The laws of Lambert and Beer

Lambert s law states that the proportion of radiation absorbed by a substance is independent of the intensity of the incident radiation. This means that each successive layer of thickness dx of the medium absorbs an equal fraction —dl/J of the radiation of intensity I incident upon it. In other words, 

The proportionality constant b, which is characteristic of the absorbing medium, is known as the Napierian absorption coefficient, or the Napierian extinction coefficient. 

Integration of equation (2.1), for passage of light through a thickness I, proceeds as follows  

The constant of integration d can be evaluated using the boundary condition that when I 0, J = Iq, where Iq is the intensity of the radiation before passage through the medium. Thus the constant d is equal to In Iq, and equation (2.3) thus becomes 

The product bl may be written as B, which is known as the Napierian absorbance or Napierian extinction  

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State the laws of Lambert and Beer, and write down an equation comprising the two laws. 

Due to its principle and instrumental realisation, atomic absorption spectrometry is a technique for quantitative analysis and is practically unsuitable for qualitative analysis. Quantitative response is governed by the law of Lambert and Beer, i. e. the absorption A is proportional to the optical pathlength I, the absorption coefficient k at the observed wavelength, and the concentration c of the species. 

The absorption of light follows the law of Lambert and Beer. The transmitted intensity of light I with respect to the intensity of the entering light Iq is... 

The laws of Lambert and Beer are also found to be well satisfied for colloidal systems, provided that by the change of concentration nothing is really changed except the number of particles per cm (thus no changes in solubility, swelling etc.) and provided one does not go to such high concentrations that the results are affected by multiply scattered light ... 

This method is used for the gas and liquid phase in the same way. The law of Lambert-Beer is valid between 0.1 < Ig /// < 1. So that the concentration range for this method depends on the width of the absorption cell. Systems exist that can measure up to 150 mg L- in the liquid phase, and up to 600 mg L 1 in the gas phase. 

IR adsorption at 3400 cm 1 following the law of Lambert-Beer with molar extinction coefficient e = 60. The microinterferometric technique h2 is evaluated by correcting the equivalent water thickness with 3.6 nm. Here, the hydrophilic heads are incorporated in the aqueous core. Fig. 3.53 presents the compared h2(Cei) and d2(Cei) dependences. It is seen that within the whole electrolyte concentration range studied h2 is higher than d2. 

The photosedimentometer (Figure 35) uses a stabilized, often monochromatic, light source and a simple lens and aperture system. The light beam thus defined penetrates the suspension in a thermostatically controlled cuvette its intensity is reduced according to the law of Lambert-Beer. The photo element produces a current that is proportionate to light intensity / and is recorded as a function of time. The percentage passing D is calculated with... 

Ambient air is sucked through a cell, where the ozone concentration is measured by its UV-absorption at 253.7 nm. At this wavelength the ozone absorption coefficient is close to its maximum. The UV light is produced by a Hg resonance lamp at 253.7 nm and the intensity of the light is measured by a photomultiplier. The ozone concentration is determined fi om the absorbance making use of the law of Lambert Beer ... 

The rate of the absorption is related to the extinction coefficient or to the absorption cross-section. It obeys the law of Lambert-Bouguer and Lambert-Beer. The density of the photons [photon], and the concentration or density of the chromophore [D], thus define the rate of the concentration or density change of the excited chromophore [ > ], here appearing during the excitation (equation 6). 

The most ubiquitous method of transmission spectroscopy, in which the amount of light passing through a sample is determined. Very often the influence of reflection and scattering is neglected and the ratio of incident and transmitted intensity ( / ) is linked to the absorption coefficient (a) and the sample thickness (d) by Lambert-Beer s law (see Eq. (9.11)). 

In view of the above, the initiation of polymerization and cross-linking reactions in polymeric systems is different from events induced by photons. Most of the events initiated by UV irradiation occur near the surface, because the absorption is governed by the Beer-Lambert Law, whereas the reactive species produced by electron beam are dispersed randomly throughout the entire thickness of the material (see Figure 5.1). 

