In the Laboratory
An Experiment in the Sampling of Solids for Chemical Analysis Robert D. Guy, Louis Ramaley, and Peter D. Wentzell* Trace Analysis Research Centre, Department of Chemistry, Dalhousie University, Halifax, NS, B3H 4J3 Canada
In the course of their undergraduate training, most chemistry students will be exposed to a wide variety of chemical analysis techniques, both classical and instrumental. In the vast majority of cases, these will involve the use of homogenous liquid or solid samples. Because of this, the contribution of the sampling process to the overall analytical uncertainty is usually ignored, even though errors introduced by the sampling of “real world” materials, such as ores, pharmaceutical preparations, and food products, can be considerable. While many students will advance to careers that involve such practical problems, it is rare for them to have experienced the basic principles of sampling in the analytical laboratory. For some time we have been attempting to develop an undergraduate experiment that would adequately illustrate the basic principles of the uncertainty associated with the sampling of solids. A fundamental difficulty in this endeavor is that it requires an examination of the systematic trends in variance, which itself has a high degree of variability when a limited number of measurements are made. One way around this problem is to employ simple models of the solid sample, such as that described by Clement (1), which uses colored candies to represent solid particles. A similar strategy, employing colored cork stoppers, was described as a lecture demonstration by Bauer (2). Although this approach is useful for illustrating the features of the binomial distribution, students do not always see the relevance for real chemical samples. As an alternative, a laboratory experiment to introduce the role of sampling uncertainty in chemical analysis was described by Kratochvil et al. (3). This experiment involved the titration of mixtures of potassium hydrogen phthalate (KHP) and sugar with sodium hydroxide. Although carefully designed to permit the calculation of sampling variance, the procedure is experimentally intensive, requiring sieving and thorough mixing of the components and 18 titrations (not including standardizations); and a back-titration is recommended in each case for high precision. Good precision was required because of the high analyte concentration (25 to 70% KHP), which tends to reduce sampling errors. Furthermore, the sampling variance is calculated for only one mixture, making it difficult for students to directly observe the effects of sample size. The experiment presented here was developed to show students taking the analytical chemistry laboratory at the third-year level the limitations imposed by sampling in a real chemical analysis problem. The requirements of the analytical procedure were (i) it should be capable of determining a small amount of analyte in the presence of a large amount of concomitant material so that the errors introduced by sampling are significant, and (ii) it should permit the analysis of a large number of samples in one laboratory period so that an adequate statistical analysis can be performed. To satisfy the first
requirement, the sensitive molybdenum-blue method was employed for the determination of phosphate in the presence of sodium chloride. To facilitate rapid analysis, the method was implemented as a flow injection analysis (FIA) technique (4), which permits the handling of more than 100 samples in a typical four-hour laboratory period. From a pedagogical perspective, several important objectives are met through this experiment: 1. Students are taught the importance of sampling as a potential source of imprecision. 2. The utility of a semi-automated analytical procedure (FIA) is demonstrated. 3. Multiple sources of variance in an experiment are identified and quantified. 4. Propagation of error is used to determine the variance of sample preparation. 5. Students are exposed to the use of spreadsheets for the analysis of large amounts of data.
Basic Principles Figure 1 shows a breakdown of the overall process of chemical analysis and some of the sources of uncertainty. In the first step, the “gross sample” is obtained from the bulk population (e.g., the ore on a railway car or soil in a field). This step is critical in that it can determine the validity of the interpretation of the chemical analysis, but the proper procedure depends on the nature of the problem and is often outside the control of the chemist. For these reasons, this step will not be addressed here. Once the sample is in the lab, it may be preprocessed (e.g., grinding, mixing) and is then further divided into “subsamples” or “laboratory samples”. Variance introduced at this stage should technically be referred to as “subsampling error”, but most analytical chemists will simply call it “sampling error” because, from the perspective of the laboratory, the gross sample represents the “population” to be analyzed. (This view is not unusual, since in the overall process there can be multiple subsampling steps. For example, a railway car filled with ore can be considered to be a sample of the mine, which in turn is a sample of the planet’s crust.) We will adopt this convenient, if somewhat inaccurate, convention throughout this paper. The analysis of variance is a widely studied problem in statistics and, in this experiment, the random sources of variance are studied using a nested (or hierarchical) design (5). The total variance in the analysis of the laboratory sample involves errors introduced at every step: sampling, sample preparation, and measurement. Mathematically, we have σtotal2 = σsamp2 + σprep2 + σmeas2
where σ2 represents the variance. Obviously, to determine the
*Corresponding author.
