month period, from the two methods are not statistically different. For the automated method, the mean value was 24.4 ppm N and the standard deviation was k 1 . 4 ppm N. The mean error was 0.1 ppm N. The relative error was 0.41 %. Similar relative errors were found for other standards in the 0 to 100 ppm N range. Many types of petroleum distillates, feed-stocks, and finished products have been analyzed by both methods and compared (see Table 11). The mean difference ( d = MG - AEK) is +OS6 ppm N. Since the standard deviation of the mean ( g / d N )is 0.58 ppm N, the results of either method cannot be distinguished from the other. N o problems have been encountered by the automated extraction Kjeldahl method with these samples even though they include volatile, low-boiling distillates, olefinic and aromatic materials, and the denser, more viscous oils and lube stock. Time Requirement. The automated extraction Kjeldahl
method is rapid and requires very little sample preparation, careful attention, or calculations. After the 2 hr needed for warm-up and calibration, samples are processed at the rate of 10/hr. As many as 40 samples can be completed in 1 day. A priority sample can be analyzed from a “cold start” (idle equipment) in less than 2.5 hr. Total time elapse for a single sample being processed during a sample run is about 30 min. ACKNOWLEDGMENT
The author thanks D. A. Nickey and W. C. Shaner for performing the numerous modified Gouverneur nitrogen determinations needed to calculate the many extraction efficiencies to produce paired data for the many petroleum samples.
RECEIVED for review March 4,1970. Accepted May 4,1970.
Numerical Deconvolution of Overlapping Stationary Electrode Polarographic Curves with an On-Line Digital Computer W. F. Gutknecht and S. P. Perone Department of Chemistry, Purdue University, Lafayette, Ind. 47907 The numerical deconvolution of overlapping stationary electrode polarographic (SEP) curves has been accomplished with a small, on-line digital computer. An empirical equation which describes the SEP curves for a wide variety of electroactive species has been developed for this purpose. During the first part of a typical analysis procedure, this empirical equation is fit to the standard SEP curves, and the constants of the equation, as specifically determined for each species, are stored in a library. When a mixture is analyzed, these standard constants are used to regenerate the standard curves which are then fit to the components of the “unknown” SEP signal. During the course of fitting the regenerated curves to the mixture signal, corrections are automatically made for slight lateral shifts of the individual component signals. This approach to deconvolution was evaluated for a number of different mixtures, including a system of well separated SEP curves which have significantly different magnitudes and a system of severely overlapping SEP curves which have similar magnitudes. This approach has enabled the quantitative resolution of overlapping SEP curves of similar magnitudes, the peak potentials of which are separated by less than 40 mV.
A FUNDAMENTAL PROBLEM with general analytical techniques is the achievement of quantitative resolution from overlapping data bands. This is a particularly formidable problem in the application of the electroanalytical technique of stationary electrode polarography (SEP). Because this technique has other attractive analytical advantages-such as speed, instrumental simplicity, extensive theoretical background, and applicability to a wide variety of systems-we have put some effort into diminishing the problems of quantitative resolution of overlapping reduction waves ( I , 2). A recent publication ( I ) describes previous work in this laboratory where on-line (1) S. P. Perone, D. 0.Jones, and W. F. Gutknecht, ANAL.CHEM., 41,1154(1969). (2) S. P. Perone and T. R. Mueller, ibid., 37,2 (1965). 906
ANALYTICAL CHEMISTRY, VOL. 42, NO. 8, JULY 1970
computer control and optimization of SEP experiments (using analog derivative read-out) were employed to achieve quantitative determination of up to 1OOO:l mixtures with widely separated peaks. However, the computer-optimized approach failed for peaks separated by less than 150 mV. The work presented here demonstrates the application of the small on-line computer for the resolution of severely overlapping electroanalytical data. The approach involves extracting the analytical information from SEP data using mathematical deconvolution techniques. An empirical equation was developed which describes the general stationary electrode polarogram for a wide variety of electroactive species. The function is fit to a number of standard polarograms, and the constants of the function, as specifically determined for each species, are stored in a library. Upon analysis of an unknown mixture, these constants are used to regenerate the standard curves, a composite of which is then fit to the unknown signal. The small computer was used to perform several different functions in the development of the on-line electroanalytical system. Primary among these were experimental control, timing, synchronization, and data acquisition functions. In addition, with the aid of the oscilloscopic display system, offline simulation studies were carried out to evaluate empirical equations developed for later on-line data processing. Finally, the computer was used to process SEP data acquired online for qualitative and quantitative information. The processing approach proved especially valuable for the analysis of mixtures of similar concentrations where the overlap was so severe as to preclude visible recognition of the individual signals. EXPERIMENTAL
Instrumentation. The electrochemical instrument used in this work was designed to be used for either stationary
I
I
Mech
Y IOK
I
10 Turn
K L 4 5 ;Q T J
453K
Refme cell
~K 22.M .
f &
3 9 q
output to
Anolop-to-DigiRI Converter
Figure 1. Schematic diagram of electroanalytical instrumentation The modules shown are: (Al, A4) voltage followers, P65A’s (A2, AS) voltage adders, P35A’s (A3, A6) current-to-voltageconverters, P35’s (A7) differential ampliier, P35 (A8, A10) multiple feedback active filters, P35A, P S A , respectively (A9, A l l ) output amplifiers, P35A, P85A, respectively (A12) integrator, P35A (RS1, RS2) low voltage power supplies (All amplifiers are Philbrick “P’, series amplifiers, Philbrick-Nexus, Dedham, Mass.) Note. All capacitances are in picofarads unless specified otherwise electrode polarography (SEP) (3) or derivative voltammetry (2). To minimize base-line interferences, a differential cell system and corresponding measurement electronics (4-6) were constructed. The electronic system used is shown in Figure 1. The two potentiostats used were of conventional design. Two low-pass active filters were included in the voltage amplification circuitry. Both were constructed with a gain of 1.0. These filters were of the Butterworth, multiple feedback type (7) and each had an fc value of 25 Hz. The first stage of output amplification was constructed with a gain of 10.0 (Amplifier A9, Figure 1). The gain of the second stage of amplification was varied by changing the feedback resistor. (3) Richard S. Nicholson and Irving Shah, ANAL.CHEM., 36, 706 (1964). (4) H. M. Davis and R. C. Rooney, J. Polurogr. SOC.,8, (2), 22 (1962). ( 5 ) H. M. Davis and Joyce E. Seaborn, “Proceedings of the 2nd International Congress of Polarography,” Cambridge, 1959, Vol. I, Pergamon Press, London, 1961, pp 239-250. (6) H. M. Davis and H. I. Shalgosky, ibid., Vol. 11, pp 618-627. (7) “Handbook of Operational Amplifier Active RC Networks,” Burr-Brown Res. Corp., Tucson, A r k , 1966.
