9.7: Exercises
- Page ID
- 290717
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)1. For each of the following equations determine the optimum response using a one-factor-at-a-time searching algorithm. Begin the search at (0,0) by first changing factor A, using a step-size of 1 for both factors. The boundary conditions for each response surface are 0 ≤ A ≤ 10 and 0≤ B ≤ 10. Continue the search through as many cycles as necessary until you find the optimum response. Compare your optimum response for each equation to the true optimum. Note: These equations are from Deming, S. N.; Morgan, S. L. Experimental Design: A Chemometric Approach, Elsevier: Amsterdam, 1987, and pseudo-three dimensional plots of the response surfaces can be found in their Figures 11.4, 11.5 and 11.14.
(a) R = 1.68 + 0.24A + 0.56B – 0.04A2 – 0.04B2 μopt = (3,7)
(b) R = 4.0 – 0.4A + 0.08AB μopt = (10,10)
(c) R = 3.264 + 1.537A + 0.5664B – 0.1505A2 – 0.02734B2 – 0.05785AB μopt = (3.91,6.22)
2. Use a fixed-sized simplex searching algorithm to find the optimum response for the equation in Problem 1c. For the first simplex, set one vertex at (0,0) with step sizes of one. Compare your optimum response to the true optimum.
3. A 2k factorial design was used to determine the equation for the response surface in Problem 1b. The uncoded levels, coded levels, and the responses are shown in the following table. Determine the uncoded equation for the response surface.
| A | B | A* | B* | response |
|---|---|---|---|---|
| 8 | 8 | +1 | +1 | 5.92 |
| 8 | 2 | +1 | –1 | 2.08 |
| 2 | 8 | –1 | +1 | 4.48 |
| 2 | 2 | –1 | –1 | 3.52 |
4. Koscielniak and Parczewski investigated the influence of Al on the determination of Ca by atomic absorption spectrophotometry using the 2k factorial design shown in the following table [data from Koscielniak, P.; Parczewski, A. Anal. Chim. Acta 1983, 153, 111–119].
| [Ca2+] (ppm) | [Al3+] (ppm) | Ca* | Al* | response |
|---|---|---|---|---|
| 10 | 160 | +1 | +1 | 54.92 |
| 10 | 0 | +1 | –1 | 98.44 |
| 4 | 16 | –1 | +1 | 19.18 |
| 4 | 0 | –1 | –1 | 38.52 |
(a) Determine the uncoded equation for the response surface.
(b) If you wish to analyze a sample that is 6.0 ppm Ca2+, what is the maximum concentration of Al3+ that can be present if the error in the response must be less than 5.0%?
5. Strange [Strange, R. S. J. Chem. Educ. 1990, 67, 113–115] studied a chemical reaction using the following 23 factorial design.
| factor | high (+1) level | low (–1) level |
|---|---|---|
| X: temperature | 140oC | 120oC |
| Y: catalyst | type B | type A |
| Z: [reactant] | 0.50 M | 0.25 M |
| run | X* | Y* | Z* | % yield |
|---|---|---|---|---|
| 1 | –1 | –1 | –1 | 28 |
| 2 | +1 | –1 | –1 | 17 |
| 3 | –1 | +1 | –1 | 41 |
| 4 | +1 | +1 | –1 | 34 |
| 5 | –1 | –1 | +1 | 56 |
| 6 | +1 | –1 | +1 | 51 |
| 7 | –1 | +1 | +1 | 42 |
| 8 | +1 | +1 | +1 | 36 |
(a) Determine the coded equation for this data.
(b) If \(\beta\) terms of less than \(\pm 1\) are insignificant, what main effects and what interaction terms in the coded equation are important? Write down this simpler form for the coded equation.
(c) Explain why the coded equation for this data can not be transformed into an uncoded form.
(d) Which is the better catalyst, A or B?
(e) What is the yield if the temperature is set to 125oC, the concentration of the reactant is 0.45 M, and we use the appropriate catalyst?
