ANALYST, JULY 1992. VOL. 117 Determination of Thallium in Cement Dust and Sediment Samples Differential-pulse Anodic Stripping Voltammetry: A Chemometric Approach to Linear Calibration Mahmoud A. Allus and Richard G. Brereton" School of Chemistry, University of Bristol, Cantock's Close, Bristol BS8 1 TS, UK 1075 bY Differential-pulse anodic stripping voltammetry (DPASV) was used to determine TI in cement kiln dust and sediment samples. The separation of TI from the sample matrix was achieved by the extraction of TP+ into diethyl ether from an HBr-Br2 medium. Initially, a mixture of HN03 and HF was used to digest the cement dust samples to cause volatilization of silica as SiF4. The Tl3+ was then reduced to TI+ using hydrazine sulfate prior to determination by DPASV. A calibration graph based on a deterministic relationship between the measured response y and the analyte concentration x (so that y = Po + Px + r ) is also discussed using classical (or reverse) regression.The precision of parameter estimates of the model (bo and b: p = bo + bx) can be achieved by display of the diagonal elements of the hat matrix. Methods for computing the standard uncertainty in the predicted response s$. and in the concentration s$ are also provided. Thallium concentrations of 700 k 3.07,395 k 2.91 and 850 k 2.59 ng 9-1 were found in the cement dust and sediment samples, respectively. Keywords: Differential-pulse anodic stripping voltammetry; chemometrics; linear calibration The quantitative determination of T1 in cement kiln dust (collected from Libya) and sediment samples is described in this paper using differential-pulse anodic stripping voltam- metry (DPASV).Thallium is concentrated as a trace element in these samples. It is a very toxic element, and local pollution problems with T1 emitted from cement producing installations have been reported.'-3 Cement production in Libya is one of the major heavy industrial projects that the country has achieved recently. The cement is produced from more than six factories in agricultural areas, namely Khomes, Zaletin, Benghazi, Sauk El-Khameasl Tarhona and Derna. The emission of trace amounts of toxic metals produced as a by-product from the roasting of the raw materials used in cement manufacture may affect the quality of the agriculture in the vicinity of these cement producing installations and thus increase environmental pollution that has already become a serious problem.4~5 In Germany the TI content of the cement dust emitted by one factory constituted an environmental hazard.The source of T1 was shown to be the pyrite cinders and slag which had been added to the raw mixture for the production of the sulfate resistant Portland cement. Pyrite cinders are the iron oxide residue which remains after roasting sulfur-containing ores for the production of sulfuric acid. When added to a cement raw mix and burned in a kiln, it combines with other materials to form a calcium aluminoferrite phase with the approximate molecular formula Ca4A1Fe. At the same time, TI is volatilized and a portion of the recirculating load in the kiln system is emitted to the atmosphere in the stack gases.1-3 The experiments described in this paper involve the determination of T1 in cement kiln dust samples and the quantification of T1 in sediment samples. The separation of Tl from the sample matrix was achieved by the extraction of Tl3+ into diethyl ether from an HBr-Brz medium. Initially, a mixture of HN03 and HF was used to digest the cement dust samples to cause volatilization of silica as SiF4. No loss of T1 by volatilization has been found by other workers using this dissolution procedure.616 The TP+ was then reduced to Tl+ using hydrazine sulfate prior to determination by DPASV. A calibration graph based on a deterministic relationship between the measured response y and the analyte concentra- * To whom correspondence should be addressed.tion x is also discussed:17-33 the precision of parameter estimates of the model can be achieved by the application of experimental design and display of the diagonal elements of the hat matrix h,. Methods for computing the standard uncertainty in the predicted response s;,, and in the concentra- tion sz are also provided. Experimental All chemicals used were of AnalaR grade (where possible). Concentrated Aristar HN03 and HF were used, whereas AnalaR H2S04 and HBr were employed unless specified otherwise. All glassware was soaked in H2S04-H20 (40 + 60) before use. Dilutions were performed using triply distilled water. Reagents and Sample Preparation The following reagents were prepared as de~cribed.6.~ Stock standard TI solution (lo00 pg cm-3).Prepared by dissolving 1.303 g of TlN03 in water, adding a suitable volume of HN03 to bring the pH to about 1 and diluting the mixture to 1 dm3 with water. TI calibration solution (10 pg cm-3). Prepared by diluting 1 cm3 of 1000 pg cm-3 TI stock solution to 100 cm3 in a calibrated flask. Base electrolyte [O. 1 mol dm-3 ammonium tartrate44 mol dm-3 ethylenediaminetetraacetic acid (EDTA) J. Prepared by dissolving 18.4 g of ammonium tartrate and 14.9 g of EDTA (disodium dihydrogen salt) in water and diluting to 1 dm3. Reducing solution. Prepared by cautiously adding 15 cm3 of H2S04 to 15 cm3 of water and then dissolving 10 g of anhydrous NaZSO4 in the mixture. The resulting solution was then saturated with hydrazine sulfate. HBr-Br2 mixture.Prepared by mixing 50 cm3 of Br, and 450 cm3 of HBr (density 1.46-1.49 g cm-3). The solution was stored in a brown glass reagent bottle. HBr (0.1 mol dm-3). Prepared by diluting 115 cm3 of HBr (density 1.46-1.49 g cm-3) to 1 dm3. The above reagents were used in the preparation of samples for analysis by DPASV.1076 ANALYST, JULY 1992, VOL. 117 The cement dust sample (10 g) was transferred into a 150 cm3 Teflon beaker and placed in an electric oven at 50°C for 48 h. A 1.0 g amount was weighed accurately from the dried