Fourier Transform Faradaic Admittance MeasurementsOn the Use of High-Precision Data for Characterization of Very Rapid Electrode Process Kinetic Parameters Sam C. Creasonl and Donald E. Smith2 Department of Chemistry, Northwestern University, E vanston, 111. 60207
Recent studies have demonstrated that outstanding faradaic admittance measurement efficiency can be provided by minicomputerized, on-line Fast Fourier Transform (FFT) data processing (1-5). Using an appropriately designed multiple-frequency applied potential waveform, a faradaic admittance frequency response profile has been obtained in a single measurement-data processing pass of 2 seconds (approx.) a t a precision level comparable to that obtained by the conventional, repetitive, single-frequency measurement (3). By ensemble averaging a series of replicate measurements of this type, unprecedented precision is obtained in the frequency response profile. For example, in one case a k,-value of 0.40 cm sec-l was measured with an apparent precision of &0.3% ( * one relative standard deviation) on the basis of the cot @ - w1 profile observed after 64 measurement passes ( 4 ) . One of the areas where this enhanced measurement precision might be of considerable help is in extending the range of rate constants accessible to electrochemical relaxation measurements. An important case in point and a useful example is the so-called quasi-reversible process (6, 7) represented by the simple reaction scheme 0 + ne s R. For such systems, the response to an electrical perturbation is kinetically influenced by diffusion and the heterogeneous charge transfer step. As the standard heterogeneous charge transfer rate constant, k,, becomes larger, its kinetic manifestations become more difficult to separate from those arising from diffusion, other factors unchanged. Ultimately, only diffusion control is discernible, and the electrode reaction is said to be "reversible" or "nernstian." When measurement of k , is of interest, the standard strategy employed when heterogeneous charge transfer kinetic effects are not visible under a given set of circumstances has been to increase diffusion kinetics by raising the frequency (or lowering the time scale) of the observation. When taking this step, it is hoped that heterogeneous charge transfer kinetic effects will be accentuated to a point where they can be accurately measured before reaching the upper frequency (lower time) limit imposed by instrumental problems (6, 7). These concepts may be illustrated more quantitatively in terms of the fundamental harmonic phase angle cotangent obtained in faradaic admittance (ac polarographic) measurements. For quasi-reversible processes, the theoretical rate law is (6, 7): 'Present
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To w h o m correspondence s h o u l d b e addressed. (1) H. Kojirna and S. Fujiwara, Bu//. Chem. SOC.Jap.. 44, 2158 (1971). (2) S. C. Creason and D. E. Smith, J. ElecfroanaL Chem. 36, A1 ( 1972). (3) /bid., 40, A1 (1972). ( 4 ) / b i d . , 47, 9 (1973). ( 5 ) D. E. Glover and D. E. Smith, Anal. Chem., 45, 1869 (1973). (6) D. E. Smith, in "Electroanalytical Chemistry,"Vol. 1, A . J. Bard, Ed., M. Dekker, Inc.. NewYork. N.Y., 1966. (7) M . Sluyters-Rehbach and J. H. Sluyters, ibid,. Vol. 4. 1970.