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(1)
Journal of Chemical Education • Vol. 75 No. 8 August 1998 • JChemEd.chem.wisc.edu
In the Laboratory
σ
σ
the same size distribution, often a reasonable assumption. Detailed treatments of more general cases where such assumptions don’t hold can be found in the literature (see, for example, ref 19). Equation 3 is not necessarily valid in these cases, but is often a convenient working approximation. In practice, Ks can serve as a useful sampling guide for a given system, but accurate estimation of this value is difficult without performing a very large number of measurements. In this experiment, students are asked to obtain an approximate value for Ks for the mixture they are examining.
σ
The Experiment
Chemicals All chemicals used in this experiment are analytical reagent grade or better and can be obtained from a variety of commercial suppliers. Students are provided with the following solutions:
Figure 1. Sources of uncertainty in a chemical analysis.
contribution of sampling to the overall variance, the other sources of variance have to be taken into account. The measurement variance can be estimated simply by running replicate measurements on the same solution. Likewise, if the sample preparation step is relatively simple, the variance of sample preparation can be approximated by error propagation. If the overall variance is known from replicate samples, the sampling variance is easily obtained by subtraction. The theory relating the physical characteristics of the solid material to the expected sampling variance (or vice versa) can become quite complex and a comprehensive discussion is beyond the scope of this article. The interested reader is referred to a number of books and articles on the subject (6–20). In one of the simplest models, the solid can be considered to be a binary mixture of two types of particles, which contain the analyte at different fixed weight percentages, PA and PB. This model is often a reasonable approximation to real systems (e.g., a pharmaceutical preparation that contains particles of active ingredient and filler). If we make the further (often unrealistic) assumption that all the particles are identical in size, the binomial distribution leads directly to the Benedetti-Pichler equation (6 ):
R=
σP 100 P A – P B d d = × A B 2 P P d
p(1 – p) n
(2)
where R is the relative standard deviation of sampling (in percent), P is the overall percentage of analyte in the mixture, dA and dB are the densities of particles A and B, d¯ is the density of the mixture, p is the fraction of type A particles in the mixture, and n is the number of particles in the sample. If we make the (reasonable) approximation that the mass of the sample, m, is directly proportional to the number of particles and assume that, for a given mixture, all other quantities remain fixed, we can write mR 2 = Ks
Molybdate reagent: 0.005 M ammonium molybdate ((NH4)6Mo2O24 ⴢ 4H2O) in 0.40 M HNO3 (1 L). Ascorbic acid reagent: 0.7% ascorbic acid in 1% glycerin (1 L). Glycerin is used as a surfactant in the FIA system. Phosphate stock solution: 100 ppm phosphate prepared from KH2PO 4 (100 mL). Phosphate unknown: Between 0.40 and 0.64 g of solid KH2PO 4 is added to about 80 g of solid NaCl.
Students are instructed to prepare phosphate standards of 10, 20, 30, 40, and 60 ppm PO43᎑ from the stock solution in 25-mL volumetric flasks. For this experiment, a graduated pipet provides sufficient accuracy and precision.
