The Beer Lambert law

The basis for the measurement of the strength of light absorption by a molecule at various wavelengths is shown in Fig. 3.11. A parallel monochromatic light beam of wavelength A and power P ] or intensity / , defined as the energy per second striking a unit area, 

TABLE 3.2 Typical Wavelengths, Frequencies, Wavenumbers, and Energies of Various Regions 

Name Typical wavelength or range of wavelengths (nm) Typical range of frequencies v (s -1) Typical range of wavenumbers 0 (cm-1) Typical range of energies (kJ einstein - )  

TABLE 3.3 Some Relationships between Commonly Used Energy Units 

As the light passes through a thickness of sample dl, it undergoes a fractional reduction in intensity proportional to the absorbing path length, i.e., 

The fraction of light transmitted through an absorbing system is very frequently found to be represented by the equivalent relations 

The law embodied in Eq. 10 was originally known as Lambert s law a second law, Beer s law, stated that if C and d were altered but the product Cd was constant, then the fraction of light transmitted remained the same. Since this latter law follows in any case from Lambert s law, Eq. 10 is now known as the Beer-Lambert law. A logarithmic form of the equation, for example 

A proof of the Beer-Lambert law may be derived if it is assumed that the rate of loss of photons is proportional to the rate of bimolecular collisions between photons and the absorbing species. The decrease, -dI, in intensity I at any point x in the system (Fig. 6) for a small increase in x, dx, is given by 

The intensity of radiation absorbed, 4bs is. of course, /(l - ft, so that the fraction absorbed is given by 

An important approximate expression results when and or aCd is small expansion of the exponential and rejection of second- and higher-order terms leads to the conclusion that 

This is the Beer Lamhert laiu, stating that the absorbance is proportional to the concentration of absorbers in the sample. 

We note that the absorbance A, and the optic al density D, previously introduced (Sect. 6.4.2) are synonymous concepts. 

We will now consider the accuracy of a concentration determined, by a measurement of the transmission through the sample. In analytical instruments based on absorption measurements the transmittance or absorbance value is read off directly on a scale or is given digitally. In Fig. 6.56 a linear scale for transmittance and a corresponding iogaritlimic scale for absorbance are given (compare with Table 6.2). BVom (6.46) and (6.47) we find 

The relative accuracy in the concentration determination Ac/c depends on the error in reading the scale (or the error in digitizing). Clearly, a small error AT in the T reading results in a large uncertainty in the absorbance if T is small (then also A - T s small). In the same way it is important that the full-scale deflection (T = 100%) is suitably set in order to be able to determme small concentrations. If we assume that the imcertainty in the full-scale setting and in measurements is the same, we obtain the resulting relative concentration error by quadratic addition of the errors  

the error in the relative concentration depends on AT and on the factor 0.43(1 -1- T y f AT. The variation in this factor as a function of A is given in Fig. 6.57. The factor is large for small and large values of A and has a minimum for A = 0.48. The analysis performed above reflects the almost self-evident fact that it is much more difl cult to measure accurately a concentration corresponding to very little absorption or very httie transmission than to measure it in a situation for which about half of the light is absorbed. 

A = cl B, where A = absorbance of the sample c = concentration of the sample, in moles/hter I = length of the hght path through the sample, in centimeters e = molar absorptivity (ivr cm ) 

The molar absorptivity (e) is a constant that is characteristic of the compound at a particular wavelength. It is the absorbance that would be observed for a 1.00 M solution in a cell with a 1.00-cm path length. (The abbreviation e comes from the fact that molar absorptivity was formerly called the extinction coefficient.) 

For example, the molar absorptivity of acetone dissolved in hexane is 9000cm at 195 nm. TTie solvent in which the sample is dissolved is reported because molar absorptivity is not exactly the same in all solvents. Therefore, the UV spectrum of acetone in hexane would be reported as = 195 nm (e ax = 9000, hexane). 

A solution of a compound in ethanol shows an absorbance of 0.52 at 236 nm in a cell with a 1-cm light path. Its molar absorptivity in ethanol at that wavelength is 12,600M cm What is the concentration of the compound  

The solution whose UV or visible spectrum is to be taken is put into a cell, such as one of those shown here. Most cells have 1-cm path lengths. Either glass or quartz cells can be used for visible spectra, but quartz cells (made of high-purity fused silica) must be used for UV spectra because glass absorbs UV light. 

Consider a sedimentation container of width L measured in the direction of the light beam, containing the suspension of powder under analysis. 

Consider the small fall (AS ) in the optical density as the sedimentation time changes from t to r +At so that the maximum Stokes diameter in the beam changes from with an average value d.  

The cumulative distribution undersize by surface, assuming that K is constant for the restricted size range under consideration, is  

It is therefore necessary to know how K varies with d in order to determine the size distribution. If this correction is not applied, the method is only valid for comparison purposes. Theoretical values of K may be used but this will also introduce errors, since the effective K values depend upon the optical geometry of the system. Calibration may also be against some external standard. The cumulative distribution undersize by weight is given by  

The surface area of the powder is derivable from the initial concentration of the suspension and the maximum optical density 

As shown in Fig. 2.6, when a parallel monochromatic light of wavelength A with an intensity Iq (energy or number of photons) irradiates and passes through media with concentration C (mol L ) and length I (cm), the intensity of the light after the passing can be expressed as 

This relationship is called the Beer-Lambert law. The proportionality coefficient k is generally referred to absorption coefficient (L moL 

The Beer-Lambert law is also often expressed taking the base 10 of the logarithm 

The concentration unit of the media is usually taken as molecular number density n (molecules cm ) in atmospheric chemistry, and in this case the Beer-Lambert law is customarily expressed with the natural logarithm as, 

the proportionality coefficient, o (cm molecule ) has the dimension of area and is called the absorption cross section. Also, the dimensionless number in the above equation cnl is denoted by r and is called optical depth. 

