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# The Beer-Lambert Law and Its Limitation Essay

Objective :

1. To determine the linear relationship between absorbance and concentration of an absorbing species. 2. To study the effects of molecular dissociation complex formation on the applicability of the Beer-Lambert Law. 3. To investigate the derivation and limitation of Beer-Lambert Law.

Introduction:

In optics, the Beer–Lambert law, also known as Beer’s law, the Lambert–Beer law, or the Beer–Lambert–Bouguer law relates the absorption of light to the properties of the material through which the light is traveling. The general Beer-Lambert law is usually written as: A = a() * b * c

where A is the measured absorbance, a() is a wavelength-dependent absorptivity coefficient, b is the path length, and c is the analyte concentration. When working in concentration units of molarity, the Beer-Lambert law is written as: A = * b * c

where is the wavelength-dependent molar absorptivity coefficient with units of M-1 cm-1. The law states that there is a logarithmic dependence between the transmission (or transmissivity), T, of light through a substance and the product of the absorption coefficient of the substance, α, and the distance the light travels through the material (i.e., the path length), ℓ. The absorption coefficient can, in turn, be written as a product of either a molar absorptivity (extinction coefficient) of the absorber, ε, and the molar concentration c of absorbing species in the material, or an absorption cross section, σ, and the (number) density N’ of absorbers. Experimental measurements are usually made in terms of transmittance (T), which is defined as: T = I / Io

where I is the light intensity after it passes through the sample and Io is the initial light intensity. The relation between A and T is: A = -log T = – log (I / Io).
Absorption of light by a sample

Modern absorption instruments can usually display the data as either transmittance, %-transmittance, or absorbance. An unknown concentration of an analyte can be determined by measuring the amount of light that a sample absorbs and applying Beer’s law. If the absorptivity coefficient is not known, the unknown concentration can be determined using a working curve of absorbance versus concentration derived from standards. The Beer-Lambert law can be derived from an approximation for the absorption coefficient for a molecule by approximating the molecule by an opaque disk whose cross-sectional area, , represents the effective area seen by a photon of frequency w.

If the frequency of the light is far from resonance, the area is approximately 0, and if w is close to resonance the area is a maximum. The linearity of the Beer-Lambert law is limited by chemical and instrumental factors. Causes of nonlinearity include: deviations in absorptivity coefficients at high concentrations (>0.01M) due to electrostatic interactions between molecules in close proximity scattering of light due to particulates in the sample

fluoresecence or phosphorescence of the sample
changes in refractive index at high analyte concentration
shifts in chemical equilibria as a function of concentration non-monochromatic radiation, deviations can be minimized by using a relatively flat part of the absorption spectrum such as the maximum of an absorption band stray light

Beer’s law can be applied to the analysis of a mixture by spectrophotometry, without the need for extensive pre-processing of the sample. An example is the determination of bilirubin in blood plasma samples. The spectrum of pure bilirubin is known, so the molar absorption coefficient is known. Measurements are made at one wavelength that is nearly unique for bilirubin and at a second wavelength in order to correct for possible interferences.The concentration is given by c = Acorrected / ε.

Materials :
Benzene, Cyclohexane, Unknown solution U1.2, 0.1 M potassium thiocynate,
Acetone, 3 M Sulphuric acid, 10-3 M ferric sulphate in 1 M Sulphuric acid

Procedure:
A. Determination of benzene
The spectrum of cyclohexane alone is recorded over the range 230-270nm. A stock solution is made by making 0.5cm3 benzene up to 25cm3 with cyclohexane, taking 0.5cm3 of this solution and again made it up to 25cm3. From this stack solution, the following standard solutions are prepared for calibration: Solution 1. 8cm3 stock solution, made up to 10cm3

2. 6cm3 stock solution, made up to 10cm3
3. 4cm3 stock solution, made up to 10cm3
4. 2cm3 stock solution, made up to 10cm3
5. 1cm3 stock solution, made up to 10cm3
The spectra of these standard solutions over the range 230-270nm are recorded and the absorbance at the wavelength of strongest absorption is noted. The absorbance value of the unknown solution U1.2 is measured.

B. Effect of complex formation
100mL of the potassium thiocynate solution is mixed with 10mL of the the ferric sulphate solution and 100mL of water is added. Stirred briskly and the spectrum is recorded. Then, the absorbance is measured at 480nm at the beginning of mixing. The solution is exposed to sunlight and the absorbance is measured at 5,10,15 and 20 minutes later. Another mixture of the thiocynate and iron solutions are prepared. 100mL of acetone is added instead of water. The spectrum is recorded. The absorbance is measured at 5, 10, 15 and 20 minutes later at 480nm.

Treatment of data:
1. Graphical determination of benzene part A:
i) A graph of absorbance versus concentration is plotted and a straight line is drawn to pass through the origin. The concentration of benzene is determined based on the calibration graph. 2. From the results in part B, graph of absorbance versus time is plotted for complex formation with (i) water and (ii) acetone.

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