FIA System To facilitate rapid analysis of the prepared solutions, the FIA system shown in Figure 2 was used. A peristaltic pump (Ismatec MS Reglo model 7331-10, Cole-Parmer, Chicago, IL) with standard Tygon pump tubing was used to transfer reagent solutions through the 0.8-mm-i.d. PTFE (Teflon) tubing making up the rest of the manifold (individual flow rates ca. 0.5 mL/min). Sample solutions containing phosphate were injected into the reagent stream with a six-port, twoway, low-pressure sample injection valve (Rheodyne model 5020, Cotati, CA) with a 50-µ L sample loop. After reacting for a sufficient time in a mixing coil (ca. 50 cm), the product was detected at 650 nm using a simple spectrophotometer (Turner model 330, Palo Alto, CA) equipped with a flow cell. The output of the spectrophotometer was connected to a data acquisition computer, but a chart recorder could also be used. The color-forming reaction for the molybdenum blue method
(3)
where Ks is called Ingamells’ sampling constant (19). Ks can be regarded as the minimum mass of sample needed to reduce the sampling uncertainty to within ±1% of the mean concentration. In practical situations, the particles cannot generally be considered to be of uniform size, but eq 3 can be shown to remain valid as long as each type of particle has
µ
Figure 2. Flow-injection analysis system for phosphate. Flow rates are in mL/min.
JChemEd.chem.wisc.edu • Vol. 75 No. 8 August 1998 • Journal of Chemical Education
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In the Laboratory
is normally given as H3PO4 + 12 H2MoO4 → H3P(Mo12O40) + 12 H2O Mo(VI) + ascorbic acid → Mo(V) It should be noted that the presence of high levels of sodium chloride did not noticeably change the sensitivity of this procedure. However, the reaction was found to produce a film on the flow cell, which slowly increased the background absorbance over a few hours. This problem was minimized by periodically washing the lines with distilled water or dilute nitric acid.
Procedure To save time, students worked in pairs. Typically, one student was in charge of preparing the samples (Student A) and the second was responsible for the FIA apparatus (Student B). (Roles can be switched halfway through the experiment.) Student A is given the unknown solid mixture and instructed to mix and grind it with a mortar and pestle for at least 10 minutes. Following this, the mixture is transferred to a clean sheet of white paper to form a small, pie-shaped pile. A spatula is used to divide the “pie” into six equal wedges. A portion of each wedge, accurately weighed with a nominal mass of 0.10 g, is then transferred to a 10 mL volumetric flask and dissolved in distilled water. The remaining salt mixture is transferred back to the mortar and briefly mixed. The process is then repeated using a different nominal mass of material and volume of solution. The masses and volumes used are listed in Table 1 (along with estimated uncertainties for propagation of error calculations described below). In the end, the student should have five sets of six solutions, each set with the same nominal concentration, but different masses of sample. After preparing the phosphate standards, Student B collects data for the calibration curve by injecting each standard three times. At this point, Student A normally has the first set of samples prepared. Each set of samples is run in turn, making triplicate injections for each solution. In our experience, one set of samples (18 injections) requires about 20 minutes to run, usually leaving the students more than enough time to complete the lab. For the final solution in the last set, 10 replicate injections are made so that a reliable estimate of the measurement uncertainty can be obtained. Figure 3 shows typical traces from the calibration run of five phosphate standards and for the first and last set of six samples. In the case of the 0.1-g samples, students immediately notice the poor reproducibility among the six solutions compared to the reproducibility of the triplicate measurements on the same solution. Likewise, they observe the improved reproducibility for the larger samples. Although they complete the experiment with a thorough analysis of sources of variance (see below), it is this visual image that has perhaps the biggest impact.
Table 1. Measurement Uncertainties for Solutions of Phosphate–NaCl Mixture Nominal Mass/g
Mass Range/g
Solution Volume/mL
σmass /ga
σvol /mL
0.10
0.09– 0.11
10
0.0001
0.02
0.25
0.20– 0.30
25
0.0001
0.03
0.50
0.45– 0.55
50
0.0001
0.05
1.00
0.90 –1.10
100
0.001
0.08
2.50
2.25–2.75
250
0.001
0.12
N OTE: Uncertainties in masses were taken from documentation for the balances. Volumetric uncertainties are Class A tolerances specified in the Fisher Scientific catalog. aThis value also applies to the tare. The first three samples were weighed on the same balance; a different balance was used for the two largest samples.
Figure 3. Typical data from the FIA system for three sets of samples.
Data Analysis Students first measure all the peak heights (conversion from transmittance to absorbance is assumed) and calculate the mean values for each set of triplicates. If a sloping baseline is observed, students are instructed to draw a smooth line through it before making the peak height measurements. 1030
Figure 4. Distribution of results from a typical experiment (see data in Table 2). The solid lines show one standard deviation boundaries expected for Ks = 350.
