25, and the general relationships (Equations 5, 8-11, 13) must be used for these ac responsive terms. However, the Nernstian limit for the dc responsive functions, F ( t ) and F 2 ( t ) , is still applicable. AC quasi-reversibility yields phase angle shifts, attenuated alternating current amplitudes, and broadened ac waves, relative to the pure Nernstian case. These effects are quantitatively identical to those described for conventional ac polarography (26-28), so that familiar, well-established data analysis procedures are applicable to evaluating k , and CY.Aside from these alterations in wave shape and magnitude, other properties described above for the pure Nernstian case are retained-Le., the ac responses are independent of scan direction, scan rate, number of preceding scans, and the switching and initial potentials. In other words, one concludes that a cyclic ac voltammogram of a system which is Nernstian i n t h e dc sense will yield overlapping ac waves on successive scans, even if switching potentials, and/or scan rates are altered in the course of the dc potential cycling, regardless of the kinetic status of the ac process. Non-Nernstian DC Behavior. The foregoing discussion indicates that the rate laws for ac cyclic voltammetry with the specified conditions are identical to those which have been presented long ago for normal ac polarography. The conclusion that the cyclic sweep mode generates nothing new in ac response properties is quickly altered when one considers conditions where the dc process is non-Nernstian. Under these conditions, the ac responses’ independence of the dc potential program is lost, and one observes that the ac wave shape, position, and magnitude can de-
pend markedly on scan rate, scan direction, number of preceding scans, the switching potential, the initial potential, and electrode sphericity. As a consequence, an entirely new dimension is introduced into ac voltammetric observables, providing some novel possibilities for system kinetic characterization. These new aspects of the ac response arise because dc quasi-reversibility invalidates the simple relationships for the rate law terms which respond to the dc process, F ( t ) and F2(t). Equations 24, 25, and 26 must be replaced by the general relationships of Equations 7, 9-12 in which the dc surface concentrations are explicitly represented. These surface concentrations and, more importantly, their deviations from Nernstian values, are markedly dependent on the manner whereby the dc potential is scanned. Under these conditions, the time-scale duality of the ac voltammetric experiment is abundantly manifested. Figures 1-3 provide illustrations of the predicted dependence of the fundamental harmonic response on dc scan direction and rate for a variety of k, and a values. A marked divergence between the fundamental harmonic ac voltammetric wave on the forward and reverse dc scans is evident, and is dependent significantly on CY (Figure l), k , (Figure 2), scan rate (Figure 3), and switching potential (not shown). Despite the diversity of results shown in Figures 1-3, one common feature is manifested in each computational result. Specifically, a point is observed where the ac waves associated with the forward and reverse scans cross over. This “cross-over point,” E,,, is observed to depend markedly on a (Figure I), but is totally insensitive to essentially all other experimental and electrochemical system variables. The E,, value is k,-independent (Figure 2) and the latter rate parameter influences it only to the extent that E,, becomes indiscernible if k, becomes too large (Nernstian dc process). Scan rate independence of E,, is shown in Figure 3. Its switching potential independence is not illustrated explicitly, although Figure 3 serves the purpose of showing that the switching point, relative to the ac peak, does not influence E,,. The choice of switching potentials in Figure 3 should be considered relatively poor in that the entire ac wave is not revealed for either the forward or reverse scans. Nevertheless, E,, is uninfluenced. E,, is related to a by the simple expression
E,, =
+ RT In nF 1 - C Y CY
Thus, E,, is a unique observable for characterizing CY from the viewpoint of both experimental convenience and theoretical simplicity. The origin of Equation 29, as well as Ecok independence of experimental conditions and system properties is readily understood by recognizing that an equivalent form of Equation 7 for planar diffusion is (26, 40 1
(30)
Equation 7 can be transformed to Equation 30 with the aid of the absolute rate expression (26)
