scope. Essentially any input-response measurement scheme, electrochemical or nonelectrochemical, can be serviced by the device. Thus, electrochemical applications to pulse or square-wave polarography, cyclic voltammetry, etc. are readily invoked. Finally, the 500-kHz data rate upper limit is a result primarily of our conservative attitude in approaching higher frequency measurement. Having encountered none of the anticipated special problems, such as solid electrode frequency dispersion, insidious electronic component nonlinearities, and the like, it now appears that a faster SYDAGES would have been worth considering. With proper substitution of solid state components, the basic SYDAGES should adapt to data rate increases of about an order-of-magnitude. This possibility is being considered.
(8) A. A. Pilla, in “Computers in Chemistry and Instrumentation”, Vol. 2, J. (9) (10) (1 1) (12) (13) (14) (15) (16) (17) (18)
LITERATURE CITED
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S. C. Creason and D. E. Smith, J . Electroanal. Chem., 36, A 1 (1972). S .C. Creason and D. E. Smith, J . Electroanal. Chem., 40, A 1 (1972). S. C. Creason, J. W. Hayes, and D. E. Smith, J . Hectroanal. Chem.. 47, 9 (1973). S. C. Creason and D. E. Smith, Anal. Chem., 45, 2401 (1973). D. E. Smith, Anal. Chem.. 48, 221A (1976). M. Sluyters-Rehbach and J, H. Sluyters, in “Electroanalytical Chemistry”, Vol. 4, A. J. Bard, Ed., M. Dekker, New York, N.Y., 1970, Chap. 1. R. delevie, J. W. Thomas, and K . M. Abbey, J . Electroanal. Chem., 62, 111 (1975).
(20)
S. Mattson, H. D. MacDonald, Jr., and H. E. Mark, Jr.. Ed., M. Dekker, New York, N.Y., 1972, pp 139-181. K. Matsuda and R. Tamamushi, J . Electroanal. Chem., 75, 193 (1977). R. J. Schwall, A. M. Bond, and D. E. Smith, J . Nectroanal. Chem., in press. D. E. Smith, in “ElectroanaMIcal Chemistry”, Vol. 1, A. J. Bard, Ed.. M. Dekker, New York, N Y , 7966. Chap. 1 . E 0 Brigham, “The Fast Fourier Transform“, Prentice-Hall. Englewood Cliffs. N J . 1974 K. F. Drake, Doctoral Dissertation, Northwestern University, Evanston, Ill., 1978. R. J. Schwall, Doctoral Dissertation, Northwestern University, Evanston, Ill., 1977. S.C. Creason, Doctoral Dissertation, Northwestern University, Evanston, Ill., 1973. “Applications Manual for Computing Amplifiers”, George A. Philbrick Researches, Inc., Boston, Mass., 1966. A. M. Bond, R. J. Schwall, and D. E. Smith, J , Electroanal. Chem.. in press. J. W. Hayes, D. E. Glover, D. E. Smith, and M. W. Overton, Anal. Chem., 45, 277 (1973). R. Saucedo and E. E. Schiring, “Introduction to Continuous and Digital Control Systems”, Macmillan, New York, N.Y., 1968, Chap. 4. R. J. Schwall, A. M. Bond, and D. E. Smith, Anal. Chem., following paper in this issue.
RECEIVED for review March 11, 1977. Accepted July 1, 1977. The authors are indebted to the National Science Foundation (Grant No. MPS74-14597) and the Australian Research Grants Committee for support of this work.
On-Line Fast Fourier Transform Faradaic Admittance Measurements: Real-Time Deconvolution of Heterogeneous Charge Transfer Kinetic Effects for Thermodynamic and Analytical Measurements Richard J. Schwall, Alan M. Bond,” and Donald E. Smlth’ Department of Chemistry, Northwestern University, Evanston, Illinois
A recently-suggested concept whereby FFT faradalc admlttance spectral measurements are used as a basis for real-time monitoring and deconvolutlon of heterogeneous charge transfer kinetic effects is further discussed and subject to experimental test. Data obtained uslng an automated procedure for data acquisition, FFT data processing, and the kinetic effect deconvolution fully support the belief that the reversible response can be extracted routinely from observed quasi-reversible faradalc admittance spectra. A particular surfactant-sensitive heterogeneous process Is shown to yield essentially identical reversible spectra for a range of k , values via the correction procedure. Four additional systems Investigated encompassed aqueous and nonaqueous media, organic and Inorganic electroactive species, and Pt as well as Hg electrodes. All yielded satisfactory results.