Journal of Chemical Education • Vol. 75 No. 8 August 1998 • JChemEd.chem.wisc.edu
In the Laboratory
For the calibration data, the peak heights for the five standards are entered into a spreadsheet and linear regression is used to calculate a calibration curve (slope + intercept). The measurements for the five sets of six samples are then entered into the spreadsheet as shown in Table 2. The regression parameters are used to calculate the results in the final column, which are expressed in terms of the weight percentage of KH2PO4 in the original mixture. Although this calculation is relatively simple, it offers the student an opportunity to gain experience with spreadsheets and demonstrates their general utility. Students are instructed to use the results in the table to generate a plot of %KH2PO4 vs sample mass, such as that shown in Figure 4 (plotted from the results in Table 2). These results are typical in the visual contrast they provide for the reproducibility of small and large samples. Although there will obviously be some statistical variability in these results, the importance of sample size is clearly indicated in virtually all cases. This is the central point of the exercise, and the figure allows it to be made in a powerful way. Further analysis of the data allows the students to decompose the variance exhibited in Figure 3 into its various sources and actually estimate sampling errors and Ingamells’ sampling constant, Ks . The results of such an analysis (for the data in Table 2) are shown in Table 3. Columns 1 and 2 contain the average mass and concentration of KH 2PO4 determined for each group of samples. Column 3 contains the total standard deviation of the concentrations determined within each group, evaluated directly from the last column of Table 2. (Note that some spreadsheets calculate the standard deviation using N rather than N – 1 degrees of freedom and the result must be adjusted accordingly.) Column 4 gives the measurement standard deviation; that is, it indicates the reproducibility of replicate measurements on the same solution. This is determined from the 10 replicate measurements made on the final solution. However, the standard deviation in peak heights for this sample, speaks, must be converted to a standard deviation in %KH2PO4 using the following formula:
s peaks volume (L) s meas = 1 × × × 3 slope av mass (g)
Table 2. Typical Results from FIA of Mixture Samples Nominal Sample Actual Vol/ Peak Concn Mass/g No. Mass/g mL Height/cm %KH2PO4 0.10 1 0.1039 10.00 1.14 0.085
0.25
0.50
1.00
2.50
2
0.1015
10.00
9.20
1.078
3
0.1012
10.00
3.78
0.413
4
0.1010
10.00
10.53
1.248
5
0.1060
10.00
5.99
0.654
6
0.0997
10.00
4.48
0.507
1
0.2515
25.00
7.26
0.847
2
0.2465
25.00
5.16
0.598
3
0.2770
25.00
4.26
0.431
4
0.2460
25.00
7.07
0.842
5
0.2485
25.00
8.11
0.964
6
0.2590
25.00
10.21
1.178
1
0.5084
50.00
8.65
1.009
2
0.4954
50.00
7.95
0.947
3
0.5286
50.00
5.67
0.618
4
0.5232
50.00
6.67
0.744
5
0.4965
50.00
4.98
0.572
6
0.4995
50.00
6.11
0.709
1
1.027
100.0
6.16
0.696
2
0.987
100.0
7.10
0.843
3
0.991
100.0
4.68
0.535
4
0.998
100.0
6.43
0.750
5
0.997
100.0
6.11
0.711
6
1.001
100.0
5.56
0.639
1
2.496
250.0
6.56
0.766
2
2.504
250.0
6.60
0.769
3
2.496
250.0
5.89
0.682
4
2.496
250.0
5.30
0.609
5
2.557
250.0
5.26
0.589
6
2.509
250.0
5.39
0.617
(4)
MW KH2 PO4 3᎑ PO4
᎑3
× 10 g/mg × 100%
MW _ The factor of √3 arises from the fact that the “measurement” is actually the average of triplicate injections, and the variance of this mean is the variance of the population divided by 3. The remainder of the equation is the result of the conversion from peak height to % KH2PO4 (note that, because this is an equation for uncertainty, the intercept term in the calibration model does not propagate through). Of course, eq 4 assumes that the uncertainty of the FIA peak height is independent of its magnitude—an assumption that should be reasonable in this range. Since there is a small variation in the individual sample masses within each group, an average is used. However, because the mass per unit volume is nearly the same for every sample, all values in the fourth column of Table 3 appear the same. (The slope of the calibration curve