and the well-known relationship (valid for planar diffusion) Neither Equation 31 nor 32 is limited to any particular dc potential function, so in this context Equation 30 is general (26, 40). Clearly, Equation 30 predicts that F ( t ) will reduce to unity, making the observed fundamental harmonic response independent of details of t h e dc polarization proANALYTICAL CHEMISTRY, VOL. 48, NO. 6, MAY 1976
* 875
I 0C
!I 4
i40
!/
d
0 20
0 IO
0 00
EOC- E:,2
-010
-020
iwltil
320
0 00
0 10
E ,
-E:,,
Figure 1. Predicted fundamental harmonic ac cyclic voltammograms with
~~~-
- 0 10
-020
0 0
3 20
i~ollil
000
EDC -E:,
-010
-020
lrollsi
non-Nernstian dc behavior: Illustration of effect of a
cm2 s-', v = 50 mV s-', w = 500 X 2 T s-', A€ = M, DO = 4, = 1.00 X Parameter values: n = 1.00, T = 2 9 8 K, A = 0.30 cm2, Cb = 1.00 X cm s-', CY = 0.50 ( A ) , 0.30 (B), 0.70 (C). (-) = forward (cathodic) scan. (- -) = reverse (anodic) scan. Alternating current units = 5.00 mV, ks = 4.4 X RT/(wt)/r?PA(2w/Jo)1'2 Cb A€ (normalized total fundamental harmonic faradaic current: planar diffusion)
-
cess a t the potential where ae-J
- /3
=0
(33)
Thus, Equation 33 defines the cross-over point, E c w Solving Equation 33 for the dc potential yields Equation 29. The identity and common origin of E,, to the cross-over point with conventional ac polarograms recorded a t different dropping mercury electrode column heights (40,44, 4 7 ) should be recognized. The potential defined by Equation 29 also is coincident with the c o u - Ed, profile's peak potential (26). Whereas with dc quasi-reversibility, sphericity effects on the ac response magnitudes and wave shapes will be significant (Figure 4), one finds that the E,, value is negligibly influenced by this perturbation. Rigorously, a spherical effect on E,, should exist because, with spherical diffusion, the expression for F ( t ) with dc quasi-reversibility becomes ( 4 4 )
tial separation, AE,, on log k , for a variety of conditions is illustrated in Figure 5 . Knowledge of a,from the E,, value, and the experimental conditions allows one to use plots such as shown in Figure 5 as working curves to deduce k, from AE,. This procedure for evaluating k, should be useful and accurate, unless k , becomes so large that dc reversibility is approached and AE, 0. Then, the more conventional methods for obtaining k, (26-28) must be applied. The sensitivity of AE, to sphericity effects is shown to be rather small from our computations. Figure 4 illustrates this point. Another observable made potentially useful by the onset of dc quasi-reversibility is the ratio of ac peak magnitudes for the forward and reverse scans, R,, where
-
R,
$dc(t - U ) exp (DRrZ'u) erfc(DA" r;' u112)du (34)
Thus, when Equation 33 is valid, Equation 27 will describe F ( t ) for spherical diffusion. Consequently, as indicated in the previous discussion of Equation 27, the spherical correction term will be negligible under most conditions by virtue of the requirement that diffusion coefficients for Species 0 and R must differ markedly. Thus, both the location of E,, (Equation 29) and the magnitude of the response at E,, will be negligibly influenced by electrode sphericity. Figures 1-3 suggest that with dc quasi-reversibility k, values may be deduced from the separation of the fundamental harmonic ac peaks associated with the forward and reverse dc potential sweeps. The dependence of this poten876
ANALYTICAL CHEMISTRY, VOL. 48, NO. 6, MAY 1976
peak current on forward scan peak current on reverse scan
(35)
Figure 2 indicates that a plot of R , vs. log k, will pass through a minimum, with R , = 1 for large k , (dc reversibility) and R , assuming an a-dependent limiting value as k, 0. Consequently, a given R , value may be consistent with two k, values. This duality can be resolved by using some additional observable, like AE,,but the necessity for resorting to this measure indicates that R , is a more complicated criterion for deducing k, than several other options. In addition, it is possible to observe two ac peaks on the reverse dc scan under appropriate conditions (Figure 6), further complicating this data analysis mode. We have emphasized in this section fundamental harmonic characteristics which are novel to ac cyclic voltammetry with non-Nernstian dc behavior. However, one should keep in mind that, for these conditions, most of the more conventional observables remain potentially useful for purposes of deducing k , and a. In particular, the ac cyclic voltammetry rate law shows that cot4 (Equation 8) follows the normal ac polarographic relationship, regardless of the status of the dc process. Thus, the cot4 cyclic voltammogram will have a fixed shape and magnitude for a particular set of rate parameters, regardless of scan rate (within slow scan limit), direction, switching potential, etc., even
-
S,'
=
c z n W 3
I
I
I
E ,,
- E:,2
(volts)
,E
-
(volts)
C
