Page 200 - Academic Press Encyclopedia of Physical Science and Technology 3rd Analytical Chemistry

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Encyclopedia of Physical Science and Technology EN007C-340 July 10, 2001 14:45

Infrared Spectroscopy 811

spectrum, the total absorbance is equal to the sum of the This full-spectrum method improves the precision over
abc values for each component. One can specify that all those that use only a few wave numbers. Corrections can
the measurements will be done in the same cell, so the be added for Beer’s law deviations or fitting spectral base-
thickness is constant for the whole analysis. This means lines. However, all components present must be included
that b, the cell thickness, can be combined with a, the in the calibration mixtures.
absorbtivity, to give a new constant k, which replaces ab In a variation of this type of analysis, the concentra-
in Beer’s law. If there are three components, 1, 2, and 3, tions are expressed as functions of the various absorbances
absorbing at a specific wave number, then Beer’slaw is rather that vice-versa as before. This is called the inverse
least squares (ILS) (or the Pmatrix method). An advantage
A = k 1 c 1 + k 2 c 2 + k 3 c 3 . (24)
of this method is that a quantitative analysis can be per-
In order to measure these three concentrations in an un- formed on some components using calibrated standards,
known, three absorbances are needed at three different even if some other components with unknown concentra-
wave numbers. tions are present in the standards in amounts bracketing

A 1 = k 11 c 1 + k 12 c 2 + k 13 c 3 (25a) those in the samples. A disadvantage is that it is not a full-
spectrum method. In the analysis, there must be at least
A 2 = k 21 c 1 + k 22 c 2 + k 23 c 3 (25b)
as many standards for calibration as there are analytical
A 3 = k 31 c 1 + k 32 c 2 + k 33 c 3 . (25c) wave numbers used.
Two factor analysis methods that are used are the prin-
Here, A 1 , A 2 and A 3 absorbances at wave numbers 1, 2,
cipal components regression (PCR) and the partial least
and 3, where concentrations of components 1, 2, and 3
squares (PLS). In the PRC method, the concentrations are
are best measured. In k 12 for example, the first subscript
expressed as functions of the principal components (PC)
is for wave number 1 and the second subscript is for the
instead of absorbances as in ILS. The PC are orthogonal
concentration of component 2. Three standards (std.) are
vectors that are linear combinations of the original spectral
prepared with known concentrations with suitable ranges
data of the standards. Here, PC1 accounts for the maxi-
for the analysis. These are all run at the first analytical
mum variability in the data, and PC2 accounts for the
wave number to give three A 1 equations.
maximum variability not accounted for by PC1, etc. The
(std. 1) A 11 = k 11 c 11 + k 12 c 21 + k 13 c 31 (26a) other method PLS, is similar to the PCR method except
(std. 2) A 12 = k 11 c 12 + k 12 c 22 + k 13 c 32 (26b) that the PCs are weighted. The weighting is based on the
correlation of the PCs with concentration. These are full-
(std. 3) A 13 = k 11 c 13 + k 12 c 23 + k 13 c 33 . (26c)
spectrum methods like CLS, but like ILS, one can analyze
Here, the second subscript on A and c is for the standard one component at a time. These methods are most often
number. For these three equations, where the A values used for quantitative analysis in the near infrared region
and the nine concentrations (C 11 etc.) in the standards because of the broadness and overlapping nature of the
are known, the three unknown k values can be evaluated. bands here.
This same procedure is used for the A 2 and A 3 equations
(25b and c) to evaluate all nine of the unknown k values. VI. GROUP FREQUENCIES
Once these are known, the three equations (25a, b, and c)
can be used to calculate all the unknown concentrations
A. Concept of Group Frequencies
(c 1 , c 2 , and c 3 ) in a sample from the measured ab-
sorbencies. This is called the method of simultaneous Bands at certain frequencies in the IR spectra have been
equations. While this method is straightforward, the dis- related to the presence of certain functional groups in
advantages are that Beer’s law nonlinearities are difficult the chemical structures. For example, in the spectra of
to handle and all the components in the mixture must be a series of unconjugated ketones, a band common to
−1
accounted for. Also, only a limited number of analytical all is a strong band near 1715 cm . This has been as-
wave numbers can be used. signed to the stretching of the carbony1 bond and is
For use in repetitive analyses, these quantitative meth- a group frequency for unconjugated ketones. There are
ods have been computerized. In the example discussed, many other bands in these ketone spectra, which differ
−1
the absorbances are expressed as functions of the various from molecule to molecule, especially below 1300 cm .
concentrations. A computerized version of this is called These are fingerprint-type bands that can be used to distin-
the classical least squares (CLS) (or the Kmatrix method), guish one ketone from another. A large body of empirical
as it gives the least squares prediction for the concentra- knowledge has been built up about the characteristic group
tions. This method can use many more than the minimum frequencies, which has proved to be very useful to the
number of analytical wave numbers, or even all of them. chemist. The vibrations that give rise to group frequencies