Alternating Current Polarography in the Harmonic Multiplex Mode Observations on the Use of Digital Signal Conditioning with the Fast Fourier Transform Algorithm Donald E. Glover’ and Donald E. Smith2 Department of Chemistry, Northwestern University, Evanston, 111. 60207
Digital signal conditioning based on the Fast Fourier Transformation (FFT) is applied to ac polarography in the harmonic multiplex mode. The FFT is used in two capacities. First, it is applied to the discrete, digital representation of the cell current time domain waveform, which is obtained by analog-to-digital conversion. Through this operation, the digital FFT provides the means for separation and quantitative characterization of the direct current, fundamental harmonic alternating current, and second harmonic alternating current polarographic responses. Second, polarograms resulting from measurement of the response as a function of the dc potential are smoothed by a FFT smoothing technique. Results show that the foregoing digital signal conditioning scheme provides at least comparable measurement fidelity, together with superior efficiency and convenience, compared with a previously-described procedure which relies more heavily on analog signal conditioning.
The simultaneous measurement of the direct current, fundamental harmonic alternating current, and second harmonic alternating current under ac polarographic conditions has been referred to as “A.C. Polarography in the Harmonic Multiplex Mode” ( I , 2). Recently, the feasibility of this technique for quantitative characterization of these response components has been demonstrated ( I ) . The instrumentation used for this purpose relied exclusively on analog signal conditioning networks (tuned amplifiers, lock-in amplifiers, full-wave rectifiers, etc.) to separate and convert to an amplitude-proportional dc format the cell current signals of interest. An on-line digital computer and peripherals were used to implement analogto-digital (A/D) conversion; store and calibrate data, and control certain experimental parameters such as mercury drop life and the dc potential. Except for enabling signal averaging, the on-line computer played a completely passive role regarding separation of signal components and signal enhancement which are among the most crucial measurement steps. An alternative measurement strategy in which the computer plays a more active role in signal processing involves the use of digital Fourier analysis. Kojima and Fujiwara ( 3 ) invoked the latter concept for simultaneous measurement of the fundamental and second harmonic ac polarographic responses. They used an 8-bit A/D converter to acquire a digital representation of the cell current time-domain waveform whose Fourier Present address, Department of Chemistry, California Institute of Technology, Pasadena, Calif. 91109. Author to whom correspondence should be addressed. (1) D. E . Glover and D. E. S m i t h , Anal. Chem., 44, 1140 (1972). (2) D. E. Smith, in “Applicationsof Computers in Analytical Chemistry,” Vol. 2, H . B. Mark, Jr., J . S. Mattson, and J. C. MacDonald, E d . , Marcel Dekker, New York. N.Y. 1972. (3) H. Kojima and S.Fujiwara, Bull. Chem. SOC.Jap. 44, 2158 (1971).
transform provided the amplitude and phase characteristics of the frequency components comprising the waveform. Although data precision did not match that of good quality analog circuitry, presumably because of the limited A/D converter word length, Kojima and Fujiwara’s observations definitely established the feasibility of digital Fourier analysis as an approach to ac polarography in the harmonic multiplex mode. Digital signal conditioning via the Fourier transform is particularly appealing because it enables one to replace with digital computer software and hardware, the sometimes troublesome frequency-selective analog circuits whose operation places nontrivial demands on operator skill. In effect, the digital Fourier transform simulates the action of a large, parallel array of lock-in amplifiers. If effective, digital signal conditioning can reduce investments in operator time and analog circuitry to an extent which might compensate for initial costs of the computer system. Moreover, the present proliferation of on-line minicomputers is unlikely to be reversed, so that eventually these devices will be standard components in most scientific laboratories. Consequently, decisions regarding acquisition of ac polarographic measurement capabilities in a research or service laboratory often will be made in a context where a minicomputer already is available. In such situations, any reduction in analog circuitry requirements made possible by digital signal conditioning, will definitely make more appealing the acquisition and use of ac polarographic equipment from both the economic and ease of operation viewpoints. Such advantages stimulated us to carefully evaluate this digital approach to ac polarography in the harmonic multiplex mode using high-precision data acquisition components and special digital signal enhancement procedures. Particular attention was given to comparing the digital and analog signal conditioning approaches with regard to measurement convenience and quantitative data fidelity. Measurement procedures, typical results, and conclusions are presented here.
