Methods to Include Beer's Law Nonlinearities in Quantitative Spectral Analysis

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Abstract
Multivariate least-squares methods have been recently applied to the quantitative infrared analysis of multicomponent samples. Many deviations from Beer's law can be approximated in linear least-squares analyses by using a calibration set which consists of spectra of known mixtures and by adding a nonzero intercept to Beer's law. More recently, methods which specifically include nonlinear terms in the relationship between concentration and absorbance have been applied to samples whose spectra exhibit Beer's law deviations. Using these methods, reductions in relative errors have been achieved for several nonlinear systems compared to the results of least-squares methods which include only the linear terms. An alternate method to improve the quantitative accuracy in the presence of Beer's law deviations involves using full-spectrum residuals. Since the multivariate least-squares analysis generates estimated pure-component spectra from a calibration set of known mixture spectra, residual spectra can be generated by subtracting the appropriate amounts of the estimated spectra from the known mixture spectra. The residual spectra, which represent the nonlinearities from Beer's law that are not accurately approximated in the multivariate least-squares analysis, are then used as additional terms to improve the least-squares fit of the unknown sample spectrum. Empirically this method has been also found to improve relative errors. These methods have been applied to two systems which exhibit nonlinear Beer's law behavior, that is, binary mixtures of esters and low-resolution gas-phase spectra of nitrous oxide (N2O) in nitrogen (N2). Demonstration of the methods and a comparison of the accuracies of the different techniques are made for these two systems.