In the Laboratory
Analysis of Iron in Lawn Fertilizer: A Sampling Study
W
Michael A. Jeannot Department of Chemistry, St. Cloud State University, St. Cloud, MN 56301;
[email protected] The errors associated with the sampling process are often overlooked when performing a chemical analysis. This is particularly true for the analysis of solids, where the soughtafter component may be present only in certain types of particles or may be present at different concentrations in different particles. Even under “homogeneous” conditions where the solid sample is thoroughly mixed (sought-after component is not segregated), sampling error can still be large, particularly if the sought-after component is present at trace levels (1, 2). The laboratory activity described in this article is designed to illustrate these concepts and is suitable for students in a beginning analytical chemistry course. For the simple case of two types of particles (one type of particle contains the component of interest and the other types of particles do not contain the component of interest), the sampling standard deviation can be derived from the binomial distribution (2) and represented as, σ =
n p (1 − p )
(1)
where σ is the standard deviation for the number of either type of particle, n is the total number of particles, and p is the probability of drawing the particle containing the component of interest (i.e., the fraction of particles that contain the component of interest). Thus, the relative standard deviation, expressed as a percent, can be represented as,
RSD =
n p (1 − p ) np
100% =
(1 − p ) 100% np
(2)
Thus, as p → 1, RSD → 0 (sampling is not an issue). However, as p → 0, RSD → ∞. Furthermore, as n → ∞, RSD → 0 (sampling is not an issue when a very large number of particles are sampled, as is the case for a homogeneous liquid solution). However, as n → 0, RSD → ∞. Thus, both the total number of particles and the “concentration” of the sought-after component determine the precision that can be obtained. For a given concentration, the inverse relationship between n and RSD is sometimes expressed empirically using Ingamells’s sampling constant (3),
m RSD2 = KS
(3)
where m is the total mass of sample and KS is the sampling constant. The sampling constant is numerically equal to the mass required to obtain one RSD. Several excellent examples of laboratory and classroom exercises have been published recently in this Journal (4–11). Although pedagogically valuable, these exercises use “artificial” chemical samples or objects to illustrate sampling and statistical concepts and, in some cases, are labor intensive. The experiment described here uses a real-world sample of lawn fertilizer in a simple exercise to illustrate problems aswww.JCE.DivCHED.org
•
sociated with the sampling step of a chemical analysis. A mixed-particle fertilizer containing discrete particles of iron oxide (magnetite, Fe3O4) mixed with other particles (e.g., coated or uncoated urea, muriate of potash) provides an excellent demonstration of sampling uncertainty and reasonably approximates the assumptions that lead to eq 2. It may be noted that iron is present in some fertilizers as a trace nutrient since it is necessary in several enzymatic processes, including the production of chlorophyll (12). The separation of the iron oxide particles is conveniently achieved using a small magnetic stir bar, since the other components are nonmagnetic, and there is no observable residue from the other particles on the iron oxide particles. The analysis of iron in the fertilizer is then simply a matter of weighing the iron oxide particles and calculating the mass of iron from the formula (Fe3O4). This mass is then compared to the mass of the total sample. Since the masses can be measured with a high degree of precision, the precision of the analysis reflects only sampling uncertainty. Pooled student data for different sample sizes can be used to compare with the theoretical RSD from eq 2. Experimental Procedure A granular, discrete-particle type fertilizer containing iron oxide may be purchased from a local retailer. We used Lesco’s Professional Turf Fertilizer. Students sample two 10-mL portions and two 40-mL portions of the fertilizer. The iron oxide particles are easily separated from the rest using a Teflon-coated magnetic stir bar. The fraction of iron oxide particles, p (needed for eq 2), is obtained by counting the total number of iron oxide particles, compared to the total number of all particles. Rather than count all the particles, the students may be given an average “particle density” for the fertilizer. For the Lesco fertilizer, the particle density is 49 ± 6 (SD) particles兾mL, based on 5 trials, using a 1-mL volumetric flask. The percent Fe (w兾w) is determined simply by weighing the iron oxide particles (72.36% Fe in Fe3O4) and comparing with the total mass of the sample. The students pool their data using a sheet posted in the laboratory. From the class data, students are able to determine the observed RSD based on the mass measurements and compare to the theoretical RSD from eq 2. This process is repeated for different sample sizes (e.g., 10-mL and 40-mL samples). Hazards There are no significant hazards associated with this laboratory experiment. Results and Discussion Results of the analysis from groups of 16 and 17 students over two semesters are shown in Table 1. The effect of n (total number of particles sampled) on the RSD is clearly
Vol. 83 No. 2 February 2006
•
Journal of Chemical Education
243
In the Laboratory
Table 1. Results of the Analysis of Iron in Fertilizer Sample
na
Ave pb
Ave Fe Fraction (%)c
SDc
95% Confidence Intervald
Observed RSDc (%)
Theoretical RSDe (%)
40 mL
f
1960
0.0859
5.93
1.03
5.9 ± 0.4
17.4
7.37
40 mL
g
1960
0.0934
6.03
1.18
6.0 ± 0.4
19.6
7.04
10 mL
f
0490
0.0937
5.73
1.58
5.7 ± 0.6
27.6
14.0
0490
0.1010
6.04
1.79
6.0 ± 0.6
29.6
13.5
10 mL g a
Estimate using average particle density of 49 particles/mL.
b
From average student count of number of iron oxide particles/n.
c
From pooled class data of mass measurements.
d e f
Average Fe fraction ± t(SD)/N1/2, where t = 2.04 and N is the number of measurements (34 or 32).
