In the Laboratory
Two Experiments Illustrating the Importance of Sampling in a Quantitative Chemical Analysis
W
David Harvey Department of Chemistry, DePauw University, Greencastle, IN 46135;
[email protected] 2 , and that due to the that due to sample preparation, σ prep 2 measurements, σ meas.
The goal of a quantitative analysis is to determine the amount of analyte in a sample with accuracy and precision. In the traditional quantitative analysis laboratory course, where samples are well characterized and homogeneous, it is easy for a student to evaluate his or her success in achieving these goals. In particular, students recognize that an unusually large variance occurs when they fail to reproducibly carry out the steps of an analysis, and that results that are consistently too high or too low indicate the presence of a significant determinate error. For example, when determining the amount of CO32᎑ in an unknown by an acid–base titration, a student might assign a positive determinate error to over-titrating the end point, or recognize that failing to titrate to a consistent end point color decreases the precision of the analysis. Many instructors of quantitative analysis courses now emphasize the analysis of more realistic samples (so-called “real-world” samples), often as part of an open-ended assignment. For example, Williams describes an analytical lab in which students collect and interpret forensic data for use in a mock trial (1). Including the use of more realistic samples, however, introduces additional sources of error, complicating the evaluation of accuracy and precision. A student who obtains an unexpected result can no longer assume that he or she made a determinate error in following the experiment’s procedure, but must also consider whether the analyzed samples were representative. An unusually large variance may reflect poor reproducibility in the student’s analytical technique, but may also result from a significant variation in the amount of analyte in different samples. Analytical chemists know that the sources of variance in 2 an analysis are additive; thus, the total variance, σ total , for an 2 , analysis can be partitioned into that due to sampling, σ samp
2 2 2 2 σ total = σ samp + σ prep + σ meas
(1)
If desired, individual components of the total variance often can be partitioned into smaller parts. In a spectrophotometric analysis, for example, the measurement variance can be partitioned into the variance due to the spectrometer’s source, 2 , and the variance due to the posidetector, and optics, σ spect 2 (2). tioning of the sample cell within the spectrometer, σ pos In this case, eq 1 becomes 2 2 2 2 2 σ total = σ samp + σ prep + σ pos + σ spect
(2)
The importance of sampling and sampling variance is addressed in current analytical textbooks (3–6 ), and several classroom demonstrations (7–10) and laboratory experiments (11–17) have been published. Only three of the experiments address the partitioning of the total variance into its component parts. Kratochvil et al. proposed an experiment in which students determine the relationship between the sampling variance and the concentration of KHP in a mixture of KHP and sucrose (13). The experiment, however, is lengthy, and the authors suggest 18 acid–base back-titrations. Guy et al. modified this experiment, replacing the titrations with a flow-injection analysis (15). In addition to determining the sampling variance, students verify the following relationship between sampling variance and sample size: mR 2 = Ks
(3)
where m is the sample’s mass, R is the percent relative standard deviation due to sampling, and Ks is Ingamells’s sampling
Gross Sample Level I
Level II
Level III
II
I
IA
IB
IA1
IIA
IIA1
IB1 IA2
IB2
IB1a
IA1a IA1b
IIB1 IIA2
IIB2
IIA1b
IIIA2
IVB1 IVA2
IIIB2
IIIB1a IIIA1b
IVB
IVA1
IIIB1
IIIA1a IIB1b
IVA
IIIB
IIIA1
IIB1a
IIA1a IB1b
IIIA
IIB
Figure 1. Four-level nested design showing the relationship between samples at Levels I, II, III, and IV. Samples are coded using a Roman numeral for Level I, an uppercase A or B for Level II, the number 1 or 2 for Level III, and a lower case a or b for Level 4.