A recent publication ( 1 ) has speculated on the possibility of using Fast Fourier Transform (FFT) faradaic admittance spectral measurements to enhance data quality used in On leave from Department of Inorganic Chemistry, University of Melbourne, Parkville. Victoria 3052, Australia.
6020 1
electrochemical assay procedures. T h e essential concept involves use of the detailed kinetic-mechanistic information content of admittance spectra to effect a highly sensitive computerized monitoring of the statui5 of electrode reactions employed for assay work. Through such procedures the operator can be informed of deviations from expected response characteristics, which might arise from an unusual sample background constituent, improper cell preparation, or a subtle instrument malfunction. Such response characteristic perturbations can invalidate a n assay procedure, but many will go unrecognized with the conventional single frequency (ac polarogram) or single time point (pulse polarogram) readout modes which are employed in most instruments used for electroanalysis. The possibility of using the observed response spectrum as a basis for automatically correcting undesirable kinetic-induced fluctuations in the course of a n assay procedure has been advanced ( I ) as an important component of the overall procedure. The reader is referred to the cited publication (I) for further discussion on the details and rationale behind t h e kinetic status monitoring and correction concept. The purpose of this article is to present data which demonstrate the feasibility of this concept in the context of the widely-encountered quasi-reversible mechanism. Both mercury and platinum electrode results were obtained in this initial study. The data ANALYTICAL CHEMISTRY, VOL. 49, NO. 12, OCTOBER 1977
1805
insensitive to fluctuations in sample background constituents or electrode surface properties because of the corresponding insensitivity of the rate-controlling step, diffusion. Likewise, the thermodynamically relevant quantities, Eo, n, etc., are simply related to the peak potential and magnitude of the admittance polarogram ( 2 , 3). Finally, factors such as electrode geometry, and mode of control of Edc are of negligible or minor importance in this case (2). Obtaining a pure reversible response directly often is difficult with the relatively short time scale conditions associated with the small amplitude relaxation techniques in question. More frequently one will achieve an electrode relaxation which is reversible on the dc time scale, but quasi-reuersible in the ac sense. For this case the faradaic admittance magnitude becomes (2-4)
I'
D.C. POTENTIAL
Figure 1. AC polarograms of the faradaic in-phase component for Cd2+/Cd(Hg) system in 1.0 M ZnSO with varying n-butanol concentrations. System: 1.0 X M CdZt at DME-aqueous 1.0 M ZnSO, M (A), 22 X interface with n-butanol concentrations of 0.00 X M (B), 44 X M (C), and 88 X M (D), 25 'C. Applied: 200 Hz, 24 mV peak-to-peak sinusoidal component superimposed on a linear dc ramp voltage of 50 mV/min scan rate. Measured: 200 Hz in-phase component using conventional analog circuitry with positive feedback iR compensation ( 4 )
obtained clearly show that an electrode reaction's mechanistic character can be checked, the heterogeneous rate monitored, and the analytically-preferred reversible (diffusion-controlled) response accurately computed, using FFT faradaic admittance spectra. In addition to assay work, the results presented here are of significance for thermodynamic applications of electrochemical relaxation techniques, where the reversible response also is the observable of choice.
THEORY The preferred observable normally is the reversible current or admittance response in analytical and thermodynamic applications of the highly-sensitive small amplitude relaxation techniques, such as fundamental harmonic ac polarography (phase-selective mode), differential pulse polarography, and square-wave polarography. For the simple electrode reaction (in the phenomenological sense)
O + ne+ R the expressions (see Notation below)
n 2 F 2 AC:(wDO)l Are, =
j 4R T cash' -
'
F(t)rev
2
cot Q = 1
(4) define t h e small amplitude faradaic admittance magnitude and phase angle for the reversible process (2-4). F(t),,, is unity for planar systems and differs negligibly from unity for most nonplanar systems, except amalgam-forming cases where it slightly alters wave magnitude and shape ( I , 5 ) . Equations 1-4 are exact for small amplitude faradaic admittance measurements, and differ from the appropriate result for the other techniques cited above only by a n Edc-independent proportionality constant. Perusal of Equations 1-4 clearly suggests t h a t the proportionality factor relating reversible response magnitude and concentration will be relatively 1806
ANALYTICAL CHEMISTRY, VOL. 49, NO. 12, OCTOBER 1977
(5)
i A ( w t ) I = A,,, G ( u )
fiz
cotp=l+-
x
G ( w ) = [1+ (cot Q)*
f = fo"'I2"
1';
(7)
(9)
