ANALYST, JANUARY 1993, VOL. 118 59 Use of a Quadratic Response Surface in the Polarographic Determination of Lead B. Lopez Ruiz Laboratorio de Tecnicas Instrumentales, Facultad de Farmacia, Universidad Complutense, 28040 Madrid, Spain G. Frutos and P. Sanz Pedrero De pa rta m en to de Qu im ica - Fisica Farm aceu tica, Facu ltad d e Farm acia, U n ive rsidad Co m plu tense, 28040 Madrid, Spain J. P. Martin lnstituto de Fermentaciones, CSIT, Madrid, Spain By investigating the influence of the instrumental and chemical factors affecting the differential-pulse polarographic measurements of Pbll and subsequently developing suitable factor and orthogonal designs, a surface response was obtained that allowed the optimization of the response variable, viz., the peak current. In this manner, the polarographic technique was improved in two respects, by increasing its sensitivity (thereby lowering its detection limit) and by shortening the analysis time (the lead concentration was obtained from a single measurement). The results provided by the proposed method are comparable to those afforded by the traditional standard additions method.Keywords: Quadratic response surface; differential-pulse polarograph y; lead determination The determination of meta!s in matrices of various natures is attracting growing interest, particularly as regards biological, environmental and foodstuff samples, on account of the toxic character of some metals even at very low concentrations (of the order of a few pg cm-3 or ng cm-3) and of the essential nature of others which must be ingested in small amounts (usually a few pg ml-'), above which they also become toxic.1 4 Choosing an appropriate technique for the determination of metals is thus no easy task. The technique in question must afford detection limits of a few ng cm-3, be highly sensitive, selective and reproducible, allow small differences in concen- tration to be distinguished and, if possible, be economic to implement. In many instances, particularly in cases of intoxication, the technique of choice must also be rapid. In this work, the determination of lead was examined on account of this metal's toxic character and ubiquity in environmental and foodstuff samples. For this purpose, differential-pulse polarography (DPP) was chosen, as it is a sensitive and reproducible technique that affords low determi- nation limits with low analytical costs.s--") In rapid studies, because of the many instrumental and chemical factors affecting polarographic measurements, determinations required the use of the standard additions method.As this involves at least four measurements on each sample, a significantly decreased sample throughput results in comparison with alternative techniques. Studies were therefore made on the effect on the polaro- graphy of Pb" of instrumental factors such as the scan rate and pulse amplitude and of chemical variables such as the electrolyte pH and concentration. 11712 The influence of these parameters on the sensitivity was investigated in order to determine whether they must be taken into account in applying the DPP technique.The basic aim of this work was to optimize the instrumental and chemical parameters affecting the determination of PbII by DPP. For this purpose, factor designs were used that were in turn employed to develop orthogonal designs for the generation of response surfaces that allow the optimum experimental conditions to be established. The response variable (viz., the peak current) was optimized by selecting these experimental conditions. 13-15 Once optimized, the technique for the DPP determination of lead should be more sensitive and, more important, faster as a result of allowing the metal to be assayed in a single, direct measurement. Experimental Apparatus and Reagents Polarographic measurements were made on a Princeton Applied Research Model 384 analyser that was connected to an RE0082 digital recorder and a Model 303 polarographic vessel module furnished with a static mercury drop (SMD) working electrode, a silver-silver chloride reference electrode and a platinum auxiliary electrode. Lead(") standards were prepared from Pb(N03)Z (Titrisol, Merck).Those of the other metal ions [Co", Cu", Ni", Znl[ and FeIII)] were also prepared from the required volumes of Merck Titrisol solutions. Other reagents used included analytical-reagent grade LiC104 and Suprapur HClO4 (Merck). The mercury used was distilled three times and the water used throughout was distilled twice in Pyrex glass and subsequently de-ionized in a Milli-Q system (Millipore). Nitrogen gas (SEO N-48, 99.98% N,) was also used.Procedure The determination of Pb" involved passing