An operational amplifier configured in the integrator mode was used as the voltage ramp source. The output of the integrator was calibrated to 1.000 volt/second. The ramp was initiated with the opening of a mechanical relay, driven by a Digital Equipment Corporation (DEC) W051 relay driver. The ramp was found to be reproducible to within =tl mV/V. The voltage applied to the cell at time zeroi.e., the “initial” cell voltage-was provided by a second lowvoltage power supply. The general power supply used with this instrument was a Philbrick Model 300, *15 V power supply. Cells, Electrodes, and Cell Environment. The cells used in this work were 50-ml borosilicate glass centrifuge bottles with threaded mouths (No. 5-5558-5, Fisher Scientific Co.). Threaded lids for these containers were machined from Teflon (Du Pont) and holes were drilled in the lids for the electrodes and N Pdispersers. The cell lids were supported with adjustable clamps made of’ Plexiglas. The working electrodes used were conventional dropping mercury electrodes. In this work, the voltammetric experiments were performed during the latter parts of the lives of the mercury drops wherein the change in the drop areas with time is minimal. So that the areas of the two electrodes might be close to equivalent at the time of the experiment ANALYTICAL CHEMISTRY, VOL. 42, NO. 8, JULY 1970
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drop synchronization was provided. This was done with a drop dislodgment device patterned after one developed by Stephens, Behrin, and Harrar (8). The drop dislodgment lever system, which was constructed of Plexiglas, was driven with a Guardian Electronics No. 28,6-voit solenoid, activated with the closing of a mechanical relay. The ability to consistently reproduce experimental SEP peak heights for a given solution to within i1 % indicated reproducible drop areas. The two capillaries used in the cells were the results of cutting one 12-inch DME capillary exactly in half. However, it was not possible to find a pair which had identical characteristics, and it was necessary to further match the two dropping electrodes with a technique described below. Coleman 3-510 saturated calomel electrodes (Coleman Instruments, Maywood, Ill.) were used as reference electrodes. Extensions were added to the reference electrode reservoirs SO that they would fit into the cells. Electrodes were compared until an approximately matched pair was found. Remaining difl‘erences were compensated by an adjustment procedure described below. The counter electrodes were constructed from 4-cm pieces of 22-gauge Pt wire bent into spirals and sealed into 3-mm glass tubes. The cells and the drop dislodgment device were mounted in a steel box 14” X 14” x 18/’. The box provided shielding from electronic noise, and was supported by four thick pads of folded paper, which provided protection against vibration. The reservoirs of the SCE reference electrodes were refilled daily with fresh, saturated KC1 solution. The platinum counter electrodes were washed regularly with a 50 HNO, solution. In order to maintain reliable working electrodes, the DME’s were allowed to run continuously. The analog electronics were balanced daily using two matched dummy cells. The two polarographic cells were balanced with a procedure described earlier by Davis and Rooney (4). Using this procedure, each cell was filled with a solution of 5 X 10-5M Cd(I1) in 1.OM KC1. The initial potential of the sample cell was then set to a desired level. Voltage scans were made and the outputs of the two cells were observed with an oscilloscope. The initial potential of the reference cell was adjusted until the peak potentials for the two systems matched. Next the height of the mercury column for the sample cell was adjusted so as to give a desirable drop growth rate. The output of the differential system was then monitored and the height of the mercury column for the reference cell was adjusted so as to give a minimum difference signal. Solutions. All chemicals used were reagent grade. Water was purified by distillation and passage through a mixedbed cation-anion exchange resin. This work was performed in an air conditioned room wherein the temperature was maintained at 23.0 f 0.5 “C. All solutions were deaerated with high purity nitrogen which was “scrubbed” with a Cr2+/Cr3+ solution (9) before entering the cells. Three separate, but presumably identical, solutions were prepared for each standard species and each unknown mixture. Four SEP runs were made on each sample and each run consisted of an ensemble average of eight voltage scans. Computer Hardware and Interface. The computer hardware used in this work is essentially identical to that described in detail previously ( I ) . The computer used was a Hewlett-Packard 2115A with 8192 words of core memory and an extended arithmetic unit. Peripherals included a teletype, a high-speed paper tape reader, and a Tektronix 601 scope display. The analog-to-digital converter (ADC) used was a DEC Model COO2 with 10 bits and a 33 microsecond conversion time. Experimental timing was provided (8) F. B. Stephens, E. A. Behrin, and J. E. Harrar, U. S. At. Energy Comm. R e p . UCRL-50374(1968). 18, 367 (1958). (9) L. Meites, ANAL.CHEM., 908
ANALYTICAL CHEMISTRY, VOL. 42, NO. 8, JULY 1970
by a DEC Model B405 10-MHz crystal clock counted down to 1 KHz (10). The interface was also constructed with DEC Flip-Chip “R” series logic cards. Several output bits were made available from the computer for various control functions. The voltage ramp was initiated by a signal from bit 10 and reset to zero with a signal from bit 11. A voltage pulse of sufficient duration for drop dislodgment was made available with a DEC R302 “one-shot” logic card. A signal from bit 12 of the computer activated the one-shot which in turn controlled the solenoid used for drop dislodgment. Control was applied through a DEC W051 relay driver card just as for the ramp generator. A pulse width of 30 milliseconds was sufficient to dislodge the mercury drops with a minimum of stirring. THEORY AND PROCEDURES
Numerical deconvolution has been used extensively and there are several different approaches which can be taken. One approach, which has been used in gamma ray spectral analysis (11-13), is that of fitting standard spectra stored in a library to the unknown spectrum. A second approach is that of fitting a computed curve to the unknown spectrum wherein the computed curve is derived from a summation of individual functions which describe the contributions of the individual components of the spectrum. This technique has been used for the analysis of overlapping peaks in gas chromatography (14), mass spectroscopy (15), NMR spectroscopy (16), and EPR spectroscopy (17). It has also been used to separate the individual bands of IR spectra (18, 19). In this work a combination of the two procedures described above was used to perform the analysis of overlapping electrochemical signals. An empirical equation was developed which fits a large variety of SEP waveforms by adjustment of appropriate constants. The function is first fit to a number of different standard polarograms, and the constants of the function for each species are stored in a library. Upon analysis of an unknown mixture, these constants are used to regenerate the standard curves, a composite of which is then fit to the unknown signal. Stationary electrode polarography, the experimental electroanalytical technique used here, was chosen for this work for several reasons: the measured output is a smooth, continuous function of time; extensive theoretical descriptions have been developed and evaluated ( 3 ) ; and it has a number of experimental characteristics which made it suitable for automated analysis. As stated previously (20), the technique is rapid and repetitive and therefore digital data acquisition, processing, and control are particularly advantageous. Also, the instrumentation is relatively simple and reliable and the technique can be compatible with the dropping mercury electrode. The SEP technique is especially suited to the (10) “Logic Handbook,” Digital Equipment Corporation, Maynard, Mass., 1967. (11) Oswald U. Anders and William H. Beamer, ANAL.CHEM., 33, 226 (1961). (12) Felix J. Kerrigan, ibid., 38, 1677 (1966). (13) E. Shonfeld, Nucl. Instrum. Methods, 52, 177 (1967). (14) Peter D. Klein and Barbara A. Kunze-Falkner, ANAL.CHEM., 37, 1245 (1965). (15) David G. Luenberger and Ulric E. Dennis, ibid., 38, 715 (1966). (16) Warren D. Keller, T. R. Lusebrink, and C. H. Sederholm, J. Chem. Phys., 44,782 (1966). (17) Alfred Bauder and Rollie J. Meyers, . . J. Mol. Soectros., 27, ’ llO(1968). (18) Henry Stone, J. Oot. SOC.Amer., 52,998 (1962). (i9j D. papousel; and J. Pliva, coliect.. Czech. c i e m . Commun., 30, 3007 (1965). (20) S . P. Perone, J. E. Harrar, F. B. Stephens, and Roger E. Andersen, ANAL.CHEM., 40,899 (1968).
