Chapter 5. Standardizing Analytical. Methods. The American Chemical Society s Committee on Environmental Improvement defines.

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1 Chapter 5 Standardzng nalytcal Chapter Overvew Methods 5 nalytcal Standards 5B Calbratng the Sgnal (S total ) 5C Determnng the Senstvty (k ) 5D Lnear Regresson and Calbraton Curves 5E Compensatng for the Reagent Blank (S reag ) 5F Usng Excel and R for a Regresson nalyss 5G Key Terms 5H Chapter Summary 5I Problems 5J Solutons to Practce Exercses The mercan Chemcal Socety s Commttee on Envronmental Improvement defnes standardzaton as the process of determnng the relatonshp between the sgnal and the amount of analyte n a sample. 1 In Chapter 3 we defned ths relatonshp as S k n + S or S k C + S total reag total r eag where S total s the sgnal, n s the moles of analyte, C s the analyte s concentraton, k s the method s senstvty for the analyte, and S reag s the contrbuton to S total from sources other than the sample. To standardze a method we must determne values for k and S reag. Strateges for accomplshng ths are the subject of ths chapter. 1 CS Commttee on Envronmental Improvement Gudelnes for Data cquston and Data Qualty Evaluaton n Envronmental Chemstry, nal. Chem. 1980, 5,

2 154 nalytcal Chemstry.0 See Chapter 9 for a thorough dscusson of ttrmetrc methods of analyss. The base NaOH s an example of a secondary standard. Commercally avalable NaOH contans mpurtes of NaCl, Na CO 3, and Na SO 4, and readly absorbs H O from the atmosphere. To determne the concentraton of NaOH n a soluton, t s ttrated aganst a prmary standard weak acd, such as potassum hydrogen phthalate, KHC 8 H 4 O 4. 5 nalytcal Standards To standardze an analytcal method we use standards contanng known amounts of analyte. The accuracy of a standardzaton, therefore, depends on the qualty of the reagents and glassware used to prepare these standards. For example, n an acd base ttraton the stochometry of the acd base reacton defnes the relatonshp between the moles of analyte and the moles of ttrant. In turn, the moles of ttrant s the product of the ttrant s concentraton and the volume of ttrant needed to reach the equvalence pont. The accuracy of a ttrmetrc analyss, therefore, can be no better than the accuracy to whch we know the ttrant s concentraton. 5.1 Prmary and Secondary Standards We dvde analytcal standards nto two categores: prmary standards and secondary standards. prmary standard s a reagent for whch we can dspense an accurately known amount of analyte. For example, a g sample of K Cr O 7 contans moles of K Cr O 7. If we place ths sample n a 50-mL volumetrc flask and dlute to volume, the concentraton of the resultng soluton s M. prmary standard must have a known stochometry, a known purty (or assay), and t must be stable durng long-term storage. Because of the dffculty n establshng the degree of hydraton, even after dryng, a hydrated reagent usually s not a prmary standard. Reagents that do not meet these crtera are secondary standards. The concentraton of a secondary standard must be determned relatve to a prmary standard. Lsts of acceptable prmary standards are avalable. ppendx 8 provdes examples of some common prmary standards. 5. Other Reagents Preparng a standard often requres addtonal reagents that are not prmary standards or secondary standards. Preparng a standard soluton, for example, requres a sutable solvent, and addtonal reagents may be need to adjust the standard s matrx. These solvents and reagents are potental sources of addtonal analyte, whch, f not accounted for, produce a determnate error n the standardzaton. If avalable, reagent grade chemcals conformng to standards set by the mercan Chemcal Socety should be used. 3 The label on the bottle of a reagent grade chemcal (Fgure 5.1) lsts ether the lmts for specfc mpurtes, or provdes an assay for the mpurtes. We can mprove the qualty of a reagent grade chemcal by purfyng t, or by conductng a more accurate assay. s dscussed later n the chapter, we (a) Smth, B. W.; Parsons, M. L. J. Chem. Educ. 1973, 50, ; (b) Moody, J. R.; Greenburg, P. R.; Pratt, K. W.; Rans, T. C. nal. Chem. 1988, 60, Commttee on nalytcal Reagents, Reagent Chemcals, 8th ed., mercan Chemcal Socety: Washngton, D. C., 1993.

3 Chapter 5 Standardzng nalytcal Methods 155 can correct for contrbutons to S total from reagents used n an analyss by ncludng an approprate blank determnaton n the analytcal procedure. 5.3 Preparng Standard Solutons It s often necessary to prepare a seres of standards, each wth a dfferent concentraton of analyte. We can prepare these standards n two ways. If the range of concentratons s lmted to one or two orders of magntude, then each soluton s best prepared by transferrng a known mass or volume of the pure standard to a volumetrc flask and dlutng to volume. When workng wth larger ranges of concentraton, partcularly those extendng over more than three orders of magntude, standards are best prepared by a seral dluton from a sngle stock soluton. In a seral dluton we prepare the most concentrated standard and then dlute a porton of t to prepare the next most concentrated standard. Next, we dlute a porton of the second standard to prepare a thrd standard, contnung ths process untl all we have prepared all of our standards. Seral dlutons must be prepared wth extra care because an error n preparng one standard s passed on to all succeedng standards. (a) (b) Fgure 5.1 Examples of typcal packagng labels for reagent grade chemcals. Label (a) provdes the manufacturer s assay for the reagent, NaBr. Note that potassum s flagged wth an astersk (*) because ts assay exceeds the lmts establshed by the mercan Chemcal Socety (CS). Label (b) does not provde an assay for mpurtes, but ndcates that the reagent meets CS specfcatons. n assay for the reagent, NaHCO 3 s provded.

