Now that we have shown the relationships between different units for concentration, we continue by demonstrating their effects on the data we collected and used for our examples. We also begin our discussion on the ramifications and consequences of our findings.
This column is the next continuation of our discussion of units on calibration, as described in part I of this series (1) and examined through the use of the classical least squares (CLS) approach to calibration (2–13). In this column, we continue the numbering of equations, figures, and tables from where we left off in part I (1).
In our previous columns (1,9) we confirmed that the volume percent is the physical quantity that agrees with the spectroscopic evaluation of the contribution of the components of a mixture to the spectrum of the mixture. We also demonstrated that the nature of the CLS algorithm allowed us to determine two important properties of the conversion of "concentration" between different units.
We determined in Table I from part X of the previous subseries (11), as well in Table IV from part XI (12), that there is not a unique conversion between concentration values expressed in different units. We also showed, in Figures 1–3 from part I of this subseries (1), that there is not a linear relationship between concentration when expressed in different units.
Thus, the data (Figures 1–3 from part I of this subseries and Figure 7–9 here) show that different units of measurement have different relationships to the spectral values, for reasons having nothing to do with the spectroscopy. One of the conclusions of this finding is that it disproves the usual, although inevitably unstated, assumption that different measures of concentration are equivalent except, perhaps, for a constant scaling factor. Furthermore, it is clear that if two measures of concentration are nonlinear with respect to each other, then a third measurement, such as a spectroscopic measurement, that is linear with respect to one measure must be equally nonlinear with respect to the other measure.
Figure 7: Plot of CLS values versus weight percent and mole percent, for toluene. (a) Weight percent versus CLS values; (b) volume percent versus CLS values.

We also showed how a concentration measurement unit can be constructed so that it is indeed equivalent to the volume percent, except for a scaling factor. The key to this conversion is to multiply the volume percentage by a unit that has a volume in the denominator — examples include weight/unit volume and molarity (moles/unit volume).
Figure 8: Plot of CLS values versus weight percent and mole percent, for dichloromethane. (a) Weight percent versus CLS values; (b) volume percent versus CLS values.

One of the more common measures of concentration used in conjunction with spectroscopic analysis, however, is weight percent (that is, weight/unit weight, which is not included among the measurement units that are equivalent to volume percent). Table I from part X (11), as we previously noted, shows that a value for volume percent of a given component can correspond to a wide range of weight percentages and vice versa: A given value of weight percent can correspond to a correspondingly wide range of volume percents. Here, Figures 7–9 show the same effect: Many values for concentration calculated from the spectra using the CLS algorithm, which stands as a surrogate for the volume fraction, correspond to a given value of the concentration expressed as weight percent units. This variability does not depend on the concentration of the analyte, but instead on the composition of the rest of the mixture, affecting the volume fraction and spectroscopic concentrations, whereas the weight percents are essentially constant at each level (note that they are not exactly constant, because of the use of the "dispense approximately then measure exactly" method of making the samples). This nonequivalency has nothing to do with the spectroscopy; it is purely a matter of elementary physical chemistry, and is the source (or at least one of the sources) of what we used to call matrix effect in undergraduate analysis courses. In those courses, matrix effect was typically considered small, but we see in Table I from part X (11) and in Figures 7–9 that it can be, and in our experiments indeed is, very large — being, in our case, a source of errors as large as 5–10%. This error source is larger than any other we typically encounter in spectroscopic analysis. It is much larger than any of the usual laboratory error sources (a laboratory showing such poor performance would be rejected for consideration as a reference laboratory), larger than virtually any instrumental error, and larger than any other error source we normally encounter.
Figure 9: Plot of CLS values versus weight percent and mole percent, for nheptane. (a) Weight percent versus CLS values; (b) volume percent versus CLS values.

Yet this error source has previously been hidden and undetected, despite its role as the largest error source in our chemometric calibration work. When you think about it, you realize that one of the problems a calibration algorithm has to solve when confronted with data where the "wrong" units are used for the analyte values, is how to determine the value of that analyte when two (or more) samples are described as having different values of the analyte by the scientist performing the calibration (that is, different reference values) while the spectroscopic data are telling the algorithm that they are the same. And, of course, the opposite situation invariably also occurs, that the spectroscopy indicates that the analyte concentrations are the same, while the reference values indicate that they are different.