In the Laboratory
A Laboratory Exercise To Demonstrate the Theory and Practice of Analytical Sampling Brian A. Logue* and Stephanie L. Youso Department of Chemistry and Biochemistry, South Dakota State University, Brookings, South Dakota 57007 *
[email protected] When determining the chemical makeup of a bulk material, the material is normally too large to be analyzed in its entirety. Therefore, gathering a representative sample (i.e., a small quantity of material that is characteristic of the bulk material) is typically the first step toward a successful chemical analysis. If a sample is not representative of the bulk material, even the best analytical method will fail to produce accurate results. Uncertainty involved in producing a representative sample from a bulk material arises from the random nature of drawing particles from a mixture. As the number of particles sampled increases, the uncertainty will decrease (1). Therefore, sampling error is usually particularly small for homogeneous mixtures of liquids and gases because there are approximately 1022 and 1019 molecules per cm3 of liquid and gas (at 25 °C and 1 atm), respectively. Conversely, error caused by sampling solids may be great because the volume of individual particles may be large. Therefore, sampling error should be considered when sampling solid bulk materials and special care must be taken when the individual particles are large. For the analysis of a bulk material, the overall error is generated from a number of sources. Typically, the two main sources of error are the sampling procedure and the analytical method. The relationship between the error of the overall method, so, the sampling error, ss, and the error of the analytical method, sa, in the simplest form is so 2 ¼ sa 2 þ ss 2
(1Þ
Normally, the error of an analytical method can be easily determined, but quantifying sampling error can be difficult. Determining sampling error is important, but it is often more desirable to estimate the quantity of bulk material necessary to achieve a desired error. This estimation can be accomplished by applying the Bernoulli principles of population sampling (1). To apply these principles to sampling for chemical analysis, the bulk material must be composed of two particle types (particles A and B) of similar size and density with only one particle containing the analyte. If these requirements are met and the number fraction of the desired particle in the bulk material, fA, can be estimated, the number of particles, n, necessary to achieve a desired relative error (fractional), rs, can be calculated 1 - fA n¼ 2 (2Þ r s fA For most situations where sampling error must be considered, particles will have different sizes, different densities, or 316
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more than one particle type contains the analyte. If this is the case, eq 3, the Benedetti-Pichler equation (2, 3; some of the symbols have been modified for clarity), can be used to incorporate some properties of the bulk material, !2 !2 dA dB PA - PB (3Þ n ¼ fA fB rs Pavg davg 2 where fB is the number fraction of particle B (i.e., fB = 1 - fA), dA and dB are the densities of the individual particles, PA and PB are the percentages of analyte in the respective particle, and davg and Pavg are the average density and the average percent analyte in the bulk material, respectively. Equations 2 and 3 are used to calculate the number of particles in the sample, but for chemical analysis, it is more useful to calculate mass of the sample. Therefore, if the sample particles are assumed to be spherical, the mass (m) of the sample can be estimated using eq 4, where ravg is the average radius of the particles (equations for calculating davg, Pavg, and ravg are provided in the supporting information). 4 m ¼ πravg 3 davg n (4Þ 3 Introducing analytical chemistry students to sampling theory in the classroom can provide them with an idea of the importance of obtaining a representative sample, but the addition of hands-on exercises adds depth to sampling concepts. A number of classroom (4-7) and laboratory exercises (8-11) that introduce basic sampling concepts have been developed. These highlight the factors that influence sampling error (e.g., mixing), but generally neglect to demonstrate one of the most important uses of sampling theory: the ability to determine the quantity of sample to gather to achieve a desired error. The laboratory presented here focuses on the use of sampling theory for this purpose by initially presenting simple “unrealistic” exercises and then a more realistic scenario of sampling that involves determination of the financial feasibility of a multimillion dollar nickel mining operation. In addition, this laboratory introduces students to the principles of gravimetric analysis by requiring analysis of Ni2þ by precipitation with dimethylgloxime (DMG). The gravimetric analysis uses essential laboratory techniques, including quantitative transfers, precipitation and filtration techniques, and analytical weighing practices. This laboratory is meant for undergraduate analytical chemistry (quantitative analysis) courses and takes two 3-h laboratory periods to complete.