4 156 nalytcal Chemstry.0 See Secton D.1 to revew how an electronc balance works. Calbratng a balance s mportant, but t does not elmnate all sources of determnate error n measurng mass. See ppendx 9 for a dscusson of correctng for the buoyancy of ar. 5B Calbratng the Sgnal (S total ) The accuracy of our determnaton of k and S reag depends on how accurately we can measure the sgnal, S total. We measure sgnals usng equpment, such as glassware and balances, and nstrumentaton, such as spectrophotometers and ph meters. To mnmze determnate errors affectng the sgnal, we frst calbrate our equpment and nstrumentaton. We accomplsh the calbraton by measurng S total for a standard wth a known response of S, adjustng S total untl S total S Here are two examples of how we calbrate sgnals. Other examples are provded n later chapters focusng on specfc analytcal methods. When the sgnal s a measurement of mass, we determne S total usng an analytcal balance. To calbrate the balance s sgnal we use a reference weght that meets standards establshed by a governng agency, such as the Natonal Insttute for Standards and Technology or the mercan Socety for Testng and Materals. n electronc balance often ncludes an nternal calbraton weght for routne calbratons, as well as programs for calbratng wth external weghts. In ether case, the balance automatcally adjusts S total to match S. We also must calbrate our nstruments. For example, we can evaluate a spectrophotometer s accuracy by measurng the absorbance of a carefully prepared soluton of mg/l K Cr O 7 n M H SO 4, usng M H SO 4 as a reagent blank. 4 n absorbance of ± absorbance unts at a wavelength of nm ndcates that the spectrometer s sgnal s properly calbrated. Be sure to read and carefully follow the calbraton nstructons provded wth any nstrument you use. 5C Determnng the Senstvty (k ) To standardze an analytcal method we also must determne the value of k n equaton 5.1 or equaton 5.. S k n + S total reag 5.1 S k C + S total reag 5. In prncple, t should be possble to derve the value of k for any analytcal method by consderng the chemcal and physcal processes generatng the sgnal. Unfortunately, such calculatons are not feasble when we lack a suffcently developed theoretcal model of the physcal processes, or are not useful because of nondeal chemcal behavor. In such stuatons we must determne the value of k by analyzng one or more standard solutons, each contanng a known amount of analyte. In ths secton we consder 4 Ebel, S. Fresenus J. nal. Chem. 199, 34, 769.

5 Chapter 5 Standardzng nalytcal Methods 157 several approaches for determnng the value of k. For smplcty we wll assume that S reag has been accounted for by a proper reagent blank, allowng us to replace S total n equaton 5.1 and equaton 5. wth the analyte s sgnal, S. S S k n 5.3 k C 5.4 5C.1 Sngle-Pont versus Multple-Pont Standardzatons The smplest way to determne the value of k n equaton 5.4 s by a sngle-pont standardzaton n whch we measure the sgnal for a standard, S, contanng a known concentraton of analyte, C. Substtutng these values nto equaton 5.4 k S 5.5 C gves the value for k. Havng determned the value for k, we can calculate the concentraton of analyte n any sample by measurng ts sgnal, S samp, and calculatng C usng equaton 5.6. C Ssamp 5.6 k sngle-pont standardzaton s the least desrable method for standardzng a method. There are at least two reasons for ths. Frst, any error n our determnaton of k carres over nto our calculaton of C. Second, our expermental value for k s for a sngle concentraton of analyte. Extendng ths value of k to other concentratons of analyte requres us to assume a lnear relatonshp between the sgnal and the analyte s concentraton, an assumpton that often s not true. 5 Fgure 5. shows how assumng a constant value of k may lead to a determnate error n the analyte s concentraton. Despte these lmtatons, sngle-pont standardzatons fnd routne use when the expected range for the analyte s concentratons s small. Under these condtons t s often safe to assume that k s constant (although you should verfy ths assumpton expermentally). Ths s the case, for example, n clncal labs where many automated analyzers use only a sngle standard. The preferred approach to standardzng a method s to prepare a seres of standards, each contanng the analyte at a dfferent concentraton. Standards are chosen such that they bracket the expected range for the analyte s concentraton. multple-pont standardzaton should nclude at least three standards, although more are preferable. plot of S versus Equaton 5.3 and equaton 5.4 are essentally dentcal, dfferng only n whether we choose to express the amount of analyte n moles or as a concentraton. For the remander of ths chapter we wll lmt our treatment to equaton 5.4. You can extend ths treatment to equaton 5.3 by replacng C wth n. Lnear regresson, whch also s known as the method of least squares, s one such algorthm. Its use s covered n Secton 5D. 5 Cardone, M. J.; Palmero, P. J.; Sybrandt, L. B. nal. Chem. 1980, 5,

6 158 nalytcal Chemstry.0 assumed relatonshp actual relatonshp S samp Fgure 5. Example showng how a sngle-pont standardzaton leads to a determnate error n an analyte s reported concentraton f we ncorrectly assume that the value of k s constant. S C (C ) reported (C ) actual C s known as a calbraton curve. The exact standardzaton, or calbraton relatonshp s determned by an approprate curve-fttng algorthm. There are at least two advantages to a multple-pont standardzaton. Frst, although a determnate error n one standard ntroduces a determnate error nto the analyss, ts effect s mnmzed by the remanng standards. Second, by measurng the sgnal for several concentratons of analyte we no longer must assume that the value of k s ndependent of the analyte s concentraton. Constructng a calbraton curve smlar to the actual relatonshp n Fgure 5., s possble. ppendng the adjectve external to the noun standard mght strke you as odd at ths pont, as t seems reasonable to assume that standards and samples must be analyzed separately. s you wll soon learn, however, we can add standards to our samples and analyze them smultaneously. 5C. External Standards The most common method of standardzaton uses one or more external standards, each contanng a known concentraton of analyte. We call them external because we prepare and analyze the standards separate from the samples. Sngle External Standard quanttatve determnaton usng a sngle external standard was descrbed at the begnnng of ths secton, wth k gven by equaton 5.5. fter determnng the value of k, the concentraton of analyte, C, s calculated usng equaton 5.6. Example 5.1 spectrophotometrc method for the quanttatve analyss of Pb + n blood yelds an S of for a sngle standard whose concentraton of lead s 1.75 ppb What s the concentraton of Pb + n a sample of blood for whch S samp s 0.361?