Figure 2. Predicted fundamental harmonic ac cyclic voltammograms with non-Nerstian dc behavior: Illustration of effect of ks Parameter values and notation: Same as Figure 18, except w = 400 X cm s-l ( A ) , 4.4 X cm s-l (B), 4.4 X
Prrs-’, and ks = 4.4 X cm s-l (C)
with non-Nernstian dc behavior. Consequently, cotd’s utility for characterizing k , and CY (26) is generally applicable in the present case, and represents an always-available alternative for kinetic characterization. Other observables which should be of use in this regard are the shape of the I ( & ) - u1/2profile at the peaks derived from the forward and/or reverse scans, the ac peak half-widths, and the like (26-28). One concludes that there are numerous independent bases in the fundamental harmonic ac cyclic voltammetric response for evaluating the relevant rate and ther-
modynamic parameters for Mechanism I. Cross-checking parameter values. obtained by several procedures using the fundamental harmonic response provides ample means for assuring that a correct mechanistic diagnosis has been achieved. From the precedent provided by rate laws for conventional ac measurements (26, 28, 48),it should come as no surprise that predicted second harmonic ac cyclic voltammograms can assume a wide variety of appearances with a quasi-reversible dc process. Such waves exhibit high sensitivity to experimental conditions and system parameters. A selection of examples is provided by Figure 7 , which supports the latter remark. As with the fundamental harmonic, onset of dc quasi-reversibility is accompanied by nonoverlapping waves on forward and reverse scans, sensitivity to scan rate (Figure 8),switching potential, etc. All of these effects are more pronounced in the second harmonic, relative to the fundamental, as is the dependence on k , and a (e.g., compare Figures 1A and 7 A , 1B and 7 B ) .One important consequence is a greater sensitivity of the second harmonic voltammogram to dc quasi-reversibility. This is illustrated in Figure 9, where conditions were selected which lead to almost indiscernible deviations of the fundamental harmonic voltammogram from behavior predicted for a Nernstian dc process (overlapping waves on forward and reverse sweeps, etc.). For these conditions, the second harmonic voltammogram shows distinct effects of dc quasireversibility. The greater sensitivity of the second harmonic response to the electrode reaction’s kinetic status and experimental conditions, and its relative insensitivity to double-layer charging current contributions, makes this observable a more accurate basis for assessing k , and a (26, 28, 48). However, it is also a more complicated data base from the viewpoint of fitting observables to the rate law. For example, with non-Nernstian dc behavior, we have shown above that the fundamental harmonic’s crossover point, E,,,, is a simply-interpreted, unique measure of a. This is not the case with the second harmonic response. A crossover of the second harmonic voltammograms associated with forward and reverse sweeps is observed. However, in most situations where it is easily discernible, this point will be a function of k , (Figure 7 A , D ) ,scan rate (Figure 7 C , E ) and other relevant parameters. Similarly, the second harmonic cot& lacks the insensitivity to the status of the dc process which characterizes the fundamental harmonic case. Treatment of second harmonic data in most cases must rely on the use of theoretical working curves (48). Once computed, working curves will provide sensitive and varied strategies for assessing k , and a from the second harmonic voltammograms. Among the useful observables for the purpose will be: a) peak-to-peak dc potential separations on either forward or reverse scans; b) peak magnitudes (relative or absolute) from either total or phase-selective second harmonic observations; c) the shift of the total second harmonic minima relative to a reference potential (dc E1/2, fundamental harmonic peak potential); d ) cot& a t a second harmonic peak. As with the fundamental harmonic, these observables can be plotted vs. log k , for various a values and a range of selected experimental conditions. Comparison between Theory and Experiment. Hendrickson, et al. (49, 5 0 ) , have shown that a t platinum, the electrode processes
+ e + M(R2dtc)sM(Rzdtc)a + M(RZdtc)a+ + e
M(R2dtc)3
are reversible or quasi-reversible in acetone with respect to a range of electroanalytical methods [M = Mn or Fe, Rzdtc ANALYTICAL CHEMISTRY, VOL. 48, NO. 6, MAY 1976
877
OS0 r
I
040 I
Figure 3.
Predicted fundamentalharmonic ac cyclic voltammograms with non-Nernstian dc behavior: Illustration of scan rate effect
Parameter values and notation: Same as Figure 1A, except w = 400 X 2a s-’, ks = 4.4 X
cm s-’, and v = 5.0 mV s-l (A), 50 mV s-l (B), 500 mV s-l
( C)
050r
E ,,
Figure 4.