cement dust sample in a 150 cm3 poly(tetrafluoroethy1ene) container. Then, HF (5 cm3) and HN03 (5 cm3) were added to the sample and the mixture was evaporated to dryness. A 10 cm3 volume of the HBr-Br2 mixture was added and the solution warmed gently until most of the bromine had been expelled.The beaker was covered after the addition of 25 cm3 of hot water and the whole digested on a hot-plate until the residue had dissolved, then cooled to room temperature. The solution was transferred quantitatively into a 100 cm3 separat- ing funnel (marked at 50 cm3) and diluted to the mark with water, after which 15 cm3 of diethyl ether were added and the whole system was shaken vigorously for 2-3 min. The aqueous layer was run off into a similar separating funnel with the addition of a further 15 cm3 of diethyl ether and the process was repeated. All the diethyl ether layers were combined and a further 15 cm3 of 1 rnol dm-3 HBr added. After shaking for 1-2 min, the acid layer was discarded and the combined organic extracts were transferred into a 100 cm3 conical glass flask and evaporated to dryness.Then, HN03 (2 cm3) and H2SO4 (2 cm3) were added to the residue and the mixture was heated until strong white fumes appeared, whereupon 2 cm3 of reducing solution were added and the contents evaporated to dryness. The residue was then dissolved in 15 cm3 of hot base electrolyte, cooled to room temperature and diluted with 2.5 rnol dm-3 soldium hydroxide until the final pH lay in the range 5 k 0.5.6 The solution was transferred into a 25 cm3 calibrated flask and diluted to the mark with water. The same procedure (described above) was used to prepare the reagent blank solutions but without any sample. Many cement dust samples were weighed out and the preparation procedure described above was followed both for these samples and for the sediment samples.For the cement dust samples, the final dilution was to 25 cm3, whereas for the sediment samples it was to 50 cm3. A 10 cm3 volume of the prepared sample solution was transferred into the polarographic cell containing a magnetic stirrer and purged with purified nitrogen for 10 min. The determination of the TI content was carried out using the experimental conditions described in Table 1. Differential-pulse Anodic Stripping Voltammetry The determination of TI was carried out using anodic stripping voltammetry involving two discrete steps.34-37 Firstly, after preparation of a solution of a T1 containing sample in the base electrolyte, TI is preconcentrated by deposition on a working electrode (which, in this instance, is a hanging mercury drop electrode) by maintaining its potential more negative than the reduction potential of TI.Secondly, the Tl is transferred back into the solution (which, in this instance, is the base electrolyte solution at pH 5 k O S ) , i.e., TI is stripped by applying a steadily increasing positive potential across the electrodes and the current generated by oxidation of the deposited TI produces a peak, the height of which is measured. The current is proportional to the concentration of TI in the sample solution and the potential at which the peak occurs is known as the element peak potential. The working electrode used was a mercury drop extruded automatically from a glass capil- From a study of the peak potentials of the elements known to be commonly present in the cement dust sample, it is clear that Cd2+, Pb2+ and Fe3+ are the most likely to interfere in the determination of T1+ by DPASV.However, complexing agents, especially chelating agents such as EDTA, can be used to separate the reduction potential of elements during the preliminary electrolysis stage or to alter the potential at which their peaks occur during the stripping process. Experiments using this method showed that Tl+ could be determined in the i a y . * 5 3 Table 1 Instrumental conditions used during sample analysis. 15~16,34 The base electrolyte was 0.1 mol dm-3 ammonium tartrate4.4 mol dm-3 EDTA (disodium dihydrogen salt) at pH 5 f 0.5 Source* Working electrode Reference electrode Model 303- Auxiliary electrode Conditioning potential Model 315- Initial potential Final potential Purge time Conditioning time Equilibration time Deposition time Model 174- Scan rate Scan direction Modulation amplitude Current range Drop time Display direction Low pass filter Operation mode Initial potential Potential scan rate Potential range Model 7040- Condition Hanging mercury drop (medium size) Ag-AgC1 (saturated with 3 mol dm-3 Platinum wire KCl) +0.20 V versus Ag-AgC1-3 mol dm-3 -0.20 V versus Ag-AgC1-3 mol dm-3 -0.20 V versus Ag-AgC1-3 mol dm-3 KCl KCI KCI 10.00 min 0 min 30.00 s 300.00 s 5.00 mV s-1 (+) 25.00 mV 10.00 pA 0.50 s (-) (Off) Differential pulse 0.0 v 5.00 mV s-1 3.00 V x-y recorder? * Model 303 (static mercury drop electrode), Model 315 (auto- mated electroanalytical controller), Model 174 (polarographic ana- lyser) and Model 305 (stirrer) were all from Princeton Applied Research.? The x-y recorder (Model 7040) was from Hewlett-Packard. presence of both Cd2+ and Pb*+ in a tartrate-EDTA medium. However, the presence of Fe3+ in the sample to be analysed had a suppressive effect on the recovery of TI+, which indicated that the separation of TI+ from the matrix would be necessary.614 In this work the determination of T1 was performed in a tartrate-EDTA base electrolyte after extraction from an HBr-Br2 medium into diethyl ether. However, the extraction from acidic medium could be carried out by using other solvents instead of diethyl ether; for instance, diisopropyl ether has been shown to be more selective for TP+, but, for safety reasons, diethyl ether was used in the extraction procedure.6,7,'5,'6.34 The DPASV analyses were carried out by transferring 10 cm3 aliquots of the prepared sample solution into the polarographic cell34 under the instrumental conditions given in Table 1.Results and Discussion Typical