cot@ = 1
G +x
where
p = 1 - a
D =
=
(4)
Do0DR"
(-)(-)DD Ro
RT fR EO - - In nF fo
lI2
(6)
@ is the phase angle of the fundamental harmonic faradaic alternating current relative to the applied alternating potential at frequency w, and the other symbols have their usual significance (6). The quantities unity and G / A in Equation 1 can be viewed as the contributions of diffusion and heterogeneous charge transfer kinetics, respectively, to the cot 4 magnitude. With quasi-reversible systems, k , is usually evaluated from cot 4 data using the linear cot 4 - w1/2 profile's slope (6). It is evident from Equations 1-6 that
(7) and, when Edc = E1/zr (the admittance peak potential with rapid processes),
For any finite frequency range, it is apparent that a sufficiently large k, value will yield a slope whose magnitude is negligible, relative to experimental noise, thus precluding quantitative k , assessment. Extending the frequency range accentuates the charge transfer kinetic term, d Z / A, but this strategy is limited by the double layer charging current (6, 7) which predominates at sufficiently high frequencies ( > l o kHz in most cases), making faradaic current and cot 4 evaluation inaccurate. Faradaic response measurements a t extremely high frequencies often employ observables resulting from faradaic nonlinearity [faradaic rectification effect, second harmonic currents, etc. (6, 7)] where double layer charging current contributions are greatly minimized. This is accomplished at the expense of greater theoretical complexity and more stringent demands on instrument and cell design (6-9). When very rapid rate processes are encountered, an alternate and complementary strategy to that of trying to employ ultrahigh frequencies is to somehow improve mea(8) R. deleeuwe, M. Sluyters-Rehbach. and J. H. Sluyters, Eiectrochim. Acta, 12, 1593 ( 1967). (9) /bid.. 14, 1183 (1969).
A N A L Y T I C A L C H E M I S T R Y , V O L . 45, N O . 1 4 , D E C E M B E R 1 9 7 3
2401
1.10
1.08
1.06 COT
9 1. 04
1.02
1.00
Figure 1. Fundamental harmonic cot 4 frequency response data from 100-pass Fourier transform measurement System: 5.0 X 10-3M F e ( C ~ 0 4 ) 3 in~ -1.OM K 2 C 2 0 4 + 0.05M H Z C 2 0 4 at Hg, 25 "C. Applied: Multiple-frequency, computer-generated waveform (see text for frequencies) with 2.0-mV amplitude for individual components, dc potential of -0.180 V vs. Ag/AgCI. (faradaic admittance peak potential). Measured: Fundamental harmonic cot 0 after ensemble averaging 100 measurement passes. Notation: Solid line = least-squares f i t 0 = experimental points
surement precision so that even small contributions of kiin Equation 1, can be accunetic terms, such as =/A rately evaluated. Except for taking all possible precautions in instrument design, this strategy for handling fast rate processes has not been given much attention, presumably because measurement precision has not been a parameter which could be manipulated conveniently (except in the wrong direction!). However, by virtue of the short time invested per measurement pass, the Fourier transform faradaic admittance measurement concept now permits selection of the data precision level over a reasonable range, simply by varying the number of measurement replicates which are ensemble averaged. Consequently, enhancing data precision is now a practical alternative to raising the applied perturbation frequency in the quest to characterize faster rate processes. Presentation of an example which supports the latter statement is the purpose of this Note. Although the foregoing discussion and the example given below focus on evaluation of k , with a quasi-reversible process, it should be recognized that the basic principles are generally applicable to kinetic parameter characterization for any rate process type normally associated with electrode reactions ( e . g . , coupled chemical reactions).
Details of instrumentation, data acquisition, and data processing procedures have been provided elsewhere (3, 4, 10). Measurements were made a t the frequencies 29.9, 49.8, 69.7, 89.6, 129.4, 189.0, 248.7, 328.3, 408.0, 507.4, 606.9, 726.2, 865.7, and 985.0 Hz on the basis of FFT analysis of the faradaic response to an applied potential waveform comprised of these frequency components. Detailed properties of the waveform in question have been discussed ( 4 , 10). A solution of Fe(C204)33- in 1.OM K2C204 + 0.05M H z C ~ 0 4was prepared using conventional precautions (11). The cell employed a dropping mercury working electrode, a saturated Ag-AgCI reference electrode, and a platinum wire auxiliary electrode.
RESULTS AND DISCUSSION In the media employed here, the electrode process Fe(C204)s3- + e F e ( C ~ 0 4 ) 3 ~is- a relatively facile reaction in which heterogeneous charge transfer kinetic effects are barely detectable in the faradaic admittance cot 4 - u1I2 plot using conventional measurement procedures at low and moderate frequencies (
