Lambert-Beer model

Figure 2.4 PCA model and real chemical Beer-Lambert model for a Raman emulsion image.

A Modified Beer-Lambert Model of Skin Diffuse Reflectance for the Determination... 

By far, the most widely used model in calculating hemodynamic response is based on the classic Beer-Lambert law. The Beer-Lambert law is derived from solution to radiation transport equation under several simplifying assumptions [91]. It describes a linear relationship between absorbance, A, of light through a medium and wavelength dependent extinction coefficient, e(A). This relationship is given by Equation (1)... 

Only multivariate (e.g. multi-wavelength) data are amenable to model-free analyses. While this is a restriction, it is not a serious one. The goal of the analysis is to decompose the matrix of data into a product of two physically meaningful matrices, usually into a matrix containing the concentration profiles of the components taking part in the chemical process, and a matrix that contains their absorption spectra (Beer-Lambert s law). If there are no model-based equations that quantitatively describe the data, model-free analyses are the only method of analysis. Otherwise, the results of model-... 

Calibration Most process analyzers are designed to monitor concentration and/or composition. This requires a calibration of the analyzer with a set of prepared standards or from well-characterized reference materials. The simple approach must always be adopted first. For relatively simple systems the standard approach is to use a simple linear relationship between the instrument response and the analyte/ standard concentration [27]. In more complex chemical systems, it is necessary to adopt either a matrix approach to the calibration (still relying on the linearity of the Beer-Lambert law) using simple regression techniques, or to model the concentration and/or composition with one or more multivariate methods, an approach known as chemometrics [28-30]. 

Where C is a matrix of component concentrations or sample properties, S is a matrix of basis vectors (pure component spectra, or spectral profiles reflecting a pure sample property), and E and Ec are model residuals. The direct model expresses the analyzer responses (X) as a function of concentrations, whereas the inverse model expresses concentrations as a function of the analyzer responses. Because the former is more in line with the Beer-Lambert Law (absorbance = concentration x absorptivity), it is given the label direct . 

In contrast to MLR, CLS is a direct calibration method that was designed specifically for use with spectroscopic data, and whose model formula is a reflection of the classical expression of the Beer-Lambert Law for a system with mnltiple analytes ... 

The first problem is deciding on which of these two common models to use. It has been argued that for spectrophotometric methods where the Beer-Lambert Law is known to hold, Y = bX + e, the force through zero model is the correct model to choose if the absorbance values are corrected for the blank." The correct way to carry out the calibration regression is to include the blank response at assumed zero concentration and use the model Y = bX + a + instead. This may be a nicety from a practical standpoint for many assays but there are instances where a force through zero model could produce erroneous results. Note that the e denotes the random error term. Table 15 contains a set of absorbance concentration data from a UV assay. 

Classroom exercise to derive Beer s law R. W. Ricci, M. A. Ditzler, and L. P. Nestor, Discovering the Beer-Lambert Law, J. Chem. Ed. 1994, 71, 983. An alternate derivation W. D. Bare, A More Pedagogically Sound Treatment of Beer s Law A Derivation Based on a Corpuscular-Probability Model, J. Chem. Ed. 2000, 77, 929. 

The aim of the multivariate evaluation methods is to fit a reaction model to the measured reaction spectrum on the basis of the Beer-Lambert law and thus identify the kinetic parameters of the model. The general task can be described by the non-linear least-squares optimisation described in Equation 8.20 ... 

Next, we will look into various kinetic examples of increasing complexity and determine solely concentration profiles (C). This can be seen as kinetic simulation, since the calculations are all based on known sets of rate constants. Naturally, in an iterative fitting process of absorbance, data on these parameters would be varied until the sum of the squared residuals between measured absorbances (Y) and Beer-Lambert s model (C x A) is at its minimum. 

As the goal is to build a calibration model based on spectral data, we assume that Beer-Lambert s law is valid,... 

For lignin model compounds and lignin samples of known molecular weight, c is expressed in moles per liter. When the path length (b) is in centimeters, the Beer-Lambert equation becomes... 

Data Assume that the reactors are long enough for the dispersion model to be applied and that laminar flow prevails at all points. The Beer Lambert law of light intensity, I, is applicable I/Io = exp(aCL), where a is absorptivity of the reactant gas mixture at a concentration, C, which absorbs the light of the CO2 laser and L is the path length. 

Figure 2.1 The underlying Beer-Lambert law (bilinear) model of hyperspectral images.

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