The multiplying factor which reflecis the extent of quasireversibility, G ( u ) , complicates the relationship between thermodynamic quantities and response characteristics, introduces influence of the heterogeneous charge transfer rate parameters on the response, and reduces assay sensitivity [ G ( w ) 5 11. Most troublesome for assay work is the fact that the k,- and a-values can be quite sensitive to varying amounts (even traces) of sample background constituents, such as surfactants (6-10) and a variety of anions in the case of certain metal cation reactions (6, 7, 10). If an electrochemical assay is based on the observation of a single frequency quasi-reversible admittance peak, a single sample time pulse polarogram, etc., as is usually the case, fluctuating amounts of such interfering background components cannot be tolerated. Additional sample preparation steps must be introduced t o either remove or buffer the interference. These additional steps complicate to varying degrees the overall assay procedure without one's achieving reasonable assurance that the effort was indeed successful. In principle, the latter assurance could be realized by careful examination of the concentration-independent shape characteristics of the admittance or polarographic peak, t~ assure that the electrode reaction kinetics met some specific criteria. However, for the kinetic regime in question this is a rather insensitive observation. The G ( o ) factor can be made to vary substantially in magnitude due to a k , change, particularly at the analytically significant peak potential, without noticeably altering wave shape to the casual inspection ("eyeball" approach). Consequently, even an alert, well-trained analyst may fail to recognize that a low admittance peak magnitude is the result of a suppression in h,-value, relative to the norm, rather than a low concentration. One could suggest a more quantitative, higher resolution wave shape examination to check for heterogeneous kinetic fluctuations in the context of conventional single frequency waves. However, this would require a computerized instrument for convenient implementation in an analytical lab, in which case one must conclude that the more versatile, informative FFT approach recommended here is to be preferred.
I t is evident (Equation 6) t h a t the cot $J value is a concentration-independent observable which provides direct and sensitive insights into the status of the heterogeneous charge transfer step (2, 4 ) . Fluctuations in h , and a are readily detected through this observable, and deviations from the predicted linear cot Q - w1 profile provide good evidence for the existence of more complex mechanisms. By inserting the observed cot 4 value in Equation 7 , G(w) is calculated. From this result one may achieve a heterogeneous charge transfer kinetic correction simply by dividing the observed admittance magnitude, IA(ut)l,by G ( u ) ,to obtain A,,, (see Equation 5 ) , i.e.,
’
In principle, this operation can be done using single frequency faradaic admittance data, but it would involve the assumption t h a t the electrode reaction is indeed quasi-reversible. It is better t o eliminate this assumption by using the cot Q spectrum t o first verify t h a t the linear cot @ - u1 profile is observed, and then invoke the kinetic correction defined by Equations 7 and 12, using the IA(ot)l and G ( w ) spectra. I t is the required IA(wt)l and cot $J spectral data which the on-line FFT admittance measurement yields with speed, convenience, and accuracy, thus providing the basis for both kinetic monitoring and confident implementation of Equation 12. A final double check on the fidelity of the kinetic correction is realized by checking for linearity in the computed A,,, - w 1 profile (Equation 1). The foregoing procedure always has been possible, but never was it conveniently implementable on an analytically-palatable time scale until FFT measurements were developed and refined. T h e dc reversible, ac quasi-reversible kinetic regime discussed in the previous paragraph is of primary concern in this work. It is well known, for analytical substances which follow Mechanism R1, that media can be found in the normal course of method development which will achieve this kinetic regime. However, should a process which also is quasi-reversible in the dc sense be encountered, the ideas presented here are applicable with only minor restrictions. T h e general phenomenological effect accompanying onset of dc quasi-reversibility on the rate law is that the dc responsive term, F ( t ) , deviates from its reversible value, F(t),,, and becomes responsive to h, and a (2, 4 ) . Direct correction for this effect is more complicated than the ac quasi-reversibility correction, but it is possible ( I ) . HouTever, we submit that in the analysis context such a correction often will be unnecessary. With the vast majority of systems one is likely to encounter under such conditions, there is one point along the wave where F ( t ) = J’(t),,\, regardless of the h,-value. This condition will exist for either stationary electrodes or the dropping mercury electrode (2,11, 12). I t usually is referred to as the “crossover point” (2,11,12),is often located near the peak potential, and is readily identified (11, 12). Consequently, the current magnitude a t this point provides a perfectly acceptable observable for assay work, unless the a-value is fluctuating (which is readily detected). One concludes t h a t the dc quasi-reversible situation normally is encompassed by the strategy embodied in Equation 12, provided that the analysis uses the admittance magnitude a t the crossover point, rather than the peak value. For thermodynamic measurements, one must be less frivolous about dismissing the perturbations attending dc quasi-reversibility. In this situation, direct compensation for the F ( t ) effect ( I , 2) may be considered. However, heterogeneous processes which are sufficiently sluggish t o produce a quasi-reversible dc response allow rather convenient implementation of an alternative plan for evaluating E?, etc. based on use of high-frequency limiting response