mathematical deconvolution approach as the signal is simple in shape and is easier to describe than derivative voltammetric ( 2 , 2 1 ) signals for example. Development of Mathematical Function. It is imperative to develop a descriptive mathematical function so that it is consistent with the overall analysis system. The small digital computer generally has limited memory (4K to 8K of core memory) and limited numerical precision (12 to 18 bits/word of memory). Also, as on-line analysis is required for automation, the numerical analysis should be as rapid as possible. Thus, the function can not be too complex without its implementation exceeding the computer’s capabilities or causing processing delays. Thought was given to using one of the rigorously derived theoretical functions which describe the stationary electrode polarogram (3). However, these are not of closed form, and any approximation technique woud both consume too much computer memory and too much time. Moreover, the theoretical equations rarely describe accurately the polarographic behavior of a wide variety of electroactive species. The philosophy used in developing a function to describe the stationary electrode polarogram is similar to that used by Kowalski and Isenhour (22). Various geometric and algebraic terms were combined, and, by a process of trial and error, a combination was found which gave a curve having a shape almost identical to a stationary electrode polarogram. Moreover, the shape of this general function could be readily adjusted to accommodate a wide variety of electroactive systems. While developing this function, extensive use was made of the Hewlett-Packard BASIC programming language (23) and the computer-interfaced oscilloscopic display system. The trial and error procedure was greatly facilitated by the ability to edit and re-run programs immediately and the ability to display the results of‘ each new trial on the oscilloscope. Casual analysis would indicate that the normal stationary electrode polarogram has the appearance of a skewed Gaussian distribution. Several functions describing skewed Gaussians have been presented in the literature (24, 25). However, the one developed by Levy and Martin (26) was found to most closely fit the leading edge of a reversible polarographic curve. The function is: For this function, X, determines the center of the curve, B determines the sharpness of the peak, and C determines the extent of skew present in the curve. The amplitude is determined by a coefficient A , which is defined below. This function was also observed to fit, approximately, the shape of the peak region of the test polarogram. For any uncomplicated reduction process, the tail of the polarogram should decay at a rate proportional to l / d where t is time (27). This was not observed to be strictly so when polarograms of several different species were examined. Thus, an inverse power o f t decay term, wherein the exponent was a variable rather than the square root, was deemed a necessary part of the function. So that the decay term might have continuity along the entire “x” axis, it was (21) C. V. Evins and S . P. Perone, ibid., 39,309 (1967). (22) B. R. Kowalski and T. L. Isenhour, ibid., 40, 1186 (1968). (23) “A Pocket Guide to Hewlett-Packard Computers,” HewlettPackard Co., Palo Alto, Calif. (24) R. D. B. Fraser and Ellichi Suzuki, ANAL.CHEM.,41, 37 (1969). (25) Eli Grushka, Marcus N. Meyers, Paul D. Schettler, and J . Calvin Giddings, ibid., p 889. (26) E. J. Levy and A. J. Martin “Pittsburgh Conference on Analytical Chemistry and Applied Spectroscopy,” Cleveland, Ohio, March 3-8,1968. (27) A. Sevcik, Coltect. Czech. Chem. Commun., 13, 349 (1948).
&(volt11
Figure 2. Distribution of algebraic and geometric terms of the synthetic function over a typical stationary electrode polarogram Refer to text for complete description of different terms
necessary to set the decay term equal to 1.0 until that point was reached in the voltammetric curve where the decay started. Thus the decay function is
where D and E combine to determine the magnitude and rate of the decay. In order to gradually merge the two terms-the Gaussian leading edge and the exponential trailing edge-a third term was utilized:
- TANH { G ( X - X , ’ ) } ] (3) This term goes from 1.0 to 0.0 as G ( X - X,’)goes from a F
= 0.5 [1.0
large negative value to a large positive value. The term G determines the rate at which this change takes place. By comparison with the general polarographic curve, the center of the merging process did not necessarily occur at the center of the Gaussian (i.e., X,’ f X,). The distribution of the respective terms over the SEP waveform is shown in Figure 2. The total function is:
Y
=
A[FPi
+ (1.0 - F)P2]
(4)
where A determines the height of the peak. This function, which has eight constants, was found to fit polarograms for a number of different reducible species. The requirement of several constants is not unexpected as the rigorously-derived equation for the simplest reversible case requires four independent constants in addition to the numerically tabulated at) function (3). More constants are required as complicating processes are considered (3). Program Description. There are two major programming functions performed in this work. The first involves fitting the empirical function to the standard curves. The second involves analyzing mixtures. These two functions were carried out with two different programs. Both programs do, however, contain many of the same subfunctions such as data acquisition routines, peak locating routines, etc. Determination of Standard Coefficients. The program which fits the empirical function to the standard curves is executed as follows. The program first requests, via the teletype, those parameters associated with taking data. These are loaded via the teletype. Next called for and loaded is the initial cell voltage. At this point one can choose to load previously acquired data which are stored on paper tape ANALYTICAL CHEMISTRY, VOL. 42, NO. 8, JULY 1970
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standard curve is nonlinear, residual least squares (29, 30). In this technique, one determines those values of A j (where A j represents each of the constants in the function) for which the summation
Figure 3. Display of synthetic and experimental SEP Curves for Oth, Znd, and 5th iterations of typical curve fitting process 4.80 X 10-5M In(III), 1.OM HCl Voltage range shown: -0.100 + -0.900 V us. SCE Experimental peak current: 3.2pA
Upper trace: 0th iteration (Synthetic curve calculated with initial estimates of constants) Middle trace: 2nd iteration Lower trace: 5th iteration
or one can take real-time experimental data. If real data are collected, a number of replicate scans can be run with the data averaged and normalized. The data are next displayed on the oscilloscope. The program determines the approximate location of the peak of the polarographic signal and then fits a quadratic equation to the top of the peak using linear least squares (28). The first derivative of the quadratic is set equal to 0.0 and solved for the true location of the peak. The technique used to fit the empirical function to the