From eq 2, using calculated values of n and p from this table.
Data from 17 students (two measurements each) in fall 2003.
g
Data from 16 students (two measurements each) in spring 2004.
observed and in accordance with sampling theory. However, the magnitude of the error is considerably larger than that predicted from eq 2. Students are asked to speculate as to why this might be. Clearly, some of the assumptions that went into eq 2, such as uniform particle size and density, and complete mixing are not possible in practice. In particular, the iron oxide particles are of varying size and shape, and they tend to be somewhat segregated in spite of thorough mixing before sampling. Thus, while the numerical results obtained do support the basic premise of sampling theory (better precision with a larger sample), they also point to some limitations of eq 2 when real samples are used. It is also of interest to compare the measured Fe fraction with the value stated by the manufacturer (3.00%). Table 1 shows the 95% confidence intervals for the measured Fe. The measured results are clearly statistically higher than 3.00%. Students are asked to think about why their results are “inaccurate”. The explanation for the difference most likely lies in the fact that the students sample their fertilizer from a 2-L beaker containing only a portion of the entire bag of fertilizer (which, in turn, is only a portion of the entire lot from the manufacturer). Thus, it appears that the fertilizer that the students sampled from the 2-L beaker contained a higher Fe fraction than the average value reported by the manufacturer. This is a nice illustration of segregation (imperfect mixing or heterogeneity) within a larger portion of the fertilizer. As an additional exercise, students may also calculate a sampling constant for this analysis using their measured mass of fertilizer and observed RSD. This calculation serves to demonstrate the unacceptably high mass that would be needed to obtain one RSD. KS is on the order of 10,000 g for this system. The product mRSD2 is not truly constant for this system (decreasing the mass by a factor of 4 should ideally result in double the RSD). Once again, the model only approximately describes a real system where there are different particle sizes and some segregation. Students may be asked to think about how a smaller mass could be used while still obtaining good precision. Since the precision depends on the number of particles, grinding the particles to a smaller size would enable better precision with less mass. Overall, this has been a valuable exercise that has helped to reinforce important sampling concepts in an introductory 244
Journal of Chemical Education
•
analytical chemistry course. While the use of a real sample results in some deviation from ideal behavior, the theoretical effect of sample size is clearly observed. The deviation from ideal behavior is thought provoking for the students and encourages them to think carefully about the constraints placed into the theoretical equation, and how these constraints may not hold perfectly in a real system. Student feedback about this experiment has been positive. Students typically comment that the experimental procedure is simple and fast (data can be collected in about one hour). However, they often struggle initially with the data analysis and interpretation. Some students need guidance in working with either a spreadsheet or using the statistical functions on their calculators. While the data analysis is more involved than some other quantitative analysis labs (where the focus is on technique and accuracy of results), students have typically commented that this lab has helped immensely in understanding sampling error and that it ties in well with the lecture and real-world analysis in general. W
Supplemental Material
Instructions for the students and notes for the instructor are available in this issue of JCE Online. Literature Cited 1. Kratochvil, B.; Taylor, J. K. Anal. Chem. 1981, 53, 924A– 938A. 2. Harris, W. E.; Kratochvil, B. Anal. Chem. 1974, 46, 313–315. 3. Ingamells, C. O.; Switzer, P. Talanta 1973, 20, 547–568. 4. Harvey, D. J. Chem. Educ. 2002, 79, 360–363. 5. Ross, M. R. J. Chem. Educ. 2000, 77, 1015–1016. 6. Hartman, J. R. J. Chem. Educ. 2000, 77, 1017–1018. 7. Vitt, J. E.; Engstrom, R. C. J. Chem. Educ. 1999, 76, 99–100. 8. Guy, R. D.; Ramaley, L.; Wentzell, P. D. J. Chem. Educ. 1998, 75, 1028–1033. 9. Kratochvil, B.; Reid, R. S. J. Chem. Educ. 1985, 62, 252. 10. Bauer, C. F. J. Chem. Educ. 1985, 62, 253. 11. Kratochvil, B.; Reid, R. S.; Harris, W. E. J. Chem. Educ. 1980, 57, 518–520. 12. Longacre, A.; Papendick, R. I.; Parr, J. F. Fertilizer. In McGrawHill AccessScience Home Page. http://www.accessscience.com (accessed Nov 2005).
Vol. 83 No. 2 February 2006
•
www.JCE.DivCHED.org
