A Short Review
In the first pair of columns in the previous subseries (1,2) we looked at the CLS analysis of water–methanol–acetic acid mixtures
and found that the values computed using the CLS algorithm did not agree with the known amounts of the mixture components
introduced in the mixtures. In that experiment, we also found that the reconstruction of the mixture spectrum from the spectra
of the pure components was rather poor. Based on the poor reconstructions of the mixture spectra, we attributed the inaccurate
analytical results obtained to the known changes in hydrogen bonding because of intermolecular interactions between the three
components, interactions that distorted their spectra.
Therefore, starting with part III of the previous subseries (3), we redid the experiment using hydrocarbons that would not
interact as the materials in the previous experiment did because the materials used for the second experiment did not contain
any –OH in their structure. With these mixture components we found that indeed, we were able to reproduce the spectra of the
mixtures fairly well.
Lo and behold, despite this ability to reproduce the spectra, we still found large disagreements between the CLS calculations
and the accurately known values for the mixture compositions. Considerable sweat was expended trying to reconcile the differences,
a process that was recounted over the last few columns.
The starting point for thinking about this is the behavior of the CLS algorithm when applied to data that follow Beer's law.
The CLS algorithm is the nearest thing we have in spectroscopy to an absolute method, in that we start with the spectra of
the pure components of a mixture and use those to recreate the spectrum of the mixture; we described this above. The coefficients
obtained directly represent the contribution of each of those components to the spectrum of the mixture. There is no room
for fudging here. A given set of spectra (that is, pure components and the mixture) admits of only one solution, where each
coefficient represents the fraction of the corresponding pure component spectrum that contributes to the final mixture spectrum.
Thus, the CLS algorithm is concerned only with the relationships between spectra; it says nothing about the relationships
between the spectra and the concentrations of the components of the mixture. Therefore, while the CLS algorithm is developed
from Beer's law, and requires that Beer's law holds for the materials involved, CLS is not Beer's law.
There is also the question of units. Absorption is dimensionless. A counterargument that has been raised is that in Beer's
law, any units for the concentration get taken up in the absorption coefficient. This is true, but beside the point. The point
is that the absorption is dimensionless, because it is the logarithm of another dimensionless quantity, the sample transmittance,
and it is the absorbance that is used in the CLS algorithm. This is another aspect of the difference between Beer's law and
the CLS algorithm. It is immaterial that if an arbitrary unit is used for the concentration, the absorbance coefficients must
have units that cancel those concentration units. Because the absorbance is dimensionless, the coefficients calculated for
the absorbances by the CLS algorithm are also dimensionless quantities, and those are the fractional amounts of the corresponding
purecomponent spectra that each contributes to the mixture spectrum.
The question then becomes, what physical quantities do those spectral fractions correspond to? In this view, being linearly
related to the spectral absorbance is a necessary, but insufficient, condition. The quantity that the concentration is expressed
in and the numerical values for the concentration the CLS algorithm provides for the spectroscopic contribution of each component
spectrum to the mixture spectrum have to be in agreement. From the experiments we did, that physical quantity is the volume
fraction. It is true that for purposes of conventional calibration you can scale those values by multiplying them by the density,
molecular weight, or some other quantity, but that does not change the nature of the fundamental interaction between the light
and the samples, which does seem to be related to the volume fraction.
What we found was that the physical property that agreed with the spectral results is the volume fractions of the components
of the mixtures. This leads to some insights into what is happening.
Figure 4: Relationship between weight fractions and volume fractions for toluene in ternary mixtures. The multiple lines represent
different relative amounts of the other two components (dichloromethane and nheptane with compositions at 20% intervals).

We noted in the previous subseries in part X, Table I (9) (as well as in Table IV in part XI [11]) that there is not a unique
conversion between concentration values expressed in different units.
Figure 5: Relationship between weight fractions and volume fractions for dichloromethane in ternary mixtures The multiple
lines represent different relative amounts of the other two components (toluene and nheptane with compositions at 20% intervals).

There is another aspect to the relationships between different units, which we demonstrate in Figures 4–6. In those figures,
we see that the relationship between the different units (weight fraction and volume fraction) is not a linear relationship.
Furthermore, those plots also show that the relationship is not unique. For example, a vertical line drawn at any value of
volume fraction can intersect any of the lines, and therefore can correspond to different weight fractions of the analyte,
depending on the composition of the remainder of the sample (that is, relative amounts of the other components). Similarly,
a horizontal line drawn at any value of weight fraction can intersect any of the lines, showing the correspondence to the
different volume fractions, again depending on the composition of the rest of the sample. This is the same effect observed
in Table I from part IX and Table II from part XI (from the previous columns); the main difference is that the numeric values
presented in those tables correspond to the actual samples that were made up for the experiment, while Figures 4–6 indicate
the nature and magnitude of the effects for essentially all possible samples.
Furthermore, there is another, more subtle effect that can be observed in these figures, although it is perhaps easier to
see this in Figures 4 and 6 than in Figure 5. Note that the curved lines in each figure are not equally spaced despite the
fact that the compositions of the remainder of the samples differ by even 20% intervals. This underscores the fact that not
only is the main analyte in each graph exhibiting a nonlinear relationship between the different units, but this nonlinearity
exists even between the different solutions of the sample.
Figure 6: Relationship between weight fractions and volume fractions for nheptane in ternary mixtures. The multiple lines
represent different relative amounts of the other two components (toluene and dichloromethane with compositions at 20% intervals).

We will have more to say about these effects in our next column.
