D. Harvey - Modern Analytical Chemistry (794078), страница 37
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A standard containing 1.75ppb Pb2+ and 2.25 ppb Cu2+ yields a ratio of SA/SIS of 2.37. A sample of blood isspiked with the same concentration of Cu 2+, giving a signal ratio of 1.80.Determine the concentration of Pb2+ in the sample of blood.SOLUTIONEquation 5.11 allows us to calculate the value of K using the data for thestandard C S 2.25K = IS A =× 2.37 = 3.05CS1.75 A IS standThe concentration of Pb2+, therefore, is C S 2.25C A = IS A =× 1.80 = 1.33 ppb Pb 2+3.05 K SIS samp1400-CH05 9/8/99 3:59 PM Page 117Chapter 5 Calibrations, Standardizations, and Blank CorrectionsA single-point internal standardization has the same limitations as a singlepoint normal calibration.
To construct an internal standard calibration curve, it isnecessary to prepare several standards containing different concentrations of analyte. These standards are usually prepared such that the internal standard’s concentration is constant. Under these conditions a calibration curve of (SA/SIS)stand versusCA is linear with a slope of K/CIS.EXAMPLE 5.9A seventh spectrophotometric method for the quantitative determination ofPb2+ levels in blood gives a linear internal standards calibration curve for which SA = (2.11 ppb –1 ) × C A – 0.006 SIS standWhat is the concentration (in ppb) of Pb2+ in a sample of blood if (SA/SIS)samp is 2.80?SOLUTIONTo determine the concentration of Pb2+ in the sample of blood, we replace(SA/SIS)stand in the calibration equation with (SA/SIS)samp and solve for CACA =(S A / SIS )samp + 0.0062.11 ppb –1=2.80 + 0.006= 1.33 ppb2.11 ppb –1The concentration of Pb2+ in the sample of blood is 1.33 ppb.When the internal standard’s concentration cannot be held constant the data mustbe plotted as (SA/SIS)stand versus CA/CIS, giving a linear calibration curve with a slopeof K.5C Linear Regression and Calibration CurvesIn a single-point external standardization, we first determine the value of k bymeasuring the signal for a single standard containing a known concentration ofanalyte.
This value of k and the signal for the sample are then used to calculatethe concentration of analyte in the sample (see Example 5.2). With only a singledetermination of k, a quantitative analysis using a single-point external standardization is straightforward. This is also true for a single-point standard addition (see Examples 5.4 and 5.5) and a single-point internal standardization (seeExample 5.8).A multiple-point standardization presents a more difficult problem. Consider thedata in Table 5.1 for a multiple-point external standardization.
What is the best estimate of the relationship betweenTable 5.1 Data for Hypothetical MultipleSmeas and CS? It is tempting to treat this data as five separatePoint External Standardizationsingle-point standardizations, determining k for each stanCSSmeasdard and reporting the mean value. Despite its simplicity,this is not an appropriate way to treat a multiple-point0.0000.00standardization.0.10012.36In a single-point standardization, we assume that0.20024.83the reagent blank (the first row in Table 5.1) corrects for0.30035.91all constant sources of determinate error. If this is not0.40048.79the case, then the value of k determined by a single0.50060.42point standardization will have a determinate error.1171400-CH05 9/8/99 3:59 PM Page 118118Modern Analytical ChemistryTable 5.2CASmeas(true)k(true)Smeas(with constant error)k(apparent)1.002.003.004.005.001.002.003.004.005.001.001.001.001.001.001.502.503.504.505.501.501.251.171.131.10mean k(true) =1.00mean k (apparent) =1.2380Smeas60402000.00.10.20.3CAFigure 5.8Normal calibration plot of hypothetical datafrom Table 5.1.0.4Effect of a Constant Determinate Error on the Valueof k Calculated Using a Single-Point Standardization0.50.6Table 5.2 demonstrates how an uncorrected constant erroraffects our determination of k.
The first three columns showthe concentration of analyte, the true measured signal (noconstant error) and the true value of k for five standards. Asexpected, the value of k is the same for each standard. In thefourth column a constant determinate error of +0.50 hasbeen added to the measured signals. The corresponding values of k are shown in the last column. Note that a differentvalue of k is obtained for each standard and that all values aregreater than the true value. As we noted in Section 5B.2, thisis a significant limitation to any single-point standardization.How do we find the best estimate for the relationship between the measured signal and the concentration of analyte ina multiple-point standardization? Figure 5.8 shows the data inTable 5.1 plotted as a normal calibration curve.
Although thedata appear to fall along a straight line, the actual calibrationcurve is not intuitively obvious. The process of mathematically determining the best equation for the calibration curve iscalled regression.5C.1 Linear Regression of Straight-Line Calibration CurvesA calibration curve shows us the relationship between the measured signal and theanalyte’s concentration in a series of standards.
The most useful calibration curve isa straight line since the method’s sensitivity is the same for all concentrations of analyte. The equation for a linear calibration curve isy = β0 + β1xlinear regressionA mathematical technique for fitting anequation, such as that for a straight line,to experimental data.residual errorThe difference between an experimentalvalue and the value predicted by aregression equation.5.12where y is the signal and x is the amount of analyte.