7 Chapter 5 Standardzng nalytcal Methods 159 So l u t o n Equaton 5.5 allows us to calculate the value of k for ths method usng the data for the standard. k S ppb C 175. ppb Havng determned the value of k, the concentraton of Pb + n the sample of blood s calculated usng equaton 5.6. C Ssamp ppb -1 k ppb -1 Multple External Standards Fgure 5.3 shows a typcal multple-pont external standardzaton. The volumetrc flask on the left s a reagent blank and the remanng volumetrc flasks contan ncreasng concentratons of Cu +. Shown below the volumetrc flasks s the resultng calbraton curve. Because ths s the most common method of standardzaton the resultng relatonshp s called a normal calbraton curve. When a calbraton curve s a straght-lne, as t s n Fgure 5.3, the slope of the lne gves the value of k. Ths s the most desrable stuaton snce the method s senstvty remans constant throughout the analyte s concentraton range. When the calbraton curve s not a straght-lne, the S C (M) Fgure 5.3 Shown at the top s a reagent blank (far left) and a set of fve external standards for Cu + wth concentratons ncreasng from left to rght. Shown below the external standards s the resultng normal calbraton curve. The absorbance of each standard, S, s shown by the flled crcles.

8 160 nalytcal Chemstry.0 method s senstvty s a functon of the analyte s concentraton. In Fgure 5., for example, the value of k s greatest when the analyte s concentraton s small and decreases contnuously for hgher concentratons of analyte. The value of k at any pont along the calbraton curve n Fgure 5. s gven by the slope at that pont. In ether case, the calbraton curve provdes a means for relatng S samp to the analyte s concentraton. Example 5. second spectrophotometrc method for the quanttatve analyss of Pb + n blood has a normal calbraton curve for whch S -1 ( ppb ) C What s the concentraton of Pb + n a sample of blood f S samp s 0.397? So l u t o n To determne the concentraton of Pb + n the sample of blood we replace S n the calbraton equaton wth S samp and solve for C. C S samp ppb ppb ppb It s worth notng that the calbraton equaton n ths problem ncludes an extra term that does not appear n equaton 5.6. Ideally we expect the calbraton curve to have a sgnal of zero when C s zero. Ths s the purpose of usng a reagent blank to correct the measured sgnal. The extra term of n our calbraton equaton results from the uncertanty n measurng the sgnal for the reagent blank and the standards. The one-pont standardzaton n ths exercse uses data from the thrd volumetrc flask n Fgure 5.3. Practce Exercse 5.1 Fgure 5.3 shows a normal calbraton curve for the quanttatve analyss of Cu +. The equaton for the calbraton curve s S 9.59 M 1 C What s the concentraton of Cu + n a sample whose absorbance, S samp, s 0.114? Compare your answer to a one-pont standardzaton where a standard of M Cu + gves a sgnal of Clck here to revew your answer to ths exercse. n external standardzaton allows us to analyze a seres of samples usng a sngle calbraton curve. Ths s an mportant advantage when we have many samples to analyze. Not surprsngly, many of the most common quanttatve analytcal methods use an external standardzaton. There s a serous lmtaton, however, to an external standardzaton. When we determne the value of k usng equaton 5.5, the analyte s pres-

9 Chapter 5 Standardzng nalytcal Methods 161 S samp standard s matrx sample s matrx (C ) reported (C ) actual Fgure 5.4 Calbraton curves for an analyte n the standard s matrx and n the sample s matrx. If the matrx affects the value of k, as s the case here, then we ntroduce a determnate error nto our analyss f we use a normal calbraton curve. ent n the external standard s matrx, whch usually s a much smpler matrx than that of our samples. When usng an external standardzaton we assume that the matrx does not affect the value of k. If ths s not true, then we ntroduce a proportonal determnate error nto our analyss. Ths s not the case n Fgure 5.4, for nstance, where we show calbraton curves for the analyte n the sample s matrx and n the standard s matrx. In ths example, a calbraton curve usng external standards results n a negatve determnate error. If we expect that matrx effects are mportant, then we try to match the standard s matrx to that of the sample. Ths s known as matrx matchng. If we are unsure of the sample s matrx, then we must show that matrx effects are neglgble, or use an alternatve method of standardzaton. Both approaches are dscussed n the followng secton. The matrx for the external standards n Fgure 5.3, for example, s dlute ammona, whch s added because the Cu(NH 3 ) 4 + complex absorbs more strongly than Cu +. If we fal to add the same amount of ammona to our samples, then we wll ntroduce a proportonal determnate error nto our analyss. 5C.3 Standard ddtons We can avod the complcaton of matchng the matrx of the standards to the matrx of the sample by conductng the standardzaton n the sample. Ths s known as the method of standard addtons. Sngle Standard ddton The smplest verson of a standard addton s shown n Fgure 5.5. Frst we add a porton of the sample, V o, to a volumetrc flask, dlute t to volume, V f, and measure ts sgnal, S samp. Next, we add a second dentcal porton of sample to an equvalent volumetrc flask along wth a spke, V, of an external standard whose concentraton s C. fter dlutng the spked sample to the same fnal volume, we measure ts sgnal, S spke. The followng two equatons relate S samp and S spke to the concentraton of analyte, C, n the orgnal sample.