voltammograms for a series of reagent blanks and TI standard solutions are shown in Fig. 1. Several reagent blank solutions were prepared in the same way as for the TI standard solutions at pH 5 k 0.5 and examined using the same practical and instrumental conditions as for the TI standard solutions. It is evident from the traces shown in Fig. 1 that the response of the TI standard solution (20 ng cm-3) produces a peak height significantly greater than that of the reagent blank solution.A number of runs were made of these reagent blank samples to ascertain that the concentration of TI in the reagents used to prepare the samples to be analysed for their TI content was negligible. It is clear that the TI concentration in the base electrolyte is very low and probably not at a significant levelANALYST, JULY 1992, VOL. 117 1077 A 1- ll ' I -1.20 -0.80 -0.40 -1.00 -0.60 -0.20 PotentialN versus Ag-AgC1-3 mol dm-3 KCI Fig. 1 Voltammograms of (a) a series of reagent blank solutions and (b) TI standard solutions added to the base electrolyte. The base electrolyte is tartrate-EDTA at pH 5 k 0.5 [the symbols A, B, C, D, E and F are the observed responses for the reagent blank plus 20,40,60, 80 and 100 mm3 of T1 standard solution (each 1 mm3 = 1 ng ~ m - ~ ) , respectively] when compared with those responses obtained from the standard TI solutions (2G100 ng cm-3) under the same experimental conditions.The nature of the standard additions procedure was considered in order to validate the calibration graph for the determination of T1 by DPASV. This is because calibration in general is a fundamental step for accurate and more informa- tive quantitative trace analysis.38239 The calibration graph of T1 in this instance is based on a deterministic relationship between an independent factor x (TI concentration in the base electrolyte) and a dependent response y (observed peak height). The relationship between x and y can be described by the linear model as: y = po + px + r i) = bo + bx (1) (2) where bo and b are the best fit linear regression coefficients for the calibration model, and p are the true linear calibration coefficients and r is the error; eqn.(2) is the least-squares estimated regression for eqn. (1). The use of matrix least- squares solutions for such models has been discussed in many text-books.4w2 The precision of estimating eqn. (2) depends on the measurement process used to obtain the observed response y and the nature of the experimental design. Hence, our experiments aimed to achieve such a goal (precise estimation of bo and b ) . In this paper only approaches for classical (or reverse) calibration are considered, in which y is regressed on x , i.e., it is assumed that all the errors are in the response and none in Table 2 Results for Method A: x = standard Tl solution added to the base electrolyte, h, is as discussed in the text, y and 9 = the measured and predicted responses (peak height in cm), respectively, r = residual and s = standard error n X hrl Y 9 r S 1 20 0.7 4.1 3.80 0.30 0.517 2 40 0.3 7.8 8.15 -0.35 0.452 3 60 0.3 12.3 12.50 -0.20 0.452 4 80 0.7 17.1 16.85 0.25 0.517 Table 3 Results for Method B: x, h,, y, 9 , r and s are as defined in Table 2 n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 X h n Y 20 0.14 4.1 20 0.14 4.6 20 0.14 3.6 20 0.14 4.0 20 0.14 4.2 40 0.06 7.0 40 0.06 8.6 40 0.06 7.8 40 0.06 8.0 40 0.06 7.6 60 0.06 12.5 60 0.06 13.0 60 0.06 11.0 60 0.06 12.7 60 0.06 12.3 80 0.14 17.1 80 0.14 17.5 80 0.14 16.7 80 0.14 17.0 80 0.14 17.2 9 3.80 3.80 3.80 3.80 3.80 8.15 8.15 8.15 8.15 8.15 12.50 12.50 12.50 12.50 12.50 16.85 16.85 16.85 16.85 16.85 r 0.30 0.80 -0.20 0.20 0.40 -1.15 0.45 -0.35 -0.15 -0.55 0.00 0.50 - 1 S O 0.20 -0.20 0.25 0.65 0.15 0.35 -0.15 S 0.625 0.625 0.625 0.625 0.625 0.603 0.603 0.603 0.603 0.603 0.603 0.603 0.603 0.603 0.603 0.625 0.625 0.625 0.625 0.625 the independent variable.There are a number of other approaches to linear calibration, including inverse (or for- ward) calibration, and a large amount of statistical literature comparing the merits of these approaches. It is, however, usual for most user-friendly microprocessor software (e.g., for graphics and spreadsheets) to be based on classical calibra- tion. It is beyond the scope of this paper to contrast the merits of these and related approaches, but the interested reader is referred to the text by Martens and Nzs,43 and references cited therein, for an up-to-date chemometric review of this large area.Four aliquots of the prepared TI standard solution (10 pg cm-3) were added to 10 cm3 of base electrolyte in the polarographic cell to give TI concentrations of 20, 40, 60 and 80 ng (3111-3 (at pH 5 k 0.5). The observed responses of these aliquots were measured (as the peak height in cm). The process was repeated five times. Hence 20 measurements were obtained for these factor levels (five at each level). In matrix terms, the design matrix D has 20 rows (n = 20) and one column (there is only one independent factor), while the model matrix X has 20 rows and two columns ( n = 20 andp = 2, where n = total number of experiments performed and p = number of parameters in the model). The measured peak heights were recorded resulting in 20 responses (five at each TI concentration; hence the observed response row vector Y has 20 rows and one column).A matrix least-squares regression was performed using the SAS (Statistical Analysis System) software on the resulting measurements. 