(28) Wendel E. Groves, “Brief Numerical Methods,” PrenticeHall, Inc., Englewood Cliffs, N. J., 1966. 910
ANALYTICAL CHEMISTRY, VOL. 42, NO. 8, JULY 1970
is a minimum. At this minimum, A i , the best value for the constant, is approximated by the summation of Ai,,, an estimate of the constant, plus AAj, a small correction term. Here Yc represents the experimental data and Wi represents the weights of the residuals (30). The minimum is determined by calculating the partials of Equation 5 with respect to each of the correction terms, AAj, and equating these particals to zero. The initial estimates of A j (Le., A j J are entered by the operator; the value of the peak location, along with a visual display of the data, aids considerably in determining these values. The calculations of the partials of Equation 5 for the empirical function described above result in eight simultaneous equations. These equations are solved for the correction terms, AAj, using matrix decomposition (31). The values of A A j determined here are, however, only approximations of the final corrections since the term b f ( A A j ) of Equation 5 constitutes a truncated Taylor i = bAj l series (18). Thus the calculated values of the correction terms are added to the previous estimates of the constants, these sums are used in new estimates for the constants, and the entire procedure is repeated until the values of the correction terms become insignificantly small, (about 0.1 % of the current value of each constant). During the course of the first and subsequent iterations, both the experimental data points and the synthetic data points are displayed on the oscilloscope. Thus one can monitor qualitatively how well the fit is going and/or how good the initial estimates of the constants were. Figure 3 shows the output during three different iterations for a typical fitting process. The polarographic signal being fit is that for 4.8 X lO-5M In(II1) in 1.OM HCl. During the course of fitting the standards, it was assumed that the weights of the residuals for any one curve were equivalent. Thus a weight of 1.0 was used. (It is realized that this assumption is not totally valid as data points on the rapidly changing portions of an SEP curve generally have a greater standard deviation than those data points on the flatter portions of the same curve. However, it does not appear practical to calculate the standard deviations for each of the 500 data points usually acquired for each standard curve. Moreover, the analytical results discussed below do show that the overall precision of the fits are approximately equal to the precision of the experimental data.) Generally five to ten iterations were required to fit each standard. Each iteration for a 500-data point run took approximately two minutes including print out time. At the conclusion of this procedure, one has the option of dumping the experimental data on paper tape for later use. Analysis of Overlapping Polarographic Signals. The program which analyzes overlapping signals has two main parts. The first part allows for loading standard information, and the second controls the electroanalytical experiment, acquires the data, and performs the actual analysis.
(29) R. H. Moore and R. K. Zeigler, “The Solution of the General Least Squares Problem with Special Reference to High-speed ComDuters.” LA-2367,. (1959). . ,. U. S. Government Printing Office, Washington D. C. (301 John Mandel. “The Statistical Analysis of ExDerimental . Data,” Interscience, New York, 1964. (31) Anthony Ralston, “A First Course in Numerical Analysis,” McGraw-Hill Book Co., New York, 1965, Chap. 9.
~~
Table I. Standard Constants of Empirical SEP Function for Several Electroactive Species Standards 4.94 X 10-6M Cd(II), 5.02 X 10+M Pb(II), 4.96 X 10-'M SWIII), 4.78 X 10-6M Tl(I), 1 .OM KCl 1 .OM KCl 1 .OM HCL 1 .OM KCl Re1 std Re1 std Re1 std Re1 std Value dev, Z Value dev, 'Z Value dev, 'Z Value dev, % 0.55 0.81 620.0 675.5 697.2 0.75 339.0 0.72 6.763 0.35 0.13 1.601 4.708 1.1 5.408 0.79 0.3713 2.9 0.2450 1.6 2.9 0.3595 0.6401 1.4 6.6 0.2242 0.1627 5.5 0.2387 0.1864 4.4 8.3 2.534 4.5 4.4 13.89 3.283 1.072 2.8 11.6 0.4579 1.9 3.3 0.3603 0.4187 0.4451 1.9 3.2 2.017 8.1 2.4 2.863 1.920 3.3 1.009 2.9 6.749 0.59 0.74 1.663 4.696 5.391 1.1 2.1
Constants A XO
B C
D E G Xo
Re1 std dev of fit, %
2.0
3.3
2.4
2.6
%
0.90
0.58
0.85
0.64
Re1 std dev of peak height,
The standard information is stored in a multifunctional subroutine. This routine can be used for storing standard information, retrieval of standard information, or identification searches. Room is available within this routine for the storage of standard information for five different species, but this capacity can be expanded as needed. Information stored for each species includes a code name (e.g., C D for cadmium), the standard peak location, the standard concentration, and eight standard constants of the empirical function. Experimental data to be analyzed can be input uia paper tape or can be acquired directly from the experiment. Detection and identification of peaks is not as direct with mixtures as it is with single component systems. When two polarographic curves overlap severely, the peak of one of the signals may appear as a very slight peak or only as a shoulder. A simplified version of the technique presented by Morrey (32) was used here to detect peaks. This involves calculating the second derivative of the polarographic signal and locating the resulting negative peaks of the second derivative. The second derivative is calculated here using an eleven-point quadratic convolution as presented by Savitsky and Golay (33). The qualitative identification of each reduction step is made by comparing the 2nd-derivative peak location of the unknown to the standard peak locations. A positive identification is made when the 2nd-derivative peak potential of the unknown species is found to be within =t15 mV of a standard peak potential. The identification window used was large enough to accommodate the slight difference between the second-derivative peak location and the standard peak potential (2). The magnitude of the identification window also permits slight potential shifts which can arise from several experimental sources. For example, shifts in the peak potential can arise from drifts in the electronics and small changes in the potential of the calomel reference electrodes. Moreover, slight changes in solution conditions, such as pH, electrolyte concentration, complexing agent concentration, etc., from sample to sample can cause nonequivalent shifts in peak potentials for various mixture components. In addition, acute overlap of current-voltage peaks can itself cause apparent shifts in the peak potentials. For the systems and instrumentation used here, maximum experimental peak shifts of the order of 7 mV were observed. If there were no uncertainty in the experimental peak potentials, quantitative analysis for the individual components of a multicomponent system could be carried out using a set
*
(32) J. R. Morrey, ANAL.CHEM., 40,905 (1968). (33) A. Savitsky and M. J. R. Golay, ibid., 36, 1627 (1964).