10 16 nalytcal Chemstry.0 add V o of C add V of C Fgure 5.5 Illustraton showng the method of standard addtons. The volumetrc flask on the left contans a porton of the sample, V o, and the volumetrc flask on the rght contans an dentcal porton of the sample and a spke, V, of a standard soluton of the analyte. Both flasks are dluted to the same fnal volume, V f. The concentraton of analyte n each flask s shown at the bottom of the fgure where C s the analyte s concentraton n the orgnal sample and C s the concentraton of analyte n the external standard. Concentraton of nalyte C dlute to V f Vo C V f V V o + C V V f f The ratos V o /V f and V /V f account for the dluton of the sample and the standard, respectvely. S k C V o 5.7 V samp f V V o S k C C spke f + V p V 5.8 f f s long as V s small relatve to V o, the effect of the standard s matrx on the sample s matrx s nsgnfcant. Under these condtons the value of k s the same n equaton 5.7 and equaton 5.8. Solvng both equatons for k and equatng gves S samp C V V o f C V V o f S spke + C whch we can solve for the concentraton of analyte, C, n the orgnal sample. Example 5.3 thrd spectrophotometrc method for the quanttatve analyss of Pb + n blood yelds an S samp of when a 1.00 ml sample of blood s dluted to 5.00 ml. second 1.00 ml sample of blood s spked wth 1.00 ml of a 1560-ppb Pb + external standard and dluted to 5.00 ml, yeldng an V V f 5.9

11 Chapter 5 Standardzng nalytcal Methods 163 S spke of What s the concentraton of Pb + n the orgnal sample of blood? So l u t o n We begn by makng approprate substtutons nto equaton 5.9 and solvng for C. Note that all volumes must be n the same unts; thus, we frst covert V from 1.00 ml to ml. C ml 500. ml C ml ml ml ppb 500. ml C 0. 00C ppb C ppb C 0.045C ppb C 1.33 ppb The concentraton of Pb + n the orgnal sample of blood s 1.33 ppb. It also s possble to make a standard addton drectly to the sample, measurng the sgnal both before and after the spke (Fgure 5.6). In ths case the fnal volume after the standard addton s V o + V and equaton 5.7, equaton 5.8, and equaton 5.9 become add V of C V o V o Concentraton of nalyte C C V o V o + V + C V o V + V Fgure 5.6 Illustraton showng an alternatve form of the method of standard addtons. In ths case we add a spke of the external standard drectly to the sample wthout any further adjust n the volume.

12 164 nalytcal Chemstry.0 S k C samp V V o S k C C spke f + V V p + V + V 5.10 o o S samp C C V o V o + V S spke + C V o V + V 5.11 Example 5.4 fourth spectrophotometrc method for the quanttatve analyss of Pb + n blood yelds an S samp of 0.71 for a 5.00 ml sample of blood. fter spkng the blood sample wth 5.00 ml of a 1560-ppb Pb + external standard, an S spke of s measured. What s the concentraton of Pb + n the orgnal sample of blood. V o + V 5.00 ml ml ml So l u t o n To determne the concentraton of Pb + n the orgnal sample of blood, we make approprate substtutons nto equaton 5.11 and solve for C C 500. ml ml C ppb ml ml C C ppb C ppb 1.546C C 1.33 ppb The concentraton of Pb + n the orgnal sample of blood s 1.33 ppb. Multple Standard ddtons We can adapt the sngle-pont standard addton nto a multple-pont standard addton by preparng a seres of samples contanng ncreasng amounts of the external standard. Fgure 5.7 shows two ways to plot a standard addton calbraton curve based on equaton 5.8. In Fgure 5.7a we plot S spke aganst the volume of the spkes, V. If k s constant, then the calbraton curve s a straght-lne. It s easy to show that the x-ntercept s equvalent to C V o /C.

13 Chapter 5 Standardzng nalytcal Methods 165 (a) y-ntercept k C V o V f S spke slope k C V f 0.10 (b) S spke V x-ntercept -C (ml) V o C y-ntercept k C V o V f x-ntercept -C V o V f C V Vf slope k (mg/l) Fgure 5.7 Shown at the top s a set of sx standard addtons for the determnaton of Mn +. The flask on the left s a 5.00 ml sample dluted to ml. The remanng flasks contan 5.00 ml of sample and, from left to rght, 1.00,.00, 3.00, 4.00, and 5.00 ml of an external standard of mg/l Mn +. Shown below are two ways to plot the standard addtons calbraton curve. The absorbance for each standard addton, S spke, s shown by the flled crcles. Example 5.5 Begnnng wth equaton 5.8 show that the equatons n Fgure 5.7a for the slope, the y-ntercept, and the x-ntercept are correct. So l u t o n We begn by rewrtng equaton 5.8 as S spke kcv kc o + V V V whch s n the form of the equaton for a straght-lne f Y y-ntercept + slope X f

14 166 nalytcal Chemstry.0 where Y s S spke and X s V. The slope of the lne, therefore, s k C /V f and the y-ntercept s k C V o /V f. The x-ntercept s the value of X when Y s zero, or kcv kc o 0 + x-ntercept V V f f kcv o V CV f x-ntercept kc C V f o Practce Exercse 5. Begnnng wth equaton 5.8 show that the equatons n Fgure 5.7b for the slope, the y-ntercept, and the x-ntercept are correct. Clck here to revew your answer to ths exercse. Because we know the volume of the orgnal sample, V o, and the concentraton of the external standard, C, we can calculate the analyte s concentratons from the x-ntercept of a multple-pont standard addtons. Example 5.6 ffth spectrophotometrc method for the quanttatve analyss of Pb + n blood uses a multple-pont standard addton based on equaton 5.8. The orgnal blood sample has a volume of 1.00 ml and the standard used for spkng the sample has a concentraton of 1560 ppb Pb +. ll samples were dluted to 5.00 ml before measurng the sgnal. calbraton curve of S spke versus V has the followng equaton S spke ml 1 V What s the concentraton of Pb + n the orgnal sample of blood. So l u t o n To fnd the x-ntercept we set S spke equal to zero ml 1 V Solvng for V, we obtan a value of ml for the x-ntercept. Substtutng the x-nterecpt s value nto the equaton from Fgure 5.7a 4 CV C 100. ml o ml C 1560 ppb and solvng for C gves the concentraton of Pb + n the blood sample as 1.33 ppb.