17241 744 In order to study the influence of replication on the resulting analysis, the experiments were analysed using the following two methods. Method A . The regression was performed on only four points (the average at each concentration so that D and Y are 4 x 1 matrices; X is a 4 x 2 matrix). Method B. All 20 experimental points were included in the regression analysis so that D, Y and X are 20 x 1 , 20 x 1 and 20 x 2 matrices, respectively.1078 ANALYST, JULY 1992, VOL.117 The results of both methods of analysis are given in Tables 2 and 3. We definef= levels of T1 concentration: for method A, n = 4 , p = 2andf= 4, andformethodB, n = 20,p = 2andf= 4. The fitted model for both methods is the same in this instance and of the form: 9 = -0.55 + 0.2175~ (3) with variance s = 0.397 and 0.586 for methods A and B, respectively. The variance s (in this instance) is the square root of the mean square error for the residual, being smaller in value for method A, as some of the variability is reduced by averaging out the replicates: (4) where SS, is the sum of squares of residuals, and n - p is the number of degrees of freedom associated with SS,. The variance of the estimated parameters 60 and b is not the same: i.e., sbO = 0.4861 and sb = 0.0087 for method A, and for sbg = 0.321 and sb = 0.0059 for method B. The diagonal elements h, (often called leverage) of hat matrix H and h (where h is a parameter related to h,), defined by44-51 H = X(XTX)-lXT, h, = xn(XTX)-lxT and h = x(XTX)xT (some workers add a factor of lln to the definition of leverage, but this makes no difference to the over-all properties of this function: the definition used in this work is preferred as it has the property that h varies between 0 and 1 for experimental points) can be used to study the influence of experimental points on the regression parameters.The h, values for both designs are given in Tables 2 and 3. Graphically, h, could be displayed as a function of x as illustrated in Fig.2. The higher the value of h, the less well the experimental point can be estimated. However, in spite of the smaller value of h and estimated parameter variances for method B (n = 20), the standard variance for method B is still greater than that of method A. However, the 95% variance for method B is smaller than that of method A as compared graphically in Fig. 3(a) and (b). This is because although method A shows a low variance value for the fitted model, the degrees of freedom associated with the residual are very small indeed. Hence, as a result, the critical Fvalue will be higher [where Fis the Fisher variance with one degree of freedom in the numerator and ( n - p ) degrees of freedom in the denominator because sj.,, is based on s?, as j , = X,B when used for obtaining the desired level of confidence]. For example, at the 95% confidence level, F(l, - or F ( l , 2 ) = 18.51 when method A was used compared with F ( l , 18) = 4.41 for method B.The confidence bands for both results are also displayed graphically in Fig. 1 .o 0.8 & 0.6 2 > 3 0.4 0.2 1 1 I I I 0 20 40 60 80 100 TI concentrationhg cm-3 Fig. 2 Value of h as a function of the factor level x (T1 concentration in the base electrolyte) for the model y = Po + Px + r , Methods A and B 4(a) and (b). These are the 95% upper and lower confidence intervals for the following.40>44 1. Mean predicted response values: accounts for the variations caused by estimating the parameter coefficients only (bo and b): En = E n VP(1, n - p ) x s: x h,I 2.Individual predicted response values: these add in the variability of the error term, and hence they are wider than the former type: En = En !E VRl, n - p ) x s: x (1 + hn)l 3. Working-Hotelling confidence bands: used for predicting the entire response values: En = En k V[W x s; X hn] where W = p x F(p, - p ) and F values refer to 95% confidence intervals (of course a similar exercise can be performed using any desired level of confidence). The analysis of variance (ANOVA) for the results from both designs reveals important information. The ANOVA of both results is shown in Tables 4 and 5 from which it is clear that the regression results using both methods show a high significance for the goodness-of-fit (which is also called the significance of the regression): this means that the factor x (or T1 concentration in the base electrolyte) does have an effect on the predicted response 9.On the other hand, it is not possible to test the lack-of-fit for the results from method A: this is because there are no replicated experiments in the analysis. Unlike the results for method A, those for method B do have replicated experiments and the lack-of-fit in this instance can be tested by ANOVA as shown in Table 5. The ANOVA is not significant for the latter design results; hence the model fits the data we11.40>44,52 Further statistical analysis from both designs shows that r2 (the coefficient of determination) is 0.997 and 0.987 for methods A and B, respectively, indicating that the factor x, as I I a, CO 0.4 1 1 I 1 I .- 5 2.4 > 2.0 1.6 1.2 B 0 20 40 60 80 100 Fig.3 ( a ) Standard uncertainty s as a function of the factor level x (TI concentration in the base electrolyte) for the model y = Po + Px + r for Methods A and B. (6) 95% confidence level for s as a function of the factor level x for the model y = Po + fix + r for Methods A and B TI concentrationhg ~ r n - ~ANALYST, JULY 1992, VOL. 117 1079 0 20 40 60 80 100 TI concentrationhg ~ r n - ~ Fig. 4 (a) Confidence bands (95% level) for predicting individual value of response (outer bands), entire value of response (middle bands) and mean value of response (inner bands) (Method A). The least-squares regression line is in the centre of the seven bands. Note that there is a crossover between the inner and middle bands at high and low values of concentration.