D. Harvey - Modern Analytical Chemistry (794078), страница 35
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When working with larger concentration ranges, particularly those extending over more than three orders ofmagnitude, standards are best prepared by a serial dilution from a single stock solution. In a serial dilution a volume of a concentrated stock solution, which is the firststandard, is diluted to prepare a second standard.
A portion of the second standardis then diluted to prepare a third standard, and the process is repeated until all necessary standards have been prepared. Serial dilutions must be prepared with extracare because a determinate error in the preparation of any single standard is passedon to all succeeding standards.5B.2 Single-Point versus Multiple-Point Standardizations*single-point standardizationAny standardization using a singlestandard containing a known amount ofanalyte.The simplest way to determine the value of k in equation 5.2 is by a singlepoint standardization.
A single standard containing a known concentrationof analyte, CS, is prepared and its signal, Sstand, is measured. The value of k is calculated ask =SignalAssumedrelationshipActualrelationshipSstandCsConcentrationreportedCAActualconcentrationFigure 5.2Example showing how an improper use ofa single-point standardization can lead to adeterminate error in the reportedconcentration of analyte.SstandCS5.3A single-point standardization is the least desirable way to standardizea method. When using a single standard, all experimental errors, both determinate and indeterminate, are carried over into the calculated value fork.
Any uncertainty in the value of k increases the uncertainty in the analyte’s concentration. In addition, equation 5.3 establishes the standardization relationship for only a single concentration of analyte. Extendingequation 5.3 to samples containing concentrations of analyte differentfrom that in the standard assumes that the value of k is constant, an assumption that is often not true.6 Figure 5.2 shows how assuming a constant value of k may lead to a determinate error. Despite these limitations,single-point standardizations are routinely used in many laboratories whenthe analyte’s range of expected concentrations is limited. Under these conditions it is often safe to assume that k is constant (although this assumption should be verified experimentally).
This is the case, for example, inclinical laboratories where many automated analyzers use only a singlestandard.The preferred approach to standardizing a method is to prepare a series of standards, each containing the analyte at a different concentration.Standards are chosen such that they bracket the expected range for the*The following discussion of standardizations assumes that the amount of analyte is expressed as a concentration. Italso applies, however, when the absolute amount of analyte is given in grams or moles.1400-CH05 9/8/99 3:59 PM Page 109Chapter 5 Calibrations, Standardizations, and Blank Correctionsanalyte’s concentration. Thus, a multiple-point standardization should use at leastthree standards, although more are preferable. A plot of Sstand versus CS is known asa calibration curve.
The exact standardization, or calibration relationship, is determined by an appropriate curve-fitting algorithm.* Several approaches to standardization are discussed in the following sections.109multiple-point standardizationAny standardization using two or morestandards containing known amounts ofanalyte.5B.3 External StandardsThe most commonly employed standardization method uses one or more externalstandards containing known concentrations of analyte. These standards are identified as external standards because they are prepared and analyzed separately fromthe samples.A quantitative determination using a single external standard was described atthe beginning of this section, with k given by equation 5.3.
Once standardized, theconcentration of analyte, CA, is given asCA =Ssampkexternal standardA standard solution containing a knownamount of analyte, prepared separatelyfrom samples containing the analyte.5.4A spectrophotometric method for the quantitative determination of Pb2+ levelsin blood yields an Sstand of 0.474 for a standard whose concentration of lead is1.75 ppb. How many parts per billion of Pb2+ occur in a sample of blood ifSsamp is 0.361?SstandEXAMPLE 5.2SOLUTIONEquation 5.3 allows us to calculate the value of k for this method using the datafor the standardk =0.474Sstand== 0.2709 ppb –11.75 ppbCSCA(a)CA =Ssamp0.361== 1.33 ppbk0.2709 ppb –1A multiple-point external standardization is accomplished by constructing acalibration curve, two examples of which are shown in Figure 5.3.
Since this isthe most frequently employed method of standardization, the resulting relationship often is called a normal calibration curve. When the calibration curve is alinear (Figure 5.3a), the slope of the line gives the value of k. This is the most desirable situation since the method’s sensitivity remains constant throughout thestandard’s concentration range.
When the calibration curve is nonlinear, themethod’s sensitivity is a function of the analyte’s concentration. In Figure 5.3b,for example, the value of k is greatest when the analyte’s concentration is smalland decreases continuously as the amount of analyte is increased. The value ofk at any point along the calibration curve is given by the slope at that point. In*Linear regression, also known as the method of least squares, is covered in Section 5C.SstandOnce k is known, the concentration of Pb2+ in the sample of blood can becalculated using equation 5.4CA(b)Figure 5.3Examples of (a) straight-line and (b) curvednormal calibration curves.normal calibration curveA calibration curve prepared usingseveral external standards.1400-CH05 9/8/99 3:59 PM Page 110110Modern Analytical Chemistryeither case, the calibration curve provides a means for relating Ssamp to the analyte’s concentration.EXAMPLE 5.3Colorplate 1 shows an example of a set ofexternal standards and their correspondingnormal calibration curve.A second spectrophotometric method for the quantitative determination ofPb2+ levels in blood gives a linear normal calibration curve for whichSstand = (0.296 ppb–1) × CS + 0.003What is the Pb2+ level (in ppb) in a sample of blood if Ssamp is 0.397?SOLUTIONTo determine the concentration of Pb2+ in the sample of blood, we replaceSstand in the calibration equation with Ssamp and solve for CACA =Ssamp – 0.0030.296ppb –1=0.397 – 0.003= 1.33 ppb0.296 ppb –1It is worth noting that the calibration equation in this problem includes anextra term that is not in equation 5.3.
