1920
Anal. Chem. 1984, 56, 1920-1923
Kinetic Convolution Analysis of Cyclic Voltammetric Data Floyd E. Woodard,* Richard D. Goodin, and Patrick J. Kinlen Monsanto Company, 800 North Lindbergh Blvd., St. Louis, Missouri 63167
A new method for determlnlng the homogeneous rate constant from cyclic voltammetric data for EC irr systems is presented. The method, whlch Involves the numerlcal evaluatkm of an “EC kinetk convolutkn Integral”, does not requlre reversible charge transfer nor “total klnetlc” control as requlred by the correspondlng semiintegral method.
Nadjo and Saveant (1) have studied the variation with sweep rate of peak potentials in linear sweep voltammetry as a method for mechanistic analysis and for obtaining kinetic parameters for various reaction schemes including the EC irr scheme of interest here. O+ne+R (1) k
R-P
(2) They noted that the reaction can be assigned to various “kinetic zones” depending on which of the following phenomena control the overall reaction rate: diffusion, heterogeneous charge transfer, or the rate of the first-order homogeneous reaction. Assuming charge transfer is described by the Butler-Volmer equation Nadjo and Saveant have defied two dimensionless parameters [i.e., A = ko(DnFu/RT)-1/2and X = RTk/nFv] which can be used to specify kinetic zones. Their classification scheme for EC irr reactions will be used to compare the conditions under which various analysis techniques can be used to determine the homogeneous rate constant. Nadjo and Saveant (I) noted that in certain cases (where the charge transfer reaction is sufficiently rapid) the homogeneous rate constant can be extracted from a plot of peak potentials vs. log v (sweep rate). At low scan rates the electrochemical system must be operating in the KP zone (Le., controlled by the chemical reaction with Nernstian charge transfer), while at fast scan rates the operation point must switch to the DO zone (i.e., diffusion controlled Nernstian behavior). The homogeneous rate constant can be determined from the sweep rate corresponding to the intersection of the straight line plots of peak potential vs. log Y obtained in each kinetic zone. Imbeaw and Saveant (2)have shown that the homogeneous rate constant for an EC irr reaction can be obtained from convolutive potential sweep voltammetric data. Their equations (18 and 19 in ref 2) were derived for Nernstian charge transfer with total kinetic control by the homogeneous reaction (i.e., the KP kinetic zone). Woodard (3) has shown that the reversible charge transfer requirement can be removed allowing the homogeneous rate constant to be determined for a system operating in the KI zone (i.e., a stationary state in the concentration of R with quasi-reversible charge transfer). Whether the systems is operating in the KP or KI kinetic zones, before the homogeneous rate constant can be determined certain parameters describing the heterogeneous charge transfer step must be known: Eo if charge transfer is reversible and Eo, a,and ko with quasi-reversible charge transfer. These parameters can be obtained from CV data obtained at sweep rates sufficiently fast that the follow-up reaction occurs to a negligible extent (as evidenced by a semiintegral value which returns to zero during the reverse scan). 0003-2700/84/0358-1920$01.50/0
Both of the methods outlined above require data from several experiments in which the system is operating in more than one kinetic zone. At least one cyclic voltammogram must be obtained under stationary state conditions as concerns the concentration of R. We report here a method for analyzing cyclic voltammetric data which does not require Nernstian charge transfer nor “pure kinetic” conditions. Moreover, the homogeneous rate constant can be determined from a single cyclic voltammogram. A8 shown in Figures 1, 2, and 3 the method has been applied to nearly every kinetic zone (i.e,,DO, IR, QR, and KO) except those corresponding to a stationary state in the concentration of R. In practice, the technique is applicable to any CV in which an appreciable amount of R reacts to give P, provided at least a slight peak is evident in the return scan. This new method involves a numerical evaluation of an “EC kinetic convolution integral”, IEc. This integral is similar to the semiintegral except that the current data are not convolved with just a diffusion term (Le,, t-1/2) but also a kinetic expression (i.e., exp(-kt)). The treatment here is for a reduction process. The derivation of the oxidative analogue is straightforward. Theory. The EC kinetic convolution integral, IEc,is a function with a value proportional to the concentration of species R at the electrode (Le., CR(O,t). An outline of the derivation of this integral is given below. The differential equation describing the concentration of R as a function of time and distance from the electrode is