(b) Confidence bands (95% level) for predicting individual value of response (outer bands), entire value of response (middle bands) and mean value of response (inner bands) (Method B). The least-squares regression line is in the centre of the seven bands. Note that there is a crossover between the inner and middle bands at high and low values of concentration Table 4 Analysis of variance (ANOVA) for the data results presented in Table 2; * indicates a significance level of 99.9%. In these data results there is no replication in the design, hence it is not possible to calculate the analytical error; therefore, the lack-of-fit could not be tested Mean Sum of Degrees of Mean square Source squares freedom square ratio Regression on- Mean Factor effect Total regression Residual Pure error Lack-of- fit Corrected Total 426.4225 1 426.4225 2707.44* 94.6125 1 94.6125 600.71* 521.0350 2 260.5 175 1654.08* 0.3150 2 0.1575 - 0.00 0 0.00 - 0.3150 2 0.1575 - - - 94.9275 3 521.350 4 - - it appears in the fitted model: j = bo + bx, explains the data well.The coefficient of determination for method A is better than that for method B; however, it is important to realize that the coefficient of determination gives no indication of whether the lack of perfect prediction is caused by an inadequate model or by analytical error. The r2 term is not a good measure of the effectiveness of the factor x as it appears in the model, primarily because it does not take into account the degrees of freedom.From the statistical analysis of the results from both methods, it is evident that method B is more informative than method A. Hence a good method for analysis is one that allows for the analytical error to be estimated and utilized in a statistical test for the lack-of-fit, in addition to the improved Table 5 Analysis of variance (ANOVA) for the data results presented in Table 3; * indicates a significance level of 99.9%. The lack-of-fit is not highly significant although there are only 2 degrees of freedom and it is tested against pure analytical error mean square, which is very small (s& = 0.2875) because there are 16 degrees of freedom Mean Sum of Degrees of Mean square Source squares freedom square ratio Regression on- Mean Factor effect Total regression Residual Pure error Lack-of-fit Corrected Total 2132.1125 473.0625 2605.1750 6.1750 4.600 1.5750 479.2375 26211.3500 1 2132.1125 6215.06* 1 473.0625 1387.97* 2 1302.5875 3797.02* 18 0.3431 - 16 0.2875 - 19 20 2 0.7875 2.74 - - - - precision for estimating the model parameters (bo and b) and confidence bands.Having constructed the calibration graph for standard Tl solutions with decreased variance in the parameter estimates (bo and b) and narrower confidence bands, the next step is to utilize this graph for chemical analysis and in the estimation of confidence intervals for the T1 concentration in the samples to be analysed. The detection limit can be calculated from the confidence intervals as the amount of x associated with the predicted response j .The fitted model, given by eqn. (2), which describes the relationship between x and y , can be rewritten as: (j, - bo) x=- b ( 5 ) The linear regression analysis used to solve eqn. ( 5 ) assumes that there are errors associated with the dependent variable y only and that the errors associated with the independent factor x are negligible. However, there are many instances where the regression analysis is performed with errors associated with both the dependent and independent variables: for two variables these methods are ,equivalent to principal com- ponents analysis rather than conventional linear ~ a l i b r a t i o n . ~ ~ In the real world, experiments fit into this latter class as the standards are often prepared by weighing, dilution, and so on; as a result, their concentrations are rarely known exactly.31J9,44 In the case reported here, errors in the independent variable x are ignored and attention is paid only to the errors associated with the dependent variable y and the fitted model estimates y and also x with the same variance.A confidence interval for the predicted response j is found at any given x value. Similarly, a confidence interval for an estimated x concentration is found for a given j value [as shown in Figs. 4(a) and (b)]. The intersection of the regression bands and the x-axis (concentration) defines the range of response obtained for an analyte-free sample (x = 0) at a specific confidence level [sometimes responses less than the point of intersection result in a negative x value (lower interval). In this instance, the detection limit can be con- sidered as the upper value of the intervals and the negative value is rejected].Hence, estimation of the detection limit from the fitted linear calibration graph for any predicted response j or concentration x values can be achieved.26~39 It is also possible to utilize the linear calibration graph to estimate the amount of the analyte in the sample using the instrumental response obtained for the sample by interpolation and the associated confidence intervals. The standard additions method of analysis is a method where the standards are added to the sample matrix and the observed response y of the analyte plus the standard is monitored as a function of the added concentration of the standard. A graphical representation of the standard additions1080 ANALYST, JULY 1992, VOL.117 3 6 -6 - 3 0 3 Added concentrations 6 Fig 5 (a) Graphical representation of the standard additions method: each triangle (A) point represents a single response measurement, c is the concentration of the sample without addition of standards with response readings, cl, c2 and c3 are the added concentrations with c + cl, c + c1 + c2 and c + c1 + c2 + c3 response measurements, respectively. The solid line represents the regression line described by: 9 = bo + bx. The vertical broken line represents the centre of experimentation which is estimated with most confidence. (b) Estimation procedure by standard additions: the dotted line is the estimated regression line and the two enveloping curves are the 95% confidence bands for any given x value in the prediction equation shown in Fig.5(a). UL is the upper and lower confidence interval for c method and its estimation procedure is presented in Fig. 5.27 Although the standard additions method of analysis is a tool well known to electrochemists, the estimation procedure in this method, as shown in Fig. 5(a), has at least one