15 Chapter 5 Standardzng nalytcal Methods 167 Practce Exercse 5.3 Fgure 5.7 shows a standard addtons calbraton curve for the quanttatve analyss of Mn +. Each soluton contans 5.00 ml of the orgnal sample and ether 0, 1.00,.00, 3.00, 4.00, or 5.00 ml of a mg/l external standard of Mn +. ll standard addton samples were dluted to ml before readng the absorbance. The equaton for the calbraton curve n Fgure 5.7a s Snce we construct a standard addtons calbraton curve n the sample, we can not use the calbraton equaton for other samples. Each sample, therefore, requres ts own standard addtons calbraton curve. Ths s a serous drawback f you have many samples. For example, suppose you need to analyze 10 samples usng a three-pont calbraton curve. For a normal calbraton curve you need to analyze only 13 solutons (three standards and ten samples). If you use the method of standard addtons, however, you must analyze 30 solutons (each of the ten samples must be analyzed three tmes, once before spkng and after each of two spkes). Usng a Standard ddton to Identfy Matrx Effects We can use the method of standard addtons to valdate an external standardzaton when matrx matchng s not feasble. Frst, we prepare a normal calbraton curve of S versus C and determne the value of k from ts slope. Next, we prepare a standard addtons calbraton curve usng equaton 5.8, plottng the data as shown n Fgure 5.7b. The slope of ths standard addtons calbraton curve provdes an ndependent determnaton of k. If there s no sgnfcant dfference between the two values of k, then we can gnore the dfference between the sample s matrx and that of the external standards. When the values of k are sgnfcantly dfferent, then usng a normal calbraton curve ntroduces a proportonal determnate error. 5C.4 Internal Standards S V What s the concentraton of Mn + n ths sample? Compare your answer to the data n Fgure 5.7b, for whch the calbraton curve s S C (V /V f ) Clck here to revew your answer to ths exercse. To successfully use an external standardzaton or the method of standard addtons, we must be able to treat dentcally all samples and standards. When ths s not possble, the accuracy and precson of our standardzaton may suffer. For example, f our analyte s n a volatle solvent, then ts concentraton ncreases when we lose solvent to evaporaton. Suppose we

16 168 nalytcal Chemstry.0 have a sample and a standard wth dentcal concentratons of analyte and dentcal sgnals. If both experence the same proportonal loss of solvent then ther respectve concentratons of analyte and sgnals contnue to be dentcal. In effect, we can gnore evaporaton f the samples and standards experence an equvalent loss of solvent. If an dentcal standard and sample lose dfferent amounts of solvent, however, then ther respectve concentratons and sgnals wll no longer be equal. In ths case a smple external standardzaton or standard addton s not possble. We can stll complete a standardzaton f we reference the analyte s sgnal to a sgnal from another speces that we add to all samples and standards. The speces, whch we call an nternal standard, must be dfferent than the analyte. Because the analyte and the nternal standard n any sample or standard receve the same treatment, the rato of ther sgnals s unaffected by any lack of reproducblty n the procedure. If a soluton contans an analyte of concentraton C, and an nternal standard of concentraton, C IS, then the sgnals due to the analyte, S, and the nternal standard, S IS, are S k C S k C IS IS IS where k and k IS are the senstvtes for the analyte and nternal standard. Takng the rato of the two sgnals gves the fundamental equaton for an nternal standardzaton. S S IS kc K C 5.1 kc C IS Because K s a rato of the analyte s senstvty and the nternal standard s senstvty, t s not necessary to ndependently determne values for ether k or k IS. Sngle Internal Standard In a sngle-pont nternal standardzaton, we prepare a sngle standard contanng the analyte and the nternal standard, and use t to determne the value of K n equaton 5.1. IS C S IS K f p # f C S Havng standardzed the method, the analyte s concentraton s gven by C IS C S IS # f p K S IS p samp IS 5.13

17 Chapter 5 Standardzng nalytcal Methods 169 Example 5.7 sxth spectrophotometrc method for the quanttatve analyss of Pb + n blood uses Cu + as an nternal standard. standard contanng 1.75 ppb Pb + and.5 ppb Cu + yelds a rato of (S /S IS ) of.37. sample of blood s spked wth the same concentraton of Cu +, gvng a sgnal rato, (S /S IS ) samp, of Determne the concentraton of Pb + n the sample of blood. So l u t o n Equaton 5.13 allows us to calculate the value of K usng the data for the standard C IS p K f C S #. 5 ppb Cu ppb Cu + f p SIS ppb Pb + # ppb Pb + The concentraton of Pb +, therefore, s C C IS S #. 5 ppb Cu + f p # ppb Cu + K SIS samp ppb Cu ppb Pb + Multple Internal Standards sngle-pont nternal standardzaton has the same lmtatons as a snglepont normal calbraton. To construct an nternal standard calbraton curve we prepare a seres of standards, each contanng the same concentraton of nternal standard and a dfferent concentratons of analyte. Under these condtons a calbraton curve of (S /S IS ) versus C s lnear wth a slope of K/C IS. Example 5.8 seventh spectrophotometrc method for the quanttatve analyss of Pb + n blood gves a lnear nternal standards calbraton curve for whch S f S IS p -1 (. 11 ppb )# C lthough the usual practce s to prepare the standards so that each contans an dentcal amount of the nternal standard, ths s not a requrement. What s the ppb Pb + n a sample of blood f (S /S IS ) samp s.80? So l u t o n To determne the concentraton of Pb + n the sample of blood we replace (S /S IS ) n the calbraton equaton wth (S /S IS ) samp and solve for C.