Ideally, we expect the calibration curve togive a signal of zero when CS is zero. This is the purpose of using a reagentblank to correct the measured signal. The extra term of +0.003 in ourcalibration equation results from uncertainty in measuring the signal for thereagent blank and the standards.SignalAn external standardization allows a related series of samples to be analyzed using a single calibration curve.
This is an important advantage in laboratories where many samples are to be analyzed or when the need for a rapidCalibration curve obtainedin standard’s matrixthroughput of samples is critical. Not surprisingly, many of the most commonly encountered quantitative analytical methods are based on an externalCalibration curve obtainedstandardization.in sample’s matrixThere is a serious limitation, however, to an external standardization.The relationship between Sstand and CS in equation 5.3 is determined whenthe analyte is present in the external standard’s matrix. In using an external standardization, we assume that any difference between the matrix ofthe standards and the sample’s matrix has no effect on the value of k.
Aproportional determinate error is introduced when differences between thetwo matrices cannot be ignored. This is shown in Figure 5.4, where the reReportedActuallationship between the signal and the amount of analyte is shown for bothAmount of analytethe sample’s matrix and the standard’s matrix. In this example, using aFigure 5.4normal calibration curve results in a negative determinate error. WhenEffect of the sample’s matrix on a normalmatrix problems are expected, an effort is made to match the matrix of thecalibration curve.standards to that of the sample. This is known as matrix matching. Whenthe sample’s matrix is unknown, the matrix effect must be shown to be negligimatrix matchingble, or an alternative method of standardization must be used.
Both approachesAdjusting the matrix of an externalare discussed in the following sections.standard so that it is the same as thematrix of the samples to be analyzed.method of standard additionsA standardization in which aliquots of astandard solution are added to thesample.5B.4 Standard AdditionsThe complication of matching the matrix of the standards to that of the samplecan be avoided by conducting the standardization in the sample. This is knownas the method of standard additions. The simplest version of a standard addi-1400-CH05 9/8/99 3:59 PM Page 111Chapter 5 Calibrations, Standardizations, and Blank CorrectionsAdd Vo of CAAdd Vo of CADilute to Vf111Add VS of CSDilute to VfFigure 5.5Total concentrationof analyteCAVoVfIllustration showing the method of standardadditions in which separate aliquots ofsample are diluted to the same final volume.One aliquot of sample is spiked with aknown volume of a standard solution ofanalyte before diluting to the final volume.Total concentrationof analyteCAVoVS+ CSVfVftion is shown in Figure 5.5.
A volume, Vo, of sample is diluted to a final volume,Vf, and the signal, Ssamp is measured. A second identical aliquot of sample isspiked with a volume, Vs, of a standard solution for which the analyte’s concentration, CS, is known. The spiked sample is diluted to the same final volume andits signal, Sspike, is recorded. The following two equations relate Ssamp and Sspike tothe concentration of analyte, CA, in the original sampleSsamp = kCAVoVfVV Sspike = k C A o + C S s VfVf 5.55.6where the ratios Vo/Vf and Vs/Vf account for the dilution. As long as Vs is small relative to Vo, the effect of adding the standard to the sample’s matrix is insignificant,and the matrices of the sample and the spiked sample may be considered identical.Under these conditions the value of k is the same in equations 5.5 and 5.6.
Solvingboth equations for k and equating givesSsampSspike=C A (Vo /Vf ) C A (Vo /Vf ) + CS (Vs /Vf )5.7Equation 5.7 can be solved for the concentration of analyte in the original sample.aliquotA portion of a solution.1400-CH05 9/8/99 3:59 PM Page 112112Modern Analytical ChemistryEXAMPLE 5.4A third spectrophotometric method for the quantitative determination of theconcentration of Pb2+ in blood yields an Ssamp of 0.193 for a 1.00-mL sample ofblood that has been diluted to 5.00 mL. A second 1.00-mL sample is spikedwith 1.00 µL of a 1560-ppb Pb2+ standard and diluted to 5.00 mL, yielding anSspike of 0.419. Determine the concentration of Pb2+ in the original sample ofblood.SOLUTIONThe concentration of Pb2+ in the original sample of blood can be determinedby making appropriate substitutions into equation 5.7 and solving for CA.
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