disadvantage, viz., that the confidence interval (UL) for the unknown x concentration at the point C is large.27 However, by carefully considering the method for data analysis and calculating h,, the disadvantage of such a problem can be overcome. The cement dust samples analysed by DPASV were: (1) a cement dust sample from Zaletin, Libya (sample 1); and (2) a cement dust sample from Lepda, Libya (sample 2).Typical voltammograms for these samples are shown in Figs. 6 and 7. For the determination of T1 in these samples, 13 experimen- tal points with five replicated experiments at x = 0 and two replicated experiments at x = 20, 40, 60 and 80 were used. Using a linear model as above [eqns. (1) and (2)], the number of experiments n = 13, the number of factor levelsf = 5 and the number of parameters p = 2. This provides 11 degrees of freedom for the estimation of the residuals error with 8 and 3 degrees of freedom for the estimation of the analytical error and the lack-of-fit, respectively. The results of these two experiments are presented in Tables 6-9. The least-squares estimate regression equations are given as: Sample (1): j j = 5.237 + 0.1876~ Sample (2): 9 = 3.290 + 0.208~ (6) (7) with variance = 0.505 and 0.539, respectively.The standard error for parameter estimates is sbo = 0.201 and sb = 0.00467 for sample 1 and sbo = 0.214 and sb = 0.00499 for sample 2. I I I I I - 1.20 - 1 .oo- 0.80 - 0.60- 0.40- 0.20 PotentialN versus Ag-AgCI-3 mol dm-3 KCI Fig. 6 Current-voltage curves (voltammograms) for cement kiln dust samples from Zaletin (Libya) [the symbols are for (a): A = blank response, B = sample signal and C, D, E, F and G are sample plus 20, 40, 60, 80 and 100 mm3 of the added standard Tl concentration (10 pg cm-3 so that each mm3 = 1 ng cm-3); and for (b): A = blank response, B, C, D and E = sample responses] The h, value at x = 0 with this type of design is reduced to h, = 0.158 compared with h, = 0.284 at x, = 80.Both x, = 0 and x, = 80 points are outliers and will have the same h, value if the x, = 0 point is not replicated and treated in the same way as the x, = 80 point. This will reduce the variance for 9 and the unknown concentration will be estimated with improved precision. The variance (s) for 9 (and the estimated TI concentration in the base electrolyte as a result of sample solution) is a function of both s, and h,, as the standard variance s (or uncertainty) is defined as: s = vs; x (1 + h,), where: s2= ( - nyp) and h, = x,(XTX)-lx;T (8) The 95% confidence bands are displayed graphically in Fig 8. The standard variance for each analysed sample is smaller at x, = 0 due to the replication of this point, as demonstrated in Fig. 8(c).The ANOVA results for the cement dust samples 1 and 2 are given in Tables 7 and 9. The coefficients of determination are 0.993 and 0.994, respectively. The significance of the coefficient of determination is contained in F@ - 1, , - = F(1, 11) (ie., mean square for residuals divided by the meanANALYST, JULY 1992, VOL. 117 1081 I -1.20 -0.80 -0.40 -1.00 -0.60 -0.20 PotentialN versus Ag-AgCI-3 mol dm-3 KCI Fig. 7 Current-voltage curves (voltammograms) for cement kiln dust samples from Lepda (at Khomes, Libya): [the symbols are for (a): A = blank response, B = sample signal and (C, D, E, F and G are sample plus 20, 40, 60, 80 and 100 mm3 of the added standard TI concentration (10 pg cm-3 so that each 1 mm3 = 1 ng cm-3); and for (b): A = blank response, B, C, D and E = sample responses] Table 6 Results from the analysed cement dust sample (1) from Zaletin, Libya: x, h,, y, 9 , r and s are as defined in Table 2 n 1 2 3 4 5 6 7 8 9 10 11 12 13 X 0 0 0 0 0 20 20 40 40 60 60 80 80 h n 0.158 0.158 0.158 0.158 0.158 0.087 0.087 0.084 0.084 0.150 0.150 0.284 0,284 Y 5.5 4.9 5.7 5.2 5.8 8.7 8.9 12.5 11.5 17.0 16.4 20.5 20.5 E 5.237 5.237 5.237 5.237 5.237 8.988 8.988 12.739 12.739 16.490 16.490 20.241 20.241 r 0.263 0.463 0.563 -0.336 -0.036 -0.287 -0.087 -0.238 - 1.239 0.510 -0.090 0.258 0.258 S 0.542 0.542 0.542 0.542 0.542 0.526 0.526 0.525 0.525 0.541 0.541 0.571 0.571 square for factors effect) and in both experiments this value is 0.001 and 0.1, respectively, which is not significant.This means that the factor x explains the response well.In this case the sum of the squares due to the residuals is very small and that of the factor is relatively large. In contrast, when the factor has very little effect on the response, then the sum of squares due to the factor effect would be very small and, therefore, SS, would be large. The lack-of-fit is not significant in both cases. The method used for the determination of trace amounts of TI in cement dust samples was also applied to the determina- tion of TI in sediment samples (collected from the Haw-Wood area, Avonmouth, Bristol, UK). These samples were pre- pared in the same way as described above. The instrumental Table 7 Analysis of variance (ANOVA) for the analysed cement dust sample collected from Zaletin, Libya; * indicates a significance level of 99.9%.Both the mean response (intercept) and the factor effect (TI concentration) are highly significant as tested against the residual error (total error). The lack-of-fit is not highly significant as tested against the pure error mean square. In the original experiment: n = 13,p = 2, f = 5 Mean Sum of Degrees of Mean square Source squares freedom square ratio Regression on- Mean Factor effect Total regression Residual Pure error Lack-of-fit Corrected Total 1575.201 411.289 1986.490 2.801 1.248 1.553 414.089 1989.290 1 1 2 11 8 3 12 13 1575.201 6186.66* 411.289 1615.35* 993.245 3901 .OO* 0.255 - 0.156 - 0.518 3.32 Table 8 Results from the analysed cement dust sample (2) from Lepda, Libya: x, h,, y, 9 , r and s are as defined in Table 2 n 1 2 3 4 5 6 7 8 9 10 11 12 13 X 0 0 0 0 0 20 20 40 40 60 60 80 80 h n 0.158 0.158 0.158 0.158 0.158 0.087 0.087 0.084 0.084 0.150 0.150 0.284 0.284 Y 9 3.6 3.290 2.8 