IV
III
IVA1a IIIB1b
IVA1b
IVB2
IVB1a IVB1b
Level IV IA2a
360
IIA2a
IB2a IA2b
IB2b
IIB2a IIA2b
IIIA2a IIB2b
IVA2a
IIIB2a IIIA2b
IIIB2b
IVB2a IVA2b
IVB2b
Journal of Chemical Education • Vol. 79 No. 3 March 2002 • JChemEd.chem.wisc.edu
In the Laboratory
constant (18). For well-mixed samples, Ks is equivalent to the mass of sample giving a percent relative standard deviation due to sampling of 1%. Although this experiment is more manageable in a single laboratory period, many undergraduate institutions do not have access to the necessary instrumentation. In determining the salt content of snack foods by titrating Cl᎑ with AgNO3, Settle and Pleva use a three-level nested design to obtain the relative importance of the variances due to sampling, sample preparation, and the titrations (17 ). Again, the analysis is lengthy, requiring 16 titrations. Furthermore, the absence of an accurately known true value for the salt content makes an evaluation of accuracy more difficult. The experiments described here are modifications to the experiments of Settle and Pleva, and Guy et al. The gross sample is a nominally 0.1–0.2% w/w mixture of the acid– base indicator erythrosin B (Sigma) and crystalline NaCl (unsieved), prepared by placing appropriate masses in a glass jar and shaking to achieve a uniform mixture in which the erythrosin B adheres to the salt crystals. Samples are collected
according to an appropriate sampling plan and diluted to volume in volumetric flasks. Because erythrosin B is in its base form for all pH levels greater than 3, dilutions can be made using distilled water instead of a buffer, which simplifies sample preparation. The concentration of erythrosin B is determined spectrophotometrically at a wavelength of 526 nm, providing a rapid analysis using commonly available instrumentation. Although these experiments use erythrosin B, other acid–base indicators can be used provided that the indicator is water soluble and has a pKa significantly removed from the pH of distilled water. An experiment illustrating the importance of sampling is most effective when it provides an unexpected result. At first glance, the sample appears to be uniform and students are surprised to discover that sampling is the largest source of variance. This discrepancy between a student’s expectation that sampling uncertainty is insignificant and experimental evidence to the contrary helps emphasize the importance of sampling.
Table 1. Typical Results for Analysis of a 0.116 wt % Mixture of Er ythrosin B and NaCl Using a Four-Level Nested Design Sample
Level IV Wt % EBa
IA1a
0.1232
IA1b
0.1234
IA2a
0.1236
IA2b
0.1238
IB1a
0.1288
IB1b
0.1288
IB2a
0.1290
IB2b
0.1290
IIA1a
0.1423
IIA1b
0.1423
IIA2a
0.1427
IIA2b
0.1427
IIB1a
0.1445
IIB1b
0.1447
IIB2a
0.1447
IIB2b IIIA1a
0.1447 0.1163
IIIA1b
0.1161
IIIA2a
0.1165
IIIA2b
0.1165
IIIB1a
0.1106
IIIB1b
0.1106
IIIB2a
0.1104
IIIB2b
0.1106
IVA1a
0.0938
IVA1b
0.0936
IVA2a
0.0936
IVA2b
0.0938
IVB1a
0.0943
IVB1b
0.0943
IVB2a
0.0941
IVB2b
0.0941
Level III dIV
Wt % EBa
᎑0.0002
0.1233
᎑0.0002
0.1237
0.0000
0.1288
0.0000
0.1290
0.0000
0.1423
0.0000
0.1427
᎑0.0002
0.1446
0.0000
0.1447
0.0002
0.1162
0.0000
0.1165
0.0000
0.1106
᎑0.0002
0.1105
0.0002
0.0937
᎑0.0002
0.0937
0.0000
0.0943
0.0000
0.0941
Level II dIII
Wt % EBa
᎑0.0004
0.1235
᎑0.0002
0.1289
᎑0.0004
0.1425
᎑0.0001
0.1447
᎑0.0003
0.1164
0.0001
0.1105
0.0000
0.0937
0.0002
Level I dII
Wt % EBa
᎑0.0054
0.1262
᎑0.0021
0.1436
0.0059
0.1135
᎑0.0005
0.0939
0.0942
aEB is abbreviation for erythrosin B. The results are shown with an extra significant figure to aid in calculating the wt % EB for the next higher level.
JChemEd.chem.wisc.edu • Vol. 79 No. 3 March 2002 • Journal of Chemical Education
361
In the Laboratory
2 =s2 = s IV spect
Σi
d IV
(4)
8n
2 s spect
2
=
Σi
d III 2i
=
s2 2 + pos s prep 2
+
2 s spect
4
=
Σi
d II 2n
2 i
(6)
where dII is the difference between related Level II samples (e.g., IA and IB). The factors of 2 and 4 in the terms for the variances due to the sample cell’s positioning and the spectrometer, respectively, account for the two Level III samples and the four Level IV samples used to determine the result for each Level II sample. Finally, the variance for Level I, sI2, is determined using the standard equation for the variance. It includes contributions
362
Av Mass/ ga
Av Wt % EBa,b
2 s total
ssamp
Ks
0.10
10
0.110
0.101
3.72 × 10᎑4
0.0193
40.3
0.25
25
0.258
0.092
3.57 × 10᎑4
0.0189 108.1
0.50
50
0.486
0.103
1.44 × 10᎑4
0.0120
66.1
1.00
100
0.997
0.092
6.08 × 10᎑5
0.0078
71.0
2.50
250
2.508
0.092
3.36 × 10᎑5
0.0058 100.1
aAverage bEB
of six replicates. is abbreviation for erythrosin B.