- E2,:
(volts)
Predicted fundamental harmonic ac cyclic voltammograms
with non-Nernstian dc behavior: Illustration of electrode sphericity
effects Parameter values and notation: Same as Figure 1A. except w = 400 X 2 a cm s-‘, and electrode sphericity (*Tho) = 0.448 (A), 0.142 (B),0.0819 (C). where T = period of cyclic sweep in seconds
s-‘, k, = 4.4 X
= dithiocarbamate]. In the present work, we have examined a number of these dithiocarbamates and found that the oxidation process for the Mn(B2dtc)a (B = benzyl) complex to be completely reversible in the dc sense (dc cyclic peak-to-peak separation of 56 f 2 mV) for scan rates of 20 to 500 mV s-1. Figure 10 shows that this system exhibits complete overlap of the forward and reverse fundamental harmonic ac cyclics for many cycles. The electrode process is readily assigned to be reversible in the ac sense from the observations that the fundamental harmonic peak width a t half-height (“half-width”) is 90 f 1 mV for both forward and reverse scan directions a t all dc scan rates (20 to 200 mV s-l) and ac frequencies (100 to 800 Hz)employed. The prediction that the pure Nernstian response is independent of scan rate a t a given frequency was verified with this system. Similarly, the observed peak-to-peak separation in the second harmonic voltammogram exhibited the Nernstian value (68 f 2 mV) for the same range of conditions. Further, the peak heights of the fundamental and second harmonic voltammograms were both linear functions of @, the fundamental harmonic peak height is directly proportional to AE, and the second harmonic to A E 2 , as predicted. 878
ANALYTICAL CHEMISTRY, VOL. 48, NO. 6. MAY 1976
For the other dithiocarbamate electrode processes, observations were totally consistent with predictions for systems which are Nernstian in the dc sense, but not on the ac time scale. The quasi-reversible nature of the ac process is readily observed by broadening of the fundamental harmonic voltammogram half-width, an increase in peak-topeak separation of the second harmonic waves, and alternating current magnitudes which were notably smaller than for the pure reversible case. At the same time, these compounds yield forward and reverse scan ac voltammograms which overlap and are independent of scan rate manifesting the dc Nernstian condition except, in a few cases, a t the higher end of the scan rate spectrum where non-Nernstian dc behavior begins to appear. These observations are illustrated in Figure 10 by the ac cyclic voltammetric wave associated with the Mn(B2dtc)s reduction process, and by results obtained with the Fe(P2dtc)s (P = pipyridyl) oxidation process shown in Figure ll. Methods for calculating the k , values from these data are numerous, as indicated earlier. We employed: a) the ratio of the ac cyclic voltammetric peak heights for these compounds to that observed for the pure reversible response provided by the Mn(B2dtc)s complex oxidation; b) fundamental harmonic half-widths; c) second harmonic peak-to-peak separations. A summary of results of k , values calculated by these different approaches, and with various experimental conditions are presented in Table I for several dithiocarbamates. Mean values and uncertainties are given for the fundamental and second harmonic modes. Good agreement for k , values deduced from this range of procedures and experimental conditions is indicated by the acceptably small 12, value uncertainties and by the consistency of fundamental and second harmonic results. Bond et al. (51) recently have shown that a series of acetylacetone complexes exhibit quasi-reversible behavior using dc cyclic voltammetry a t platinum electrodes. Reversible dc cyclic behavior is found a t mercury. These systems, therefore, represent ideal examples with which to compare ac cyclic voltammetric experimental results under dc quasi-reversible conditions (platinum electrode) with theoretical predictions, because the system E$Z is known from the mercury data, thus reducing the unknown system parameters to k , and a. Figure 12 shows the fundamental harmonic cyclic voltammogram for reduction of Fe(acac)s a t a platinum electrode left standing in 1:l nitric acid for
0160
300
y
L
W
:
I \\
I
om.
200
Q
% 1I 0040
100 Epc -E:,,
,."