used for these calculations was 0.168 cm/ppm.) Furthermore, measurement errors are typically much smaller than the overall standard deviation for this method, making the subtleties of eq 4 mainly academic. The standard deviation in the results due to the preparation of the samples is given in the fifth column of Table 3. This is calculated directly by propagation of the measurement uncertainties estimated in Table 1. The relevant equation is
s prep = av %KH2 PO4
2σmass
2 2
(av mass)
+
σvol2 2
(vol)
(5)
Since the only quantifiable sources of uncertainty in preparation are the mass and volume, and since these are carried through the calculations in a multiplicative way, the relative variance in the % KH2PO4 is the sum of the relative variance in the two measurements. The factor of 2 in front of the mass
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In the Laboratory
unusual to observe outlier peaks (arising, for example, from an air bubble) and the use of tripliStandard Deviation (units of %KH2PO4) Av Mass/ Av cate measurements makes it easier to identify %RSDsamp Ks /g g %KH2PO4 s smeas sprep ssamp total these. It has also been suggested that a randomized design for the standards, or interspersing the 0.1020 0.664 0.432 0.012 0.0016 0.432 64.3 422 samples with the standards, would be better from 0.2548 0.810 0.264 0.012 0.0011 0.264 39.3 394 a statistical perspective. Although this is true, it may 0.5086 0.766 0.176 0.012 0.0008 0.176 26.1 348 be difficult to implement in practice, owing to the 1.0002 0.696 0.104 0.012 0.0011 0.103 15.4 236 timing of sample preparation and analysis and because of the delays introduced when purging the 2.5097 0.672 0.080 0.012 0.0005 0.079 11.8 350 sample loop with a new sample. Identification of sample peaks may also become less reliable. uncertainty arises from the fact that the variance in the tare Finally, one source of variance that was not considered in this and the measurement itself are additive. By multiplying the experiment was that introduced by the calibration itself. Alrelative standard deviation (RSD) in preparation by the average though this may have a small influence on the quality of the %KH2PO4 for each group of samples, the absolute standard predicted concentration, in this experiment the effect is condeviation due to preparation is obtained. It is clear from Table stant and will influence the accuracy but not the precision 3 that this source of variation is even less important than the of the result. It was therefore not included in the variance measurement standard deviation. calculations. Column 6 of Table 3 is completed by subtracting the Conclusions variance due to measurement and sample preparation from The very nature of sampling uncertainty makes it difficult the total variance (see eq 1) and taking the square root. It will to design experiments that efficiently and clearly demonstrate be noted that the result hardly differs from the total standard the characteristics of this source of variance. This probably deviation, so a simplified analysis could normally eliminate accounts for the dearth of literature in this area. We have the calculation of measurement and preparation uncertainties. found the procedure reported here to be nearly ideal in many Nevertheless, we feel that these calculations are useful in respects. It involves simple and inexpensive apparatus and demonstrating to the students the relative magnitude of each chemicals, and is easily completed by a pair of students in source of variance. Column 7 is obtained by taking the ratio one afternoon. Students are exposed to the analysis of a large of the entries in column 6 to the last entry in column 2 and data set using a spreadsheet and learn to isolate sources of multiplying by 100. Note that the last entry of column 2 variance. Most importantly, the experiment vividly illustrates was used here because this provides the best estimate of the the effect of sample size on sampling uncertainty at several mean phosphate concentration, but a weighted average or the points: (i) when the FIA peaks are first measured, (ii) when “known” concentration (if it is provided