a stream of nitrogen through the polarographic solution for a pre-set time (10 min). A nitrogen atmosphere was maintained throughout each experiment. Potential scans were performed between -0.1 and -0.5 V. The pulse amplitude, drop time and potential increment between pulses used in each experiment were dictated by the instrumental set-up used. The influence of the mercury drop size on the peak current could not be studied because the experimental set-up used could not afford some of the sizes required by the factor designs. A fixed drop size was therefore used throughout. Despite the well known influence of temperature on the peak current and because of the difficulty involved in controlling this parameter accurately, all recordings were obtained at controlled room temperature (between 22 and 24 "C) .60 ANALYST, JANUARY 1993, VOL.118 Statistical Analysis The experimental method developed in this work is based on the execution of a series of factor and fractionated factor designs that allow the influences of the factors studied on the response variable to be determined. The quadratic surface was determined by using a second- order design to obtain a mathematical model descriptive of the measurement procedure. 16 Results and Discussion The factors studied and their levels are listed in Table 1. The first design developed was a Factor 27-2 with resolution IV, i.e. , F = ABCD and G = ABDE (definition relation, I = ABCDF = ABDEF = CEFG). The experimental matrix and the corresponding results of the response variables are summarized in Table 2.The stepwise regression analysis of this experimental design (Table 3) allows us to state that variables R, D and F have no significant influence on the response variable; however, because of the significant influence of the DE interaction, only the drop time ( B ) and pure time ( F ) have been eliminated as Table 1 Levcls of the factors involved in Design 27-2: potential increment between pulses, drop time, pulse amplitude, LiCIOj concentration, pH, purge time and Pb" concentration Lcvels Factors - 1 +1 A = Potential increment bctween pulses (mV) 3 7 B = Drop time (s) 2.5 6.5 C = Pulsc amplitude (mV) 35 75 D = LiCIOl concentration (mol dm-3) 0.01 0.20 E = p H 1 2 F = Purge time (min) 7 20 G = Pbll Concentration (ng cm-3) 60 140 Table 2 Experimental matrix and results of the rcsponse variable for Design 2'p2 A -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 B -1 1 -1 1 -1 1 -1 1 - 1 1 -1 1 -1 1 -1 1 - 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 C -1 -1 1 1 -1 - 1 1 1 - 1 -1 1 1 -1 - 1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 D - 1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 - 1 -1 -1 1 1 1 1 -1 -1 -1 - 1 1 1 1 1 E -1 -1 - 1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 F 1 -1 -1 1 -1 1 1 -1 1 -1 -1 I -1 1 1 -1 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 1 -I -1 1 G 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 ilnA 15.36 9.50 27.82 16.15 6.66 16.76 13.33 26.08 5.86 12.46 9.37 20.90 14.50 7.24 27.24 12.17 6.08 11.29 12.17 22.03 9.76 5.07 17.68 10.45 8.00 3.62 15.12 7.94 5.50 11.09 11.58 25.60 factors, the LiC104 concentration ( D ) being a dubious factor, the influence of which on the peak current cannot be readily established. Solving this ambiguity calls for supplementary experimentation or a more powerful design. Reducing the overall number of factors LO five allowed a Complete Factor Design 25 to be applied. The features of such a design allow one to obtain non-mixed information on the principal factors A, C, D, E and G, and on their interactions. These five factors were kept at the levels given in Table 1. The drop and purge times were set to 1 s and 15 min, respectively. This short drop time was chosen in order to diminish over-all determination time after checking through parallel experi- ments that it did not influence the response variable of the design.The purge time chosen was an intermediate value suited to the sample volume used (10 cm3). Table 4 shows the experimental matrix and results of the response variable obtained with Design 25, and Table 5 gives the results obtained in the regression analysis by using the Table 3 Results of the stepwise regression analysis applied to Design Z7-2 Increrncnt F No. Variable R R' in R2 to enter 1 7G 0.6458 0.4147 0.4171 24.3248 2 3 c 0.8744 0.7646 0.3475 48.7056 3 1A 0.9155 0.8382 0.0737 14.5724 12.3895 4 22 DE 0.9404 0.8844 0.0462 12.4426 5 21 CG 0.9583 0.9183 0.0339 6 5 E 0.9671 0.9352 0.0169 7.5767 7 14AG 0.9728 0.9463 0.0111 5.7926 Table 4 Experimental matrix and results of the response variable for Design 2' A C D E G i,lnA A C D E C; i,,lnA -1 -1 -1 -1 -1 -1 1 -1 -1 -1 -1 -1 1 -1 -1 -1 1 1 - 1 -1 -1 -1 -1 1 -1 -1 1 -1 1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 -1 1 -1 1 -1 -1 1 -1 -1 1 -1 1 -1 1 1 -1 1 -1 -1 - 1 1 1 -1 1 -1 1 1 -1 -1 1 1 1 - 1 1 1 1 1 5.67 9.00 6.00 11.20 6.00 15.65 6.81 13.91 13.22 23.70 14.10 23.33 12.60 28.40 13.76 26.55 1 -1 -1 - I -1 10.80 1 1 -1 -1 -1 7.25 1 -1 1 -1 -1 3.20 1 1 1 -1 -1 7.53 1 -1 -1 1 -1 6.04 1 1 -1 1 -1 13.48 1 -1 1 1 -1 5.15 1 1 1 1 -1 8.70 1 -1 -1 -1 1 8.50 1 1 -1 -1 1 16.50 1 -1 1 -1 1 8.18 1 1 1 -1 1 16.65 1 -1 -1 1 1 10.14 1 1 -1 1 1 23.91 1 -1 1 1 1 10.86 1 1 1 1 1 19.00 Table 5 Results of the stepwise regression analysis applied to Design 2s Variable Coefficient Student's t Intercept A C D E G AC AD AE AG CD CE CG DE DG EG 12.67 -1.68 8.86 -0.51 1.12 4.15 -0.72 -0.58 0.05 -0.92 -0.20 1.01 1.54 -0.23 0.20 0.18 * Statistically significant value of Student's t-test -5.65* 12.99* -1.71 3.77" 3.96" -2.43" -1.94 0.16 -3.11" -0.67 3.41" 5.19* 0.69 0.59 -0.77ANALYST, JANUARY 1993, VOL.118 61 program BMDPIR. I 7 The effects of the principal variables and their interactions are given by the coefficients of the regression equation. Taking into account the Student's t values given in Table 5 , the following factors and interactions were significant at the a < 0.05 level: G, C, A; CG, E, CE, AG and AC. These results allow one to rule out the initially dubious influence of the LiCI04 concentration on the peak current under the conditions associated with the design.For this reason, the same concentration (0.1 mol dm-3) for the background electrolyte (LiC104) was used in all subsequent experiments. Taking into account that the concentration of lead would be unknown, the response of this factor was used in all subsequent experiments. Taking into account that the concentration of lead would be unknown, the response of this factor should be made independent. For this purpose three designs involving con- stant Pb" concentrations of 20, 140 and 5000 ppb in which the new response variable was the ratio between the measured peak current (ip) and the lead concentration in the sample were investigated. The choice of the new variable was justified by the Parry and Osteryoung equation.18 The peak current in DPP is given by D 0 - 1 i, = nFAC ,/-(-) JCtp O + 1 where o = exp(nF/RT)(AE/2) n being the number of electrons involved in the process, F = 96500 C, A the area of the electrode surface, C the concentration of the electroactive species (Pb"), D its diffusion coefficient, t, the interval between application of two consecutive pulses and AE the pulse amplitude.Eqn. (1) can be rewritten as The choice of the i,: [Pbli] ratio as the response variable allows the design of the Pb" concentration factor to be made independent. This has some advantages, as such a concentra- tion is unknown in the sample. In order to check the behaviour of the factors studied on this new response variable over a wide concentration range, three designs at constant Pb" concentrations of 20, 140 and 5000 ng cm-3 were investigated.The influences of the factors (pulse increment between scans, pulse amplitude and pH) were found to be significant with all three designs. Once the influence of different factors had been deter- mined, the response variable was optimized by developing a second-order orthogonal design. This corresponding matrix was assayed and the results it provided are given in Table 7; Table 6 lists the experimental counterparts of the scaled values given in Table 7. Table 8 shows the results of multiple regression analysis performed by using the program BMDP2R. 17 The quadratic equation obtained by taking the significant terms A , C, CT and El into account resulted in no fitting failure, as can be seen Table 6 Levels of the factors involved in the orthogonal design Variable -1.52 -1 0 1 1.52 A = Potential increment C = Pulseamplitude 25 25 55 75 85 E = pH 0.74 1.0 1.5 2.0 2.26 between pulscs 2 3 5 7 8 from Table 9.Thus, the response surface obtained for a lead concentration of 1000 ng cm-3 was -- I' - 0.1372 - 0.02183A + 0.00649G - 0.00833E2 (2) [Pb"] From eqn. (2) it follows that the optimum value of the response variable is defined by the conditions C = 2.9 and E = 0, the value of A being uncertain. This critical point corresponds to a pulse amplitude of 113 mV and pH 1.5 Fig. 1 shows a three-dimensional representation of the response surface defined by eqn. (2) for the coded value A = 1.52. Fig. 2 shows the contour lines for A = - 1.52,O and 1.52. Within the limits studied, the most favourable experimental conditions would be A = -1.52, C = 1.52 and E = 0 , which Table 7 Experimental matrix and results of thc response variable for thc orthogonal design Experiment No.A 1 -1 2 1 3 -1 4 1 5 -1 6 1 7 -1 8 1 9 1.52 10 -1.52 11 0 12 0 13 0 14 0 15 0 16 0 17 0 18 0 19 0 20 0 C -1 -1 I 1 -1 -1 1 1 0 0 1.52 - I .52 0 0 0 0 0 0 0 0 E -1 -1 -1 -1 1 1 1 1 0 0 0 0 1.52 -1.52 0 0 0 0 0 0 i,/n A 98.60 60.45 174.50 124.45 109.85 69.95 189.00 139.25 11 3.10 177.55 186.20 63.65 118.35 122.95 131.45 138.80 135.25 134.20 137.90 131.20 Table 8 Results of the stcpwise regression analysis applied to the orthogonal design Variable Coefficient Student's t Indepcndent term A C E A A A C A E CC CE EE 0.13576 0.03757 0.00340 0.00228 - 0.0027 1 - 0.0001 9 -0.00649 0.00 107 -0.00833 -0.02 183 * Statistically significant value of Student's r-test.