D. Harvey - Modern Analytical Chemistry (794078), страница 41
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Asecond 10.00-mL aliquot was spiked with 10.00 mL of a 1.00ppm standard solution of the analyte and diluted to 25.00 mL.The signal for the spiked sample was found to be 0.502.Calculate the weight percent of analyte in the original sample.8. A 50.00-mL sample containing an analyte gives a signal of11.5 (arbitrary units). A second 50-mL aliquot of the sample,which is spiked with 1.00-mL of a 10.0-ppm standard solutionof the analyte, gives a signal of 23.1.
What is the concentrationof analyte in the original sample?9. An appropriate standard additions calibration curve based onequation 5.8 plots Sspike(Vo + Vs) on the y-axis and CsVs onthe x-axis. Clearly explain why you cannot plot Sspike on the yaxis and Cs[Vs/(Vo + Vs)] on the x-axis. Derive equations forthe slope and y-intercept, and explain how the amount ofanalyte in a sample can be determined from the calibrationcurve.10. A standard sample was prepared containing 10.0 ppm of ananalyte and 15.0 ppm of an internal standard.
Analysis of thesample gave signals for the analyte and internal standard of0.155 and 0.233 (arbitrary units), respectively. Sufficientinternal standard was added to a sample to make it 15.0 ppmin the internal standard. Analysis of the sample yielded signalsfor the analyte and internal standard of 0.274 and 0.198,respectively. Report the concentration of analyte in thesample.11. For each of the pairs of calibration curves in Figure 5.13on page 132, select the calibration curve with the betterset of standards. Briefly explain the reasons for yourselections. The scales for the x-axes and y-axes are the samefor each pair.12. The following standardization data were provided for a seriesof external standards of Cd2+ that had been buffered to a pHof 4.6.14[Cd2+] (nM)Smeas (nA)15.44.830.411.444.918.259.026.672.732.386.037.7(a) Determine the standardization relationship by a linearregression analysis, and report the confidence intervals for theslope and y-intercept.
(b) Construct a plot of the residuals,and comment on their significance.At a pH of 3.7 the following data were recorded[Cd2+] (nM)Smeas (nA)15.415.030.442.744.958.559.077.072.710186.0118(c) How much more or less sensitive is this method at thelower pH? (d) A single sample is buffered to a pH of 3.7 andanalyzed for cadmium, yielding a signal of 66.3. Report theconcentration of Cd2+ in the sample and its 95% confidenceinterval.13. To determine the concentration of analyte in a sample, astandard additions was performed.
A 5.00-mL portionof the sample was analyzed and then successive0.10-mL spikes of a 600.0-ppb standard of the analyte1400-CH05 9/8/99 3:59 PM Page 132SignalSignalModern Analytical ChemistryCACASignalSignal(a)CACASignal(b)Signal132CA(c)Figure 5.13CA1400-CH05 9/8/99 3:59 PM Page 133133Chapter 5 Calibrations, Standardizations, and Blank Correctionswere added, analyzing after each spike. The followingresults were obtainedVolume of Spike(mL)Signal(arbitrary units)0.000.100.200.300.1190.2310.3390.442Construct an appropriate standard additions calibrationcurve, and use a linear regression analysis to determine theconcentration of analyte in the original sample and its 95%confidence interval.14. Troost and Olavesen investigated the application of aninternal standardization to the quantitative analysis ofpolynuclear aromatic hydrocarbons.15 The following resultswere obtained for the analysis of the analyte phenanthreneusing isotopically labeled phenanthrene as an internalstandardSA/SISCA/CIS0.501.252.003.004.00Replicate 1Replicate 20.5140.9931.4862.0442.3420.5221.0241.4712.0802.550(a) Determine the standardization relationship by a linearregression, and report the confidence intervals for the slopeand y-intercept.
(b) Based on your results, explain why theauthors of this paper concluded that the internalstandardization was inappropriate.15. In Chapter 4 we used a paired t-test to compare two methodsthat had been used to independently analyze a series ofsamples of variable composition. An alternative approach is toplot the results for one method versus those for the other. Ifthe two methods yield identical results, then the plot shouldhave a true slope (β1) of 1.00 and a true y-intercept (β0) of0.0. A t-test can be used to compare the actual slope and yintercept with these ideal values.
The appropriate test statisticfor the y-intercept is found by rearranging equation 5.18t exp =β 0 – b0s b0=b0s b0Rearranging equation 5.17 gives the test statistic for the slopet exp =β1 – b1s b1=1.00 – b1s b1Reevaluate the data in problem 24 in Chapter 4 using thesame significance level as in the original problem.*16. Franke and co-workers evaluated a standard additionsmethod for a voltammetric determination of Tl.16 Asummary of their results is tabulated here.ppm Tladded0.0000.3871.8515.734Instrument Response for Replicates(µA)2.538.4229.6584.82.507.9628.7085.62.708.5429.0586.02.638.1828.3085.22.707.7029.2084.22.808.3429.9586.42.527.9828.9587.8Determine the standardization relationship using a weightedlinear regression.5I SUGGESTED READINGSIn addition to the texts listed as suggested readings in Chapter 4,the following text provides additional details on regressionDraper, N.