0.14 0.13 0.12 0.11 0.10 0.09 0.08 0.07 0.06 1
2
3
Mass of Sample / g Figure 2. Results used to compile the information in Table 2. The solid lines represent boundaries of one standard deviation about the overall mean of 0.096% w/w using an average Ks of 77.
from sampling, sample preparation, the positioning of the sample cell, and the spectrometer:
(5)
4n
where dIII is the difference between related Level III samples (e.g., IA1 and IA2). The factor of 2 in the term for the spectrometer’s variance accounts for the two Level IV samples used to determine the result for each Level III sample. The variance for the Level II samples, sII2 , includes contributions from the spectrometer, the positioning of the sample cell, and sample preparation; thus
s II2
Volumetric Flask/ mL
Nominal Mass/g
0
2 i
where n is the number of Level I samples (4 in this case). 2 The variance for the Level III samples, sIII , includes contributions from both the spectrometer and the positioning of the sample cell; thus 2 =s2 + s III pos
Table 2. Typical Results for Determination of Ingamells’s Sampling Constant for a 0.095 Wt % Mixture of Er ythrosin B and NaCl
Weight Percent Erythrosin B
Experiment 1. Finding the Weakest Link Using a Nested Design This experiment, which is adapted from that of Settle and Pleva (17 ), uses a four-level nested design to determine the variances due to sampling, sample preparation, the spectrometer, and the sample cell’s positioning in the spectrometer (Fig. 1). The first level consists of four samples collected randomly from the gross sample. After grinding the Level I samples, duplicate samples are obtained from each and diluted to volume in volumetric flasks, providing the eight Level II samples. Each Level II sample is divided in half, yielding the 16 Level III samples. Finally, each Level III sample is placed in the spectrometer and its absorbance is measured twice without repositioning the sample, providing the 32 Level IV samples. Only the 32 Level IV samples are analyzed spectrophotometrically and the wt % of erythrosin B is calculated for each using Beer’s law with an absorptivity of 0.0916 ppm᎑1 cm᎑1. The wt % of erythrosin B for each Level I–III sample is the average result for the corresponding Level IV samples. The result for sample IA, for example, is the average result for samples IA1a, IA1b, IA2a, and IA2b. Typical results are shown in Table 1. 2 , which is The variance for the Level IV samples, sIV equivalent to the spectrometer’s variance, is determined from the differences, d IV, between related Level IV samples (e.g., IA1a and IA1b),
s I2
=
2 s prep 2 s samp +
2
+
2 s pos
4
+
2 s spect
8
=
Σi
Xi – X
2
(7)
n –1 –
where Xi is the result for each Level I sample and X is the average result for all Level I samples. The factors of 2, 4, and 8 in the terms for the variances due to sample preparation, the sample cell’s positioning, and the spectrometer, respectively, account for the two Level II samples, four Level III samples, and eight Level IV samples used to determine the result for each Level I sample. A typical set of results is shown in Table 1 for the analysis of a sample that is 0.116 wt % in erythrosin B. Using these 2 2 results gives σ spect = 9.65 × 10᎑9 (df = 16), σ pos = 3.18 × 10᎑8 2 ᎑6 2 (df = 8), σ prep = 8.48 × 10 (df = 4), and σ samp = 4.34 × 10᎑4 2 (df = 3). A one-tailed F test at α = .05 shows that σ samp is 2 significantly greater than σ prep (Fexp = 51.21 vs Fcrit = 6.591), 2 2 σ prep is significantly greater than σ pos (Fexp = 266.6 vs Fcrit = 2 2 3.838), and σ pos is significantly greater than σ spect (Fexp = 3.296 vs Fcrit = 2.591). Clearly sampling is the weakest link. Students can also evaluate the accuracy of their analysis. Using the data for the eight Level II samples gives a 95%