’
data (13). The latter plan also enables a two-step method for computing A,ev. Of course, if an electrode process is totally irreversible, or close to it, the ideas advanced here are neither applicable in a pragmatic sense, nor is there a K,-sensitive response with which to be concerned1 ( 4 , 14). T h e fact that the heterogeneous charge transfer kinetic effects are manifested as a frequency domain multiplication of the reversible response (Equation 5 ) implies that they act as a convolution on the reversible time domain waveform ( I ) . Thus, the division operation suggested by Equation 1 2 amounts to a time domain deconvolution of the heterogeneous kinetic effects. This deconvolution operation not only eliminates sensitivity to fluctuating k,- and cu-values,but also enhances response sensitivity. Decon.volution operations to enhance data quality are hardly unique in chemical measurements. However, the particular operation espoused above does differ from many in the sense t h a t the deconvolution function magnitude [ G(o)-value] and validity (i.e., mechanism double-checks) are continuously monitored via the cot 4 observation.
EXPERIMENTAL Although FFT faradaic admittance measurements have been employed frequently for the past 4-5 years, only recently (15,16) have data acquisition rates, error correction procedures and measurement routines been programmed which provide sufficient convenience for use in frequently-invoked analysis protocols. In our laboratory the approach is centered around a high speed synchronous data generation and sampler (SYDAGES) system, whose characteristics are described in detail elsewhere (15). With SYDAGES, on-line FFT measurements provide accurate total cell admittance measurements to 125 kHz., from which sensible faradaic admittance data have been extracted t o 60 kHz. This admittance measurement bandwidth is more than adequate to automatically effect simultaneous measurement of both faradaic and nonfaradaic components on an electrolytic cell under typical assay conditions. The key steps in our SYDAGES-assisted FFT faradaic admittance measurement program are (15): (a) a brief dummy cell calibration procedure which yields an instrument “error transfer function spectrum“; (b) broadband m’easurement of the raw electrolyticcell admittance spectra at a series of dc potentials along the faradaic admittance wave (staircase scan); (c) correction of the raw admittance data for instrumentation error using the “error transfer function spectrum”; (d) high frequency ext.rapolation of the corrected admittance data to obtain the nonfaradaic ohmic resistance and double-layer admittance; (e) subtraction of the nonfaradaic effects from the cell admittance spectra to reveal the faradaic admittance spectra (complex .plane format); (f) computation of A(&) and cot I#I from complex plane faradaic data; (9) computation of G ( o ) from cot I#I and impiementation of Equation 12 to obtain A,,,; (h) computation of Arev/ul vs. w ” ~ . The option to obtain a scope display or hard copy plot following any of the foregoing key steps was available and frequently implemented. Electrolytic cell response data could be plotted versus either Edc,w, or a’,’.Scope displays provided for efficient interactive decision-making on matters such as mechanistic tests (e.g., is cot c$ - o1 profile linear) and efficiency of nonfaradaic component compensation. In the hands of operators with only casual interests in electrochemistry, these interactive decisions might also have to be automated. Options also existed for implementing FFT smoothing algorithms following the key steps cited. Finally, an option existed for implementing the nonfaradaic compensation using data acquired in the absence of the faradaic component, as a preliminary step. This is deemed advisable t o provide checks of double-layer admittance data obtained with and without the faradaic component, particularly when the validity of the a priori separability concept (3, 4 ) is in question. One should note that the on-line FFT measurement protocol outlined provides substantial data on aspects of instrument and cell behavior in addition t o the primarily sought faradaic admittance (2 components) vs. Edc vs. o’ z profile. Characteristics of the analog instrumentation are revealed in the instrument error transfer function. Deviations from normalcy in this observable
’
ANALYTICAL CHEMISTRY, VOL. 49, NO. 12, OCTOBER 1977
4
1807
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; Y
B
A
7,
\\-
!
