Beer s law

Relationships between the intensity of incident light, sample thickness, concentration and intensity of transmitted light are embodied in Beer s law and Lambert s law. ... 

Beer s law This states that the proportion of light absorbed depends on the thickness d) of the absorbing layer, and on the molecular concentration (c) of the absorbing substance in the layer. It is an extension of Lambert s law, and may be written in the form... 

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 ... 

The absorbance A is proportional to 1 through Beer s law (see Computer Projeet 2-1). The analytical problem is to solve the matrix equation... 

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. 

Beer s law can be extended to samples containing several absorbing components provided that there are no interactions between the components. Individual absorbances, A , are additive. For a two-component mixture of X and Y, the total absorbance, Atot, is... 

Calibration curves showing positive and negative deviations from Beer s law. 

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. 

Chemical Limitations to Beer s Law Chemical deviations from Beer s law can occur when the absorbing species is involved in an equilibrium reaction. Consider, as an example, an analysis for the weak acid, HA. To construct a Beer s law calibration curve, several standards containing known total concentrations of HA, Cmt, are prepared and the absorbance of each is measured at the same wavelength. Since HA is a weak acid, it exists in equilibrium with its conjugate weak base, A ... 

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... 

Stray radiation is the second contribution to instrumental deviations from Beer s law. Stray radiation arises from imperfections within the wavelength selector... 

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

For small concentrations of analyte, Pstray is significantly smaller than Pq and Py, and the absorbance is unaffected by the stray radiation. At higher concentrations of analyte, however, Pstray is no longer significantly smaller than Py and the absorbance is smaller than expected. The result is a negative deviation from Beer s law. 

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. 

The applications of Beer s law for the quantitative analysis of samples in environmental chemistry, clinical chemistry, industrial chemistry and forensic chemistry are numerous. Examples from each of these fields follow. 

Quantitative Analysis for a Single Analyte The concentration of a single analyte is determined by measuring the absorbance of the sample and applying Beer s law (equation 10.5) using any of the standardization methods described in Chapter 5. The most common methods are the normal calibration curve and the method of standard additions. Single-point standardizations also can be used, provided that the validity of Beer s law has been demonstrated. 

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... 

An important assumption in the slope-ratio method is that the complexation reaction continues to completion in the presence of a sufficiently large excess of metal or ligand. The slope-ratio method also is limited to systems that obey Beer s law and in which only a single complex is formed. 

We can derive an expression between precision and transmittance by applying the propagation of uncertainty as described in Chapter 4. To do so we write Beer s law as... 

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. 

The emission spectrum from a hollow cathode lamp includes, besides emission lines for the analyte, additional emission lines for impurities present in the metallic cathode and the filler gas. These additional lines serve as a potential source of stray radiation that may lead to an instrumental deviation from Beer s law. Normally the monochromator s slit width is set as wide as possible, improving the throughput of radiation, while being narrow enough to eliminate this source of stray radiation. 

The relationship between transmittance and the concentration of the scattering particles is similar to that given by Beer s law... 

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. 

This experiment demonstrates the chemical limitations to Beer s law using the NO2-N2O4 equilibrium as an example. Blanco, M. Iturriaga, H. Maspoch, S. et al. A Simple Method for Spectrophotometric Determination of Two-Components with Overlapped Spectra, /. Chem. Educ. 1989, 66, 178-180. 

One instrumental limitation to Beer s law is the use of polychromatic radiation instead of monochromatic radiation. Consider a radiation source that emits two wavelengths of... 

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