6. Pharmaceutical tablets coated with lactose often develop a brown discoloration. The primary factors that affect the discoloration are temperature, relative humidity, and the presence of a base acting as a catalyst. The following data have been reported for a 23 factorial design [Armstrong, N. A.; James, K. C. Pharmaceutical Experimental Design and Interpretation, Taylor and Francis: London, 1996 as cited in Gonzalez, A. G. Anal. Chim. Acta 1998, 360, 227–241].
| factor | high (+1) level | low (–1) level |
|---|---|---|
| X: benzocaine | present | absent |
| Y: temperature | 40oC | 25oC |
| Z: relative humidity | 75% | 50% |
| run | X* | Y* | Z* | color (arb. unit) |
|---|---|---|---|---|
| 1 | –1 | –1 | –1 | 1.55 |
| 2 | +1 | –1 | –1 | 5.40 |
| 3 | –1 | +1 | –1 | 3.50 |
| 4 | +1 | +1 | –1 | 6.75 |
| 5 | –1 | –1 | +1 | 2.45 |
| 6 | +1 | –1 | +1 | 3.60 |
| 7 | –1 | +1 | +1 | 3.05 |
| 8 | +1 | +1 | +1 | 7.10 |
(a) Determine the coded equation for this data.
(b) If \(\beta\) terms of less than 0.5 are insignificant, what main effects and what interaction terms in the coded equation are important? Write down this simpler form for the coded equation.
7. The following data for a 23 factorial design were collected during a study of the effect of temperature, pressure, and residence time on the % yield of a reaction [Akhnazarova, S.; Kafarov, V. Experimental Optimization in Chemistry and Chemical Engineering, MIR Publishers: Moscow, 1982 as cited in Gonzalez, A. G. Anal. Chim. Acta 1998, 360, 227–241].
| factor | high (+1) level | low (–1) level |
|---|---|---|
| X: temperature | 200oC | 100oC |
| Y: pressure | 0.6 MPa | 0.2 MPa |
| Z: residence time | 20 min | 10 min |
| run | X* | Y* | Z* | % yield |
|---|---|---|---|---|
| 1 | –1 | –1 | –1 | 2 |
| 2 | +1 | –1 | –1 | 6 |
| 3 | –1 | +1 | –1 | 4 |
| 4 | +1 | +1 | –1 | 8 |
| 5 | –1 | –1 | +1 | 10 |
| 6 | +1 | –1 | +1 | 18 |
| 7 | –1 | +1 | +1 | 8 |
| 8 | +1 | +1 | +1 | 12 |
(a) Determine the coded equation for this data.
(b) If \(\beta\) terms of less than 0.5 are insignificant, what main effects and what interaction terms in the coded equation are important? Write down this simpler form for the coded equation.
(c) Three runs at the center of the factorial design—a temperature of 150oC, a pressure of 0.4 MPa, and a residence time of 15 min—give percent yields of 8%, 9%, and 8.8%. Determine if a first-order empirical model is appropriate for this system at \(\alpha = 0.05\).
8. Duarte and colleagues used a factorial design to optimize a flow-injection analysis method for determining penicillin [Duarte, M. M. M. B.; de O. Netro, G.; Kubota, L. T.; Filho, J. L. L.; Pimentel, M. F.; Lima, F.; Lins, V. Anal. Chim. Acta 1997, 350, 353–357]. Three factors were studied: reactor length, carrier flow rate, and sample volume, with the high and low values summarized in the following table.
| factor | high (+1) level | low (–1) level |
|---|---|---|
| X: reactor length | 1.3 cm | 2.0 cm |
| Y: carrier flow rate | 1.6 mL/min | 2.2 mL/min |
| Z: sample volume | 100 μL | 150 μL |
The authors determined the optimum response using two criteria: the greatest sensitivity, as determined by the change in potential for the potentiometric detector, and the largest sampling rate. The following table summarizes their optimization results.
| run | X* | Y* | Z* | \(\Delta E\) (mV) | sample/h |
|---|---|---|---|---|---|
| 1 | –1 | –1 | –1 | 37.45 | 21.5 |
| 2 | +1 | –1 | –1 | 31.70 | 26.0 |
| 3 | –1 | +1 | –1 | 32.10 | 30.0 |
| 4 | +1 | +1 | –1 | 27.30 | 33.0 |
| 5 | –1 | –1 | +1 | 39.85 | 21.0 |
| 6 | +1 | –1 | +1 | 32.85 | 19.5 |
| 7 | –1 | +1 | +1 | 35.00 | 30.0 |
| 8 | +1 | +1 | +1 | 32.15 | 34.0 |
(a) Determine the coded equation for the response surface where \(\Delta E\) is the response.
(b) Determine the coded equation for the response surface where sample/h is the response.
(c) Based on the coded equations in (a) and in (b), do conditions that favor sensitivity also improve the sampling rate?
(d) What conditions would you choose if your goal is to optimize both sensitivity and sampling rate?