of simple, simultaneous equations. As discussed above, this is not the case. Consequently, quantitative analysis for the components of the multicomponent system was accomplished here with an iterative procedure: After the individual component signals are located and identified, the approximate concentrations are calculated using simple, simultaneous equations. Next, a simplified deconvolution technique is used to refine the concentration values. It is assumed that there are only two unknowns for each component: the concentration and the initial potential of the voltage scan for each component, Considering Xiinitis, a variable unknown accounts for slight lateral shifts in peak potentials of individual components. The nonlinear least-squares curve-fitting technique is again used. The initial estimates of the concentrations come from the simple simultaneous equations calculation, and the initial estimate of the starting potential for each of the components is the experimental starting voltage for the multicomponent analysis. Various weighting schemes were employed here to improve the overall fit and these are discussed below. The results of the simple simultaneous equations calculation are printed out and the program proceeds to make one refinement calculation. After the results of this calculation are printed out, the operator is given the option of having another refinement of terminating the program. The procedure is usually stopped when the calculated concentrations d o not vary more than 0.1 from one iteration to the next. This procedure could be completely automated if desired.
RESULTS AND DISCUSSION Several different aspects of the computer-automated analytical scheme presented here were evaluated. Of primary consideration was the ability of the technique to qualitatively and quantitatively resolve severely overlapping SEP curves. This specific evaluation was based primarily on results with twocomponent mixtures wherein overlap was severe but concentrations were similar. A second situation evaluated was one wherein the components were well separated, but the concentrations of the components differed significantly. All the standard curves used in the evaluation of this technique were acquired with the instrument described above. However, several of the mixtures used in this work were synthetic. That is, mixtures were produced by simply adding, point by point, various standard curves. The only assumption to be made here is that, experimentally, linear additivity would exist between the signals of the species in question. This assumption is generally valid though some exceptions have been reported ANALYTICAL CHEMISTRY, VOL. 42, NO. 8, JULY 1970
911
I
i
0.0-
-0.203
I
-a4400
-0603
-C
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I
t
-0.103
-0.320
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-0.500
I
-a700
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-am
-I
Volts vs. %E.
Figure 4. Comparison of experimental and synthetic curves for 4.78 X 10-5MTI(1) in 1.OM KCI
- Experimental data Synthetic data Alternate points plotted at extremes of curve for clarity Voltage range shown: -0.100 -+I -1.100 V US. SCE Experimental peak current: 1.34 NA 0
Figure 5. Comparison of experimental and synthetic curves for 4.96 X lO-bM Sb(II1) in 1.OM HCI
- Experimental data 0 Synthetic data Alternate points plotted at extremes of curve for clarity Voltage range shown: -0.025 -0.725 V us.
SCE
-
Experimentalpeak current : 4.9 pA
(34). The generation of synthetic mixtures has allowed the simulation of experimental situations which would be difficult to obtain with real systems. Evaluation of Fit. First to be tested was the generality of the function-that is, its ability to describe the polarograms of a number of different species. Several criteria for the goodness-of-fit are used here. One is visual observation of the overall fit. A second is the randomness of the fit of the calculated curve to the experimental curve. If the fit is good, the synthetic data should match the experimental data randomly from point to point. If there are sections of the synthetic curve (Le., blocks of data points) which are either high or low compared to the same section on the experimental curve, the fit is not as good as it might be. Finally a comparison was made between relative standard deviations of the fit of the function to the experimental data and the relative standard deviations of the experimental peak heights. The relative standard deviation is defined as the standard deviation of the fit of the function to the data divided by the average value of all the data points of the experimental curve. This term, though not rigorously derived from statistical theory, does give a measure of the goodness-of-fit. Electroactive species used to provide a test of the generality of the fit included Tl(I), Pb(II), Sb(III), In(III), Cu(II), Bi(III), Cd(II), Zn(II), and Ni(I1). The calculated constants of the empirical SEP function for several species listed are shown in Table I. Here X, and X,' areexpressed in decivolts. The Y values used during the fitting process are in arbitrary units linearly related to the input voltage of the analog-todigital converter. The units for the other terms can be determined by rearrangement of the total function. Typical plots comparing the experimental and calculated curves are shown in Figures 4 and 5. The density of all the standard data used here was one point per two millivolts. The voltage (34) H. Dahms, J. Electroanal. Chem., 37, 1643 (1965). 912
ANALYTICAL CHEMISTRY, VOL. 42, NO. 8, JULY 1970
scan for the Tl(I), Pb(II), and Cd(I1) systems was from -0.100 to -1.100 V us SCE whereas the scan for the Sb(II1) system was from -0.025 to -0.725 V us SCE. Figure 4 shows the overall fit to be generally good for the Tl(1) system. There are several sections of the curve where the errors of the fit are not random but drift in either the positive or negative direction. Such errors occur at the foot and peak of the wave. This is an undesirable characteristic but the error here is small in magnitude. The Sb(II1) system shown in Figure 5 shows similar characteristics. Several characteristics of the function and its fit can be noted from Table I. First, the standard deviations for the constants C, D , E, and G show these terms to be fairly sensitive to small changes in the shape of the curve. Comparing the constants as a function of the species, one notes that the function better fits the broad TI(1) peak than it does the narrower Pb(I1) or Cd(I1) peaks. The instability noted for the Sb(II1) most likely reflects this trend but also shows a decrease in precision as X becomes smaller. This would be expected as small errors in the cell voltage (Le., X ) would be of greater significanceas the voltage approached zero. The relative standard deviations presented in Table I and the subsequent tables of standard constants do show that the fit is, on the average, about as good as the data. Nevertheless, the effect of a poor fit at any specific section of these standard curves is dependent upon the system to be analyzed and can be fully evaluated only for the specific case. Concentration Effects. If the synthetic functions are to be used for the analysis of mixtures of variable concentration, it is essential that the functions continue to describe the signals of the appropriate species as the concentration of these species change. Theory (3) does predict that for both reversible and irreversible systems, the shape of the stationary electrode polarogram should remain constant with change in concentration, and thus the requirement stated above should be met. This was verified for the reversible Pb(I1)-
Table 11. Standard Constants of Empirical SEP Function for Ni(I1) in 0.10M KN03. Concentration Dependence Concentration" 5.09 x 10-6M 5.09 x 10-6M 5.09 x 1 0 - 4 ~ Constants Value Re1 std dev, % Value Re1 std dev, 2 Value Re1 std dev, A 276.0 2.0 267.9 2.9 270.1 1.7 X* 10.85 0.37 10.80 0.35 10.99 0.18 B 0.5585 11.3 0.5130 3.2 0.5198 2.1 C 0.4059 12.3 0.3344 6.2 0.3187 8.4 D 1.320 16.4 0.9098 3.5 0.7238 3.9 E 0.3152 12.1 0.4286 2.8 0.4489 0.96 G 2.220 13.1 2.007 3.1 1.955 2.2 Xo ' 10.05 0.33 10.02 0.31 10.17 0.19 Re1 std dev of fit, % 2.4 1.5 1.5 Re1 std dev of peak height, % 2.9 3.0 1.6 Note: sensitivity change equal to dilution factor for each set of constants.