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c
2
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\
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~
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Figure 2. DC polarograms for the systems shown in Figure 1
will reveal insidious drifts in instrumentation performance, as well as the always obvious catastrophic failures. More important is the automated measurement and readout of the nonfaradaic components. Significant ohmic resistance variations from the normal value for a solvent-electrolyte are suggestive of error in solution preparation, bad electrode connections, etc. If the ohmic resistance is nominal, but the double-layer admittance is not, variations in electrode area or abnormal surfactant levels are suggested. In the absence of surfactant problems, assay data can be corrected for fluctuations in working electrode surface area on the basis of the double-layer capacitance. In any case, the information content of this now routine and rapid FFT admittance measurement approach significantly exceeds that which is the primary focus of this article. Other publications (15. 17, 18) describe solvent and reagent purification, and electrolytic cell preparation strategies.
:
RESULTS A N D DISCUSSION
This initial demonstration phase of the heterogeneous charge transfer kinetic correction concept described above can be broken down into two main parts. The first involved a detailed study of a particular model system, Cd’+ in acidified ZnSO?, for which the Cd*+/Cd(Hg)k,-value is significantly sensitive to the amount of a surfactant (n-butanol) added to the solution. With this system it is not only demonstrated t h a t the kinetic correction embodied in Equations 7 and 12 yields the expected linear A,,, - w1 * profile and other reversible admittance properties, but also that this reversible profile is of essentially identical magnitude and slope for a range of surfactant-controlled k,-values, as is also predicted. Success was realized in this endeavor using both the DME (ac polarographic conditions) and the hanging mercury drop electrode (HMDE), with the ac cyclic voltammetry measurement mode (17). T h e second phase of this work was directed toward providing a preliminary indication of the generality of the “kinetic correction” concept for quasi-reversible systems. Four additional redox systems were investigated, encompassing aqueous and nonaqueous electrolytes, platinum and mercury electrodes, organic and inorganic electroactive substances, and dc reversible as well as dc quasi-reversible electrode reactions. With these latter systems, lesser demands were made on what constituted successful implementation of Equation 12. Specifically, a linear A,,, wl” profile and appropriate wave shape (2) alone were viewed as a satisfactory conclusion. A. Cd2+in Aqueous, 1.0 M Z n S 0 4 at H g w i t h Varying Amounts of n-Butanol. The choice of Cd as a model system for t h e purpose a t hand is natural, because of the reported sensitivity of its &value to surfactants (6-10). T h e ZnS04 electrolyte was selected because of its relevance to the analysis of Cd, and other metals, in the cell feed for electrolytic Zn refining plants (19). Also, it is likely that surfactant influence on the heterogeneous rate constant will occur in such assay media. The model surfactant selected was n-butanol, mainly because i t was the first discovered in a random search of our reagent supply. Its role as a surfactant is well-documented (20-23). Figure 1 illustrates the effect of n-butanol on ac polarograms of Cd2+ in 1.0 M ZnS04. I t is seen that sufficient amounts 1808
ANALYTICAL CHEMISTRY, VOL. 49, NO. 12, OCTOBER 1977
0.5
: t-+
0
a 0.mD
20
OMEGA Y0
60
73
Figure 3. Peak faradaic admittance spectra for Cd*+/Cd(Hg) system in 1.O M ZnSO,, showing observed admittance and computed A rBv values. System: Same as Figure 1, except no butanol. Applied: Pseudo-random, odd-harmonic ac waveform ( 75)with 1.5 mV per frequency component, 32 components; superimposed on staircase dc ramp synchronized to DME. Measured: (A) Peak IA(wt)l - wl’* profile (0)and computed A,,, counterpart ( 0 ) at 3.0 s in life of DME, 10 replicates. (B) Frequency normalized presentation of data in Figure 3A