- 11.64* 20.03" 1.81 1.13 -1.15 -0.08 -3.20" 0.46 -4.11" Table 9 Contribution of each of the variables chosen to account for the response variable in the orthogonal design Entering Correlation K2 Step No. variable coefficient, R R2 increment 1 C 0.8318 0.6919 0.6919 2 A 0.9621 0.9256 0.2336 3 E2 0.9770 0.9546 0.0290 4 cz 0.9860 0.9723 0.0176 ANOVA Sum of Degrees of Mean Source squares freedom squares F Regression 0.02509 4 0.006273 131.44 Residuals 0.00072 15 0.000048 Lack of fit 0.00066 10 0.000066 6.60 5 0.000010 Error 0. 0000562 ANALYST, JANUARY 1993, VOL. 118 Table 10 Results of the prediction of the proposed mathematical model for different concentrations of PblI and comparison with the standard additions method Experi- mentNo.A C E 1 -1.52 1.52 0.00 2 -1.00 0.25 1.00 3 0.50 -0.75 0.00 4 1.00 0.75 0.00 5 0.50 -0.75 1.52 6 -0.50 -1.25 -1.52 7 0.00 -0.25 -1.00 ip( exp . )/ nA 217.57 174.75 102.60 141.90 101.35 95.60 120.30 ip( t he or. )/ nA Error(%) 212.49 +2.3 159.69 +8.6 94.95 +7.9 139.90 +1.4 75.20 +25.8 71.77 +24.9 119.07 +2.0 Fig. 1 Three-dimensional representation of the response surface correspond to actual values of 2 mV for the potential increment between pulses, 85 mV for the pulse amplitude and 1.5 for the pH. Under these conditions, the value of the response variable will thus be (3) Table 10 reflects the experiments carried out in order to contrast the predictive capacity of the mathematical model defined by eqn.(2). Every experiment was carried out at the optimum values of the instrumental and chemical variables. Lead(i1) solutions containing the concentrations given in Table 10 and arbitrary concentrations of Co", Cu", Ni", Mg", Zn" and Fell1 between 20 and 5000 ppb in 0.1 mol dm-3 LiC104 of pH 1.5 (adjusted with HC104) were used for this purpose. As can be seen in Table 10, the errors in the predictions were always less than lo%, which testifies to the feasibility of determining lead in a single, direct measurement by applying eqn. (3). The relative errors in the concentrations determined by this method were similar to those obtained by using the standard additions method, which is surpassed by the former in speed by a factor of 4. Finally, the detection and determination limits were deter- mined under the optimum conditions fixed by the experimen- tal design using the Miller method.19 The former was 7.4 ppb and the latter 12.27 ppb.As shown above, the accuracy achieved in the determina- tion of lead by the proposed method is similar to that afforded by the classical standard additions method, which it surpasses in terms of speed of analysis. Therefore, eqn. (3) can be used as a model for analytical purposes such as the determination of lead in complex samples. (a) 6.00 4.30 u 2.00 0 -2.00 (b) 6.00 -4.00 -2.00 0 2.00 4.00 4.00 u 2.00 0 -2.00 (' 6.00 -4.00 -2.00 0 2.00 4.00 4.00 u 2.00 0 -2.00 -4.00 -2.00 0 2.00 4.00 E Fig. 2 Contour lines for: ( a ) A = - 1.52, ( b ) A = 0 and (c) A = 1.52 References 1 2 Bourdon. R., in Les Oligodements en Medicine et Biofogie, ed.Chappuis, P., Lavoisier, Paris, 1991, pp. 11 1-157. Goyer. R. A., in Toxicology. The Basic Science of Poisons, ed. Klaassen, C. D., Arndur, M. 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Commun., 1984, 49, 45. Aliakbar, A., and Popl, M., Collect. Czech. Chem. Commun., 1984,49, 1140. Box, G. E. P., Hunter, J. S., and Hunter, W., Statistics for Experimenters, Wiley, New York, 1978. 14 Draper, N. P., and Smith, H., Applied Regression Analysis, Wiley, New York, 1980. 15 Morgan, E., Chemometrics: Experimental Design, Wiley, New York, 1991. 16 Wolters, R., and Kateman, G., J. Chemometr., 1990,4, 171. 17 BMDP Statistical Software Manual, ed. Dixon, W. J., Univer- sity of California Press, Berkeley, 1990. 18 Parry, E. P., and Osteryoung, R. A., Anal. Chem., 1965, 37, 1634. 19 Miller, J. C., and Miller, J. N., Statistics for Analytical Chemistry, Wiley, New York, 1984. Paper 21 03 797A Received July 16, 1992 Accepted September 21, I992