EXPERIMENTAL Instrumentation. Figure 1 provides a schematic of the instrument employed in this work. It differs from that previously reported for implementing ac polarography in the harmonic multiplex mode ( I ) in the identity of the minicomputer system employed and in the replacement of extensive frequency-selective analog circuitry by a simple analog signal conditioning network. The minicomputer used was a Raytheon 704 with a 16K-word, 16-bit core memory, a 1-psec cycle time and eight levels of priority interrupt. The central processor included a Model 72402 highspeed multiply-divide unit. The peripherals available were: two Raytheon Model 73491A 9-track magnetic tape units; an oscilloscope display based on D/A converters of 10-bit resolution and a Tektronix Model 611 storage oscilloscope; a computer-controlled X-Y plotter (Electro Instruments Model 480 X-YY‘ plotter); a 16-channel multiplexed A/D converter (Raytheon “Milliverter”) capable of digitizing signals within the limits f 1 0 V with 14-bit full-scale resolution a t a 40 kHz rate; two simultaneous sample-
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and-hold amplifiers characterized by 50 nsec aperture times; two D/A converters providing analog outputs within the limits of f 10 V with 12-bit resolution; a programmable clock with 1-Fsec resolution; and a Teletype Corp. Model ASR-33 Teletype. The potentiostat-cell networks, subtractor, initial voltage source, and drop-knocker circuit all are identical with those reported previously (I, 4-6). As before (I), the dc potential "scan" is computer-controlled via a 12-bit D/A converter. The dual potentiostat-cell-subtractor system effects analog compensation of nonfaradaic components under conditions employed in this study ( 4 4 , so that the subtractor output contains negligible nonfaradaic contributions. The drop-knockers are activated by the computer through a D/A converter-generated pulse. The ac input signal to the potentiostat is provided by a lowdistortion sinusoidal oscillator, Optimation Inc. Model RCD-10. Brown, T. G . McCord, D. E. S m i t h , and D. D. DeFord, Anal. Chem. 38,1119 (1966). (5) E. R. Brown, H. L. Hung, T. G . McCord, D. E . Smith, and G . L . Booman, Anal. Chem.. 40, 1424 (1968). (6) E . R. Brown, D. E. Smith, and G . L. Booman. Anal. Chem., 40,
The oscillator is adjusted t o output a 2.5-V peak-to-peak signal. This signal is reduced by a scaling amplifier to provide a 10-mV peak-to-peak signal for both potentiostats. The high-level (2.5-V) signal also is transmitted to the computer via the sample-andhold amplifier associated with input channel "0" of the multiplexer. To enhance potentiostat stability and subtractor accuracy, the potentiostat current amplifiers and the subtractor are operated with a low-amplification factor. As a result, the subtractor output is normally small (E300 Hz) frequencies, where average errors did not exceed 3%. The larger errors observed at low frequencies were eliminated by ensemble averaging (reduced to below 1% by 4-6 averages), indicating greater contributions of random noise a t these frequencies, probably due to the fact that only one or two complete cycles of the sine wave are digitized by the present program. Larger errors at high frequencies probably originate from small phase shifts in the potentiostat and not from problems inherent in digital signal conditioning. This hypothesis is supported by the fact that the high-frequency errors increased as greater demands were placed on the potentiostat by increasing the dummy cell capacitance. Detailed results of these tests, including similar results at dc and the second harmonic frequency, are tabulated elsewhere (11). Results of the dummy cell tests clearly indicated that the measurement system was compatible with the demands of accuracy and precision imposed by dc and ac polarographic measurements. This conclusion was confirmed by equally satisfactory results with the model electrode processes. Some typical data are shown in Figures 4-7. These data compare favorably with those obtained
using the analog signal conditioning approach to harmonic multiplexing ( I ) with regard to precision, theory-experiment agreement, and rate-parameter magnitude. Random fluctuations of data points in the dc potential profiles of the various observables are significantly smaller in the present case than with analog data processing ( I ) due to the use of Fourier transform data smoothing (14). More important than the observation that data quality at least matches that which characterizes alternate measurement methods is the relatively much greater ease with which such data are obtained by the present approach. In the measurement concept validation tests outlined here, as well as in several subsequent applications (20), the most impressive and ubiquituous advantage of the instrument based on FFT data processing has been the convenience feature. By eliminating analog lock-in amplifiers, tuned amplifiers, phase shifters, etc., all critical analog instrument adjustments were eliminated, except for the nonfar(20) K. R . Bullock, J. W. Hayes, D. E. Glover, A. M . Bond, and D. E. Smith, Northwestern University, Evanston, Ill., unpublished work, 1972.