The left-hand-side quantity is the absorbance, A, and the linear relationship between absorbance, concentration and path length is known as the Beer-Lambert law  

The Beer-Lambert law can generally be applied, except where very high-intensity light beams such as lasers are used. In such cases, a 

The units of e require some explanation here as they are generally expressed as non-SI units for historic reasons, having been used in spectroscopy for many years. 

In considering absorption of light by molecules, we have been principally concerned with transitions between electronic states. However, it is not possible to explain fully the effects of electronic excitation in molecules unless we also take into account the motions of the nuclei. 

the total energy of molecules is made up of electronic energy and energy due to nuclear motion (vibrational and rotational)  

The intensity of absorption of radiation at a particular wavelength passing through a uniform sample is related to the concentration [J] of the absorbing species J by the empirical Beer-Lambert law (Fig. 12.6, and commonly simply Beer s law we first encountered the law in In the laboratory 6.1 as a way of monitoring the concentrations of species in reactions. Ihe law is commonly written 

We think of the sample as consisting of a stack of infinitesimal slices, like sliced bread (Fig. 12.7). The thickness of each slice is dx. The change in intensity, df, that occurs when electromagnetic radiation passes through one particular shce is proportional to the thickness of the shce, the concentration of the absorber J, and the intensity of the incident radiation at that slice of the sample, so dfoc [J]7dx. Because df is negative (the intensity is reduced by absorption), we can write 

This expression appHes to each successive slice. To obtain the intensity that emerges from a sample of thickness L when the intensity incident on one face of the sample is fo, we sum aU the successive changes. Because a sum over infinitesimally small increments is an integral, we write 

If the concentration is uniform, [J] is independent of location and can be taken outside the integral, and we obtain 

Because the relation between natural and common logarithms is Inx = 

Analysis by optical techniques is frequently performed by measuring absorption. It is important to be able to correctly relate the absorption to the concentration. The relation which is called the Beer-Lambert law will now be considered. 

Consider monochromatic light of intensity Pq impinging on a sample of thickness b as illustrated in Fig.6.55. The sample can be a solution in a cuvette or atoms in a flame from a specially designed burner. An intensity is transmitted through the sample. (We disregard possible effects from the sample confinement). We now consider the conditions over a small interval Ax in the sample. Before the considered space interval, the intensity has been reduced to P, and it will be further reduced by AP in the interval Ax. The fractional attenuation AP/P is proportional to the number of absorbers, An, in the small interval Ax 

Here k and k are constants and the last equality is valid for a uniform concentration c throughout the sample. When light passes through the sample, P changes from Pq to P, n from 0 to N and x from 0 to b. By integration we obtain 

The most widespread use of UV and visible spectroscopy in biochemistry is in the quantitative determination of absorbing species (chromophores), known as spectrophotometry. All spectrophotometric methods that measure absorption, including various enzyme assa3rs, detection of proteins, nucleic acids and different metabolites, reside upon two basic rules, which combined are known as the Beer-Lambert law. Lambert s law states that the fraction ofli t absorbed by a transparent medium is independent of the incident li intensity, and each successive layer of the medium absorbs an equal fraction of the li t passing throu it. This leads to an exponential decay of the light intensity along the light path in the sample, which can be expressed mathematically, as follows  

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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 ... 

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 ... 

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). 

In the study of reactions of the types ether than exchange mentioned previously, the usual technique involves the spectrophotometric examination of reaction mixtures. The absorbance changes that occur, at a suitable wavelength where only one species (either reactant or product) absorbs, as the reaction proceeds are measured (manually or recorded). Treatment of the data via the Beer-Lambert law enables rate coefficients and laws to be found in the usual manner. Stopped flow and temperature jump techniques have been used for very rapid reactions. 

Carotenoid concentration in solution can be calculated taking in consideration the absorption coefficient (Aj ) or molar absorptivity (s) values of the specific carotenoid according to the Beer-Lambert law. The absorption coefficient and e values were determined for many carotenoids and values are available in the literature. These two values are related as shown in Equation 1. The concentration is calculated according to Equation 2. 

Both E and e are independent of concentration and cell length, provided the Beer-Lambert law is obeyed. The two quantities are related by the expression ... 

A useful measure of the strength or intensity of the colour of a dye is given by the molar extinction coefficient (e) at its 2max value. This quantity may be obtained from the UV/visible absorption spectrum of the dye by using the Beer-Lambert law, i.e. 

The rate of photolytic transformations in aquatic systems also depends on the intensity and spectral distribution of light in the medium (24). Light intensity decreases exponentially with depth. This fact, known as the Beer-Lambert law, can be stated mathematically as d(Eo)/dZ = -K(Eo), where Eo = photon scalar irradiance (photons/cm2/sec), Z = depth (m), and K = diffuse attenuation coefficient for irradiance (/m). The product of light intensity, chemical absorptivity, and reaction quantum yield, when integrated across the solar spectrum, yields a pseudo-first-order photochemical transformation rate constant. 

Absorption of radiation by solutes as a function of concentration, c, is described by the Beer-Lambert law ... 

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