R.; Smith, H. Applied Regression Analysis, 2nd. ed.Wiley: New York, 1981.Several articles providing more details about linear regressionfollow.Boqué, R.; Rius, F. X.; Massart, D. L. “Straight Line Calibration:Something More Than Slopes, Intercepts, and CorrelationCoefficients,” J. Chem. Educ. 1993, 70, 230–232.Henderson, G. “Lecture Graphic Aids for Least-Squares Analysis,”J. Chem. Educ. 1988, 65, 1001–1003.Renman, L., Jagner, D.
“Asymmetric Distribution of Results inCalibration Curve and Standard Addition Evaluations,” Anal.Chim. Acta 1997, 357, 157–166.Two useful papers providing additional details on the method ofstandard additions areBader, M. “A Systematic Approach to Standard Addition Methodsin Instrumental Analysis,” J. Chem. Educ. 1980, 57, 703–706.*Although this is a commonly used procedure for comparing two methods, it does violate one of the assumptions of an ordinary linear regression.
Since both methods areexpected to have indeterminate errors, an unweighted regression with errors in y may produce a biased result, with the slope being underestimated and the y-intercept beingoverestimated. This limitation can be minimized by placing the more precise method on the x-axis, using ten or more samples to increase the degrees of freedom in the analysis,and by using samples that uniformly cover the range of concentrations. For more information see Miller, J. C.; Miller, J.
N. Statistics for Analytical Chemistry, 3rd ed. EllisHorwood PTR Prentice-Hall: New York, 1993. Alternative approaches are discussed in Hartman, C.; Smeyers-Verbeke, J.; Penninckx, W.; Massart, D. L. Anal. Chim. Acta 1997,338, 19–40 and Zwanziger, H. W.; Sârbu, C. Anal. Chem. 1998, 70, 1277–1280.1400-CH05 9/8/99 4:00 PM Page 134134Modern Analytical ChemistryNimura, Y.; Carr, M. R. “Reduction of the Relative Error in theStandard Additions Method,” Analyst 1990, 115, 1589–1595.The following paper discusses the importance of weightingexperimental data when using linear regressionKarolczak, M. “To Weight or Not to Weight? An Analyst’sDilemma,” Curr.
Separations 1995, 13, 98–104.Algorithms for performing a linear regression with errors in bothx and y are discussed inIrvin, J. A.; Quickenden, T. L. “Linear Least Squares TreatmentWhen There Are Errors in Both x and y,” J. Chem. Educ. 1983,60, 711–712.Kalantar, A. H. “Kerrich’s Method for y = αx Data When Both yand x Are Uncertain,” J.
Chem. Educ. 1991, 68, 368–370.Macdonald, J. R.; Thompson, W. J. “Least-Squares Fitting WhenBoth Variables Contain Errors: Pitfalls and Possibilities,” Am.J. Phys. 1992, 60, 66–73.Ogren, P. J.; Norton, J. R. “Applying a Simple Linear LeastSquares Algorithm to Data with Uncertainties in BothVariables,” J. Chem. Educ. 1992, 69, A130–A131.The following paper discusses some of the problems that may beencountered when using linear regression to model data that havebeen mathematically transformed into a linear form.Chong, D.
P. “On the Use of Least Squares to Fit Data in LinearForm,” J. Chem. Educ. 1994, 71, 489–490.The analysis of nonlinear data is covered in the following papers.Harris, D. C. “Nonlinear Least-Squares Curve Fitting withMicrosoft Excel Solver,” J. Chem. Educ. 1998, 75, 119–121.Lieb, S. G. “Simplex Method of Nonlinear Least-Squares—ALogical Complementary Method to Linear Least-SquaresAnalysis of Data,” J. Chem. Educ. 1997, 74, 1008–1011.Machuca-Herrera, J. G. “Nonlinear Curve Fitting withSpreadsheets,” J. Chem.
Educ. 1997, 74, 448–449.Zielinski, T. J.; Allendoerfer, R. D. “Least Squares Fitting ofNonlinear Data in the Undergraduate Laboratory,” J. Chem.Educ. 1997, 74, 1001–1007.More information on multivariate regression can be found inLang, P. M.; Kalivas, J. H. “A Global Perspective on MultivariateCalibration Methods,” J. Chemometrics 1993, 7, 153–164.Kowalski, B. R.; Seasholtz, M. B. “Recent Developments inMultivariate Calibration,” J. Chemometrics 1991, 5 129–145.An additional discussion on method blanks is found in thefollowing two papers.Cardone, M. J.
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