z
5
Table 111. Analysis of 1 :1 [Zn(II)]-[Ni(II)] Mixtures in 0.10M KN03a Ni(II) Standard Constants Determined at Different Concentrations Case I
Standard [Ni(II)] = 5.09 X 10-eM [Ni(Wl, M
[Zn(II)I, M Expected 5.02 x 5.02 x 5.02 x 10-4 Case I1
Detd 4.92 X 10-0 4.98 x 5.21 x 10-4
Re1 std dev, Z 1.6 2.1 2.2
Re1 error, -2.0 -0.80 3.8
x
Expected 5.09 x 5.09 x 5.09 x 10-4
Re1 std dev, % 3.9 1.5 1.7
Re1 error, -2.4 -5.3 -5.1
Z
Re1 std dev, Z 2.4 1.6 1.7
Re1 error, 3.5 1.8 1.2
x
Re1 std dev, % 2.9 1.5 0.59
Re1 error, 3.9 2.6 0.39
Standard fNi(II'I1 = 5.09 X 10-5M [Ni(II)] M
[Zn(II)l, M Expected 5.02 x 5.02 X 5.02 x 10-4 Case I11
Detd 4.97 x 10-6 4.82 x 10-5 4.83 x 10-4
Detd 4.84 x 10-6 4.92 x 10-5 5.14 x 10-4
Re1 std dev, % 2.5 1.2 1.8
Re1 error, % -3.6 -2.0 2.4
Expected 5.09 x 5.09 x 10-5 5.09 x 10-4
Detd 5.27 X 5.18 x 5.15 x 10-4
Standard [Ni(II)] = 5.09 X 10-4M [Zn(II)I, M
Q
Expected Detd 5.02 x 4.59 x 10-6 5.02 X lod6 4.67 x 5.02 x 10-4 4.94 x 10-4 Synthetic mixtures.
[Ni(Wl, M Re1 std dev, Z 1.9 1.4 0.86
Re1 error, % -8.6 -6.9 -1.6
Cd(I1) system discussed below. For further verification a system wherein some change in curve shape might be expected was examined. Such a system is Ni(I1) in 0.10M KNOI (35). This system is irreversible and one might expect a change in curve shape due to a change in the degree of irreversibility (Le., 0: n,) with concentration. Table I1 presents the constants of the empirical function for three different Ni(I1) concentrations. Trends are noted in constants C, D, E, and G suggesting a change in the trailing section of the Ni(I1) curve with concentration. The effect of these and any undetected changes with concentration was evaluated with a series of synthetic Zn(I1)-Ni(I1) mixtures. Standard SEP data for 5.02 X M Zn(I1) in 0 . 1 M K N 0 3were obtained. These data were then added to the standard Ni(I1) data obtained at the three concentrations listed in Table I1 [with appropriate normalization of the magnitude of the Zn(I1) (35) S. P. Perone and C . V. Evins, ANAL.CHEM., 37,1643 (1965).
Expected 5.09 X 5.09 x 10-5 5.09 x 10-4
Detd 5.29 x 10-6 5.22 x 10-5 5.11 x 10-4
data]. The peak potential separation for this system is approximately 80 mV. Eight to ten synthetic mixtures were produced for each concentration. These mixtures were analyzed using the standard Zn(I1) constants determined for the Zn(I1) data discussed above and the three sets of standard Ni(I1) constants presented in Table 11. Synthetic mixtures were used here as it was desired to examine only the effect of NI(I1) upon the analysis of the mixtures. Using well defined Zn(I1) curves obtained at one concentration would minimize the effect of Zn(I1) upon any changes in the analytical results with change in concentration. The results of analyzing the Zn(I1)-Ni(I1) mixtures are shown in Table 111. Trends are noted in both the Zn(I1) and Ni(I1) results. The errors for the two species generally show opposite trends. The magnitudes of these errors are generally close to the relative standard deviations of experimental data, except when the standard Ni(I1) constants determined with the lowest Ni(I1) concentration are used to analyze the highest ANALYTICAL CHEMISTRY, VOL. 42, NO. 8, JULY 1970
913
Table IV. Analysis of 10: 1 [Pb(II)]-[Cd(II)] Mixture in 1.OM KC1 [Pb(II)I, M [Cd(II)I, M Weighting Re1 std Re1 std scheme Detda concn Re1 error, % Detdbconcn dev, dev, Z I 5.03 x 10-5 0.94 0.20 4.43 x 10-6 2.5 I1 5.03 x 10-5 4.52 x 1.8 0.61 0.20 I11 4.97 x 10-5 0.57 -0.99 4.69 x 4.0 = Expected value-5.02 X 10-5M Pb(I1). b Expected value-4.94 X 10-6M Cd(I1).
