Thus, we were able to confirm our experimental findings with a mathematical derivation based on the physics of Beer's law.
While we will continue our discussion about the effects of curvature of the relationship between different units for concentration
in the next column, for now we will finish off with one other point of view. As described above, only the volume fraction
agrees with the spectroscopically determined fractions. This is an empirically determined fact, backed by theory, and is a
critically important one. The matter is not discussed in the spectroscopic literature, but there seems to have been a tacit
assumption made by spectroscopists, and even analytical chemists in general, that different measures of what we consider "concentration"
are equivalent to each other, differing at most by perhaps a scaling factor.
As we see in Figures 4–6, however, different units of measure have different relationships to each other, usually relationships
that are not linear with respect to each other. As pointed out earlier, if a given unit of measure is linear with respect
to the spectroscopy, then other units of measure, which are not linear with respect to that unit of measure, will ipso facto also not be linear with respect to the spectroscopy. For this reason, knowing which unit of measure agrees with the spectroscopy
is a key point, and the empirical finding that the volume fraction is the one that agrees with the spectroscopy is crucial.
There is a difficulty in dealing with volume fractions, however, in that it is rarely used as the "natural" unit for expressing
concentrations in chemical analysis. On the other hand, the physics of the interactions between light and matter doesn't depend
on the mathematical or chemometric calculations applied to spectral data to perform the chemical analysis with. Therefore,
while the use of CLS was necessary to determine what physical sample properties conform to the spectroscopy, all other methods
of chemometric analysis are subject to the same effects.
Calibration algorithms other than CLS don't require knowledge of all the components in a given sample; most only require knowledge
of the concentration of the one component of interest (that is, the analyte). But now we see that the units that the analyte
is expressed in makes a difference. So the question arises: Are there any other units to express the analyte concentration
in that would also be linear with respect to the spectroscopic measurements, preferably while also being more familiar to
the chemist? These other quantities would not necessarily satisfy the more stringent requirements of the CLS algorithm, but
would be expected to be linearly related to the volume percent concentration, and therefore provide a satisfactory basis for
expressing the analyte concentration for the more common calibration algorithms, that do indeed require the concentration
for the calculations.

The answer is, yes, there are other units. Consider, for example, using the unit of measure of weight per unit volume (wt/vol).
Expressing volume fraction as vol/vol_{tot} we can then perform the calculation of multiplying the numerator of the volume fraction by the weight. This clearly makes
the wt/vol a scaling factor applied to the volume fraction, as expressed in equation 10:
Canceling the two volume terms in the numerator of equation 10 results in
Thus, weight per unit volume is seen to be a unit for concentration measure that will be linear with the spectroscopic data
because it is one way to scale volume fractions.
Other scaling factors can also be used. For example, molarity is a chemical measure of concentration whose units are moles
per volume. This can be similarly introduced into a volume fraction expression to form a different concentration measure that
is a scaled variation of volume fraction:
By induction, it becomes clear that any measure of concentration that is expressed as "quantity per unit volume" can be used
as a measure of analyte concentration that will be linear with respect to the spectroscopic properties of that analyte. (Note:
We extend our thanks to Jim Brown for pointing this relationship out to us.)
When these units of measure are used to describe the concentration of the analyte, then indeed, any of the conventional calibration
algorithms will implicitly incorporate whatever scaling factors are needed in the calibration coefficients, to perform the
conversions between the spectroscopic quantities measured, and the units in which the analyte concentration is expressed.
If other units are used, however, then a single "scaling factor" will not suffice to accommodate the nonlinear relations between
the different units of measure.