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Vol. 87 No. 3 March 2010 pubs.acs.org/jchemeduc r 2010 American Chemical Society and Division of Chemical Education, Inc. 10.1021/ed800087d Published on Web 02/09/2010
In the Laboratory Table 1. Student Data for the Analysis of Mixed and Unmixed Cores of Uniform Sized Particles and Estimation of the Number of Particles To Gather To Achieve a Desired Sampling Error Mixed Corea
Unmixed Corea
Number for Desired Error
Analyte (%)a,b
RSD (%)c
Analyte (%)a,b
RSD (%)c
1
64.4
11.9
35.0
105.9
14
2
57.8
13.3
28.3
112.6
18
5.6
3
48.9
28.4
23.3
75.6
26
18.9
4
46.7
37.8
13.3
135.4
29
5.1
5
48.9
31.5
18.3
123.8
26
38.2
6
37.8
36.7
20.0
115.5
41
9.8
7
46.7
24.7
15.0
116.9
29
15.6
8
40.0
33.3
30.0
112.6
38
6.2
9
64.4
21.5
11.7
135.0
14
7.5
10
60.0
11.1
11.7
117.8
17
22.2
Mean
51.6
25.0
20.7
115.1
25.1
16.6
Group
Calcd nd
RSD (%)c,e 36.7
a
A mixture of approximately 50:50 light and dark aquarium rocks. The “mixed core” is well-mixed and the “unmixed core” is layered with the light rocks (i.e., the analyte) as the bottom layer. b The mixed core and unmixed core were sampled three and four times, respectively, 15 rocks each. Means of each analysis are reported. c Relative standard deviation (RSD) of the percent analyte. d Calculated number of rocks (n) to gather from the mixed sample for a desired RSD of 20% (eq 2). e Calculated from sampling the calculated number of rocks three times from the mixed core.
Table 2. Student Data for Sampling Bulk Material with Heterogeneous Particles and Estimation of the Mass of the Bulk Necessary To Achieve a Desired Sampling Error Sampling of Heterogeneous Particlesa
Mass for Desired Errora
Number of Rocksb
Number of Marblesb
Rock (%)b
1
28.7
13.0
68.5
7.0
58.4
5.3
2
12.2
9.1
57.1
10.5
63.8
13.5
3
42.8
13.0
76.1
6.8
46.9
8.3
4
40.1
13.2
69.8
22.9
63.0
7.8
5
26.0
13.1
63.2
19.7
47.8
0.4
6
55.2
13.0
78.5
12.3
15.2
2.2
7
31.3
13.1
70.1
5.7
70.0
9.9
8
38.2
11.4
73.0
17.1
29.9
5.5
9
12.7
8.4
57.0
18.7
58.1
2.6
10
24.1
13.0
63.5
16.6
67.2
6.8
Mean
31.1
12.0
67.7
13.7
52.0
6.2
Group
RSD (%)c
Calcd Mass/gd
RSD (%)c,e
a
A mixture of approximately 70:30 aquarium rocks and marbles. This bulk material was used for initial sampling and calculation of the mass necessary to achieve a desired error, each in triplicate. b Mean of triplicate measurements. c Relative standard deviation (RSD) of the percent analyte (rocks) from triplicate analysis. d Calculated mass of the bulk material to sample for a desired RSD of 10% (eqs 3 and 4). Students typically estimated the fraction of rock (fA) by sight instead of using the percentage rock from the initial sampling. e Calculated from the percent rock in three samples of the calculated mass of bulk material.
