Beer-Lamberts Law

The Beer Lambert law (Equation 2.4) holds only for absorbing species that do not exhibit any concentration-dependent aggregation (Lambert law). 

Equation 2.4 Beer Lambert law for a single absorbing compound 

Equation 2.5 is the extension of this law for solutions containing a mixture of n absorbing species B / 1,2,. .. n. When there are no specific interactions between these 

The measurement of light absorption by a solution of molecules is governed by the Beer-Lambert Law, which is written as follows  

Calculate the percentage of the incident radiation absorbed by a sample with an absorbance 

In pharmaceutical products, concentrations and amounts are usually expressed in grams or milligrams rather than in moles and thus for the purposes of the analysis of these products, the Beer-Lambert equation is written in the following form  

BP monographs often quote a standard A (1 %, 1 cm) value for a drug which is to be used in its quantitation. 

The absorbance maxima of a 5% w/v solution of holmium perchlorate between 200 and 400 nm. 

This law states that when a monochromatic light passes through a transparent medium, the rate of decrease in intensity with the thickness of the absorbing medium is proportional to the intensity of the penetrating radiation. Let us consider a thin layer of the medium of thickness dl and let I be the intensity of the radiation entering it, then Lambert s law can be expressed by the dilferential equation as  

By changing hum natural to common logarithms the equation 1.3.1.4 can also be written as 

The extinction coefficient is generally defined as the reciprocal of the thickness (in 

When the absorbing substance is present in solution, the absorption of light also depends upon the concentration Beer s law states that the rate of decrease in intensity of radiation absorbed is proportional to the intensity of radiation and to the concentration of the solute. Mathematically 

The quantity log.Q — is generally called the optical density (O.D.) or absorbancy so that 

Equation 13-1 describes the relation between energy emitted and absorbed for a single particle. The intensity of the energy emitted or absorbed by an ensemble of particles is, of course, proportional to the number of particles per unit area in the light path and also to the probability of the occurrence of the process. This finds expression as the Beer-Lambert Law. 

The proportionality constant in Equation 13-2, a, called the absorptivity, varies with wavelength and from substance to substance. When concentration is expressed as M, mol/L, and b in cm, then a becomes e, the molar absorptivity. From quantum mechanics we learn that the e value of an absorption band at, the A, at which g is maximum, is a measure of the transition probability. Thus, the most intensely colored substances are those in which a highly probable transition, such as a 7t - ti,  

An important feature of a spectral band in evaluating its utility for analytical purposes is its selectivity. In the visible and UV part of the spectrum, involving electronic transitions, one is likely to see quite similar spectra for similar compounds. The spectra of aromatic compounds will change somewhat upon substitution which either changes the electron density in, or enhances conjugation of, the overall n electron system. 

Sinele Components. Beer s Law can be applied to analytical problems in several ways. 

Unlike titrations previously described in this book, photometric titrations are linear since absorbance is directly proportional to concentration. To be sure, we have demonstrated how to treat pH, pM, and pE data, in the context of the Gran method, to obtain linear titrations. Remember two important benefits of linear titrations (1) it is a practical way to handle dilute solutions and (2) data can be taken more rapidly, using points on either side of the equivalence point. The best line for each of the two sets (before and after the equivalence point) is obtained by linear regression and their intercept used to locate the equivalence point. 

The term logio Uo/fl is called the absorbance (A), synonymous with attenuance (D). Extinction ( ) has been used in the past but is now discouraged. Pathlength (usually in cm) and wavelength (in nm, without units) are sometimes given as subscripts, e.g. Ai cm, sso- Note that, in difference spectroscopy, a difference molar absorption coefficient (e) may be defined (2) by the equation  

Sometimes the passage of light through the cuvette is described in terms of transmittance, or transmission (T) T = 1/4 and is generally expressed as a percentage. It is important to note, however, that only absorbance, not transmittance, is linearly proportional to the chromophore concentration. In quantitative analysis, where it is required to obtain the concentration of substance, therefore, absorbance is more conunonly used. The relation between these two parameters is given by the following  

when A = 2 only 1% of the incident light is transmitted, while at A = 3, only 0.1% is transmitted. 

According to the Beer-Lambert law, absorbance should be linearly proportional to the concentration of the chromophore. However, there are systems that apparently do not obey this rule. The examples that follow are amplified in (2). 

Apparent deviations from the Beer-Lambert law arise when concentration variation causes changes in the distribution of several chromophore species in the solution. In these cases, if the absorption contribution from each species is considered, the Beer-Lambert law is obeyed by each species, but it is the variation in species concentrations which produces the apparent deviation. For example, dimerization of the chromophore molecules at higher concentrations may take place, providing a dimer with an e value other than that of the monomeric form, causing a non-linear dependence of absorbance upon the solute concentration. 

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In absorption spectroscopy, the attenuation of light as it passes tln-ough a sample is measured as a function of wavelength. The attenuation is due to rovibrational or electronic transitions occurring in the sample. Mapping out the attenuation versus photon frequency gives a description of the molecule or molecules responsible for the absorption. The attenuation at a particular frequency follows the Beer-Lambert law,... 

