Classical Least Squares, Part IV: Spectroscopic Theory Continued

In this month's column we examine, in further detail, the connection between the mathematics of classical least squares and the graphical displays that are conventionally used to present it.
Feb 11, 2011
Volume 26, Issue 2

This column is a continuation of our discussion of the classical least squares (CLS) approach to calibration (1–3). As we usually do when we continue the discussion of a topic through more than one column, we continue the numbering of equations from where we left off in the last installment.

The insight we are trying to develop hinges on Figure 1 (which first appeared as Figure 5 in Part III of this series [3]) and the meaning of it in terms of equating the concepts of the spectroscopic and mathematical views of Beer's law as it applies to spectra measured for the purpose of calibrations for quantitative analysis. Therefore, we also repeat equation 24:

where [M], [D], [T], and [H] represent the spectra (which are now vectors, in this representation) of the mixture, dichloromethane, toluene, and n-heptane, respectively, and the corresponding c 1, c 2, and c 3 represent the concentrations of dichloromethane, toluene, and n-heptane, respectively.

Figure 1: Absorbance spectra of pure dichloromethane, n-heptane, and toluene, and of a ternary mixture of the three (x-axis in wavenumbers), along with the concentration information that makes this the graphical representation of equation 24.
The reason for all this is the final sentence in our previous column (3): "Figure 5 is where the spectroscopy meets the math." We also repeat some of the prior explanation: In this figure we have taken Figure 4 from Part III of this column series and added some symbols to it. We now compare the figure (Figure 1 in this installment) to equation 24. In equation 24, we represented the spectra of each of the components of the mixture by a symbol ([D], [T], [H]), each representing the corresponding spectrum.

In Figure 1 we have effectively rewritten equation 24 by replacing the symbol representing each spectrum by the actual spectrum.

That's why Figure 1 is where the spectroscopy meets the math. Figure 1 is the same as equation 24, except that the spectra, which are indicated by the matrix symbols [D], [T], and [H] in equation 24, are shown in their conventional graphical form in Figure 1.

Table I
This can be even further emphasized by replacing the graphical presentation of the spectra with the actual numbers they represent, as we just described them. In Table I we present the numbers that make up the four spectra that concern us here: the spectra of the mixture of toluene, dichoromethane, and n-heptane, and the absorbances of the three pure materials, whose spectra are presented in Figure 1.

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