In the Laboratory
Fractional Factorial Experimental Design as a Teaching Tool for Quantitative Analysis Philip J. Oles Lebanon Valley College, 101 N. College Ave., Annville, PA 17003-0501
Quantitative analysis is traditionally designed to teach students the fundamentals of manipulations in the laboratory while maintaining precision of a few parts per thousand. The techniques used include gravimetric analysis, titrimetry, and to a lesser extent, some basic instrumental techniques such as spectrophotometry or atomic absorption spectrophotometry. The primary goal for the instructor is typically an evaluation of the ability of a student to obtain the “true” result for a series of samples having known (to the instructor) analyte content. Many of the classical methods such as the gravimetric determination of chloride and the volumetric determination of iron (II) with permanganate have been in use for several decades. These are reliable and rugged methods; therefore, it is appropriate to attribute deviations from an accepted or certified value to laboratory technique. The curriculum in undergraduate quantitative analysis provides a natural point of introduction for an additional topic, fractional factorial experimental design (FFED). FFED is a powerful tool, useful for discovering the significant experimental variables or factors in a process. It originally found application in the field of agriculture (1 ). Applications in chemistry, while relatively uncommon, have demonstrated its usefulness for optimizing a variety of laboratory (often analytical) processes (2–6 ). Properly applied, it identifies the significant or main factors and in advanced applications may uncover interactions between factors. FFED is a systematic study of several factors in a process that may all be changed
each time an experiment is performed. This differs from conventional experimentation, in which one factor is changed while all others are held fixed. The advantage to FFED is that several factors may be studied in a small matrix of experiments. A disadvantage of FFED is that an incorrect assessment of the importance of factors may be made owing to underlying assumptions. Greater care should be taken in interpretation of results obtained from FFED; however, this can be a useful teaching exercise. The application of FFED in the gravimetric determination of chloride by precipitation with Ag+ ion is described. A number of factors were chosen for study as a group laboratory exercise in an undergraduate quantitative analysis course. The factors were chosen on the basis of their likelihood of affecting the accuracy of the determination. The objective of the study was to complement the lecture discussion of the characteristics of precipitates that yield accurate gravimetric data. FFED is most often employed for examining new processes in order to optimize them. In this application, a very rugged and optimized analytical procedure was used. Factors were varied over a range, and in many cases, the determination was performed under less than optimum conditions in a systematic fashion. Our goal was to identify and measure the significant factors and their ability to affect the accuracy of the determination. The technique can also be applied to synthetic organic or inorganic chemistry to optimize yield and minimize by-products. Continued on page 358
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In the Laboratory
Experimental Procedure The method for the gravimetric determination of chloride by precipitation with silver ion was adopted from reference 7. It represents a typical laboratory procedure containing a series of steps that define an overall process: Gravimetric Determination of Chloride as a Process 1. Weigh samples 2. Dissolve in H2O (containing HNO3) 3. Add AgNO3 solution to coagulate the precipitate 4. Add additional AgNO3 after coagulation 5. Digest just below 100 °C 6. Decant supernatant liquid 7. Wash precipitate with H2O (containing HNO3) 8. Collect precipitate 9. Dry precipitate 10. Weight precipitate to constant weight
Table 1. Factors and Levels for Gravimetric Chloride Determination Factor
Level 1
Vol. 6 M HNO3, mL
1
10
2
Molarity of AgNO3
0.2
2.0
3
Vol. AgNO3 to coagulate AgCl, mL
4
Digestion Temp. ± 5 °C
5
Vol. 6 M HNO3 / L electrolyte, mL
3
10
25
100
2
20
Students were provided with a procedure containing blank spaces for the factor levels. Table 1 was attached to the procedure. Each student was assigned a cell—that is, an experimental procedure with levels assigned to all the factors. Chloride unknowns were purchased from Thorn Smith Chemist, 1023 Troy Court, Troy, MI 48084, and were certified for their chloride content.
Fractional factorial design experiments are structured in an array. The horizontal rows represent individual experiments and are referred to as cells. Vertical columns represent the values or levels for the factor assigned to that column. Genichi Taguchi (8) has simplified this approach by using an orthogonal matrix of experiments. Orthogonality means that when each factor is at one specified level, all other factors are studied at both levels an equal number of times. This simplifies the setup of the design, since the columns that show the presence of interactions are predictable. Calculations are also simplified in this design. In our design, column numbers 3 and 6 were not assigned to factors but were reserved for potential interactions between factors. Interpretation of interactions is left for more advanced applications. Each step in an experimental process contains one or more variables that may be candidates for study. Class size will determine the number of factors examined and, therefore, the number of experiments performed. Our first application was conducted with a class of 25 students. Two versions of 8-cell design experiments were conducted. Each student performed triplicate determinations, thus providing some measure of indeterminate error for each cell. Factors and their values at each of the two levels are given in Table 1. The orthogonal array used is shown in Table 2. A total of five factors were studied leaving two columns unassigned.