where planar diffusion is the sole mechanism responsible for mass transport. It is assumed that initially only species 0 is present in solution and that the initial potential has been chosen so that the reaction in eq 1 does not proceed in the forward direction. Thus the boundary conditions are cO(x,o) = Cob; CR(x,o) = 0
(4)
limX-.- co(x,t) = cob; 1irnx+- CR(X,t) = O (5) In addition, the slope of the concentration profile for R at the electrode is linearly related to the cell current i(t) = nFA&,(dCR(X,t) /ax),=, (6) Using the boundary conditions in eq 4 , 5 , and 6, differential eq 3 can be solved by use of Laplace transform techniques to obtain CR(O,t) = IEC/(nFADR112) (7) The kinetic convolution integral is defined as t
[ ( t - u)-l12i exp(-k(t
IEc=
- u))] du (8)
where the cell current i is a function of u, the integration variable. A numerical evaluation of I Ecannot ~ be obtained directly from eq 8 because of the point of singularity in the kernel at t = u;however this can be removed by an integration by parts I E C = 211-’12 t’/2 i(u = 0) exp(-kt) +
1’
2n-1/2
0
[(t - ~ ) l ’ ~ (+k idi/du) exp(-k(t -u))]du (9)
0 1984 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 50, NO. 11, SEPTEMBER 1984
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Figure 2. Simulated CV data (A) and the Corresponding kinetic conFlgure 1. Slmulated CV data (A) and the correspondlng EC kinetlc volution Integrals (6) calculated assumlng various values for the hoconvolution Integrals (E) for an EC irr system With reverslble and mogeneous rate constant k : (-----) 0.12 s-’,(-) 0.10 s-’,(---) 0.08 quasi-reversible charge transfer. General parameters: sweep rate) s-’. The simulated CV was generated assuming the same general = 0.1 V s-I, D o = D, = 1 X lo-‘ cm2s- , co = 1 X lo4 mol cmS, parameters as in Figure 1 wlth k o = 0.01 cm s-’(log A = 0.20) and c, = 0.0 mol ~ m - a~ = , 0.5, E o = 0.0 V, n = 1.0, T = 298 K, area k = 0.10 s-’ (log X = -1.59), thus corresponding to kinetic zone QR. = 1 cm2, Specific parameters: plot (-) k o = 10.0 cm s-’, log A The kinetic convolution Integral drawn with a solid line was calculated = 3.20, k = 0.1 s-‘,log X = -1.59, zone DO, k (determined from IEC) by use of the correct homogeneous rate constant. = 0,1001 s-‘: plot (---) k o = 0.001 cm si,log A = -0.80,k = 0.1 s-’,log X = -1.59, zone IR, k(determined from ZEc) = 0.1007 s-’. Figure 2, the homogenous rate constant can be determined If current measurements are made a t constant intervals (as by systematically varying k until a value is found which gives they were in this work), a numerical evaluation of IEC is IEc = 0 at ERo. In this work a one-dimensional simplex was straightforward. The dildu term can be obtained from CV used to find that optimum value for k. Note that as k passes data using the least-squares differentiation procedure dethrough the correct value, the value calculated for IECat ERO scribed by Savitzky and Golay (4). (A five-point first derivgoes through zero and changes sign. ative convolutingfunction was used in this work.) The integral Once the homogeneous rate constant is determined, pacan then be evaluated using the trapezoidal rule (e.g., ref 5). rameters describing the heterogeneous charge transfer reaction During the forward sweep of the cyclic voltammetric excan be determined from log plots involving both the semiinperiment, the electrode potential is made more negative and tegral and kinetic convolution integrals. The appropriate the concentration of R at the electrode surface increases. equations can be derived by substituting for C& = 0) and During the return scan as the potential becomes more positive, co(x = 0) in the Nernst or Butler-Volmer equations (these cR (x = 0) decreases until eventually at some potential, ERo, derivations