Linearity, Beer’s law

A spectrophotometric investigation of the ground-state absorption of benzophenone in pure SC CO2 and in the presence of 2-propanol was performed in order to determine both benzophenone solubility and the potential for solute/solute or co-solvent/solute aggregations. Linear Beer s law plots of absorbance versus concentration of benzophenone over a concentration range of 1-15 mM in SC OO2 insured that the ketone was fully disolved and in a non-associated form under the conditions of the laser experiments (- 10 mM). Analysis of the absorption spectrum of benzophenone in SC CO2 ranging from high to low pressure, resulted in absorption bands nearly... 

The assumptions of linearity (Beer s Law) and additivity (Equations 1 and 2) can be stated for UV absorption as follows ... 

The following statements hold true for what is most often termed Beer s law (1) The relationship between transmittance and concentration is nonlinear, and (2) the relationship between absorbance and concentration is linear. Beer s law is the common basis for quantitative analysis. Knowledge of Beer s law allows us to calculate the maximum theoretical dynamic range for an instrument using a few simple mathematical relationships. 

The iimnodified temi absorbance usually means this quantity, though some authors use the Napierian absorbance B = -hiT. The absorbance is so iisefiil because it nomially increases linearly with path length, /, tlirough the sample and with the concentration, c, of the absorbing species within the sample. The relationship is usually called Beer s law ... 

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. 

According to Beer s law, a calibration curve of absorbance versus the concentration of analyte in a series of standard solutions should be a straight line with an intercept of 0 and a slope of ab or eb. In many cases, however, calibration curves are found to be nonlinear (Figure 10.22). Deviations from linearity are divided into three categories fundamental, chemical, and instrumental. 

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. 

Because values of ttuA nray depend on the concentration of HA, equation 10.10 may not be linear. A Beer s law calibration curve of A versus Qot will be linear if one of two conditions is met. If the wavelength is chosen such that 8ha and 8a are equal, then equation 10.10 simplifies to... 

Effect of wavelength on the linearity of a Beer s law calibration curve. 

Spectroscopic measurements may also involve the scattering of light by a particulate form of the analyte, fn turbidimetry, the decrease in the radiation s transmittance through the sample is measured and related to the analyte s concentration through Beer s law. fn nephelometry we measure the intensity of scattered radiation, which varies linearly with the analyte s concentration. 

Fixed-time integral methods are advantageous for systems in which the signal is a linear function of concentration. In this case it is not necessary to determine the concentration of the analyte or product at times ti or f2, because the relevant concentration terms can be replaced by the appropriate signal. For example, when a pseudo-first-order reaction is followed spectrophotometrically, when Beer s law... 

Proportionality between colour and concentration. For visual colorimeters it is important that the colour intensity should increase linearly with the concentration of the substance to be determined. This is not essential for photoelectric instruments, since a calibration curve may be constructed relating the instrumental reading of the colour with the concentration of the solution. Otherwise expressed, it is desirable that the system follows Beer s law even when photoelectric colorimeters are used. 

The wavelength of maximum absorption and the molar absorptivity are very dependent on pH, buffer, temperature, solvent, and the presence of other materials that may interact with anthocyanins. In addition, anthocyanin absorption follows a linear relationship with concentration only when present at low levels therefore considerable dilution is usually necessary. Absorbance normally should vary from 0.2 to 1.0 unit in order to obey Lambert-Beer s law. However, absorbance values as high as 1.5 to 2.0 absorbance units may be valid for sophisticated new instruments. 

Measuring compound remaining in solution has the advantage that the endpoint is quantitative and similar to that in a thermodynamic solubility assay. Within reasonable experimental parameters so that Beer s law is followed a UV absorbance is linearly related to concentration in solution. Measuring compound concentration in solution is very well established technology with a wealth of available instrumentation. 

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... 

Khashaba et al. [34] suggested the use of sample spectrophotometric and spectrofluorimetric methods for the determination of miconazole and other antifungal drugs in different pharmaceutical formulations. The spectrophotometric method depend on the interaction between imidazole antifungal drugs as -electron donor with the pi-acceptor 2,3-dichloro-5,6-dicyano-l,4-benzoquinone, in methanol or with p-chloranilic acid in acetonitrile. The produced chromogens obey Beer s law at Amax 460 and 520 nm in the concentration range 22.5-200 and 7.9-280 pg/mL for 2,3-dichloro-5,6-dicyano-l,4-benzoquinone and p-chloranilic acid, respectively. Spectrofluorimetric method is based on the measurement of the native fluorescence of ketoconazole at 375 nm with excitation at 288 nm and/or fluorescence intensity versus concentration is linear for ketoconazole at 49.7-800 ng/mL. The methods... 

Walash et al. [14] described a kinetic spectrophotometric method for determination of several sulfur containing compounds including penicillamine. The method is based on the catalytic effect on the reaction between sodium azide and iodine in aqueous solution, and entails measuring the decrease in the absorbance of iodine at 348 nm by a fixed time method. Regression analysis of the Beer s law plot showed a linear graph over the range of 0.01 0.1 pg/mL for penicillamine with a detection limit of 0.0094 pg/mL. 

Thus, the band at the lower wavelengths exhibits perfect linearity, but the one at the higher wavelengths does not. Therefore, even though the underlying spectra follow Beer s law, the measured spectra not only show nonlinearity, they do so differently at different wavelengths. This is clearly shown in Figure 27-2, where absorbance versus concentration is plotted for the two peaks. 

When other types of samples are measured, the resulting data is usually known to be nonlinear (except possibly in a few special cases), so those measurements are of no interest to us here. Thus, in practice, the invocation of linearity implies the assumption that Beer s Law holds, therefore discussions of nonlinearity are essentially about those phenomena that cause departures from Beer s law. 

In practice, of course, this effect is very small, normally much smaller than any of the other sources of nonlinear behavior, and we are ordinarily safe in ignoring it, and calling Beer s law behavior linear in the absence of any of the other known sources of nonlinear behavior. However, the point here is that this completes the demonstration of our statement above, that Beer s law never exactly holds IN PRINCIPLE and that as spectroscopists we never ever really work with perfectly linear data. 

The Beer-Lambert law (also called the Beer-Lambert-Bouguer law or simply Beer s law) is the linear relationship between absorbance and concentration of an absorber of electromagnetic radiation. The general Beer-Lambert law is usually written as ... 

A continuous source can be used for atomic absorption, but since only the center part of the band of wavelengths passed by the slit will be absorbed (due to the sharp line nature of atomic absorption), sensitivity will be sacrificed, and the calibration curve will not be linear. This curvature is because even at high concentrations, only a portion of the radiation passing through the slit will be absorbed, and the limiting absorbance will approach a finite value rather than infinity. With a sharp line source, the entire width of the source radiation is absorbed and so the absorption follows Beer s law. A continuous source works best with the alkali metals because their absorption lines are broader than for most other elements. Specificity is not as great with a continuous source because nearby absorbing lines or molecular absorption bands will absorb part of the source. 

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