9: Optimizing Data

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In the presence of H₂O₂ and H₂SO₄, a solution of vanadium forms a reddish brown color that is believed to be a compound with the general formula (VO)₂(SO₄)₃. The intensity of the solution’s color depends on the concentration of vanadium, which means we can use its absorbance at a wavelength of 450 nm to develop a quantitative method for vanadium. The intensity of the solution’s color also depends on the amounts of H₂O₂ and H₂SO₄ that we add to the sample—in particular, a large excess of H₂O₂ decreases the solution’s absorbance as it changes from a reddish brown color to a yellowish color [Vogel’s Textbook of Quantitative Inorganic Analysis, Longman: London, 1978, p. 752.]. Developing a standard method for vanadium based on this reaction requires that we optimize the amount of H₂O₂ and H₂SO₄ added if we want to maximize the absorbance at 450 nm. Using the terminology of statisticians, we call the solution’s absorbance the system’s response. Hydrogen peroxide and sulfuric acid are factors whose concentrations, or factor levels, determine the system’s response. To optimize the method we need to find the best combination of factor levels. Usually we seek a maximum response, as is the case for the quantitative analysis of vanadium as (VO)₂(SO₄)₃. In other situations, such as minimizing an analysis’s percent error, we seek a minimum response. How we design experiments to optimize the response is the subject of this chapter.