3.290 3.0 3.290 3.5 3.290 4.0 3.290 8.0 7.456 7.0 7.456 11.0 11.623 11.6 11.623 15.8 15.790 15.0 15.790 20.0 19.957 20.8 19.957 r 0.311 -0.490 -0.290 0.211 0.711 0.544 -0.456 -0.623 -0.023 0.010 -0.079 0.043 0.843 S 0.580 0.580 0.580 0.580 0.580 0.562 0.562 0.562 0.562 0.578 0.578 0.611 0.611 Table 9 Analysis of variance (ANOVA) for the analysed cement dust sample collected from Lepda (at Khomes), Libya; * indicates a significance level of 99.9%.The lack-of-fit is not highly significant at F(3, 8). The remainder of the regression is highly significant (>99.9%). Also, the original experiments are the same as in Table 10 Mean Sum of Degrees of Mean square Source squares freedom square ratio Regression on- Mean Factor effect Total regression Residual Pure error Lack-of-fit Corrected Total 1223.170 507.521 1730.691 3.199 2.284 0.951 5 10.720 1733.890 1 1223.170 4206.45* 1 4507.521 1745.35* 2 856.346 2975.90* 11 0.291 - 8 0.281 - 3 0.317 1.13 - - 12 13 - - conditions were altered slightly during the analysis of the sediment samples.The results from the analysis of the sediments together with ANOVA results are shown in Tables 10 and 11. The leverage points h, for the design used in the analysis of the sediment samples is the same as that used for the analysis of the cement dusts; hence, the h, values are the same as shown in their corresponding results. However, the equation for h, which can be calculated for the inverse matrix (XTX)--I for this design, is given as: h = 0.1579 - 0.00526~ + O.ooOo86~2 (9) This equation can be used to determine the standard error for the predicted response 9 and to determine the unknown T1 concentration in the sample solution.1082 ANALYST, JULY 1992, VOL.117 Table 10 Results from the analysed sediment samples: x , h,, y, j , rand s are as defined in Table 2 Y m a -40 -20 0 20 40 60 80 100 TI concentrationhg cm-3 0.62 0.58 L >" 0.54 I. ! I 0.50 1 I I I 0 20 40 60 80 TI concentrationhg cm-3 Fig. 8 (a) Confidence bands (95% level) for the analysed cement dust sample 1 for predicting individual value of response (outer bands), entire value of response (middle bands) and mean value of response (inner bands).The least-squares regression line and the measured response (A points) are in the centre. (b) Confidence bands 95% level) for the analysed cement dust sample 2 for predicting individual value of response (outer bands), entire value of response (middle bands) and mean value of response (inner bands). The solid least-squares regression line and the measured response (A points) are in the centre. (c) Standard uncertainty (variance) for the analysed cement dust samples (1 and 2) as a function of T1 concentration levels x for the linear model: y = Po + Px + r The concentration of T1 in the solid sample (cement kiln x x v x z dust and sediment samples) may be determined as: c = (7) (10) where c is the T1 concentration in the solid sample (ng g-I), x is the T1 concentration (ng cm-3) in the actual sample solution, V is the final volume (cm3) of the analyte solution, I is the dilution factor (if any) and rn is the mass of the sample (8).The procedure for estimating the TI concentration in 10 cm3 of sample solution transferred into the polarographic cell ( x ) involves extrapolation of the original standard additions graph n 1 2 3 4 5 6 7 8 9 10 11 12 13 X 0 0 0 0 0 20 20 40 40 60 60 80 80 h n 0.158 0.158 0.158 0.158 0.158 0.087 0.087 0.084 0.084 0.150 0.150 0.284 0.284 Y E 3.1 3.047 3.7 3.047 2.8 3.047 3.5 3.047 2.7 3.047 6.4 6.751 7.0 6.751 9.8 10.456 10.3 10.456 13.8 14.160 14.2 14.160 17.9 17.864 18.5 17.864 r 0.053 0.653 0.453 -0.247 -0.347 -0.352 0.248 -0.656 -0.156 -0.360 0.040 0.036 0.636 S 0.458 0.458 0.458 0.458 0.458 0.444 0.444 0.443 0.443 0.456 0.456 0.482 0.482 Table 11 Analysis of variance (ANOVA) for the analysed sediment sample; * indicates a significance level of 99.9%. The mean response from this experiment is 8.476, and r2 = 0.9951.The lack-of-fit is not significant, while the remainder of the regression is highly significant Mean Sum of Degrees of Mean square Source squares freedom squares ratio Regression on- Mean Factor effect Total regression Residual Pure error Lack-of-fit Corrected Total 994.4377 401.0805 1395.5182 1.9919 1.3170 0.6749 403.0723 1397.5100 1 994.4377 5491.95* 1 401.0805 2215.04* 2 697.7591 3853.49* 8 3 0.2249 1.37 12 13 11 0.1811 - 0.1646 - - - - - Table 12 Thallium concentration in the analysed cement and sediment samples T1 concentration/ T1 concentration/ Sample No.ng cm-3 S* ngg-' 1 28.00 3.07 700.0 2 15.80 2.91 395.0 3 16.50 2.59 850.0 * s is the standard error in the estimation of the concentration as discussed in the text. as illustrated above or from the fitted model by putting the predicted response value equal to 0: bo + bx = 0 The value of x given by eqn. (1 1) is minus the concentration of Tl in the sample. The standard deviation (variance or uncertainty) for each fitted model is expressed in the units of the dependent variable y (or response) because, as pointed out above, the errors in the independent variable x (or concentration) are ignored using the most commonly employed methods. However, the units of variance s could be converted from the units of y (or peak height in cm) to the units of x (or concentration in ng cm-3) as follows: (11) sr s, = - b The standard deviation (or uncertainty) could be easily calculated using eqn.(8) to calculate h, which is outside the region of experimentation in this case. The T1 content in the samples is given in Table 12. The estimated Tl concentration x (ng cm-3) from the fitted model is the concentration that was present in 10 cm3 of the sample solution originally transferred into the polarographic cell. The over-all T1 concentration in the total sample solution isANALYST, JULY 1992, VOL. 117 1083 calculated by multiplying this value by the final sample solution which, when divided by the original sample mass (m), gives the TI content (in ng g-1) of the solid sample. The standard uncertainty (s,) of the estimated T1 concentration is calculated via the standard uncertainty equation.40 Hence, by converting the units of s from s,, to s, and substituting x at i, = 0 in eqn.