9. Here is a challenge! McMinn, Eatherton, and Hill investigated the effect of five factors for optimizing an H2-atmosphere flame ionization detector using a 25 factorial design [McMinn, D. G.; Eatherton, R. L.; Hill, H. H. Anal. Chem. 1984, 56, 1293–1298]. The factors and their levels were
| factor | high (+1) level | low (–1) level |
|---|---|---|
| A: H2 flow rate | 1460 mL/min | 1382 mL/min |
| B: SiH4 | 20.0 ppm | 12.2 ppm |
| C: O2 + N2 flow rate | 255 mL/min | 210 mL/min |
| D: O2/N2 ratio | 1.36 | 1.19 |
| E: electrode height | 75 (arb. unit) | 55 (arb. unit) |
The coded (“+” = +1, “–” = –1) factor levels and responses, R, for the 32 experiments are shown in the following table
| run | A* | B* | C* | D* | E* | R | run | A* | B* | C* | D* | E* | R |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | – | – | – | – | – | 0.36 | 17 | – | – | – | – | + | 0.39 |
| 2 | + | – | – | – | – | 0.51 | 18 | + | – | – | – | + | 0.45 |
| 3 | – | + | – | – | – | 0.15 | 19 | – | + | – | – | + | 0.32 |
| 4 | + | + | – | – | – | 0.39 | 20 | + | + | – | – | + | 0.25 |
| 5 | – | – | + | – | – | 0.79 | 21 | – | – | + | – | + | 0.18 |
| 6 | + | – | + | – | – | 0.83 | 22 | + | – | + | – | + | 0.29 |
| 7 | – | + | + | – | – | 0.74 | 23 | – | + | + | – | + | 0.07 |
| 8 | + | + | + | – | – | 0.69 | 24 | + | + | + | – | + | 0.19 |
| 9 | – | – | – | + | – | 0.60 | 25 | – | – | – | + | + | 0.53 |
| 10 | + | – | – | + | – | 0.82 | 26 | + | – | – | + | + | 0.60 |
| 11 | – | + | – | + | – | 0.42 | 27 | – | + | – | + | + | 0.36 |
| 12 | + | + | – | + | – | 0.59 | 28 | + | + | – | + | + | 0.43 |
| 13 | – | – | + | + | – | 0.96 | 29 | – | – | + | + | + | 0.23 |
| 14 | + | – | + | + | – | 0.87 | 30 | + | – | + | + | + | 0.51 |
| 15 | – | + | + | + | – | 0.76 | 31 | – | + | + | + | + | 0.13 |
| 16 | + | + | + | + | – | 0.74 | 32 | + | + | + | + | + | 0.43 |
(a) Determine the coded equation for this response surface, ignoring \(\beta\) terms less than \(\pm 0.03\).
(b) A simplex optimization of this system finds optimal values for the factors of A = 2278 mL/min, B = 9.90 ppm, C = 260.6 mL/min, and D = 1.71. The value of E was maintained at its high level. Are these values consistent with your analysis of the factorial design.
10. A good empirical model provides an accurate picture of the response surface over the range of factor levels within the experimental design. The same model, however, may yield an inaccurate prediction for the response at other factor levels. For this reason, an empirical model, is tested before it is extrapolated to conditions other than those used in determining the model. For example, Palasota and Deming studied the effect of the relative amounts of H2SO4 and H2O2 on the absorbance of solutions of vanadium using the following central composite design [data from Palasota, J. A.; Deming, S. N. J. Chem. Educ. 1992, 62, 560–563].
| run | drops of 1% H2SO4 | drops of 20% H2O2 |
|---|---|---|
| 1 | 15 | 22 |
| 2 | 10 | 20 |
| 3 | 20 | 20 |
| 4 | 8 | 15 |
| 5 | 15 | 15 |
| 6 | 15 | 15 |
| 7 | 15 | 15 |
| 8 | 15 | 15 |
| 9 | 22 | 15 |
| 10 | 10 | 10 |
| 11 | 20 | 10 |
| 12 | 15 | 8 |
The reaction of H2SO4 and H2O2 generates a red-brown solution whose absorbance is measured at a wavelength of 450 nm. A regression analysis on their data yields the following uncoded equation for the response (absorbance \(\times\) 1000).
\[R = 835.90 - 36.82X_1 - 21.34 X_2 + 0.52 X_1^2 + 0.15 X_2^2 + 0.98 X_1 X_2 \nonumber \]
where X1 is the drops of H2O2, and X2 is the drops of H2SO4. Calculate the predicted absorbances for 10 drops of H2O2 and 0 drops of H2SO4, 0 drops of H2O2 and 10 drops of H2SO4, and for 0 drops of each reagent. Are these results reasonable? Explain. What does your answer tell you about this empirical model?