L"
-30
-4 0
i"Oll11
Figure 6. Predicted fundamental harmonic ac cyclic voltammogram with non-Nernstinn dc behavior: Illustration of conditions yielding double peak on reverse scan Parameter values: Same as Figure 1C,except k, = 4.4 X
i
300
0
10
-10 L
W
Q -30
-50
0
cm s-' and
w = 400 X 2 7 s-'
-10
-20
-3 0
-4c
log k, Figure 5. Theoretical working curves of experimental conditions
A f p vs. log ks for
various
Parameter values: Same as Figure 1, except w . (I and ks values shown on reFigure. A€, in mV. Note: In Figure 5B, k, values smaller than about sult in forward scan peak broadening greatly, and eventually becoming a shoulder on a "new" peak which forms at more negative potentials
several hours prior to its use in acetone. The predicted nonoverlap of the forward and reverse scans with ac cyclic voltammograms, and the cross-over point are evident. The separation in forward and reverse scan peak potentials (fundamental and second harmonic) and the fundamental
harmonic cross-over point were used as direct measures of k and a , respectively, as discussed above. Using platinum electrodes cleaned by simply wiping the electrode leads to the observation of smaller k , values than with nitric acid pretreatment. Figure 13 shows results with the far more irreversible reduction of Mn(acac)s at platinum with the fundamental harmonic ac cyclic mode, which illustrate decisively the scan rate independence of Eco. Table I1 summarizes our kinetic data for both the iron and manganese acetylacetone complexes, as well as for some Fe(1dtc)s (I = isopentyl) complexes, which were obtained under conditions of dc quasi-reversibility. Table I1 also provides an illustration of the satisfactory agreement we obtain between dc and ac cyclic results. At mercury electrodes, the k, values for Cr(sacsac)s, Fe(acac)3, and Mn(acac)3 are considerably larger than a t platinum and these parameters have been calculated by conventional ac polarography (51). We have obtained ac cyclic voltammetric data on these same compounds at a slowly growing mercury drop electrode. Table I11 provides a comparison between the reported (51) ac polarographic k , values and the ac cyclic results obtained in this work. While not perfect, the agreement is satisfactory. Figure 14 gives a plot of Z ( w t ) vs. w1/2 obtained in the ac cyclic experiments (forward scan) with the complexes in question. Comparing these data to Figure 1 of Reference 51 demonstrates the comparable nature of the observed admittance frequency response with the conventional and cyclic ac modes. All of the foregoing data show that ac linear sweep and cyclic voltammetric methods can be used to rapidly generate qualitative data on the degree of reversibility of an electrode process. Quantitative calculations of k , and CY also are implemented readily via the many possibilities emanating from the dual time domain properties of the ac cyclic method. If the electrode process is quasi-reversible on the dc time scale, CY values are particularly easily calculated from fundamental harmonic ac data using the observed crossover point and Equation 29, and k , from the forward and reverse scan peak separations. As the dc process becomes reversible or near-reversible, N is not accurately evaluated by this procedure, and second harmonic measurements, or fundamental harmonic cot4 observations are preferable. We wish to emphasize in particular that, while not implemented in the present program of experimental work which focused on the newer kinds of observables provided by the ANALYTICAL CHEMISTRY, VOL. 48, NO. 6, MAY 1976
879
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ANALYTICAL CHEMISTRY, VOL. 48, NO. 6, MAY 1976
:: 0
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Figure S. Comparison of predicted fundamental and second harmonic ac cyclic voltammograms: Effects of non-Nernstian dc behavior Parameter values and notation: Same as Figure 38, except k, = 4.4 X 10+ cm s-l
0.10 0.60 VOLTS VS. Ag/AgCl
0.80
0 . 7 0.6 0.5 VOLTS VS. Ag/'AgCl
L0.8
'
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REDUCTION WAVE
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Figure I O . Fundamental harmonic ac cyclic voltammogram of Mn(Bpdtc)s showing repeatability observed with successive dc scans System: 1.0 X lo-' M M n ( B g l t ~at ) ~R-0.1 M TEAP, acetone interface: 25 OC. Applied: 10 mV peak-to-peak, 100-Hz sinusoidal waveform superlmposed on 15 repetitions of a 100 mV s-' triangular wave dc potential scan. Measured: Total alternating current at 100 Hz
----v-&
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0,
-1
"I
-0.8 0 . 7 0 . 6 0. 5 0:4+ T.2 - 0 . 3 0 . 4 - b . SOY6 VOLTS VS. Ag/AgCl
Figure 11. Fundamental and second harmonic ac cyclic voltammograms of F e ( P * d t ~ showing )~ repeatability observed with successive dc scans at varying scan rates System: 5.0 X lo-'
M Fe(P&c)s at Pt-0.1 M TEAP, acetone interface: 25
OC. Applied: 10 mV peak-to-peak, 200-Hz sinusoidal waveform superimposed on: (A) and (C) = 20 repetitions of a 100 mV s-l triangular wave dc = 5 repetitions of a 50 mV s-l triangular wave followed potential scan: (8) by 5 repetitions each of 100 mV s-l and 200 mV s-l triangular waves: (0) = 2 repetitions at each scan rate indicated. Measured: (A) and (8)= Total alternating current at 200 Hz (fundamental harmonic). ( c ) = quadrature, and (0) = In-phase alternating current at 400 Hz (second harmonic) ANALYTICAL CHEMISTRY, VOL. 48, NO.