are given in ref 3). The value for CR(X = 0) is given cR ( x = 0) becomes approximately equal to zero. We have in eq 7 while co(x = 0) is equal to found that provided Butler-Volmer conditions hold, a potential 200 mV positive of the peak potential for the return CO(X= 0) = (11 - I)/(nFADO’/’) (10) scan can be used as ERo-even if charge transfer is not rewhere I is the semiintegral and I]is the limiting value of the versible. This was determined from semiintegrals of simulated semiintegral at the cathodic plateau. CV data based on reaction 1 without complications from reaction 2. In that case the semiintegral is proportional to EXPERIMENTAL SECTION CR(X = 0). Plots of the kinetic convolution integral (e.g., see Electrochemical experiments on the cathodic cleavage of Figure 1)for simulated CV data show that this is also true tert-butyl p-toluate have been described in detail in two preceding when the consecutive homogeneous reaction is coupled to the papers (6, 7). The supporting electrolyte was 0.1 M tetraethylcharge transfer step. That is not surprising since in that case ammonium fluoroborate (Southwestern Analytical) in acetonitrile species R is not depleted just by the reverse direction of (Burdick and Jackson, Distilled in Glass). The reference electrode reaction 1but also by reaction 2. used Ag/O.l M AgNOS in acetonitrile (AgRE). The working Since I E C is proportional to CR(Z = 0), at potentials greater electrode (area = 0.091 cm2) consisted of a mercury film on a than or equal to EROwhere CR(X= 0) is approximately equal platinum substrate, while the auxiliary electrode was a coil of to zero, I E C must also equal zero. Thus, as is apparent in platinum wire in the same compartmentas the working electrode.
ANALYTICAL CHEMISTRY, VOL. 58, NO. 11, SEPTEMBER 1984
1922
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Flgure 3. Slmulated CV data (A) and the corresponding kinetic convolution integrals (9). The simulated CV's were generated assuming the same general parameters as in Figure 1 with k o = 1.00 cm s-l (Le., log A = 2.20) and k = 1.0 s-l (-), 0.1 s-' (---), and 0.01 8-l (-) corresponding to log X = -0.59 (KOzone), -1 -59 (DO zone), and -2.59 (DO zone), respectively. The values for k required to give I,, = 0 at ERowere 1.000 s-' (-), 0.1001 s-l (---), and 0.0102 s-l (-1.
- --
---
A microcomputer developed at the University of North Carolina (8)was used to obtain the experimentalcyclic voltammetric data,
to calculate EC kinetic convolution integrals, and to perform the simplex optimizations required to determine the homogeneous rate constants. Computer programs were written in interactively compiled BASIC (Digital Specialities, Chapel Hill, NC) which uses floating point numbers with six digit mantissas. Simulated cyclic voltammetric data were obtained from finite difference calculations run on an IBM 3081 computer.
RESULTS AND DISCUSSION To demonstrate the method presented above and to establish when the kinetic convolution technique can be used to determine homogeneous rate constants, simulated CV data were analyzed. A one-dimensional simplexlike optimization (9) procedure was used to determine the value for k which gives IEc = 0 a t ERO. The initial guess for k was 1 / ~where , T is the potential reversal time in the CV experiment. Shown in Figure 3A are several simulated CV's (sweep rate = 0.1 V s-l) where the value of the homogeneous rate constant was varied by 2 orders of magnitude (from 0.01 s-l to 1.0 s-l). In each case the homogeneous rate constants determined using the kinetic convolution integral (Figure 3B) were within 2% of the values used to generate the data. In terms of the dimensionless homogeneous rate parameter, X, these data correspond to -0.59 I log X I -2.59. Accurate determinations of the homogeneous rate constant are realized even when the charge transfer reaction is not reversible (see Figure 1). For a sweep rate of 0.1 V s-l and a homogeneous rate constant equal to 0.1 s-l, when the het-
-0.3 -3
-2.8
-2.6
E / V vs AgRE
-2.4
Figure 4. The cyclic voltammogram (A) and EC kinetic convolution Integral (9) obtalned for 1.05 mM tert-butyl p-toluate in 0.1 M tetraethylammonlum fluoroborate at a mercury film/platinum substrate electrode. Sweep rate (v) = 1.0 V s-'. k used to calculate the EC kinetic convolution integral = 10.6 s-'.
