Lambert- Beer approximation

This simple relationship between incident and transmitted light is well known as the Boguert-Lambert-Beer law. This expression renders positive values for Ij < Iq. In case of scattering material like TLC plates, a part of the scattered light is emitted as reflectance J from the plate surface to the top. For the hrst approximation of a parallel incident light beam with the intensity /g, some radiation may be scattered inside the layer and some radiation may be absorbed either by the sample or by the layer itself. According to the Schuster equations and with the abbreviation R (the diffuse reflectance of an infinitely thick layer). 

There are several good reasons to focus on linear models. Theory may indicate that a linear relation is to be expected, e.g. Lambert-Beer s law of the linear relationship between concentration and absorbance. Even when a linear relation does not hold strictly it can be a sufficiently good local approximation. Finally, one may try and find a transformation of the individual variables (e.g. a logarithmic transformation), in order to obtain an acceptable linear model for the transformed variables. Thus, we simplify eq. (36.1) to... 

If the rare medium exhibits absorption, the penetrating wave becomes attenuated. In the case of transmittance measurements at weak absorption the law of Lambert-Beer is valid in a linear approximation also... 

Fig.6a shows that the optical density, at the maximum optical absorbance wavelength = 376 nm, increases linearly with film thickness, up to approximately 0.4 pm. The linearity demonstrates that the film structure is independent of the number of deposited layers. When film thickness is further increased, the absorbance becomes so larg that the Lambert-Beer law is no longer satisfied. In Fig.6b the square root of the relative second-harmonic intensity is plotted versus the number of bilayers. 

Even if the results obtained for model (calibration) samples deviate fromLambert-Beer s law, the observed relationship between the A value and the concentration of the analyte may still be suitable for a fit-for-purpose practical quantitative analysis. In such a case, it is useful to express the observed results with a nonlinear relationship between the Aj value and the concentration of the target substance. This nonlinear relationship can be used for quantitative analysis, regardless of the origin of the deviation from Lambert-Beer s law. In a limited range of the concentration of the analyte, the nonhnear relationship may sometimes be approximated by a linear relationship, which is often easier to use practically. 

The penetration of visible light through foamed polystyrene has been shown to foUow approximately the Beer-Lambert law of light absorption (22). This behavior presumably is characteristic of other ceUular polymers as weU. 

Surface extended X-ray absorption fine structure (SEXAFS). In order to understand the SEXAFS approach it may first be useful to give a brief recap of X-ray absorption. In addition to the scattering process described above, an X-ray photon may instead be absorbed. X-ray absorption results in the excitation of a core electron into one of the continuum of empty levels in the lowest unoccupied band of the sample this is shown for copper in Figure 2.72. To a first approximation, the attenuation of the beam is given by a form of the Beer-Lambert expression ... 

Thus the absorption of infrared energy by solids is an approximation of the Beer-Lambert exponential law for transmission ... 

If the absorptivities or molar absorptivities are not approximately equal, the linear relationships of the Beer-Lambert Law will not hold. 

If you are sure that the Beer-Lambert law is fulfilled and a linear correlation between signal and amount is approximately given, the amount of analyte may be calculated by a simple equation ... 

The Koschmieder equation has been shown to approximate the change in contrast of an object with distance away from an observer (Middleton, 1952) note that it has a form similar to that of the Beer-Lambert law ... 

Numerous skin sites are possible for noninvasive near-infrared measurements.40 Examples proposed in the literature include the inner lip mucosa,26 finger,41 forearm, and tongue. In all cases, the principal spectral features of the noninvasive spectra correspond to the extent of scattering superimposed on absorption bands that originate from water, protein, and fatty tissue within the skin matrix.44,45 To a first approximation, noninvasive near-infrared spectra of skin can be fitted to a Beer-Lambert function that incorporates the additive features of absorption due to water, fat, (3-sheet protein, and type III collagen, with additional terms for tissue scattering (offset) and small temperature variations (a slope term). This function takes the following form ... 

The coefficient of extinction r mentioned here is approximately comparable but not identical with the coefficient of extinction in Beer-Lambert s law. r is dependent on an interaction between the absorbing molecules and the reagent carrier matrix. 

If the units of concentration are molarity (i.e. number of moles per litre), then the constant is (the Greek letter epsilon ) and is known as the molar absorptivity, with units of L mol-1 cm-1, although the units are seldom expressed, is equal to the absorbance of a 1 m solution in a cell of path length 1 cm and is usually a large number, approximately 10 000-20 000. In this case the Beer-Lambert equation is written as... 

The number of neutrons scattered out of the beam is constant for a fixed mass of a given element. It follows a law similar to the Beer-Lambert law for optical absorption and linear approximations to this law can be exploited [3] (for simplicity we shall assume that the element does not absorb neutrons). The initial neutron intensity, or flux on the sample, is Jj and the unscattered flux after the sample is Jf. Then, since the mass of the sample is the product of its thickness, dg, and concentration, C, we can write ... 

Using results established in previous sections., it can however be shown that this assertion represents only an approximation to the truth. Absorption from a white light or other broadband source in fact allows photon pairs of differing frequencies to be concertedly absorbed by molecules in close proximity (Andrews 1988). In this section, it is shown that this may result in a change to the appearance of absorption spectra, and yield an absorption law that departs from normal Beer-Lambert behavior. One particular case in which such effects may be expected to arise is where supercontinuum laser radiation is employed for spectroscopic purposes. 

Beer s law - An approximate expression for the change in intensity of a light beam that passes through an absorbing medium, viz., log(///j,) = -8c/, where is the incident intensity, I is the final intensity, e is the molar (decadic) absorption coefficient, c is the molar concentration of the absorbing substance, and / is the path length. Also called the Beer-Lambert law... 

The mathematical treatment of absorbance in such turbid samples is quite complex, and the simple Beer-Lambert law does not necessarily apply, except as a useful approximation in certain cases. 

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