18 170 nalytcal Chemstry.0 C S f p S IS samp.11 ppb ppb ppb The concentraton of Pb + n the sample of blood s 1.33 ppb. In some crcumstances t s not possble to prepare the standards so that each contans the same concentraton of nternal standard. Ths s the case, for example, when preparng samples by mass nstead of volume. We can stll prepare a calbraton curve, however, by plottng (S /S IS ) versus C / C IS, gvng a lnear calbraton curve wth a slope of K. 5D Lnear Regresson and Calbraton Curves In a sngle-pont external standardzaton we determne the value of k by measurng the sgnal for a sngle standard contanng a known concentraton of analyte. Usng ths value of k and the sgnal for our sample, we then calculate the concentraton of analyte n our sample (see Example 5.1). Wth only a sngle determnaton of k, a quanttatve analyss usng a sngle-pont external standardzaton s straghtforward. multple-pont standardzaton presents a more dffcult problem. Consder the data n Table 5.1 for a multple-pont external standardzaton. What s our best estmate of the relatonshp between S and C? It s temptng to treat ths data as fve separate sngle-pont standardzatons, determnng k for each standard, and reportng the mean value. Despte t smplcty, ths s not an approprate way to treat a multple-pont standardzaton. So why s t napproprate to calculate an average value for k as done n Table 5.1? In a sngle-pont standardzaton we assume that our reagent blank (the frst row n Table 5.1) corrects for all constant sources of determnate error. If ths s not the case, then the value of k from a sngle-pont standardzaton has a determnate error. Table 5. demonstrates how an Table 5.1 Data for a Hypothetcal Multple-Pont External Standardzaton C (arbtrary unts) S (arbtrary unts) k S / C mean value for k 1.5

19 Chapter 5 Standardzng nalytcal Methods 171 Table 5. Effect of a Constant Determnate Error on the Value of k From a Sngle- Pont Standardzaton S C k S / C (S ) e k (S ) e / C (wthout constant error) (actual) (wth constant error) (apparent) mean k (true) 1.00 mean k (apparent) 1.3 uncorrected constant error affects our determnaton of k. The frst three columns show the concentraton of analyte n the standards, C, the sgnal wthout any source of constant error, S, and the actual value of k for fve standards. s we expect, the value of k s the same for each standard. In the fourth column we add a constant determnate error of to the sgnals, (S ) e. The last column contans the correspondng apparent values of k. Note that we obtan a dfferent value of k for each standard and that all of the apparent k values are greater than the true value. How do we fnd the best estmate for the relatonshp between the sgnal and the concentraton of analyte n a multple-pont standardzaton? Fgure 5.8 shows the data n Table 5.1 plotted as a normal calbraton curve. lthough the data certanly appear to fall along a straght lne, the actual calbraton curve s not ntutvely obvous. The process of mathematcally determnng the best equaton for the calbraton curve s called lnear regresson S C Fgure 5.8 Normal calbraton curve for the hypothetcal multple-pont external standardzaton n Table 5.1.

20 17 nalytcal Chemstry.0 5D.1 Lnear Regresson of Straght Lne Calbraton Curves When a calbraton curve s a straght-lne, we represent t usng the followng mathematcal equaton y β + β x where y s the sgnal, S, and x s the analyte s concentraton, C. The constants b 0 and b 1 are, respectvely, the calbraton curve s expected y-ntercept and ts expected slope. Because of uncertanty n our measurements, the best we can do s to estmate values for b 0 and b 1, whch we represent as b 0 and b 1. The goal of a lnear regresson analyss s to determne the best estmates for b 0 and b 1. How we do ths depends on the uncertanty n our measurements. 5D. Unweghted Lnear Regresson wth Errors n y The most common approach to completng a lnear regresson for equaton 5.14 makes three assumptons: (1) that any dfference between our expermental data and the calculated regresson lne s the result of ndetermnate errors affectng y, () that ndetermnate errors affectng y are normally dstrbuted, and (3) that the ndetermnate errors n y are ndependent of the value of x. Because we assume that the ndetermnate errors are the same for all standards, each standard contrbutes equally n estmatng the slope and the y-ntercept. For ths reason the result s consdered an unweghted lnear regresson. The second assumpton s generally true because of the central lmt theorem, whch we consdered n Chapter 4. The valdty of the two remanng assumptons s less obvous and you should evaluate them before acceptng the results of a lnear regresson. In partcular the frst assumpton s always suspect snce there wll certanly be some ndetermnate errors affectng the values of x. When preparng a calbraton curve, however, t s not unusual for the uncertanty n the sgnal, S, to be sgnfcantly larger than that for the concentraton of analyte n the standards C. In such crcumstances the frst assumpton s usually reasonable. How a Lnear Regresson Works To understand the logc of an lnear regresson consder the example shown n Fgure 5.9, whch shows three data ponts and two possble straght-lnes that mght reasonably explan the data. How do we decde how well these straght-lnes fts the data, and how do we determne the best straghtlne? Let s focus on the sold lne n Fgure 5.9. The equaton for ths lne s ŷ b bx

21 Chapter 5 Standardzng nalytcal Methods 173 Fgure 5.9 Illustraton showng three data ponts and two possble straght-lnes that mght explan the data. The goal of a lnear regresson s to fnd the mathematcal model, n ths case a straght-lne, that best explans the data. where b 0 and b 1 are our estmates for the y-ntercept and the slope, and ŷ s our predcton for the expermental value of y for any value of x. Because we assume that all uncertanty s the result of ndetermnate errors affectng y, the dfference between y and ŷ for each data pont s the resdual error, r, n the our mathematcal model for a partcular value of x. r ( y yˆ ) Fgure 5.10 shows the resdual errors for the three data ponts. The smaller the total resdual error, R, whch we defne as R ( y yˆ ) 5.16 the better the ft between the straght-lne and the data. In a lnear regresson analyss, we seek values of b 0 and b 1 that gve the smallest total resdual error. ŷ 3 ŷ b 0 + bx 1 If you are readng ths aloud, you pronounce ŷ as y-hat. The reason for squarng the ndvdual resdual errors s to prevent postve resdual error from cancelng out negatve resdual errors. You have seen ths before n the equatons for the sample and populaton standard devatons. You also can see from ths equaton why a lnear regresson s sometmes called the method of least squares. r ( y yˆ ) y r ( y yˆ ) ŷ 1 ŷ r ( y yˆ ) y 3 y 1 Fgure 5.10 Illustraton showng the evaluaton of a lnear regresson n whch we assume that all uncertanty s the result of ndetermnate errors affectng y. The ponts n blue, y, are the orgnal data and the ponts n red, ŷ, are the predcted values from the regresson equaton, ŷ b 0 + bx 1.The smaller the total resdual error (equaton 5.16), the better the ft of the straght-lne to the data.