In analytical spectrometry there are many types of calibration curves which are set up by measuring spectrometric reference solutions. The measurements yield a curve of absorbance versus concentration, and the points between the data of the reference solutions are interpolated by fitting a suitable curve, which normally follows the Beer-Lambert law and which gives rise to a straight line through the origin of the coordinate system. The measurement conditions and the results of the calibration curve evaluations in the case of chromium and lead measurements by electrothermal atomic absorption spectrometry are presented in Table 1. 

Transparent materials interact with light only by absorption. This interaction is formulated quantitatively in the Bouguer-Lambert and Beer s Laws (c.f. i). In paper, however, surface reflection is the dominating type of interaction. This results in very desirable properties like high brightness and opacity, but complicates the interpretation of optical tests with regard to absorption data. The Kubelka-Munk theory attempts to separate the two types of... 

There are two fundamental laws to consider when we discuss absorption Beer s law and Lambert s law. Lambert s law states that equal thickness of materials cause equal amounts of absorption. Beer s law states that equal amounts of absorbing material cause equal absorption. Figure 2.5 illustrates these laws (Billmeyer and Saltzman, 1981). Both laws will only work in the absence of scattering and are not applicable in opaque or translucent materials. They are very useful in transparent materials, where the scattering due to the colorants and the polymers themselves (low refractive indices) is very low. 

The Lambert and Beer theory is reserved for transmittance calculations for very transparent samples. Dating back to the eighteenth and nineteenth centuries, the Lambert and Beer laws state that the absorbence [log(l/T)] for a transparent sample is proportional to the thickness and the concentration of the colorant [4]. 

The Lambert and Beer laws has been found to be valid at low and moderate concentrations in transparent applications, but it may prove to be inaccurate at higher concentrations. In order for these laws to be valid, the absorption coefficient must be a constant independent of the concentration [5]. Since all colorant layers scatter some light, these equations, even in cases of slightly turbid media, are generally not valid. 

Each compound will have particular bonding characteristics, which will generate different values of kmax- As noted earlier, bonding depends on the behaviour of electrons and electronic transitions. These electronic transitions will be observed at different wavelengths on the absorption spectrum and reveal important features of the compounds and chromophores being studied. Values of kmax and absorbance can be taken from the absorbance spectmm, and from this the maximum extinction coefficient (smax) can be determined using the Beer-Lambert law equation. Table 5.1 gives information on electronic absorption characteristics of typical chromophores in non-aromatic compounds. 

The absorption spectra of two or more drugs of interest often overlap. Subject to certain conditions, the Vierordt method of simultaneous equations can be employed to obtain the individual concentrations (A. L. Glenn, J. Pharm. Pharmac, 1960, 72, 598-608). If each of n drugs obeys the Beer-Lambert Law over the concentration range of interest, and if the law of additivity of absorbances applies, then the total absorbance, observed at any wavelength A is given by the sum... 

The initial rate was varied by a factor of 2, and the methane and bromine pressures by factors of 2 and 3 respectively. The bromine pressure could not conveniently be raised above about 50 mm Hg because the Lambert-Beer law was only known to apply accurately to bromine up to this pressure. This dependence on the square root of the bromine pressure suggests attack by bromine atoms, and the following scheme seemed a reasonable one, particularly in view of the work of Bodenstein and Jung. 

This expression is comparable with Beer-Lambert s law for the absorption of non-scattering samples. Both in the theory of Kubelka and Munk and in Beer-Lambert s law the relationship derived from the measurement value is proportional to the coefficient of absorption and hence proportional to the concentration (c) and coefficient of extinction Sn of the absorbing material ... 

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. 

The theory of Kubelka and Munk is used in calculating the concentration for the systems of Boehringer Mannheim, Hoffmann-La Roche and Bayer Diagnostic and Electronic (Ames-Miles). If modifications of the theory have been made, these are stated in the description of the instrument. If the relationship between reflection and concentration (calculated by means of the theory of Kubelka and Munk) is represented graphically, the curve obtained is similar to that for the relationship between transmission and concentration calculated by Beer-Lambert s law (Fig. 3). 

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