z
Constants A
XO B C
D E G Xo
’
Table V. Standard Constants of the Empirical SEP Function for In(II1) and Cd(I1) in 1.OM HCI Standards 4.80 X 10-5M In(II1) 4.94 x 10-5M Cd(I1) Value Re1 std dev, Value Re1 std dev, 834.9 646.2 1.2 1.7 0.14 5.852 6,267 0.064 0.49 0.75 0.2624 0.3143 2.4 0.1299 0.1566 2.6 5.9 5.9 1.406 1.535 3.3 0.5906 3.1 0.6147 3.051 1.6 2.246 2.1 0.24 0.14 6.279 5.863
Re1 std dev of fit, Re1 std dev of peak height,
z
z
1.5
2.3
1.6
1.2
Figure 6. SEP curves for 1O:l [Pb(II)]-[Cd(II)] system 5.02 X 10-5M Pb(II), 4.94 X 10-6MCd(II), 1.OM KCI Voltage range shown: -0.100 -1.100 V cs. SCE Peak current for Pb(I1): 2.8 pA
-
concentration mixtures of Zn(I1) and Ni(II), and vice L;ersa. No sweeping generalizations can be made about the results obtained with this system. If one suspects that the shape of an SEP curve will change significantly with concentration, one will be required to examine the system before using this approach. To minimize a problem of extreme nonlinearity, one could store, in memory, the standard constants as determined a t several concentrations and then, during the refinement calculations, use the standards best suited for the specific unknown mixture. Quantitative Resolution of Widely Separated Peaks. A 1 O : l [Pb(II)]-Cd(II)] mixture in 1.OM KC1 was analyzed to evaluate the ability to quantitatively analyze well separated SEP signals of dissimilar magnitude. The peak potential separation for these two components was approximately 200 mV (see Figure 6). The standard constants used for the analysis are presented in Table I. The mixture was first 914
Re1 error, - 10.3 -8.5 -5.1
ANALYTICAL CHEMISTRY, VOL. 42, NO. 8, JULY 1970
z
z
analyzed with the weights of the residuals being equal for all the data points for the multicomponent signal. The results of this aralysis are shown in Table IV under Weight Scheme I. The result for Cd(I1) is noted to be quite low. This is to be expected. The least squares technique provides the best overall fit of the function to the curve. Thus the large data points of the Pb(I1) peak cause a smoothing or flattening out of the small peak during the fitting process. To overcome this problem, several alternative schemes were tried which tended to equate the weights of the data points of the two peaks. Two of the alternate schemes are presented here. In the first scheme, each component curve of the mixture is given a weight. This weight is inversely proportional to the magnitude of the component curve relative to the largest component curve. The magnitudes of the component curves used to calculate the weights were determined using the initial concentration estimates. Another scheme tried is that of equating the weights of all of the data points making up the composite curve. This is done as follows : The program locates the largest data point in the signal and gives the residual a t this point a weight of 1.0. The rest of the residuals are then weighted inversely proportional to the magnitude of their respective data points relative to the largest data point. Table IV shows the results of using these two alternate weight schemes, referred to as Weight Schemes I1 and 111, respectively. The Cd(I1) results are still low but the error has From Figure 6, one observes been reduced to about that the ratio of the height of the Cd(I1) peak to that of the tail of the Pb(I1) curve is about 1 :5 . Thus a 1 determinate error in describing the Pb(I1) curve with the empirical function would result in a 5 error in the determination of the Cd(II), since the Pb(I1) tail provides the base line for the Cd(I1) peak. It does appear that, for two well separated peaks, the practical limit of quantitative resolution is reached when the signal ratios exceed 10 : 1. This, however, is a fundamental limitation of the SEP technique, not of the deconvolution process.
5z.
z
Table VI.
Weighting Detdbconcn 3.83 X scheme Id Real mixture. * Expected value-3.84 X 10-lM In(II1). c Expected value-3.95 X 10-5M Cd(I1). d All residuals equally weighted.
Analysis of 1:1 b(IIl)]-[Cd(II)] Mixture in 1.OM HCla Peak Potential Separation 48 mV [In(III)I, M Re1 std dev, % 2.5
Re1 error, -0.26
X
[Cd(WI, M Re1 std Detdcconcn 4.00 x 10-5
dev, Z 1.9
-
Re1 error, 1.3
Figure 7. SEP curves for real 1:l [In(III)]-[Cd(II)] system Peak potential separation 48 mV 3.84 X 10-5M In(III), 3.95 X 10-6MCd(II), 1.OM HCI -0.800 V us. SCE Voltage range shown: -0.300 Maximum peak current: 3.7 pA
-
Qualitative Resolution of Closely Spaced Peaks. The ability of the peak detection technique used here to qualitatively analyze for components having severe overlap was evaluated with an [In(III)]-[Cd(II)], 1.OM HC1 system. The normal peak potential separation for this system is approximately 48 mV (See Figure 7). A real 1 :1 mixture of these two species was analyzed using the standard constants presented in Table V. It was possible to detect the peaks for this system and to follow with a satisfactory quantitative analysis. Quantitative data are presented in Table VI. Weighting Scheme I was used for analysis. To determine the qualitative detection limits for the peak location routine, 1 : 1 In(II1)-Cd(I1) mixtures were synthesized wherein the peak potentials were brought progressively closer together until two separate peaks could no longer be detected. For composite curves having a data density of one data point/ 2 mV, the limit of detection occurred at a peak potential separation of 32-35 mV. Using a data density of one data point/millivolt did not significantly change this limit. The simultaneous equations used to approximate the concentrations did yield rough results at this separation but the least squares refinement procedure failed. Quantitative Analysis of Closely Spaced Peaks. The next situation to be examined was the quantitative analysis of severely overlapping SEP curves. Here too, synthetic In(III)-Cd(II) mixtures were used. The limit of separation for which the least squares refinement procedure worked was 38-40 mV. This was for a data density of 1 point/2 mV. The results of analyzing a synthetic 1 :1 mixture with a peak potential separation of approximately 40 mV. (see Figure 8) are shown in Table VII. The results of the analysis are quite good with relative errors in the range of 1 to 2 %. The values
Figure 8. SEP Curves for synthetic 1:1, 1:5, and 5 : l mixtures of [In(III)]-[ Cd(II)] Peak potential separations 38-42 mV Voltage range shown: -0.300 + -0.800 V us. SCE Upper trace: 4.80 X l W M In(III), 4.94 X 10-5MCd(II),l . O A 4 HCl Maximum peak current: 4.8 p A Middle trace: 0.960 X 10-5M In(III), 4.94 X 10-5M Cd(II), 1.OM HCI Maximum peak current: 3.0 p A Lower trace: 4.80 X 10-5M In(III), 0.988 X 10-6M Cd(II), 1.OM HCI Maximum peak current : 3.5 pA ANALYTICAL CHEMISTRY, VOL. 42, NO. 8, JULY 1970
915
Table VII. Analysis of 1:1, 1:5, and 5 : l [In(lII)]-[Cd(II)] Mixtures in 1.OM HCb Peak Potential Separations 38-42 mV Concn ratio-1 :1 [In(IIUl, M [Cd(II)I, M Re1 std Re1 std Detd* concn dev, % Re1 error, Z Detdc concn dev, Z
Initial estimate Weighting scheme Id
4.32 x 10-5 4.70 x
14.1
-11.1
1.3
-2.1 Concn ratio-1
Detde concn Initial estimate Weighting scheme I Weighting scheme I1
[WWI, M Re1 std dev, %
Re1 error, %
5.44 x 10-5
9.3
4.98 x lo-'
0.70
Re1 error, 10.1 0.81
:5
Detdf concn
[Cd(II)I, M Re1 std dev, %
Re1 error, %
0.705 X
35.5
-36.2
5.11 x 10-5
1.7
0.931 X
2.4
-3.1
4.92 x 10-5
0.95
-0.40
0.959 x
2.2
-0.10 4.89 x 10-5 Concn ratio-5 :1
0.96
-1.0
[WWI, M Re1 std dev, Z
Re1 error,
7.4
Detdg concn Initial estimate 4.64 x 10-5 Weighting scheme I 4.71 X Weighting scheme I1 4.69 x Synthetic mixtures. Expected value4.80 X lO-5M In(II1). Expected value-4.94 X 10-5M Cd(I1). All residuals equally weighted. e Expected value-0.960 X lO-6M In(II1). Expected value-4.94 X lO-5M Cd(I1). 0 Expected value4.80 X lO+M In(II1). Expected value-0.988 X lO-5M Cd(I1).