22 174 nalytcal Chemstry.0 Fndng the Slope and y-intercept lthough we wll not formally develop the mathematcal equatons for a lnear regresson analyss, you can fnd the dervatons n many standard statstcal texts. 6 The resultng equaton for the slope, b 1, s b 1 n x y x y n x x and the equaton for the y-ntercept, b 0, s 5.17 y b x 1 b n lthough equaton 5.17 and equaton 5.18 appear formdable, t s only necessary to evaluate the followng four summatons x y xy x See Secton 5F n ths chapter for detals on completng a lnear regresson analyss usng Excel and R. Equatons 5.17 and 5.18 are wrtten n terms of the general varables x and y. s you work through ths example, remember that x corresponds to C, and that y corresponds to S. Many calculators, spreadsheets, and other statstcal software packages are capable of performng a lnear regresson analyss based on ths model. To save tme and to avod tedous calculatons, learn how to use one of these tools. For llustratve purposes the necessary calculatons are shown n detal n the followng example. Example 5.9 Usng the data from Table 5.1, determne the relatonshp between S and C usng an unweghted lnear regresson. So l u t o n We begn by settng up a table to help us organze the calculaton. x y x y x ddng the values n each column gves x y xy x See, for example, Draper, N. R.; Smth, H. ppled Regresson nalyss, 3rd ed.; Wley: New York, 1998.

23 Chapter 5 Standardzng nalytcal Methods 175 Substtutng these values nto equaton 5.17 and equaton 5.18, we fnd that the slope and the y-ntercept are ( b 1. ) (.. ) ( ) ( ) ( ) b The relatonshp between the sgnal and the analyte, therefore, s S C For now we keep two decmal places to match the number of decmal places n the sgnal. The resultng calbraton curve s shown n Fgure Uncertanty n the Regresson nalyss s shown n Fgure 5.11, because of ndetermnate error affectng our sgnal, the regresson lne may not pass through the exact center of each data pont. The cumulatve devaton of our data from the regresson lne that s, the total resdual error s proportonal to the uncertanty n the regresson. We call ths uncertanty the standard devaton about the regresson, s r, whch s equal to s r ( y y ˆ ) n 5.19 Dd you notce the smlarty between the standard devaton about the regresson (equaton 5.19) and the standard devaton for a sample (equaton 4.1)? S C Fgure 5.11 Calbraton curve for the data n Table 5.1 and Example 5.9.

24 176 nalytcal Chemstry.0 where y s the th expermental value, and ŷ s the correspondng value predcted by the regresson lne n equaton Note that the denomnator of equaton 5.19 ndcates that our regresson analyss has n degrees of freedom we lose two degree of freedom because we use two parameters, the slope and the y-ntercept, to calculate ŷ. more useful representaton of the uncertanty n our regresson s to consder the effect of ndetermnate errors on the slope, b 1, and the y- ntercept, b 0, whch we express as standard devatons. s b1 ns n x x r sr ( x x ) 5.0 s b0 s x r n x x s r x ( ) n x x 5.1 We use these standard devatons to establsh confdence ntervals for the expected slope, b 1, and the expected y-ntercept, b 0 β 1 b 1 ± ts b 5. 1 β 0 b 0 ± ts b You mght contrast ths wth equaton 4.1 for the confdence nterval around a sample s mean value. s you work through ths example, remember that x corresponds to C, and that y corresponds to S. where we select t for a sgnfcance level of a and for n degrees of freedom. Note that equaton 5. and equaton 5.3 do not contan a factor of ( n) 1 because the confdence nterval s based on a sngle regresson lne. gan, many calculators, spreadsheets, and computer software packages provde the standard devatons and confdence ntervals for the slope and y-ntercept. Example 5.10 llustrates the calculatons. Example 5.10 Calculate the 95% confdence ntervals for the slope and y-ntercept from Example 5.9. So l u t o n We begn by calculatng the standard devaton about the regresson. To do ths we must calculate the predcted sgnals, ŷ, usng the slope and y ntercept from Example 5.9, and the squares of the resdual error, ( y yˆ ). Usng the last standard as an example, we fnd that the predcted sgnal s yˆ b + bx ( ) and that the square of the resdual error s

25 Chapter 5 Standardzng nalytcal Methods 177 ( y yˆ ) ( ) The followng table dsplays the results for all sx solutons. x y ŷ ( y yˆ ) ddng together the data n the last column gves the numerator of equaton 5.19 as The standard devaton about the regresson, therefore, s s r Next we calculate the standard devatons for the slope and the y-ntercept usng equaton 5.0 and equaton 5.1. The values for the summaton terms are from n Example 5.9. s b1 ns n x x r 6 ( ) ( ) ( 1 550) s b0 s x r ( ) n x x ( ) ( ) Fnally, the 95% confdence ntervals (a 0.05, 4 degrees of freedom) for the slope and y-ntercept are You can fnd values for t n ppendx 4. b ± ts b ± ( ) 10. 7±. 7 β b ± ts b ± ( ) 0. ± 08. β The standard devaton about the regresson, s r, suggests that the sgnal, S, s precse to one decmal place. For ths reason we report the slope and the y-ntercept to a sngle decmal place.