(8) to obtain h at that point, then the over-all s, can be easily calculated. In the analysis above, it is assumed that the errors in the data are constant throughout the experimental region. This assumption is usual in many forms of multilinear regression. In particular, standard approaches to experimental design nor- mally involve taking replicates in only one place (generally the centre of the design) and then assuming that the replicate error is constant. This assumption is often called homosced- acity of errors. Some experiments result in heteroscedacity of errors.Under these circumstances, the error distribution varies across the experimental region. In some instances the absolute errors associated with small measurements are less than those associated with larger measurements. Whether this occurs or not is related to the over-all experimental process, and the way to study this is to take replicates throughout the experimental region. In this work, in common with many practical analytical situations, the heteroscedacity of errors was not studied; instead, it was assumed that the errors are constant through- out the experimental region. It would be possible to extend the studies in this paper to obtain a fuller understanding of the errors. If such data are available, there are two main approaches to correcting for the heteroscedacity of the errors.53 (i) Use a weighted least-squares procedure for calibration, weighting each point according to the error distribution. (ii) Scale the y variable to take account of these errors; this is possible if the error distribution follows a well-defined trend, for example, if the errors are proportional to the measurements.It is easy to modify the approach described above if required. However, the analyst must carefully consider the amount of work required for the solution of any given problem. It is clearly impractical to perform a full and exhaustive study of error distributions every time a linear calibration model is computed, and the simple methods of visualizing confidence in the model as related to the design discussed above serve as a good approach.It is typical of chemometrics that there is a range of methods according to the detail required from the data. A typical example involves the classification of objects. Exploratory approaches such as principal components analysis allow graphical methods for visualizing whether there are major groups; cluster analysis will then allow the main groups to be defined better; soft modelling ( e . g . , SIMCA) provides mathematical models of the groups selected previously; hard modelling such as canonical variates analysis can be used where there is an even more rigid and well-defined class structure. No one approach is superior to any other, and the methods range from exploratory data analysis (EDA) to mathematical modelling. This paper has not presented a method for the mathematical modelling of errors, but does provide a good exploratory approach to the visualization of the influence of design on linear calibration experiments.Conclusion From the results presented here the feasibility of determining TI by DPASV has been demonstrated. The sample prepara- tion prior to DPASV analysis permits the separation of TI from the other chemical species that interfere in the analysis. Further, use of the standard additions method enhances the confidence in the estimated response. In the extraction procedure, the use of diethyl ether gave good results. However, no other solvents were used in the extraction procedure (for example, diisopropyl ether). By carefully considering the nature of the experiments and the analysis used to calibrate the observed response to the concentration of Tl, a high precision in estimating the parameters of the linear model was achieved via replicating the original experiments to provide a sufficient number of degrees of freedom to be used in the ANOVA procedure.A proper procedure involves the following. 1. Determination of the calibration graph for each experi- ment by the usual method of linear regression, ignoring the errors in the independent variable x. 2. The scatter about the calibration graph (variance or standard uncertainty) is obtained from the fitted model s,. The conversion of the variance units from that of sPn to the units of concentration s, is also demonstrated. 3. The replication error (pure error) is used to test the lack-of-fit of the model to the experimental data.4. The standard uncertainty (or variance) in the predicted response value sJn and for the estimated concentration of TI in the sample solution s, is obtained. 5. The confidence intervals and the confidence bands in the predicted response i, for each experiment can also be computed. 6. The leverage equation that can be used to calculate h should be considered: further discussion about leverage is provided elsewhere .43-48 Calibration is a very important procedure in analytical chemistry; therefore, further work is necessary in order to understand and explain the results fully: it would be of considerable interest to extend methods to situations where errors exist in both the x and y directions. In this paper, only the importance of experimental design in analytical experiments as explained by ANOVA, confidence bands and leverage has been considered.Errors in the x-axis have not been discussed. Although these are important, most designs used by experimental chemists assume that there are no errors in the x-axis. In order to incorporate these errors, signficantly different approaches need to be developed, but are outside the scope of this paper. M. A. 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