erogeneous rate constant was decreased from 10.0 cm s-l to 0.001 cm s-l, the values determined for the homogeneous rate constant were 0.1001 s-l and 0.1007 s-l, respectively. To demonstrate the usefulness of the kinetic convolution technique in the analysis of experimentally obtained data, the decomposition rate of the radical anion of tert-butyl p-toluate was determined. In previous publications (6,7)we have shown that this electrochemical reaction proceeds as predicted by an EC irr mechanism (at least in experiments with durations less than 1 s). Figure 4 shows the cyclic voltammogram of tert-butyl p-toluate after background subtraction. The values calculated for the homogeneous rate constant using the kinetic convolution technique (10.6 f 2 s-l over the range v = 0.10 to 10 V s-l) compare favorably with those obtained from double potential step experiments (9.9-10.8 s-l in ref 7) and from curve matching of simulated and experimental CV data (9.9-10.3 s-l in ref 6).
CONCLUSIONS The kinetic convolution integral appears to be an excellent tool for determining the homogeneous rate constant from cyclic voltammetric data. Solution resistance and quasi-reversible charge transfer do not affect the method. Stringent requirements are not placed on the reversal potential, though obviously the potential limit much be chosen sufficiently negative that a measurable quantity of species 0 is reduced to R. This is advantageous if other redox reactions occur at potentials associated with the diffusion tail of the CV reduction peak. On the other hand, the technique does require that background currents be accurately subtracted and that mass transfer be adequately described by the assumed model
Anal. Chem. 1984, 56:, 1923-1927
(5) Korn, G. A.; Korn, T. M. “Manual of Mathematics”; McQraw-Hill: New
(in these derivations, simple planar diffusion).
York, 1967;pp 215-216.
(6) Wagenknecht, J. H.; Goodln, R. D.; Klnlen, P. J.; Woodard, F. E. J .
Registry No. tert-Butyl-p-toluate, 98-51-1.
Nectrochem. Soc.in press.
(7) Woodard, F. E.; Goodln, R. D.; Klnlen, P. J.; Wagenknecht, J. H. Anal.
LITERATURE CITED
Chem., in press. (8) Woodard, F. E.; Woodward, W. S.; Reilly, C. N. Anal. Chem. 1081, 53, 1251A. (9) Deming, S. N.; Morgan, S. L. Anal. Chem. 1973, 45, 278A.
(1) Nadjo, L.; SavQant, J. M. J . Nectroanal. Chem. 1973, 48, 113. (2) Imbeaux, J. C.; Savbnt, J. M. J . Electroanal. Chem. 1073, 4 4 , 169. (3) Woodard, F. E. In Monsanto Report MSL-3320,1963,available from
author upon request. (4) Savitzky, A.; Golay, M. J. E. Anal. Chem. 1964, 36, 1627.
1923
RECEIVED for review March 1,1984.Accepted May 1,1984.