26 178 nalytcal Chemstry.0 Mnmzng Uncertanty n Calbraton Curves To mnmze the uncertanty n a calbraton curve s slope and y-ntercept, you should evenly space your standards over a wde range of analyte concentratons. close examnaton of equaton 5.0 and equaton 5.1 wll help you apprecate why ths s true. The denomnators of both equatons nclude the term ( x x). The larger the value of ths term whch you accomplsh by ncreasng the range of x around ts mean value the smaller the standard devatons n the slope and the y-ntercept. Furthermore, to mnmze the uncertanty n the y ntercept, t also helps to decrease the value of the term x n equaton 5.1, whch you accomplsh by ncludng standards for lower concentratons of the analyte. Obtanng the nalyte s Concentraton From a Regresson Equaton Equaton 5.5 s wrtten n terms of a calbraton experment. more general form of the equaton, wrtten n terms of x and y, s gven here. s x s 1 1 r + + b m n 1 ( Y y) ( b) ( x x ) close examnaton of equaton 5.5 should convnce you that the uncertanty n C s smallest when the sample s average sgnal, S samp, s equal to the average sgnal for the standards, S. When practcal, you should plan your calbraton curve so that S samp falls n the mddle of the calbraton curve. 1 Once we have our regresson equaton, t s easy to determne the concentraton of analyte n a sample. When usng a normal calbraton curve, for example, we measure the sgnal for our sample, S samp, and calculate the analyte s concentraton, C, usng the regresson equaton. C S b b samp What s less obvous s how to report a confdence nterval for C that expresses the uncertanty n our analyss. To calculate a confdence nterval we need to know the standard devaton n the analyte s concentraton, s C, whch s gven by the followng equaton s C ( ) s S S r 1 1 samp + + b1 m n ( b ) C C 1 ( ) 5.5 where m s the number of replcate used to establsh the sample s average sgnal ( S samp ), n s the number of calbraton standards, S s the average sgnal for the calbraton standards, and C and C are the ndvdual and mean concentratons for the calbraton standards. 7 Knowng the value of s C, the confdence nterval for the analyte s concentraton s µ C C ± tsc where m C s the expected value of C n the absence of determnate errors, and wth the value of t based on the desred level of confdence and n degrees of freedom. 7 (a) Mller, J. N. nalyst 1991, 116, 3 14; (b) Sharaf, M..; Illman, D. L.; Kowalsk, B. R. Chemometrcs, Wley-Interscence: New York, 1986, pp ; (c) nalytcal Methods Commttee Uncertantes n concentratons estmated from calbraton experments, MC Techncal Bref, March 006 (

27 Chapter 5 Standardzng nalytcal Methods 179 Example 5.11 Three replcate analyses for a sample contanng an unknown concentraton of analyte, yeld values for S samp of 9.3, 9.16 and Usng the results from Example 5.9 and Example 5.10, determne the analyte s concentraton, C, and ts 95% confdence nterval. So l u t o n The average sgnal, S samp, s 9.33, whch, usng equaton 5.4 and the slope and the y-ntercept from Example 5.9, gves the analyte s concentraton as S b C samp b To calculate the standard devaton for the analyte s concentraton we must ( ) determne the values for S and C C. The former s just the average sgnal for the calbraton standards, whch, usng the data n Table ( ) 5.1, s Calculatng C C looks formdable, but we can smplfy ts calculaton by recognzng that ths sum of squares term s the numerator n a standard devaton equaton; thus, ( C C s n ) ( C ) 1 ( ) where s C s the standard devaton for the concentraton of analyte n the calbraton standards. Usng the data n Table 5.1 we fnd that s C s and ( C C ) ( ) ( 6 1) Substtutng known values nto equaton 5.5 gves ( ) s C ( ) Fnally, the 95% confdence nterval for 4 degrees of freedom s ± 0. 41± ( ) 0. 41± µ C C tsc Fgure 5.1 shows the calbraton curve wth curves showng the 95% confdence nterval for C. You can fnd values for t n ppendx 4.

28 180 nalytcal Chemstry S 30 Fgure 5.1 Example of a normal calbraton curve wth a supermposed confdence nterval for the analyte s concentraton. The ponts n blue are the orgnal data from Table 5.1. The black lne s the normal calbraton curve as determned n Example 5.9. The red lnes show the 95% confdence nterval for C assumng a sngle determnaton of S samp C Practce Exercse 5.4 Fgure 5.3 shows a normal calbraton curve for the quanttatve analyss of Cu +. The data for the calbraton curve are shown here. [Cu + ] (M) bsorbance Complete a lnear regresson analyss for ths calbraton data, reportng the calbraton equaton and the 95% confdence nterval for the slope and the y-ntercept. If three replcate samples gve an S samp of 0.114, what s the concentraton of analyte n the sample and ts 95% confdence nterval? Clck here to revew your answer to ths exercse. In a standard addton we determne the analyte s concentraton by extrapolatng the calbraton curve to the x-ntercept. In ths case the value of C s and the standard devaton n C s C b x-ntercept 0 b 1

29 Chapter 5 Standardzng nalytcal Methods 181 s C ( ) s S r 1 + b1 n ( b ) C C 1 ( ) where n s the number of standard addtons (ncludng the sample wth no added standard), and S s the average sgnal for the n standards. Because we determne the analyte s concentraton by extrapolaton, rather than by nterpolaton, s C for the method of standard addtons generally s larger than for a normal calbraton curve. Evaluatng a Lnear Regresson Model You should never accept the result of a lnear regresson analyss wthout evaluatng the valdty of the your model. Perhaps the smplest way to evaluate a regresson analyss s to examne the resdual errors. s we saw earler, the resdual error for a sngle calbraton standard, r, s r ( y yˆ ) If your regresson model s vald, then the resdual errors should be randomly dstrbuted about an average resdual error of zero, wth no apparent trend toward ether smaller or larger resdual errors (Fgure 5.13a). Trends such as those shown n Fgure 5.13b and Fgure 5.13c provde evdence that at least one of the model s assumptons s ncorrect. For example, a trend toward larger resdual errors at hgher concentratons, as shown n Fgure 5.13b, suggests that the ndetermnate errors affectng the sgnal are not ndependent of the analyte s concentraton. In Fgure 5.13c, the resdual (a) (b) (c) resdual error resdual error resdual error C C C Fgure 5.13 Plot of the resdual error n the sgnal, S, as a functon of the concentraton of analyte, C for an unweghted straght-lne regresson model. The red lne shows a resdual error of zero. The dstrbuton of the resdual error n (a) ndcates that the unweghted lnear regresson model s approprate. The ncrease n the resdual errors n (b) for hgher concentratons of analyte, suggest that a weghted straght-lne regresson s more approprate. For (c), the curved pattern to the resduals suggests that a straght-lne model s napproprate; lnear regresson usng a quadratc model mght produce a better ft.