of butanol can force the system into the dc quasi-reversible realm (Figure 1D). Direct current polarograms (Figure 2) show no detectable influence of n-butanol on the limiting currents, and alterations in the dc polarogram wave shape occur only with sufficient butanol quantities to produce dc quasi-reversibility. Consequently, one can conclude from these observations that the amounts of surfactant employed have negligible influence on mass transport rates, and that the attenuation in the ac polarographic wave magnitudes probably arises from butanol influence on the heterogeneous charge transfer rate. This is confirmed by cot 6 values obtained by FFT faradaic admittance measurements. The cot 6 - a”’ profiles are linear and their slopes increase (decreasing k,) with increasing n-butanol concentrations. The position of the cot C$ peak value is uninfluenced by n-butanol, indicating no effect of the surfactant on a ( a = 0.28). This constancy of a as a function of surfactant concentration is not unusual ( 2 4 ) . Implementing the kinetic correction of Equations 7 and 12 produces the expected result, as illustrated in Figures 3 and 4 for the Cd2+ wave in absence of n-butanol, which also is distinctly quasi-reversible. Figure 3A shows the measured peak faradaic admittance spectrum, and the A,,, spectrum yielded via Equations 7 and 12. These same data are plotted in the frequency normalized format, A ( U ) / W ’and / ~ A,,,/w1t2, in Figure 3B which clearly shows the near-constancy of the A,,, slope as w1I2 varies. Figure 4 shows a typical measured and corrected admittance polarogram obtained a t 6013 rad
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20
Figure 5. Frequency normalized peak faradaic admittance spectra for Cd2+/Cd(Hg) system in acidified 1.0 M ZnSO,, showing observed admittance and computed A values for varying butanol concentrations. System: 1 .O X M Cd"" at DME-aqueous 1.0 M ZnSO,, 0.36 M H2S04interface with n-butanol concentrations of 0.00 X M (+), 2.43 X M (O), 5.46 X M, (0), '14.6 X M (X), and 21.9 X M (A), 25 OC. Applied: Same as Figure 3. Measured: Same as Figure 3, except notation as above. Corresponding observed I A ( o t ) l / o " 2 and computed A,,,/w"2 values (above dashed line) use same point notation. Ordinate units = rnho S - " ~ / ~ O - ~ rn
2
x
1.52 t 1.52 t 1.47 t 1.52 t 1.56 t values 1.52 t
io4 0.02a 0.01 0.02 0.02 0.03 0.02
Standard deviations.
s ', illustrating the sensitivity enhancement attending the kinetic effect deconvolution, as well as its slight, but accurate adjustment of the admittance polarograms width-at-half height. Similar results are obtained in acidified ZnS04 (0.36 M and 0.18 M H 2 S 0 4 ) . An illustration of the ability of Equations 7 and 12 to yield the same reversible response, regardless of the &value, is provided by comparing A,,, spectra obtained using varying amounts of n-butanol. Typical are the frequency normalized peak faradaic admittance data given in Figure 5 and Table I for Cd2+in acidified ZnS04. The observed spectra illustrated in Figure 5 include only the lower frequencies a t which measurements were effected because these are considered most appropriate for analytical and thermodynamic purposes. The inclusion of higher frequencies would make the kinetic correction effect appear more impressive, but the relevance for the objectives a t hand is questionable. The corrected peak A,e,/~1/2 spectra for the various n-butanol amounts overlap in a satisfactory manner, as shown by the near-horizontal array of points near the top of Figure 5 . The A,,, values a t each frequency were computed using the observed cot q5 value at that frequency. Yo attempt was made to use best linear fit cot #I values, or other forms of digital smoothing to generate the A,,, values. The cosmetic contribution of digital filtering will improve the appearance of the Figure 5 data with regard to noise, particularly a t the lower frequencies. However, we believe roughly the same statistical advantages are realized by employing the average peak A,,,/wl/* values over the observed frequency band for assay calculations (rather than the conventional single frequency point). Such average values for each n-butanol concentration are listed in Table I, together with k,-values. Corrected peak potentials are constant (&2 mV) and peak widths a t half-height equal 90 2 mV. While
K
7
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0 .-0 0.60
POTENTIAL -0.55 /'VOLTS -0.50 . . .
.
