In the Laboratory
W
Experimental Design and Multiplexed Modeling Using Titrimetry and Spreadsheets Peter de B. Harrington,* Erin Kolbrich, and Jennifer Cline Center for Intelligent Chemical Instrumentation, Department of Chemistry and Biochemistry, Ohio University, Athens, OH 45701-2979; *
[email protected] Experimental design is important in the undergraduate curriculum, especially the practical application in the undergraduate laboratory. Experimental design is the systematic construction of experiments to gain the maximum amount of information from the fewest number of experiments (1). Several excellent tutorials discuss experiment design and optimization (2, 3). This economy of experiments piques the students’ interest, but it is also important in industry because it eliminates the cost of conducting ineffective experiments. Modeling is frequently taught to undergraduates in the form of linear calibration and the calculation of the working curve. A tutorial exists on the Web for those in need of a refresher on linear and multiple regressions (4 ). Another resource is introductory analytical textbooks that present calibration. Multiplexed experimental designs are an alternative application to modeling in the context of classical quantitative analysis experiments. The calculations can be performed in MS Excel, and quantitative analysis textbooks are beginning to incorporate MS Excel calculations (5). The efficiency of multiplexed measurements can be summarized in Kaiser’s counterfeit coin problem (6 ). In this problem, a single counterfeit coin made out of lead contaminates a group of 26 silver coins. Using a pan balance, the counterfeit coin may be identified in just 3 weighings if the experiment is carefully designed. If the coins are compared two at a time, it could take 26 weighings to find the counterfeit coin. The coins are arranged into 3 groups of 9 coins. One group is placed on each pan of the balance. If one pan is heavier, it must contain the counterfeit. If the two pans balance, the counterfeit coin must be in the third group. Now, the group of 9 coins containing the counterfeit is subdivided into 3 groups of 3 coins and the same procedure is applied to detect the group that contains the counterfeit coin. In the third weighing, there is a single coin on each balance and one omitted from the measurement. The counterfeit coin is identified from the pan balance. Routine determinations may be multiplexed when the samples do not interfere with one another. The determination of vinegar acidity is simple and amenable to multiplexed designs. This determination, which is used in industry, is frequently accomplished by titrimetry or pH meter measurements. In this experiment, we will determine the acidity of 3 vinegar samples using multiplexed titrations. Typically each sample would be titrated a minimum of 3 times to yield 9 total titrations. We will use modeling to determine the acidity of the 3 samples in 5 titrations. The acidity of the 3 samples could be determined with 3 titrations, but two additional titrations will furnish 2 extra degrees of freedom so that the precision can be measured and the statistical confidence intervals obtained for the concentration estimates.
Experimental Method
Materials Distilled white vinegar with a 5% acidity was used. The vinegar was diluted 1:1 (v/v) and 1:2 (v/v) with water to furnish solutions with 3 different acidities. These are the samples for the experiment. A standardized solution of 0.1 N NaOH (CAS RN 1310-73-2) is used as the titrant and phenolphthalein (CAS RN 77-09-8) is used as the indicator. Procedure A single experimental design is evaluated here; a second is presented online.W The vinegar samples labeled A, B, and C make up the columns of the design matrix (Table 1), and the titration runs are the rows of the matrix. A 2.0-mL volumetric pipet is used to add the samples to the Erlenmeyer flasks before each titration. Each vinegar sample will be titrated three times. Two drops of indicator are added to the sample before titration. Calculations If we designate our experimental design matrix as D, one can see a simple mathematical relationship exists (i.e., model) that is given by Dc = teq
(1)
for which D, the design matrix comprised volumes of the samples delivered to the Erlenmeyer, multiplied by c, a column vector comprising the concentrations of each sample, will equal the equivalent amount of titrant (teq). Boldface upper-case symbols designate matrices; boldface lower case designates vectors. We can estimate c by using multivariate regression given by cˆ = (DT D)᎑1DTteq
(2)
for which cˆ are the estimated concentrations and the rearranged equation is equivalent to dividing teq by D. The DT refers to the transposed design matrix, which is equivalent to turning the matrix on its side and is needed to calculate the inner Table 1. Experimental Design for Multiplexed Titration Titration
Sample A
B
C
2.0 mL
2.0 mL
2.0 mL
2
—
2.0 mL
2.0 mL
3
2.0 mL
—
2.0 mL
4
2.0 mL
2.0 mL
—
5
2.0 mL
