The regression equation from Investigations 22 and 23 describes the empirical model of danshensu's extraction yield for extraction times in the range 2.18–7.82 min and for solvent-to-solid ratios in the range 10.9–39.1 mL/g. Nevertheless, it is difficult to look at the equation and predict the extraction time and the solvent-to-solid ratio that maximizes danshensu's extraction yield; it is difficult, as well, to look at the regression equation and determine how sensitive is the optimum extraction yield to a small change in extraction time or solvent-to-solid ratio.
The factor levels that give the optimum extraction yield and the sensitivity of the extraction yield to a small change in factor levels are easier to visualize if we display the results as a three-dimensional plot (or a pseudo-3D plot) with extraction yield on the z-axis and extraction time and the solvent-to-solid ratio on the x-axis and the y-axis, respectively. Figure 10 is one such a plot, which overlays a contour map of equivalent extraction yields on a heatmap that displays extraction yields using a variation in color. We call this type of plot a response surface.
Does Figure 10 agree with your results from Investigation 21 and Investigation 23? Why or why not? Estimate the optimum conditions for maximizing danshensu's extraction yield and explain your reasoning. How sensitive is the optimum extraction yield to a small change in extraction time? How sensitive is the optimum extraction yield to a small change in the solvent-to-solid ratio?
Figures 11-15 show response surfaces for lithospermic acid, salvianolic acid A, cryptotanshinone, tanshinone I, and tanshinone IIA, each based on a regression analysis of data similar to that in Table 2 for danshensu. The regression models for rosmarinic acid and for dihydrotanshinone are not significant, although the extraction of rosmarinic acid increases slightly for larger solvent-to-solid ratios and the extraction of dihydrotanshinone decreases slightly at longer extraction times; we will assume, however, that their extraction yields are independent of the extraction time and the solvent-to-solid ratio, with values of 2.317 mg/g for rosmarinic acid and 0.424 mg/g for dihydrotanshinone.
Using Figures 11–15, determine the optimum extraction time and solvent-to-solid ratio for lithospermic acid, salvianolic acid A, cryptotanshinone, tanshinone I, and tanshinone IIA. How sensitive is the extraction of each analyte to a small change in the optimum extraction time and in the optimum solvent-to-solid ratio? Considering your responses here and to Investigation 24, are there combinations of extraction times and solvent-to-solid ratios that will optimize the extraction yield for all six of these analytes?