Page:NIOSH Manual of Analytical Methods - 3800.pdf/26

ORGANIC AND INORGANIC GASES by FTIR Spectrometry: METHOD 3800, Issue 1, dated 15 March 2003 - Page 26 of 47 $$E^2={\sum_{t=1}^N}{(e_t)}^2={\sum_{t=1}^N}{[{\sum_{j=1}^M}({L_S}{a_{ij}}{D_j}) -{A_t}]}^2$$ (Equation C5)

where N represents the number of absorbance values in the analytical region. Reference 16 demonstrates that 1) for N > M there is a unique set of estimated concentrations $$D_j$$ which minimizes the estimated squared error; 2) this set of values is calculable from the known quantities in Equations C1 through C5; and 3) estimates $$\sigma_j$$ of the uncertainties in the quantities $$D_j$$ are also calculable from the same quantities. The value 3\sigma_j is generally accepted as a conservative estimate of the statistical uncertainty in the related estimated LSF concentration (see Reference 3).

The estimated LSF error at each point in the analytical region,

$${e_i}={A_i}{\sum_{j=1}^M}{L_S}{a_ij}{D_j}$$ (Equation C6)

is usually stored following the analysis as a “residual spectrum,” which can provide an estimate of the LODs for other compounds. In addition, the residual spectrum and the concentration uncertainties can allow the analyst to detect and identify compounds which are actually present in the sample gas but which were not included in the mathematical analysis. Appendix E provides an example illustrative of these procedures.

The above description illustrates a simple and easily-interpreted LSF analysis. More sophisticated LSF analytical techniques, possibly more accurate for particular types of samples, are described in the literature (see, for example, Reference 18 and references therein).

C8

Equations C1 through C6 demonstrate the importance of quantities $$L_S$$ (the absorption pathlength) and $$a_{ij}$$ (the absorptivity) in FTIR spectrometry. Accurate determinations of these quantities allow the use of reference libraries for quantitiative analyses without the necessity of compound-specific field calibrations. The system tests described in the procedures and in Appendix B are intended to ensure suitability of the system configuration for such calibration transfers, as are the requirements of obtaining CTS spectra in field. Appendix D describes procedures for recording and processing reference library spectra.

C9

Beer’s Law is based on fundamental, well-established physical principles. It holds absolutely for gas samples which are at thermal equilibrium and dominated by induced (rather than spontaneous) emission and absorption processes. (See Note A1 below). However, this is not to say that the absorbance, as measured by an FTIR spectrometer, follows Beer’s Law under all conditions. Deviations from Beer’s Law in FTIR spectra are often observed; however, they indicate inaccuracies in the FTIR spectra, not “violations” of Beer’s Law. For example, deviation from Beer’s Law is commonly exhibited by sets of single-component reference spectra recorded over a range of absorbance levels. At large enough values of the absorbance, the points $$A_i$$ of stronger absorption bands of such spectra no longer increase linearly with the concentration-pathlength product $${L_R}{C_j}$$; this is why Table 2 specifies a maximum ppm-m value for the listed reference spectra. If the assumption of detector linearity does not hold (see Appendix B), similar effects are often present in reference and sample spectra; this is the basis for the system test described in Appendix B, Section 6.

Mathematical correction of the concentration estimates $$D_j$$ derived from Beer’s Law can often reduce the error in sample analyses when either type of non-linear effect occurs. Figure C7 provides an example of such a correction. The actual and calculated ppm-m values for a set of reference spectra are plotted against each other; a “piece-wise linear” approximation to the pattern is shown by the solid line, and the dashed line indicates the ideal linear behavior based on the spectrum of lowest absorbance. At any ppm-m value indicated in a Beer’s Law sample analysis (that is, for any y-axis value up to approximately 900 ppm-m in the example), reasonably accurate values are available from the corresponding x-axis position of the solid line. If the analyst NIOSH Manual of Analytical Methods, Fourth Edition