The Effect of Mixing on Sampling This section of this experiment demonstrates the importance of sample mixing similar to the classroom exercise of Hartman et al. (6). The students are provided with aquarium rocks of two different colors and they assemble two different “cores” in small cups. Although both cores have equal quantities of the two colors, one core is mixed, and the other core is composed of two separate layers of rock. Subsequently, students sample the cores, recording the number of rocks by color. Students are then asked to calculate the number of rocks to gather from the mixed core so that a specified relative standard deviation can be obtained. Students use eq 2 to estimate the number of rocks and investigate the accuracy of this calculation
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by experimentally gathering the number of particles and determining the resulting standard deviation. Typical results for this portion of the laboratory are listed in Table 1. Students typically find that mixing greatly affects both the precision and the accuracy of the analysis and that the modified Bernoulli equation works well for determining the number of particles to sample to achieve a desired error. The Effect of Heterogeneous Particles on Sampling In this part of the laboratory, students sample from a heterogeneous mixture of one color of aquarium rock as well as standard-sized marbles (i.e., an approximate diameter of 14 mm) similar to the classroom exercises of Hartman et al. (6) and
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In the Laboratory
Figure 1. A picture of the “realistic” sample used. NiSO4 3 6H2O (particle A) is a blue-green compound pictured on the left. Silicon carbide boiling chips (particle B) are pictured on the right. The mixture of the two particles as the bulk sample is seen in the top of the picture. The penny is used for perspective. Table 3. Student Data for the Determination of the Error Involved in Sampling a Solid Bulk Material Estimation of Massa Group
Standardb
Samplea
fA dB/(g cm-3) Mass/g Ni (%) RSD (%)c Ni (%) RSD (%)c
1
0.50
4.54
4.41
0.26
2
0.50
4.04
2.18
2.85
29.5
8.51 16.5
3
0.60
4.30
2.39
1.91
20.4
4
0.60
4.32
3.16
2.25
42.2
5
0.55
4.32
3.16
3.03
6
0.60
3.78
1.99
1.40
7
0.50
4.54
4.41
5.93
4.56 12.7
8
0.45
4.09
5.01
2.87
8.60 11.4
9
0.55
4.17
3.01
1.27
13.7
10
0.60
4.54
2.58
2.14
11.5
Mean 0.55
4.46
3.23
2.39
14.8
1.76 13.0
0.53 15.0
39.5
2.94 36.3 21.6 8.33 36.8
33.6
48.5 1.85 33.8
Hazards
2.53 28.5 12.1
1.76 61.1 15.3
sampling a bulk material, gravimetrically analyzing for the desired component (i.e., Ni2þ), and calculating the financial feasibility of the mining operation. The bulk material is composed of nickel sulfate hexahydrate (NiSO4 3 6H2O) and silicon carbide (Figure 1). Students determine the mass of the bulk material to gather to obtain a desired relative standard deviation. Students gravimetrically analyze three replicate samples along with three standard solutions for Ni2þ. Typical results from the analysis of the samples and standard solutions are reported in Table 3. Students found a mean concentration of Ni2þ of 2.4% (m/m) in the standard solution, which is close to the actual concentration of 2.9% (m/m). However, only three of the 10 groups (groups 2, 5, and 8) produced satisfactory accuracy and precision. Also reported in Table 3 are concentrations of Ni2þ in the solid samples. Although the range of concentrations was large, students that accurately determined the Ni2þ concentration in the standard solution also produced good accuracy for the solid sample, implying excellent laboratory technique and accurate calculations. After determining the concentration of Ni2þ in the solid samples, students use the data to make a decision as to the feasibility of mining the proposed site. Students also evaluate the sampling error compared with analysis error. Using mean student data for the standards and the solid samples, the sampling error, rs, was approximately 0.20 (or 20% RSD) and error in the analysis was approximately 0.14 (or 14% RSD). The students' results show that the sampling error is greater than the error in analysis, which is typical of solid samples.
28.1
a
A mixture of approximately 80:20 NiSO4 3 6H2O and boiling chips (approximately 9.3% Ni2þ). The mass of the bulk material to sample for a desired RSD of 5% was calculated by eqs 3 and 4, with key parameters given to students. b The standard was 1 mL of a 0.15 M (2.94% m/m) Ni2þ standard solution. c Relative standard deviation (RSD) of the percent analyte from triplicate analysis.
