Beer s law absorption

The large inversion fraction implies that the dye jet was bleached on the pump input side, violating the Beer s law absorption profile assumed in deriving both reduction factors and showing the roughness of the present approximations. Thus, the best test of the plane-wave theory is probably the comparisons of the predicted and observed wavelength dependencies of Fig. 14(c) and (d), which... 

Fundamental Limitations to Beers Law Beer s law is a limiting law that is valid only for low concentrations of analyte. There are two contributions to this fundamental limitation to Beer s law. At higher concentrations the individual particles of analyte no longer behave independently of one another. The resulting interaction between particles of analyte may change the value of 8. A second contribution is that the absorptivity, a, and molar absorptivity, 8, depend on the sample s refractive index. Since the refractive index varies with the analyte s concentration, the values of a and 8 will change. For sufficiently low concentrations of analyte, the refractive index remains essentially constant, and the calibration curve is linear. 

The determination of an analyte s concentration based on its absorption of ultraviolet or visible radiation is one of the most frequently encountered quantitative analytical methods. One reason for its popularity is that many organic and inorganic compounds have strong absorption bands in the UV/Vis region of the electromagnetic spectrum. In addition, analytes that do not absorb UV/Vis radiation, or that absorb such radiation only weakly, frequently can be chemically coupled to a species that does. For example, nonabsorbing solutions of Pb + can be reacted with dithizone to form the red Pb-dithizonate complex. An additional advantage to UV/Vis absorption is that in most cases it is relatively easy to adjust experimental and instrumental conditions so that Beer s law is obeyed. 

Quantitative Analysis of Mixtures The analysis of two or more components in the same sample is straightforward if there are regions in the sample s spectrum in which each component is the only absorbing species. In this case each component can be analyzed as if it were the only species in solution. Unfortunately, UV/Vis absorption bands are so broad that it frequently is impossible to find appropriate wavelengths at which each component of a mixture absorbs separately. Earlier we learned that Beer s law is additive (equation 10.6) thus, for a two-component mixture of X and Y, the mixture s absorbance, A, is... 

Sensitivity The sensitivity of a molecular absorption analysis is equivalent to the slope of a Beer s-law calibration curve and is determined by the product of the analyte s absorptivity and the pathlength of the sample cell. Sensitivity is improved by selecting a wavelength when absorbance is at a maximum or by increasing the pathlength. 

Transmission, Absorption, and Beer s Law. The majority of infrared spectrometry is stiU done by the classic method of transmission spectrometry the intensity of an infrared beam passing completely through a sample is measured. The standard description of how much radiation passes through the sample is that of Beet s law (or the Bouguer-Beer-Lambertlaw) ... 

X-Ray Absorption Spectroscopy. As the excitation energy incident on a sample is increased, sharp rises in absorption occur at the K, L,..., absorption edges where the energy just matches that required for ionization by ejection of an electron from the K, L,. .., sheUs. These energies are characteristic for each element. The absorption foUows Beer s law (eq. 1), which in this region is usually written as A( A) = )px where p is the density,... 

The larger a fireball is, the stronger the absorption. An increase in absorption implies an increase in emissivity. Based on Beers s law, the following expression... 

It is often experimentally convenient to use an analytical method that provides an instrumental signal that is proportional to concentration, rather than providing an absolute concentration, and such methods readily yield the ratio clc°. Solution absorbance, fluorescence intensity, and conductance are examples of this type of instrument response. The requirements are that the reactants and products both give a signal that is directly proportional to their concentrations and that there be an experimentally usable change in the observed property as the reactants are transformed into the products. We take absorption spectroscopy as an example, so that Beer s law is the functional relationship between absorbance and concentration. Let A be the reactant and Z the product. We then require that Ea ez, where e signifies a molar absorptivity. As initial conditions (t = 0) we set Ca = ca and cz = 0. The mass balance relationship Eq. (2-47) relates Ca and cz, where c is the product concentration at infinity time, that is, when the reaction is essentially complete. 

Instrumental methods of analysis provide different measures of the progress of reaction. Consider this kinetie system as observed by absorption speetroseopy. Beer s law applied to the system gives... 

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)). 

Beer s Law. We have so far considered the light absorption and the light transmission for monochromatic light as a function of the thickness of the absorbing layer only. In quantitative analysis, however, we are mainly concerned with solutions. Beer studied the effect of concentration of the coloured constituent in solution upon the light transmission or absorption. He found the same relation between transmission and concentration as Lambert had discovered between transmission and thickness of the layer [equation (3)], i.e. the intensity of a beam of monochromatic light decreases exponentially as the concentration of the absorbing substance increases arithmetically. This may be written in the form ... 

If it is known that the compound obeys Beer s Law the molar absorption coefficient e can be determined from one measurement of the absorbance of a... 

The above considerations will be illustrated by the simultaneous determination of manganese and chromium in steel and other ferro-alloys. The absorption spectra of 0.001 M permanganate and dichromate ions in 1M sulphuric acid, determined with a spectrophotometer and against 1M sulphuric acid in the reference cell, are shown in Fig. 17.20. For permanganate, the absorption maximum is at 545 nm, and a small correction must be applied for dichromate absorption. Similarly the peak dichromate absorption is at 440 nm, at which permanganate only absorbs weakly. Absorbances for these two ions, individually and in mixtures, obey Beer s Law provided the concentration of sulphuric acid is at least 0.5M. Iron(III), nickel, cobalt, and vanadium absorb at 425 nm and 545 nm, and should be absent or corrections must be made. 

Now, starting with 0.05 mL toluene, repeat the procedure to obtain five working solutions l -5 and use solution 5 to plot the absorption curve of toluene again record the Amax values for the peaks of the curve. There is a well-developed peak at approximately 270 nm, and using the five test solutions, measure the absorbance of each at the observed peak wavelength and test the application of Beer s Law. Measure solution 5 also at the wavelength used for benzene, and solution 5 at the wavelength used for toluene. 

In the preceding discussion of Beer s Law, it was argued that x-ray absorption is a simpler process than the absorption of ultraviolet, visible, and infrared wavelengths. This greater simplicity becomes particularly obvious when x-ray absorption coefficients are examined. 

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