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Figure 7. Typical second harmonic ac polarographic data with cadmium system System: same as Figures 4C and 4D. Applied: same as Figure 4 with sine wave frequencies Of 105 ( A ) , 228 ( E ) . 373 ( C ) . and 498 ( D ) Hz. Measured: inphase and quadrature currents as a function of dc potential at 210 ( A ) , 456 (B). 746-(Cj, and 996 (0) Hz. (0) Same as Figure 4. (-) Same as Figures 4C and 4 0
adaic compensation controls. Of course, the latter also can be replaced by computerized nonfaradaic compensations. The resulting simplification of preliminary instrument adjustment procedures not only reduces total experiment time, but greatly diminishes the number of experimental runs which are aborted because of improper instrument setup. In the context of our situation (research laboratory), the economic and emotional advantages attending this enhanced measurement efficiency more than compensate for the original investment. We feel that this conclusion also will apply in the context of an analytical services 1876
laboratory, where the potentialities of ac polarographic measurements (21) still exceed their level of application. Finally, although not demonstrated in this work, the digital approach provides versatility which at least matches the analog. Digital data processing is readily extended to lower frequencies than employed here without changing instrument hardware, as mentioned earlier. Much higher frequencies (to 25 mHz) also are accessible, from the viewpoint of data acquisition, if one adds commercially(21) A. L. Woodson and D. E. Smith, Anal. Chern., 42, 242 (1970).
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available high-speed digital data acquisition systems (22) to the existing minicomputer system. Because of considerations and results such as the foregoing, as well as those arrived at in related studies on frequency multiplexing (23-25), our commitment to digital signal processing in applications of ac measurements to (22) "Model 8100 Transient Recorder," Biomation Corp., Palo Alto, Calif., 1971. 123) . , S. C. Creason and D. E. Smith. J. Electroanal. Chem.. 36. A1 (1972); 40, A1 (1972). (24) 6. J. Huebert and D. E. Smith, Anal. Chem., 44, 1179 (1972). (25) S. C. Creason, J. W. Hayes, and D. E. Smith, J. Electroanal. Chem., in press.
quantitative kinetic-mechanistic studies is now total. Ac polarographic instrumentation based on analog signal conditioning ( I , 24) is now utilized in our laboratory only for qualitative or semiquantitative survey work.
ACKNOWLEDGMENT The authors are indebted to S. C. Creason, K . R. Bullock, J. W. Hayes, and A. M. Bond for their interest and helpful suggestions.
I
Received for review February 5, 1973. Accepted April 12, 1973. Work supported by NSF Grant GP-28748X. D.E.G. was a NASA Graduate Fellow, 1971-1972.
Determination of Composition of Mixtures of Weak Acids by Potentiostatic Titration Jouko J. Kankare Department of Chemistry, University of Turku, Turku 50, Finland
A simple linear relationship between the degree of deprotonation of an acid mixture and the mole fractions, and degrees of deprotonation of the components has been derived. I t is shown that the recently developed potentiostatic titration can be successfully employed for determining the degrees of deprotonation and the composition of acid mixtures. The use of the method is illustrated by analyzing binary mixtures of acetic, tartaric, citric, phthalic, and isophthalic acids with an accuracy better than f2%. A similar analysis of the ternary mixture of acetic, tartaric, and citric acids gave inferior results.
Difficulties are often encountered in the analyses of mixtures of organic acids. Gas, liquid, or ion-exchange chromatographic methods are often employed if appropriate conditions and columns can be found. However, the conventional potentiometric titration is seldom possible because of similar strengths of the acids. Recently, Purdie, Tomson, and Cook ( I ) described two methods by which the composition of a two-component mixture of weak acids was determined by pH titration. One method was based on computer analysis of the titration curves and the other was purely empirical, utilizing titration curves measured for mixtures of known compositions. In the former method, a rather complicated mathematical formulation was necessary, because the authors preferred the use of thermodynamic ionization constants. The purpose of this paper is to show that working in constant ionic strength media considerably simplifies the mathematics and that the recently developed potentiostatic titration ( 2 ) is also applicable to the analysis of mixtures of weak acids.
THEORY A general mathematical treatment of the equilibrium system is possible without unduly complicated expres-
sions. Let the mixture contain n acids denoted by HmlA(,,.The following equilibria prevail in the solution:
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The analytical mole fraction of an m,-basic acid HmLA(,, is denoted by x , , and its cumulative acidity constants by pl,, j = 1, . . . , m,. The total amount of acids in the mixture is a (in moles), and the total volume V. The mixture is assumed to be titrated with a solution of a strong base. Let the added amount of the base be b (in moles). Then we get the following set of equations:
The mathematical complexity of the protonation equilibria is considerably reduced by introducing a quantity Z, the "degree of deprotonation" ( 2 ) : (4) Another expression for 2 is obtained from Equation 3
(5) Equations 2 give (6)
Substitution into Equation 5 gives
(1) N. Purdie, M . B. Tomson, and G. K. Cook, Anal. Chem., 44, 1525 (1972). (2) J. J. Kankare. Ana/. Chem., 44, 2376 (1972).
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