Light can also be absorbed by a material through which it passes. This leads to an attenuation in intensity of the light as it passes tlnough the material, which decays exponentially as a ftmction of distance tlnough the material and is described mathematically by the Beer-Lambert law [M] ... 

In photochemistry, we are interested in the system dynamics after the interaction of a molecule with light. The absorption specbum of a molecule is thus of primary interest which, as will be shown here, can be related to the nuclear motion after excitation by tbe capture of a photon. Experimentally, the spectrum is given by the Beer-Lambert law... 

The intensity of a spectral absorption band at a given wave length is expressed in terms of absorption or extinction coefficients, dehned on the basis of the Beer-Lambert law. The latter states that the fraction of incident light absorbed is proportional to the number of molecules in the light path, i.e., to the concentration (c) and the path length (1). The law may be expressed mathematically as ... 

Equations 10.4 and 10.5, which establish the linear relationship between absorbance and concentration, are known as the Beer-Lambert law, or more commonly, as Beer s law. Calibration curves based on Beer s law are used routinely in quantitative analysis. 

Allen, H. C. Brauers, T. Finlayson-Pitts, B. J. Illustrating Deviations in the Beer-Lambert Law in an Instrumental Analysis Laboratory Measuring Atmospheric Pollutants by Differential Optical Absorption Spectrometry, /. Chem. 

Two noncalculus-based approaches to discovering the Beer-Lambert law are found in the following papers. 

Lykos, P. The Beer-Lambert Law Revisited A Development without Calculus, /. Chem. Educ. 1992, 69, 730-732. 

In the this form the Beer-Lambert law shows that the intensity of radiation transmitted by an absorbing sample declines exponentially as the length over which the absorption takes place increases. If the radiation, travelling with the speed of light c, takes time tg to traverse the absorbing path f Equation (9.29) becomes ... 

Beer foam Beer-Lambert law Beer process Beers Beer s law... 

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. 

Colorant Mixing. A colorant, whether a dye dissolved in a medium or pigment particles dispersed in it, produces color by absorbing and/or scattering part of the transmitted light. If only absorption is present, the Beer-Lambert law appHes ... 

When both absorption and scattering are present, the Beer-Lambert law must be replaced by the Kubelka-Munk equation employing the absorption and scattering coefficients iC and S, respectively. This gives the redectivity... 

The Beer-Lambert Law of Equation (2) is a simpliftcation of the analysis of the second-band shape characteristic, the integrated peak intensity. If a band arises from a particular vibrational mode, then to the first order the integrated intensity is proportional to the concentration of absorbing bonds. When one assumes that the area is proportional to the peak intensity. Equation (2) applies. 

This is the fundamental equation of colorimetry and spectrophotometry, and is often spoken of as the Beer-Lambert Law. The value of a will clearly depend upon the method of expression of the concentration. If c is expressed in mole h 1 and / in centimetres then a is given the symbol and is called the molar absorption coefficient or molar absorptivity (formerly the molar extinction coefficient). 

When a spectrophotometer is used it is unnecessary to make comparison with solutions of known concentration. With such an instrument the intensity of the transmitted light or, better, the ratio I,/I0 (the transmittance) is found directly at a known thickness /. By varying / and c the validity of the Beer-Lambert Law, equation (9), can be tested and the value of may be evaluated. When the latter is known, the concentration cx of an unknown solution can be calculated from the formula ... 

For matched cells (i.e. I constant) the Beer-Lambert Law may be written ... 

Visual and photoelectric colorimeters may be used as turbidimeters a blue filter usually results in greater sensitivity. A calibration curve must be constructed using several standard solutions, since the light transmitted by a turbid solution does not generally obey the Beer-Lambert Law precisely. 

It is instructive to compare the sensitivity which may be achieved by absorption and fluorescence methods. The overall precision with which absorbance can be measured is certainly not better than 0.001 units using a 1 cm cell. Since for most molecules the value of emax is rarely greater than 105, then on the basis of the Beer-Lambert Law the minimum detectable concentration is given by cmin> 10 3/105= 10 8M. 

Factors such as dissociation, association, or solvation, which result in deviation from the Beer-Lambert law, can be expected to have a similar effect in fluorescence. Any material that causes the intensity of fluorescence to be less than the expected value given by equation (2) is known as a quencher, and the effect is termed quenching it is normally caused by the presence of foreign ions or molecules. Fluorescence is affected by the pH of the solution, by the nature of the solvent, the concentration of the reagent which is added in the determination of inorganic ions, and, in some cases, by temperature. The time taken to reach the maximum intensity of fluorescence varies considerably with the reaction. 

Now that we have spectra for each of the pure components, we can put the concentration values for each sample into the Beer-Lambert Law to calculate the absorbance spectrum for each sample. But first, let s review various ways of... 

Classical least-squares (CLS), sometimes known as K-matrix calibration, is so called because, originally, it involved the application of multiple linear regression (MLR) to the classical expression of the Beer-Lambert Law of spectroscopy ... 

The Beer-Lambert law (also often called Beer s law) relates Iabs to the total incident light intensity (To) (eq. 7). 

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