Results and Discussion Several unknowns were used by a laboratory section. However, the range of chloride was maintained between 48 and 60%. To compare performance among students and normalize the data, each result reported was converted to percent of true value using the expression
% true value =
xi × 100 xt
where xi = % chloride obtained by the student and xt = % chloride certified by the supplier. Converted data are summarized in Table 2. Calculations were performed using a standard approach for two-level designs (9). The effect of a factor is the change in response of the output variable that is produced by a change in the level of that factor. The results for each factor are summed for the experiments performed when the factor was equal to level 1 to produce a total T1. Likewise, a sum at level 2 is calculated to produce a total T2. The effect is calculated as follows:
effect =
T2 – T1 n 2
Table 2. Orthogonal Experimental Design For Gravimetric Chloride Determination Columns
358
Level 2
1
1
2
3
4
Cell
6 M HNO3 (mL)
AgNO3 (M)
Assigned to Interaction
AgNO3 (mL)
5
6
1
1
0.2
3
25
2
79.5
2
1
0.2
10
100
20
90.3
Temp. ± 5 Assigned to (°C) Interaction
7 HNO3 / L Average Result (mL) (% of True Value)
3
1
2.0
3
25
20
4
1
2.0
10
100
2
5
10
0.2
3
100
20
53.1
6
10
0.2
10
25
2
85.5
7
10
2.0
3
100
2
8
10
2.0
10
25
20
100 99.4
100 55.8
Journal of Chemical Education • Vol. 75 No. 3 March 1998 • JChemEd.chem.wisc.edu
In the Laboratory
the accuracy of the determination, whereas an increase in the AgNO3 concentration had a positive impact. Digestion temperature was significant as a factor and increasing the digestion temperature had a positive effect, as expected. The molarity of AgNO3 as an independent factor was not statistically significant over the range studied. Since 0.2 M is the level typically (7 ) recommended, it may demonstrate significance at much lower levels. Conclusions
Figure 1. Level average graph for factor A, volume of added 6 M HNO3.
where n is the number of cells in the design. For example, for factor A in Table 2 (volume of added 6 M HNO3) T1 = 79.5 + 90.3 + 100 + 99.4 = 369.2 T2 = 53.1 + 85.5 + 100 + 55.8 = 294.4 effect = (294.4 – 369.2)/4 = ᎑18.7 It is often useful to construct level average graphs for the factors. Such a graph is shown in Figure 1 for factor A. Values of factor effects appear in Table 3. The sign of the effect reflects a positive or negative influence upon the result when the factor was changed from level 1 to level 2. The magnitude of the effect reflects the relative ability of each factor to influence the output variable. After calculations were performed, students were encouraged to explain in terms of precipitation theory why some factors are significant or not. The data presented in Table 3 show that the concenTable 3. Factor Effects trations of HNO3 electrolyte Factor Effect and AgNO 3 are significant ᎑ 18.7 6 M HNO3 (mL) and that the electrolyte con+11.7 AgNO3 (M) centration has a greater influ᎑ 0.4 AgNO3 (mL) ence on the results than the Temp. ± 5 (°C) +5.5 concentration of AgNO3 over the ranges studied. This sup᎑ 16.3 HNO3 / L (mL) ports the textbook discussion of the influence of electrolyte and AgNO3 concentrations in achieving accurate results for this determination. However, these factors affected the results in opposite directions. An increase of HNO3 concentration had a negative impact upon
Fractional factorial experimental design is a useful tool in any laboratory exercise. It appropriate for quantitative analysis, where emphasis is placed upon accuracy and factors affecting accuracy. It is an effective team exercise because each student has some ownership in solving the overall problem. An added bonus comes from performing some of the experiments under nonideal conditions. Students performing these determinations are expected to rationalize their determinate error by documenting instances when sample precipitates failed to coagulate. I plan to expand the application of experimental design to other experimental procedures in quantitative analysis. Acknowledgments This work was carried out with the very capable assistance of my quantitative analysis class, spring semester 1995, York College of Pennsylvania, Country Club Road, York, PA. Literature Cited 1. Fisher, R. A.; Yates F. Statistical Tables for Biological, Agricultural and Medical Research, 4th ed.; Oliver and Boyd: Edinburgh, 1953. 2. Rubin, I. B.; Bayne, C. K. Anal. Chem. 1979, 51, 541–546. 3. Leggett, D. J. J. Chem. Educ. 1983, 60, 707–710. 4. Cawse, J. N.; Izadi, N. Today’s Chemist 1992, June, 24–28. 5. Otto, M.; Wegscheider, W. J. Chromatogr. 1983, 258, 11–22. 6. Strange, R. J. Chem. Educ. 1990, 67, 113–115. 7. Skoog, D.; West, D.; Holler, F. Fundamentals of Analytical Chemistry, Saunders College Publishing: Philadelphia, 1992; pp 835–837. 8. Taguchi, G. System of Experimental Design; Kraus International: White Plains, NY, 1987. 9. Montgomery, D. Introduction to Statistical Quality Control, 2nd ed.; Wiley: New York, 1991; p 484. 10. Rickmers, A.; Todd, H. Statistics; McGraw-Hill: New York, 1967; pp 154–157. 11. Wu, Y.; Moore, W. Quality Engineering Products and Process Design Optimization; American Supplier Institute: Dearborn, MI, 1986; p 63.
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