6. MAY 1976
881
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,
,
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VOLTS VI Ag1AgCI
Figure 13. Fundamental harmonic ac cyclic voltammograms of Mn(acac)s
System: 1.0 X M Mn(acac)s at Ft-0.1 M TEAP, acetone interface; 25 OC. Applied: 10 mV peak-to-peak, 100-Hz sinusoidal waveform superimposed on a triangular wave dc potential scan (single cycle) with scan rates of 20 mV s-' (A), 50 mV s-' ( E ) , 100 mV s-' (C), 200 mV s-' (D). Measured: Total alternating current at 100 Hz
ac cyclic strategy, in this context cot4 remains an excellent means for measuring k, and 01 values attending fast electrode processes, as in conventional ac polarography (2628).
NOTATION DEFINITIONS A = electrode area
Di = diffusion coefficient o f species i C, = concentration of species i
C:
= i n i t i a l concentration o f species i Ci,,, = dc component o f species i surface concentration
E o = standard redox potential in European convention Edc = dc component of applied potential
AE = amplitude o f applied alternating potential E;,* = reversible dc polarographic half-wave potential (planar diffusion)
F = Faraday's constant R = ideal gas constant T = absolute temperature n = number o f electrons transferred in the heterogeneous charge transfer step = dc faradaic current component I ( w t ) = fundamental harmonic faradaic current component I ( 2 w t ) = second harmonic faradaic current component @ = phase angle o f fundamental harmonic faradaic alternating curr e n t relative t o applied alternating potential $2 = phase angle of second harmonic faradaic alternating currentrelative t o applied alternating potential w = angular frequency u = scan rate t = time k, = apparent heterogeneous charge transfer rate constant a = charge transfer coefficient x = distance f r o m electrode surface ro = spherical electrode radius idc
ACKNOWLEDGMENT The authors thank R. L. Martin and A. F. Masters for generous donation of some of the compounds used in this investigation.
LITERATURE CITED (1) R S. Nicholson and I. Shain, Anal. Chem., 36, 706 (1964). (2) R. S. Nicholson and I. Shain, Anal. Chem., 37, 178 (1965). (3) M. L. Olmstead, R. G. Hamilton and R. S. Nicholson, Anal. Chem., 41, 260 (1969). (4) M. L. Olmstead and R . S. Nicholson, Anal. Chem., 41, 862 (1969). (5) R. S. Nicholson, Anal. Chem., 37, 667 (1965). ( 6 ) R. S. Nicholson and M. L. Olmstead, in "Computers in Chemistry and instrumentation", Vol. 2, J. s. Mattson, H. B. Mark, Jr., and H. C. MacDonald, Jr.. Ed.. M. Dekker, New York, 1972, pp 119-138.
882
ANALYTICAL CHEMISTRY, VOL. 48, NO. 6, MAY 1976
Flgure 14. Fundamental harmonic ac cyclic peak current frequency response spectra for Cr(sacsac)s, Fe(acac)3 and Mn(acac)s Systems: 1.0 X M Cr(sacsac)s (O),Fe(acac)s (W), and Mn(acac)$( 0 )at Hg-0.1 M TEAP. acetone interface; 25 OC. Applied: 10 mV peak-to-peak sinusoidal wave form with frequencies shown on Figure, superimposed on 200 mV s-' triangular wave dc potential scan. Measured: Peak total fundamental harmonic faradaic current on forward (cathodic) scan: current units = fiA, corrected for drop growth
(7) S. W. Feidberg, in "Electroanalytical Chemistry", VoI. 3, A. J. Bard, Ed., M. Dekker, New York, 1969, pp 199-296. (8) S. W. Feldberg, in "Computers in Chemistry and Instrumentation", Vol. 2, J. S. Mattson, H. E. Mark, Jr., and H. C. MacDonaid, Jr., Ed., M. Dekker, New York, 1972, pp 185-215. (9) S. W. Feldberg, J. Phys. Chem., 75, 2377 (1971). ( I O ) S. W. Feidberg and L. Jeftic. J. Phys. Chem., 76, 2439 (1972). (11) L. Nadjo and J. M. Saveant, Electrochim. Acta, 16, 887 (1971). (12) J. M. Saveant, C. P. Andrieux, and L. Nadjo, J. Electroanal. Chem., 41, 137 (1973). (13) C. P. Andrieux, L. Nadjo, and J. M. Saveant, J. Electroanal. Chem., 42, 223 (1973). (14) R. N. Adams. "Electrochemistry at Solid Electrodes". M. Dekker, New York, 1969. (15) F. A. Cotton and E. Pederson. J. Am. Chem. Soc.,97, 303 (1975). (16) F. A. Cotton and E. Pederson, horg. Chem., 14, 383 (1975). (17) F. A. Cotton and E. Pederson, lnorg. Chem., 14, 388 (1975). (18) F. A. Cotton, B. A. Frenz, E. Pederson. and T. R. Webb, lnorg. Chem., 14, 391 (1975). (19) F. A. Cotton and E. Pederson, Inorg. Chem., 14, 399 (1975). (20) K. M. Kadish, G. Larson, D. Lexa, and M. Momenteau, J. Am. Chem. Soc., 97, 282 (1975). (21) R . E. Dessy and R. L. Pohl. J. Am. Chem. SOC.,90, 1995 (1968). (22) R. E. Dessy and R. L. Pohl, J. Am. Chem. Soc., 90, 2005 (1968).