The constants β0 and β1 arethe true y-intercept and the true slope, respectively. The goal of linear regression is to determine the best estimates for the slope, b1, and y-intercept, b0. Thisis accomplished by minimizing the residual error between the experimental values, yi, and those values, ŷi, predicted by equation 5.12 (Figure 5.9). For obviousreasons, a regression analysis is also called a least-squares treatment. Several approaches to the linear regression of equation 5.12 are discussed in the followingsections.1400-CH05 9/8/99 3:59 PM Page 119119Chapter 5 Calibrations, Standardizations, and Blank Corrections5C.2 Unweighted Linear Regression with Errors in yThe most commonly used form of linear regression is based on three assumptions: (1) that any difference between the experimental data and the calculatedregression line is due to indeterminate errors affecting the values of y, (2) thatthese indeterminate errors are normally distributed, and (3) that the indeterminate errors in y do not depend on the value of x.
Because we assume that indeterminate errors are the same for all standards, each standard contributes equally inestimating the slope and y-intercept. For this reason the result is considered anunweighted linear regression.The second assumption is generally true because of the central limit theoremoutlined in Chapter 4. The validity of the two remaining assumptions is less certain and should be evaluated before accepting the results of a linear regression.In particular, the first assumption is always suspect since there will certainly besome indeterminate errors affecting the values of x.
In preparing a calibrationcurve, however, it is not unusual for the relative standard deviation of the measured signal (y) to be significantly larger than that for the concentration of analyte in the standards (x). In such circumstances, the first assumption is usuallyreasonable.Finding the Estimated Slope and y-Intercept The derivation of equations for calculating the estimated slope and y-intercept can be found in standard statisticaltexts7 and is not developed here. The resulting equation for the slope is given asb1 =n ∑ x i yi – ∑ x i ∑ yin ∑ x i2 – (∑ x i )25.13and the equation for the y-intercept isb0 =∑ yi – b1 ∑ x in5.14Although equations 5.13 and 5.14 appear formidable, it is only necessary to evaluatefour summation terms.
In addition, many calculators, spreadsheets, and other computer software packages are capable of performing a linear regression analysis basedon this model. To save time and to avoid tedious calculations, learn how to use oneof these tools. For illustrative purposes, the necessary calculations are shown in detail in the following example.EXAMPLE 5.10Using the data from Table 5.1, determine the relationship between Smeas and CSby an unweighted linear regression.SOLUTIONEquations 5.13 and 5.14 are written in terms of the general variables x and y.As you work through this example, remember that x represents theconcentration of analyte in the standards (CS), and that y corresponds to thesignal (Smeas).
We begin by setting up a table to help in the calculation of thesummation terms Σxi, Σyi, Σx i2, and Σxiyi which are needed for the calculationof b0 and b1RegressionlineŷiResidual error = yi – ŷiyiTotal residual error =∑(y – ŷ )ii2Figure 5.9Residual error in linear regression, where thefilled circle shows the experimental value yi,and the open circle shows the predictedvalue ŷi.1400-CH05 9/8/99 3:59 PM Page 120Modern Analytical Chemistryxiyixi20.0000.1000.2000.3000.4000.5000.0012.3624.8335.9148.7960.420.0000.0100.0400.0900.1600.250xiyi0.0001.2364.96610.77319.51630.210Adding the values in each column givesΣxi = 1.500Σyi = 182.31Σx 2i = 0.550Σxiyi = 66.701Substituting these values into equations 5.12 and 5.13 gives the estimated slopeb1 =(6)(66.701) – (1.500)(182.31)= 120.706(6)(0.550) – (1.500)2and the estimated y-interceptb0 =182.31 – (120.706)(1.500)= 0.2096The relationship between the signal and the analyte, therefore, isSmeas = 120.70 × CS + 0.21Note that for now we keep enough significant figures to match the number ofdecimal places to which the signal was measured.
The resulting calibrationcurve is shown in Figure 5.10.8060Smeas120402000.00.10.20.3CA0.40.50.6Figure 5.10Normal calibration curve for the hypothetical data in Table 5.1,showing the regression line.Uncertainty in the Regression Analysis As shown in Figure 5.10, the regressionline need not pass through the data points (this is the consequence of indeterminate errors affecting the signal). The cumulative deviation of the data fromthe regression line is used to calculate the uncertainty in the regression due to1400-CH05 9/8/99 3:59 PM Page 121Chapter 5 Calibrations, Standardizations, and Blank Correctionsindeterminate error. This is called the standard deviation about the regression,sr, and is given assr =∑ ni =1 (yi – yˆi )2n–25.15where yi is the ith experimental value, and ŷi is the corresponding value predicted bythe regression lineŷi = b0 + b1xiThere is an obvious similarity between equation 5.15 and the standard deviation introduced in Chapter 4, except that the sum of squares term for sr is determined relative to ŷi instead of y–, and the denominator is n – 2 instead of n – 1; n – 2 indicatesthat the linear regression analysis has only n – 2 degrees of freedom since two parameters, the slope and the intercept, are used to calculate the values of ŷi.A more useful representation of uncertainty is to consider the effect of indeterminate errors on the predicted slope and intercept.
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