to the students) could a plot like Figure 4 is generated, and (iii) when Table 3 is also be used. Column 7 shows the marked decrease in relative completed. Finally, despite the vagaries of variance measureuncertainty with increasing sample size. Finally, the value of ments, results from this procedure are remarkably consistent the sampling constant, Ks , is calculated for each group of and the same general trends are almost always observed. Given samples using eq 3 with the data in columns 1 and 7. These the importance of sampling uncertainty to measurements for values, given in column 8 of the table, should ideally be practical problems, we feel that the experiment described here constant but usually show a wide variation due to the diffican make a valuable contribution to the undergraduate culty in accurately measuring variances. The order of magnianalytical laboratory. tude of these numbers is enough to convey the appropriate message to the student, however (in this case, that a sample Literature Cited size of several hundred grams would be necessary to reduce 1. Clement, R. E. Anal. Chem. 1992, 64, 1076A–1081A. the sampling uncertainty to 1% !). Typically, a mean value 2. Bauer, C. F. J. Chem. Educ. 1985, 62, 253. for Ks is calculated from these numbers (mean = 350 in this 3. Kratochvil, B.; Reid, R. S.; Harris, W. E. J. Chem. Educ. 1980, case) and, if desired, this can be used to predict one standard 57, 518–520. deviation boundaries for the analysis, as shown in Figure 4. 4. Ruzicka, J.; Hansen, E. H. Flow Injection Analysis, 2nd ed.; Wiley: The funnel shape of these boundaries agrees remarkably well New York, 1988; pp 303–305. 5. Box, G. E. P.; Hunter, W. G.; Hunter, J. S. Statistics for Experiwith the distribution of samples. menters; Wiley: New York, 1978; p 572. It should be noted that some modifications to the pro6. Benedetti-Pichler, A. A. In Physical Methods in Chemical Analysis, cedures described here may have pedagogical or practical Vol. 3; Berl, W. G., Ed.; Academic: New York, 1956; pp 183–217. advantages, depending on the level of the students, the 7. Walton, W. W.; Hoffman, J. I. In Treatise on Analytical Chemistry, objectives of the experiment, and the time available. For Part 1, Vol. 1; Kolthoff, I. M.; Elving, P. J., Eds.; Wiley: New example, one reviewer of this paper suggested that the addiYork, 1959; Chapter 4. tional run of 10 replicates to estimate measurement uncer8. Bicking, C. A. In Treatise on Analytical Chemistry, 2nd ed.; Vol. 1, Part 1; Kolthoff, I. M.; Elving, P. J., Eds.; Wiley: New York, 1978; tainty is unnecessary, since the information can be obtained Chapter 6. by pooling variance information in the triplicate runs. This 9. Harris, W. E.; Kratochvil, B. Introduction to Chemical Analysis; should provide a better estimate of the measurement variance Saunders: Philadelphia, 1981; Chapter 21. and will reduce the time required, although it will make the 10. Gy, P. M. Sampling of Particulate Materials: Theory and Practice; calculations a little more involved. Running duplicates instead Elsevier: Amsterdam, 1979. of triplicates is also possible, but we have found that it is not 11. Kratochvil, B.; Taylor, J. K. Anal. Chem. 1981, 53, 924A–938A. Table 3. Statistical Analysis of Data from Phosphate Mixtures
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In the Laboratory 12. Smith, R.; James, G. V. The Sampling of Bulk Materials; Royal Society of Chemistry: London, 1981. 13. Kratochvil, B.; Wallace, D.; Taylor, J. K. Anal. Chem. 1984, 56, 113R–129R. 14. Woodget, B. W.; Cooper, D. Samples and Standards; Wiley: New York, 1987. 15. Principles of Environmental Sampling; Keith, L. H., Ed.; American Chemical Society: Washington, DC, 1988.
16. Kratochvil, B. Fresenius J. Anal. Chem. 1990, 337, 808–811. 17. Gy, P. M. Sampling of Heterogeneous and Dynamic Materials: Theories of Heterogeneity, Sampling and Homogenising; Elsevier: Amsterdam, 1992. 18. Thompson, M.; Ramsey, M. H. Analyst 1995, 120, 261–270. 19. Ingamells, C. O.; Switzer, P. Talanta 1973, 20, 547–568. 20. Zheng, L.; Kratochvil, B. Analyst 1996, 121, 163–168.
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