Students should exercise appropriate caution when handling any chemicals. There are significant hazards during the gravimetric analysis of Ni2þ by DMG. Specific hazards include nickel sulfate hexahydrate as a potential carcinogen. Nickel sulfate hexahydrate and DMG are harmful if swallowed and may cause irritation to skin, eyes, and respiratory tract. Silicon carbide may cause irritation to skin, eyes, and respiratory tract. Caution should also be exercised when using hydrochloric acid, which can release hazardous fumes, is caustic, and can cause severe burns to all body tissue. All work with high concentrations of this acid should be performed in a ventilated hood.
Canaes et al. (4). Students sample using a small cup and analyze the sample by counting the marbles and rocks obtained. Students then employ the modified Benedetti-Pichler equation (eq 3) to calculate the number of particles necessary to achieve the desired relative standard deviation and calculate the mass of sample using eq 4. By comparing their results to the desired results, they can evaluate the effectiveness of the Benedetti-Pichler equation. As seen in Table 2, students typically find that the Benedetti-Pichler equation does an excellent job at estimating the mass necessary to achieve a desired error. This section of the experiment introduces students to the process of determining the sample mass necessary to obtain a specific error. This process is then applied in the final section of the laboratory where a more realistic sampling scenario is presented.
Summary
Realistic Sampling Scenario
Acknowledgment
Students are charged with determining whether a multimillion dollar mining operation should commence based on
Acknowledgement is extended to the students of the Analytical Laboratory of South Dakota State University who contributed
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This laboratory was developed to help demonstrate the practical use of sampling theory for the chemical analysis of solid samples. To accomplish this goal, the laboratory initially uses unrealistic exercises and follows with a more realistic sampling exercise to build confidence and gain depth of knowledge in the theory of sampling. Gravimetric analysis concepts are also introduced or built upon in this laboratory. Students performing this laboratory indicate that it is helpful at providing a better understanding of sampling. In postlaboratory surveys, this laboratory was rated 4.3 out of 5 when students were asked how well the laboratory helped them understand lecture concepts (the eight laboratories performed during the semester received an average ranking of 3.0).
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In the Laboratory
their data. Thanks also to Chakravarthy Vinnakota, who assisted in teaching the laboratory and collecting the data from the class. Literature Cited 1. Evans, M.; Hastings, N.; Peacock, B. Statistical Distributions, 2nd ed.; Wiley and Sons: New York, 1993; pp 28-30. 2. Benedetti-Pichler, A. A. Theory and Practice of Sampling for Chemical Analysis. In Physical Methods of Chemical Analysis; Berl, W. E., Ed.; Academic Press: New York, 1956; Vol. 3; p 183. 3. Harris, W. E.; Kratochvil, B. Anal. Chem. 1974, 46, 313–315. 4. Canaes, L. S.; Brancalion, M. L.; Rossi, A. V.; Rath, S. J. Chem. Educ. 2008, 85, 1084–1088. 5. Ross, M. R. J. Chem. Educ. 2000, 77, 1015–1016.
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6. Hartman, J. R.; Bacon, D. W.; Wolsey, W. C. J. Chem. Educ. 2000, 77, 1017–1018. 7. Vitt, J. E.; Engstrom, R. C. J. Chem. Educ. 1999, 76, 99–100. 8. Jeannot, M. A. J. Chem. Educ. 2006, 83, 243–244. 9. Harvey, D. T. J. Chem. Educ. 2002, 79, 360–363. 10. Kratochvil, B.; Reid, R. S.; Harris, W. E. J. Chem. Educ. 1980, 57, 518–520. 11. Guy, R. D.; Ramaley, L.; Wentzell, P. D. J. Chem. Educ. 1998, 75, 1028–1033.
Supporting Information Available Instructions for the students; notes for the instructor. This material is available via the Internet at http://pubs.acs.org.
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