2.0 mL
2.0 mL
1
JChemEd.chem.wisc.edu • Vol. 79 No. 7 July 2002 • Journal of Chemical Education
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In the Laboratory Table 2. Entries in Spreadsheet for Volume and Milliequivalents of Titrant Titration
A
B
C
Vtit
meq
1
2
2
2
30.15
3.015
2
0
2
2
13.85
1.385
3
2
0
2
21.80
2.180
4
2
2
0
25.48
2.548
5
2
2
2
31.40
3.140
titration errors are pooled for the three samples, so the confidence intervals will be tighter than for traditional approaches that calculate individual confidence intervals for each sample. Hazards Do not get vinegar or NaOH on your skin or in your eyes. If you do, rinse with copious amounts of water. Avoid bodily contact with the phenolphthalein indicator solution. Conclusions
product of this matrix. Fortunately, eq 2 can easily be implemented in MS Excel using the LINEST function. The online help in Excel is an excellent resource for the mathematics of this function and presents an example of its use. An excellent description of this function can also be found in the textbook by Skoog et al. (5). The students construct their design matrix D and fill in a column of measured volumes of titrant. The volumes of titrant are converted to milliequivalents by multiplying the volume (milliliters) by the concentration (normality, N) of the titrant. Because the values in D are expressed in milliliters, the estimated concentrations will be in units of normality. The data are arranged in a worksheet as in Table 2. The LINEST function is called as =LINEST(F2:F5, B2:D6, False, True) (5). The first argument in the LINEST function corresponds to the milliequivalents of titrant. The second argument specifies the design matrix D. The third argument is False because we do not want to estimate an intercept in our experiment. The fourth argument is True because we would like to calculate some statistics. The output from the LINEST function is given in Table 3 and explained in the note to Table 3. The concentrations should be reported with 95% confidence intervals, obtained by multiplying the errors in the second row by two-tailed t reference values for 95% confidence and 2 degrees of freedom. The Excel function is TINV(0.05, 2). One advantage offered by multiplexed methods is that the
Experimental design and modeling may be taught in practical experiments in quantitative analysis or advanced general chemistry laboratories. However, the students should have been previously introduced to linear regression and calibration. The students can use spreadsheets to design their experiments and simulate the results before running their experiments. For example, they can optimize the titrant concentration on the basis of the range of sample concentrations and the volume of their buret. They can also explore the experiment before entering the laboratory and learn the effect of errors on their results. Students learn to mathematically model their data with something other than the linear calibration curve. Novel designs of experiments can be evaluated in the spreadsheet. For example, what would happen if the same aliquot of the three samples were combined in all the titrations? Fewer measurements are required than in the one-sample-at-a-time approach to obtain similar degrees of freedom. Multiplexed measuring methods have a role where determinations are routinely made on similar samples, such as for quality assurance. W
Supplemental Material
Documentation including an introduction to linear regression analysis, instructions for additional neutralization titrations, and questions is available in this issue of JCE Online. Literature Cited
Table 3. Output from the LINEST Function C
B
A
0.2564
0.4404
0.8379
0.0267
0.0267
0.0267
0.9959
0.0639
#N/A
163.6
2
#N/A
2.004
0.00817
#N/A
NOTE. Row 1: concentration. Row 2: standard deviation. Row 3: column 1—coefficient of determination, r 2; column 2—standard error for the concentrations. Row 4: column 1—F statistic; column 2—degrees of freedom. Larger F values indicate more reliable results. The degrees of freedom are the same as those obtained from titrating each sample 3 times and calculating the mean. Row 5: column 1—regression sum of squares; column 2—residual sum of squares.
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1. Stolzberg, R. J. Chem. Educ. 1997, 74, 216–220. 2. Morgan, E.; Burton, K. W.; Church, P. A. Chemometr. Intell. Lab. Systems 1989, 5, 283–302. 3. Lundstedt, T.; Seifert, E.; Abramo, L.; Thelin, B.; Nystrom, A.; Pettersen, J.; Bergman, R. Chemometr. Intell. Lab. Systems 1998, 42, 3–40. 4. Rao, A. B.; McAndrews, P.; Sunkara, A.; Patil, V.; Bellary, R.; Quisumbing, G.; Le, H. L.; Zhou, Z. Regression; http:// cne.gmu.edu/modules/dau/stat/regression/regression_frm.html (accessed Mar 2002). 5. Skoog, D. A.; West, D. M.; Holler, F. J.; Crouch, S. R. Analytical Chemistry: An Introduction, 7th ed.; Saunders: Fort Worth, TX, 2000. 6. Kaiser, H. Anal. Chem. 1970, 42, 24A–41A.
Journal of Chemical Education • Vol. 79 No. 7 July 2002 • JChemEd.chem.wisc.edu