- 0 . Lis
. ' " I
Figure 6. Double-layer admittance-dc potential profiles for Hg-acidified
ZnS04 interface with varying butanol concentrations. System: DME-aqueous 1.O M ZnSO,, 0.36 M H2S04interface with same butanol concentrations and notation as Figure 5 . Applied: Same as Figure 3. Measured: Double-layer admittance vs. EdC in region of Cd" wave. 6.01 X lo3 rad s-' data shown (other frequencies measured, but not shown), 3.0-sdrop life, 10 replicates. Abscissa vs. Ag/AgCI
the agreement between the average AreL./&* values (Table I) is not perfect, the 6 % range of values and 1.5% average deviation is more than acceptable for most assay purposes. This is particularly significant when one recognizes that in normal single frequency measurements, errors in calculated concentrations of the order of 100% could result if the butanol-induced k,-variation is ignored. The latter type of oversight would be expected for this system under the conditions encompassed in Figure 5 because the k,-value range is such that near-reversible appearing total or in-phase admittance waves are what one obtains. The significance of the data presented in Figure 5 and Table I in emphasizing the power of the FFT approach also is better appreciated by considering the range of nonfaradaic corrections invoked as the n-butanol concentration is varied. While the ohmic resistance is negligibly variant, the double-layer admittance is significantly influenced by n-butanol, as shown in Figure 6. Finally, the data in Figure 5 were not obtained on the same solution to which differing amounts of butanol were added, but each spectrum was obtained using separate solution preparation, degassing, etc., to better simulate an assay or thermodynamic measurement situation. ANALYTICAL CHEMISTRY, VOL. 49, NO. 12, OCTOBER 1977
1809
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Figure 7. FFT ac cyclic voltammetric faradaic admittance results for Cd2+/Cd(Hg) system in acidified 1.0 M ZnSO,. System: 1.0 X M Cd2+ at HMDE-aqueous 1.0 M ZnSO,, 0.18 M H,S04 interface, 25 O C . Applied: Pseudo-random, odd-harmonic ac waveform ( 75)with 1.5 mV per frequency component, 32 components; superimposed on staircase dc scan (5 mV per step) with triangular envelope whose scan rate is 50 mV s-'. Measured: Observed ac cyclic voltammogram at 4.30 X lo3 rad s-' with 0 = forward scan, X = reverse scan (A), and corresponding computed A,,, voltammograms with 0 = forward scan, X = reverse scan (B). Frequency normalized observed peak admittance and computed peak A,,, spectra for forward (C) and reverse scans (B)
T h e faradaic admittance under ac cyclic voltammetric conditions (HMDE) also was obtained for the Cd'+ system in ZnS04. The FFT approach to this measurement was invoked, as described elsewhere (17). Except for slightly larger noise levels attributable to the fact that ensemble averaging was not used (single pass cyclic runs), the ac cyclic data are comparable to those shown in Figures 3-5 except that two admittance peaks are acquired, one from the forward and one from the reverse dc potential scans. Examples of observed and the kinetic corrected A,,, voltammograms and peak admittance spectra in the frequency normalized format are shown in Figure 7 . B. Other Systems Studied. Four other quasi-reversible redox systems have been examined to ascertain the general applicability of the kinetic correction embodied in Equations 7 and 12. The systems in question are: (a) Cd'+ in aqueous 1.0 M N a 2 S 0 4a t the DME and HMDE; (b) c r ( c N ) 6 3 -in aqueous 1.0 M KCN a t the DME and HMDE; (c) pyrene in CH3CN-0.10 M tetrabutylammonium perchlorate (TBAP) at the DME and HMDE; (d) the tetramethylphenylenediamine cation radical (TMPD+) in CH3CN4.10 M TBAP at the DME and Pt electrode. In each case, satisfactory results were realized in the sense that the computed A,,, - d 2 spectra were linear, and the shape of the A,,, polarograms were in accord with the Nernstian predictions (2, 3 ) . Figure 8 presents examples of the observed and corrected frequency normalized admittance spectra for each of these systems. In no case was 1810
ANALYTICAL CHEMISTRY, VOL. 49, NO. 12, OCTOBER 1977
any special difficulty encountered with any aspect of the F F T faradaic admittance measurements, including the implementation of Equations 7 and 12. It should be noted that the T M P D + / T M P D couple is dc quasi-reversible at Pt and the corrected spectra in Figure 8 are at the crossover point, as recommended earlier for assay purposes. Finally, as an illustration that linear A,,, - u1 * spectra are obtained a t potentials other than the peak values for the systems studied here, Figure 9 is presented. I t provides computed A,,,/ol * results at various potentials along the faradaic admittance wave for the Cr(CN)63 reduction. By using the least-square fit to raw cot 4 - w1 results, Figure 9 also illustrates the noise reduction obtainable through this strategy. C. Some Observations. We have clearly established the capability to achieve real time correction for the effects of quasi-reversible heterogeneous charge transfer reactions on the faradaic admittance. This correction is accomplished quickly under computer control, with negligible increase in the time for analysis. The required calculations are shorter than those we already use for compensation of the solution resistance and double-layer capacitance. T h e data produced are of very high quality, even a t frequencies as high as 60 kHz, where the charging current-tofaradaic current is at least an order-of-magnitude higher than in the usual electroanalytical time scale. This gives confidence
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the parameters in the equations may be more empirical than k , and a. Once the computer has been programmed for these equations, it can serve as a “watchdog” in the analytical environment. For example, it can be set to check for nonlinearity in the cot 4 vs. plot in the case of quasi-reversible charge transfer and, if detected, sound a warning to the analyst that the nature of the electrode reaction is inappropriate. In addition, the computer’s real-time calculation of the solution resistance and double-layer capacitance gives it added opportunity t o detect drastic changes in the nature of the solution or electrode. This watchdog capability could be particularly important in a process plant environment. However, we do not propose to send the computer out alone to discover the reaction mechanisms in new analytical systems. This is the job of the analytical chemist. His task may be greatly simplified with a “watchdog” computer set to detect deviations from the pure charge transfer and diffusion phenomenology. Such an “analytical chemist’s helper” would of course be programmed to oversee the results of a standard-addition run to check for linear calibration curves. Such a helper will appreciably ease the method-development task of adjusting analytical sample conditions until a phenomenologically simple electrode response is achieved. At this point, the selected response equations are set into the instrument and analyses may run virtually automatically.