3.4
Detdh concn
tCd(II)I, M Re1 std dev, %
Re1 error, %
-3.4
1,217 x 10-5
32.2
23.2
1.1
-1.9
1.037 x 10-5
2.2
4.9
1.2
-2.3
1.069 x 10-5
0.90
8.2
'
Table VIII. Analysis of Four-Component Mixture Cu(II), Bi(III), Pb(II), and Cd(II) in 0.4M HOAc, 0.4M "PAC
Weighting scheme I I1 I11
Weighting scheme
Detd' concn 1.018 X 10-4 1.013 X 1.015 X 10-4
[CU(II)l, Re1 std dev, % 3.2 2.5 3.3 [Pb(II)I, M Re1 std dev,
Detdc concn 5.5 1.041 x 10-5 6.3 I1 1.053 x 10-5 5.4 I11 1.037 x 10-6 a Expected value-1.000 X lO-*M Cu(I1). M Bi(II1). Expected value-2.08 X Expected value-1.004 X lO-5M Pb(I1). Expected value-5.93 X 10-5MCd(I1). I
Re1 error, % 1.8 1.3 1.5
Detdbconcn 2.18 x 10-5 2.19 x 10-5 2.15 x 10-5
Z
Detdd concn 5.88 X lo-' 5.88 x 10-5 5.87 x 10-5
Re1 error, 3.7 4.9 3.3
of the concentrations initially estimated with the simultaneous equations are also shown. Both the large relative standard deviations and the large absolute errors in the initial estimates show well the effects of small lateral shifts in the individual component curves. Synthetic In(II1)-Cd(I1) mixtures having peak potential separations of 38-42 mV and concentration ratios of 1 :2 and 2 :1 were also analyzed. Again relative errors of 1-2 were 916
ANALYTICAL CHEMISTRY, VOL. 42, NO. 8, JULY 1970
[Bi(III)], M Re1 std dev, % 4.5 4.7 4.9 [Cd(II)I, M Re1 std dev, Z 0.56 0.67 0.56
Re1 error, Z 4.8 5.3 3.4 Re1 error, % -0.85 -0.85 -1.0
obtained. The limiting concentration ratios for which this same system could still be analyzed were found to be 1:5 and 5 : 1. For concentration ratios of 1 : 10 and 10: 1, two individual peaks could no longer be detected. Typical SEP curves for the 1 :5 and 5 :1 mixtures are included in Figure 8 and the results of analyzing these mixtures are included in Table VII. The residuals for all the In(III)-Cd(II) systems discussed
regards closeness of peaks and relative concentrations would have to be met in any multicomponent mixture. CONCLUSIONS
Figure 9. SEP curves for four-component system The signals observed are for: ( A ) 1.00 X 10-4M Cu(II); ( B ) 2.08 X 10-5MBi(1II); (C)1.00 X 10-6MPb(II); (0)5.93 X 10-6MCd(II) Electrolyte: 0.4MHOAc, 0.4MNH40Ac Voltagerangeshown: +0.075 + -0.925 V us. SCE Peak current For ( A ) : 2.7 pA
thus far were weighted equally. Using Weighting Scheme I1 (discussed above) for the 1 : 5 and 5: 1 [In(III)]-[Cd(II)] mixtures improved the results in only one case-the In(II1) results for the 1 : 5 mixtures. When attempts were made to use Weighting Scheme I11 (discussed above), the refinement procedure often failed and satisfactory results could not be obtained. Considering the nature of the mixtures, the results obtained with the residuals being equally weighted are quite good. Finally, a four-component mixture was analyzed. The system consisted of Cu(II), Bi(III), Pb(II), and Cd(I1) in 0.4M HOAc and 0.4M NH40Ac (Figure 9). It was necessary to start the scan for this system at 0.075 V us. SCE, as the half wave potential for Cu(I1) was about - 0.030 V us. SCE. The results of the analysis are presented in Table VIII. The results for all four components are observed to be satisfactory. This four-component mixture was also analyzed with Weighting Schemes I1 and I11 and the results are included in Table VIII. Minor improvements are noted in the Bi(II1) and Pb(I1) results. The maximum number of components which could be detected and analyzed for in any one mixture has not been experimentally determined. Certainly the limitations as
+
The most important experimental result obtained with the on-line deconvolution technique described here is the quantitative analysis (1-2 error) of severely overlapping stationary electrode polarograms. Without numerical deconvolution, quantitative analysis of closely-spaced 1 :1 mixtures would be virtually impossible with SEP. Even the major peaks of the 1st- and 2nd-derivative signals would have to be separated by approximately 500/n and 225/n mV, respectively, to present quantitatively resolved peaks for 1 :1 mixtures (36). Using numerical deconvolution, however, quantitative analysis was achieved with peak potential separations of approximately 100/n mV. This represents a tremendous enhancement of the stationary electrode technique. On the other hand, the significance of the work presented should not be based on the absolute quantitative results achieved. Derivative polarography with the dropping mercury electrode and the pseudoderivative polarographic techniques-ac, square-wave, and pulse polarography-show a limiting width at half peak height or 90.6/n mV (37,38). Thus the 1 :1 [In(III)]-[Cd(II)] mixtures discussed above could be quantitatively analyzed with a peak potential separation of approximately 45 mV, which is comparable to the results obtained here. However, for those applications where SEP techniques are preferred or required, it has been shown here that the quantitative capabilities of this technique can be considerably enhanced. Most importantly, the work presented here has provided a general mathematical and experimental approach for applying numerical deconvolution methods in on-line electroanalytical systems. The principles evaluated here can be extended advantageously to other electroanalytical techniques.
RECEIVED for review January 16,1970. Accepted May 6,1970. This work supported by Contract No. GP-8677 from the National Science Foundation. Computer programs described here are available from the authors upon request. (36) Paul E. Reinbold, Master’s Thesis, Department of Chemistry, Purdue University, Lafayette, Ind., 1968. (37) D. J. Fisher, W. L. Belew, and M. T. Kelley, Chem. Inst., 1, 181 (1968). (38) E. P. Parry and R. A. Osteryoung, ANAL.CHEM., 37, 1634 (1965).
ANALYTICAL CHEMISTRY, VOL. 42, NO. 8, JULY 1970
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