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ORGANIC AND INORGANIC GASES by FTIR Spectrometry: METHOD 3800, Issue 1, dated 15 March 2003 - Page 25 of 47 2) Measure the absorbance $$A_i$$ for the mixture of compounds (see Steps 5 and 9 above).

3) Determine the pathlength $$L_S$$ for the current measurement of $$A_i$$ (see Steps 5 and 7 above).

4) Select the analytical region—that is a set of frequencies, corresponding to the possible values of the index i—which are to be used to determine the concentration of each compound, and then mathematically invert Equation C1 to determine the desired concentrations $$C_j$$. (Appendix E addresses the topic of spectral analysis in detail.)

NOTE: The true absorptivity for a single gaseous compound is a characteristic only of the compound's structure. However, details of the FTIR system performance and operation affect the observed absorptivity and its accuracy. Similarly, FTIR measurements provide only an approximation of the true absorbance spectrum of a mixture of gaseous compounds, though it is, under many circumstances, a sufficiently accurate approximation. It is the responsibility of the analyst to verify and ensure that the reference and sample spectra provide a sufficiently accurate quantitative analysis according to Beer's Law. The following sections of this Appendix describe the mathematics of such an analysis. Appendix D addresses the topics of developing and using reference spectral libraries. Appendix E provides an illustrative example of the design and evaluation of the quantitative analytical process.

C7. .

When a sample gas contains only one absorbing compound, Equation C1 simplifies to

$$A_i={L_S}{a_{ij}}C_j$$ (Equation C2)

This means that in any analytical region where only one gas absorbs, any one (of the usually many) absorbance spectrum values $$A_i$$ can be used to yield the concentration $$C_j$$.

The absorbance area $$A_s$$ for single-component spectrum in an analytical region (from i = p to i = q) can be written as

$${A_S}={\sum_{i=p}^{i=q}}A_i={\sum_{i=p}^{i=q}}{L_S}{a{ij}}{C_j}={L_S}{C_j}{\sum_{i=p}^{i=q}}{a_{ij}}={L_S}{C_j}{A_R} $$ (Equation C3)

where $$A_R$$ is the area in the reference spectrum for that compound in the same analytical region. (This is the basis of the absorption path length $$L_S$$ calculation described in Step 7 and Appendix B, Section 1.) Because calculation of the absorbance area involves many points in the sample spectrum, Equation C3 leads to much more accurate results than the single-point calculation represented by Equation C2.

However, when many absorbing compounds are present in a sample, the absorption patterns of the various compounds often overlap. In this case, there is usually not an isolated analytical region for each compound in which only that compound absorbs infrared radiation; no single absorbance point and no simple absorbance area is suitable for determining any of the component concentrations. In this case, the simplest method for determining concentrations is to use a least square's fitting (LSF) algorithm.

LSF algorithms use the fact that there is some set of estimated concentrations $$D_j$$ which minimizes the "squared error" in Beer's Law for any given analytical region, for any set of compounds. The only requirement on the chosen analytical region is that it must contain a sufficient number of data points; since each FTIR spectrum contains many thousands of absorbance values, this requirement is nearly always fulfilled. If we use the estimated concentrations $$D_j$$ (rather that the true concentrations $$C_j$$) in Beer's law, they will lead to some estimated error $$e_i$$ at each value of i (that is, at each point in the analytical region we choose). Equation C1 becomes:

$${A_i}={e_i}+{\sum_{j=1}^M}{L_S}{a_{ij}}{D_j}$$ (Equation C4)

The estimated squared error (or "variance") in Beer's Law using the estimated concentrations is: NIOSH Manual of Analytical Methods, Fourth Edition