Table I. Comparison of k. Values Determined by Fundamental and Second Harmonic AC Cyclic Voltammetry with DC Nernstian Conditions Compound Mn"'(B2dtc)s B = benzyl Mn"'(C2dtc)a C = cyclohexyl Fe"'(P2dtc)s P = pipyridyl
k , (cm s-I)
Electrode process (at platinum electrode in acetone-0.1 M TEAP)
Fundamental harmonic"
+ +
Mn"'(B2dtc)s == M11'~(B2dtc)s+ e Mn"'(B2dtc)s e + Mn"(B2dtc)sMn1"(C2dtc)3 + MnIv(C~dtc)3+ e Mn"'(C2dtc)s e + Mn"'(C2dtc)sFe"'(P2dtc)s + FeIv(P2dtc)3+ e Fe"'(P2dtc)s e * Fe"(P2dtc)s-
+
+ +
Second harmonid' 25 (4 f 1) x 10-1 (5 f 2) x 10-1 (2 f 1) x 10-2 (1.0 f 0.5) 25
21
(3 f 1) x 10-1 (5 f 3) x 10-1 (1.5 f 0.5) X (8 f 2) x 10-1 11
+
a Computed, assuming a = 0.5, from fundamental harmonic peak half-width, and peak magnitude relative to reversible Mn(B2dtc)s oxidation with variety of scan rates and frequencies (6 individual k, determinations). Uncertainty = average deviation from mean value. b Computed, assuming a = 0.5, from second harmonic peak-to-peak separation, and peak magnitudes relative to reversible Mn(B2dtc)a oxidation with variety of scan rates and frequencies (6 individual k, determinations). Uncertainty = average deviation from mean value.
~~
~
Table 11. Comparison of ksValues Determined by DC and Fundamental Harmonic AC Cyclic Voltammetry with Non-Nernstian DC Conditions k , (cm s-l)"
Electrode process (at Pt-0.1 M TEAPacetone interface) Fe"'(I2dtc)a Fe1V(12dtc)3f+ e I = isopentyl Fe"'(I2dtc)s e Fer1(12dtc)3Fe"'(acac)s e + Fe"(aca&acac = acetylacetone Mn"'(acac)s e + Mn"(acac)s-
ffb,
DC cyclic
AC cyclic
(5 f 2) x 10-2
+
+ +
(5 f 1) x 10-2
AC cyclic 0.50 f 0.05
(1.8 f 0.5) X (1.3 .f 0.5) X
(2.0 f 0.5) X (1.3 f 0.5) X
0.50 f 0.05 0.50 f 0.05
(1.4 f 0.2) x 10-3
(1.5 f 0.2) x 10-3
0.50 f 0.05
" Computed from separations of peak potentials associated with anodic and cathodic sweeps, using Nicholson table ( 5 ) for dc case and theory presented here for ac data. Shows average values and uncertainties (av dev) derived from data at several scan rates (50,100,200 m V cm2 s-' ( 5 1 )was employed. Computed from fundamental harmonic s-l) and frequencies (200,400 Hz). Diffusion coefficient of 2.5 X cross-bver point.