NOTATION DEFINITIONS
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1812
electrode area activity coefficient of species i diffusion coefficient of species i initial concentration of species i s t a n d a r d redox potential in European convention d c component of applied potential reversible dc polarographic half-wave potential (planar diffusion) Faraday’s constant ideal gas constant absolute t e m p e r a t u r e number of electrons transferred in t h e heterogeneous charge transfer step reversible (diffusion-controlled) small amplitude faradaic a d m i t t a n c e
ANALYTICAL CHEMISTRY, VOL. 49, NO. 12, OCTOBER 1977
IA-
observed faradaic admittance magnitude
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LITERATURE CITED (1) D. E. Smith, Anal. Chem., 48, 517A (1976). (2) D. E. Smith, in “Electroanalytical Chemistry”, Vol. 1, A. J. Bard, Ed., M. Dekker, New York, N.Y., 1966, pp 1-155. (3) M. Sluyters-Rehbach and J. H. Sluyters in “Electroanalytical Chemistry”, Vol. 4, A. J. Bard, Ed., M. Dekker, New York, N.Y., 1970, pp 1-128. (4) D. E. Smith, Crlt. Rev. Anal. Chem., 2, 247 (1971). ( 5 ) I . Ruzic and D. E. Smith, Anal. Chem., 47, 530 (1975). (6) J. E. B. Randies and K. W. Somerton, Trans. fara6ay Scc., 48, 951 (1952). (7) J. E. B. Randles, Discuss. faraday Soc., 1, 11 (1947). (8) N. Tanaka, R. Tamamushi, and M. Kodama, Anal. Chim. Acta, 20, 573 (1959). (9) R . Tamamushi and N. Tanaka, 2 . Phys. Chem., 20, 573 (1959). (10) G. S. Buchanan and R. L. Werner, Aust. J . Chem.. 7, 312 (1954). (11) H. L. Hung and D. E. Smith, Anal. Chem., 36,922 (1964). (12) A. M. Bond, R. J. O’Haiioran, I . Ruzic, and D. E. Smith, Anal. Chem., 48, 872 (1976). (13) R . J. Schwail, A. M. Bond, and D. E. Smith, J . Electroanal. Chem., in press. (14) D. E. Smith and T. G. McCord. Anal. Chem., 40, 474 (1966). (15) R . J. Schwali, A. M. Bond, R. J. Loyd, J. G. Larsen, and D. E. SmRh, Anal. Chem., preceding paper in this issue. (16) R . de Levie, J. W. Thomas, and K. M. Abbey, J . Electroanal. Chem., 62, 111 (1975). (17) A. M. Bond, R . J. Schwall, and D. E. Smith, J . Electroanal. Chem., in press. (18) R. J. Schwall, Doctoral Dissertation, Northwestern University, Evanston, IiI.. 1977. (19) E. S.Pllkington, C . Weeks, and A . M. Bond. AM/. Chem.,48, 1665 (1976). (20) B. Breyer and S. Hacobian, Aust. J . Sci. Res., A5, 500 (1952). (21) W. Lorenz and F. Mockei, Z . Electrochem., 60, 507 (1956). (22) V. I . Melik-Gaikazyan, Zh. f i z . Khim., 26, 560 (1952). (23) J. O’M. Bockris, M. A. V. Devanathan, and K. Muller, Roc. R. Soc. London, Ser. A , 274, 55 (1963). (24) K. Matsuda and R. Tamamushi, J . Electroanal. Chem., 75, 193 (1977). (25) P. L. Brezonik, P. A. Bravner, and W. Stumm, Water Res., IO, 605 (1976). (26) R . de Levie and L. Pospisll, J . Nectroanal. Chem., 27, 454 (1970).
RECEIVED for review March 11, 1977. Accepted July 1, 1977. The authors are indebted to the National Science Foundation (Grant No. MPS74-14597) and the Australian Research Grants Committee for support of this work.