Journal of Chemical Education • Vol. 79 No. 3 March 2002 • JChemEd.chem.wisc.edu
In the Laboratory
confidence interval of 0.119 ± 0.018 wt % erythrosin B, which contains the gross sample’s expected value of 0.116 wt % erythrosin B. Experiment 2. Evaluating the Sampling Constant This experiment, which is adapted from that of Guy et al. (15), evaluates the relationship between sampling variance and sample size. Six replicate samples of the gross sample are prepared for each condition (nominal mass and volumetric flask) listed in Table 2. Samples are analyzed spectrophotometrically and the wt % erythrosin B calculated. A typical set of results is shown in Table 2 and Figure 2 for a gross sample that is 0.095 wt % erythrosin B. The variance for each set of nominal masses contains contributions from sampling, sample preparation, and the absorbance measurements. Students who completed the previous experiment will recognize that the analysis is dominated by sampling uncertainty and that the experimentally deter2 mined total variance provides a good estimate for σ samp . As noted by Guy et al. (15), students may verify this by using a 2 2 propagation of error to estimate σ prep and estimating σ meas by measuring the absorbance of any sample several times. Values of Ks are determined using eq 3, which gives an average value of 77. The solid curves in Figure 2 represent boundaries of one standard deviation, calculated using eq 3 and the average value for Ks, around the overall mean concentration of 0.096 wt % erythrosin B. Students can also evaluate the accuracy of their analysis for any sample size. Using data for the largest samples, for example, gives a 95% confidence interval of 0.092 ± 0.007 wt % erythrosin B, which contains the gross sample’s expected value of 0.095 wt % erythrosin B. Hazards There are no specific hazards associated with this experiment, although students should exercise appropriate caution when working with any chemicals. Erythrosin B, also known as Acid Red 51 and FD&C Red No. 3, is approved by the FDA for use in foods.
Acknowledgments The assistance of Devon Harvey in collecting data for these experiments is gratefully acknowledged. An anonymous reviewer suggested the use of the four-level nested design for the weakest-link experiment. W
Supplemental Material
Copies of laboratory handouts, notes for the instructor, and sample data sets are available in this issue of JCE Online. Literature Cited 1. Williams, T. R. Am. Lab. 2000, 32 (6), 20–24. 2. Ingle, J. R., Jr.; Crouch, S. R. Spectrochemical Analysis; Prentice Hall: Englewood Cliffs, NJ, 1988; pp 150–154. 3. Harris, D. C. Quantitative Chemical Analysis, 5th ed.; Freeman: New York, 1999. 4. Skoog, D. A.; West, D. M.; Holler, F. J. Fundamentals of Analytical Chemistry, 7th ed.; Saunders: Philadelphia, 1996. 5. Rubinson, J. F.; Rubinson, K. A. Contemporary Chemical Analysis; Prentice-Hall: Upper Saddle River, NJ, 1998. 6. Harvey, D. T. Modern Analytical Chemistry; McGraw-Hill: Boston, 2000. 7. Bauer, C. F. J. Chem. Educ. 1985, 62, 252. 8. Clement, R. E. Anal. Chem. 1992, 64, 1076A–1081A. 9. Hartman, J. R. J. Chem. Educ. 2000, 77, 1017–1018. 10. Ross, M. R. J. Chem. Educ. 2000, 77, 1015–1016. 11. Herrington, B. L. J. Chem. Educ. 1937, 14, 544. 12. Bishop, J. A. J. Chem. Educ. 1958, 35, 31. 13. Kratochvil, B.; Reid, R. S; Harris, W. E. J. Chem. Educ. 1980, 57, 518–520. 14. Dunn, J. G.; Phillips, D. N.; van Bronswijk, W. J. Chem. Educ. 1997, 74, 1188–1190. 15. Guy, R. D.; Ramaley, L.; Wentzell, P. D. J. Chem. Educ. 1998, 75, 1028–1033. 16. Vitt, J. G.; Engstrom, R. C. J. Chem. Educ. 1999, 76, 99– 100. 17. Settle, F. A.; Pleva, M. Anal. Chem. 1999, 71, 538A–540A. 18. Ingamells, C. O.; Switzer, P. Talanta 1973, 20, 547–568.
JChemEd.chem.wisc.edu • Vol. 79 No. 3 March 2002 • Journal of Chemical Education
363