Table 111. Comparison of ksValues Determined by Fundamental Harmonic AC Cyclic Voltammetry and AC Polarography
A. M. Bond, J. Nectroanal. Chem., 50, 285 (1974). H. Blutstein and A. M. Bond, Anal. Chem., 46, 1934 (1974).
D. E. Smith, in "Computers in Chemistry and Instrumentation", Vol. 2, J. S. Mattson, H. E. Mark, Jr., and H. C. MacDonald, Jr., Ed., M. Dekker, New York, 1972, pp 369-422. D. E. Glover and D. E. Smith, Anal. Chem., 44, 1140 (1972). k , (cm s-I) D. E. Glover and D. E. Smith, Anal. Chem., 45, 1869 (1973). Electrode process Y. Saito and K. Okamoto, Rev. Polarogr. (Jpn), 10, 227 (1962). (at Hg-0.1 M TEAPAC K. Okarnoto, Rev. Polarogr.(Jpn), 12, 40 (1964). acetone interface) AC cyclic" polarographyb K. Okamoto, Bull. Chem. Soc. Jpn, 36, 1381 (1963). K. Okamoto. Rev. Polarogr.(Jpn), 12, 50 (1964). Cr"'(sacsac)s e + Cr"(sacsac)321 21 K. Okamoto, Rev. Polarogr.(Jpn), 11, 225 (1964). (sacsac = dithioacetylacetone) W. L. Underkofler and I. Shain, Anal. Chem., 37, 218 (1965). (3 f 2) X 1.3 X lo-' Fe"'(acac)s e + Fe"(acac)aJ. R. Delmastro and D. E. Smith, J. Electroanal. Chem., 9, 192 (1965). I. Ruzic, D. E. Smith, and S. W. Feldberg, J. flecfroanal. Chem., 52, (acac = acetylacetone) 10-1 157 (1974). (10 f 2) X Mn"'(acac)s e == Mn"(acac)a6.8 X J. W. Hayes, I. Ruzic, D. E. Smith, G. L. Boornan. and J. R . Delmastro. J. 10-2 Electroanal. Chem.. 51. 269 (19741. H. Blutstein, A. M. Bond, and A . Norris. Anal. Chem., 46, 1754 (1974). Present work, using slowly growing mercury drop electrode. J. R. Delmastro and D. E. Smith, Anal. Chem., 38, 169 (1966). Computed using DCr(aacsac)g = 2.4 X cm2 s-', D F ~ (= ~ ~ ~ ~T. )G. ~McCord, E. R. Brown, and D. E. Smith, Anal. Chem., 38, 1615 cm2 s-l and 01 = 0.50. From Ref. 51. (1966). D M ~ = ( 2.6 ~ X~ ~ ~ ) ~ J. R. Delmastro and D. E. Smith, Anal. Chem., 39, 1050 (1967). H. L. Hung and D. E. Smith, Anal. Chem., 36, 922 (1964). T. G. McCord and D. E. Smith, Anal. Chem.. 40, 289 (1968). A. R. Hendrickson, R. L. Martin, and N. M. Rhode. lnorg. Chem., 13, 1933 (1974). R. Chant, A. R . Hendrickson. R. L. Martin, and N. M. Rohde. lnorg. (23) R. E. Dessy, R. Kornmann, C. Smith, and R. Haytor, J. Am. Chem. Soc., Chem., 14, 1895 (1975). 90, 2001 (1968). A. M. Bond, R. L. Martin, and A. F. Masters, lnorg. Chem., 14, 1432 (24) R. E. Dessy and L. Wieczorek, J. Am. Chem. Soc., 91, 4963 (1969). (1975). (25) R. E. Dessy, M. Kleiner, and S. C. Cohen, J. Am. Chem. Soc., 91, 6800 (1969). RECEIVEDfor review December 1, 1975. Accepted J a n u a r y (26) D. E. Smith, in "ElectroanalyticalChemistry", Vol. 1. A. J. Bard, Ed., M. Dekker, New York, 1966, pp 1-155. 19, 1976. This work was supported by g r a n t s from the Aus(27) M. Sluyters-Rehbach and J. H. Sluyters. in "Electroanalytical Chemistralian Research G r a n t s Committee and t h e National try", Vol. 4, A. J. Bard, Ed., M. Dekker, New York, 1970, pp 1-128. Science Foundation (MPS74-14597). (28) D. E. Smith, Crlf. Rev. Anal. Chem., 2, 247 (1971).
+
+ +
.
I
ANALYTICAL CHEMISTRY, VOL. 48, NO. 6, MAY 1976
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