Microsample Introduction into the Microwave-Induced Nitrogen Discharge at Atmospheric Pressure Using a Microarc Atomizer R. D. Deutsch and G. M. Hieftje* Department of Chemistry, Indiana University, Bloomington, Indiana 47405
A microarc atomizer has been evaluated for microsampie introduction into the recently developed microwave-induced nitrogen discharge at atmospheric pressure (MINDAP). The microarc, operating for the first tlme in a ntlrogen atmosphere, efficiently volatilizes discrete microvdumes of solution and the resulting vapor is swept into the MINDAP excitation source. The MINDAP system, unlike eariler mkrowave and rf plasmas coupled to the microarc, offers the economy of uslng nitrogen gas, possesses a high energy of excitation, and provMes good spatial and temporal stability. The combination microarcMINDAP system has been evaluated for detection limits, dynamic range, preclslon, and interelement Interferences. A comparison with other discrete sample-introduction multlelement analysis systems is made.
The recently developed microwave-induced nitrogen discharge at atmospheric pressure (MINDAP) was characterized originally with a dried-aerosol sample-introduction system (1-4). This new plasma exhibits high temperatures (2),low detection limits (3),and matrix interferences which can be overcome in a manner similar to that employed in flame spectrometry. The MINDAP system is investigated here as an analytical atomic emission source for microsampling analysis using a microarc atomizer. The microarc atomizer was developed in 1974 by Layman and Hieftje (5)as a device to convert discrete microquantities of liquid sample into atomic vapor for emission analysis in a microwave plasma. The microarc is a high-voltage, lowcurrent discharge that sequentially and efficiently desolvates, vaporizes, and atomizes sample volumes from 0.1 to 40 pL (5). This concept of separate atomization and excitation was discussed also by Falk et al. (6) who showed how it could enhance the sensitivity of other atomic emission measurements. The microarc has been successfully combined with several plasma emission sources (5, 7-9). In conjunction with the inductively coupled plasma (ICP) (7,8),the microarc yielded lower detection limits than other microsampling techniques applied to the ICP (10-13). Argon and helium microwaveinduced plasmas (MIP), when coupled to the microarc (5,9), offer the sensitivity and freedom from interferences that has 0003-2700/84/0356-1923$01.50/0
been associated with the ICP. Yet their physical size, instrumentation, and operating requirements are more compact and economical. Elemental analysis using a microwave plasma as the excitation source has been limited to samples introduced as a vapor effluent from gas chromatography (14, 15), thermal atomizers (16,17), hydride generators ( I S , 19),and laser vaporization devices (20). Nebulizer systems have also been employed (21,22) but to a lesser extent and less successfully. Analyte is preferably introduced as a vapor because the MIP lacks the thermal energy needed to decompose the sample. Another limitation of the MIP is its small physical size which restricts the amount of sample that can be introduced before overloading occurs. The MINDAP system overcomes many of the inconveniences ordinarily associated with microwave plasmas: it readily accepts aerosol samples, possesses the high thermal energy needed for sample decomposition, and is relatively unaffected by high analyte concentrations. The MINDAP system has previously been evaluated for continuous solution analysis and is now the subject of investigation with discrete microvolume aliquots from the microarc sample-introduction technique. This is the first time that the microarc has been used in a molecular-gas atmosphere (Le., nitrogen). Its operating characteristics are therefore slightly different from those in an inert monatomic-gas atmosphere and a qualitative description of its behavior is included. The microarc-MINDAP combination yields picrogram to femtogram detection limits, a broad linear dynamic range, good precision (3-7% RSD), and essentially no interference either from sodium or phosphate.
EXPERIMENTAL SECTION Connection of the microarc atomizer to the MINDAP was straightforward and is detailed in Figure 1. Initially it was attempted to introduce the microarc-atomizedsample through the central channel of the plasma torch. However, in this configuration the suspended MINDAP plasma was perturbed when the arc was struck. Consequently, in this investigation the microarc was connected to the side-on gas inlet of the torch (Figure 1).
Timing of the microarc and data collection were controlled by a laboratory computer in a manner similar to that described by Keilsohn, Deutsch, and Hieftje (7). A block diagram of the experimental setup is shown in Figure 2. The computer controls 0 1984 Amerlcan Chemical Society
