BEER’S -LAMBERT’S LAW Presented by Vyshak MG I MPharm (Quality assurance department) 08/08/07

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LAWS GOVERNING ABSORPTION OF RADIATION The two laws related to the absorption of radiation are: Beer’s law ( related to concentration of absorbing species) Lambert’s law (related to thickness/path length of absorbing species) These two laws are applicable under the following condition: I = I a + I t I = Intensity of incident light I a = Intensity of absorbed light I t =Intensity of transmitted light and No reflection/scattering of light takes place

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Beer’s law “The intensity of a beam of monochromatic light decreases exponentially with increase in the concentration of absorbing species arithmetically Accordingly, - dI / dc α I (The decrease in the intensity of incident light (I) with concentration c is proportional to the intensity of incident light (I)) -dI / dc = kI (removing and introducing the constant of proportionality ‘k’) -dI / I = k dc (rearranging terms) -I n I = kc + b ……Equation (1) (on integration , b is constant of integration) When concentration = 0, there is no absorbance. Hence I= I o Substituting in equation 1, -I n I o = k*0 + b -I n I o = b

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Substituting the value of b, in equation 1, -I n I = kc –I n I o I n I o – I n I = kc I n I o /I = kc (since log A-log B = log A/B) I o / I = e kc (removing natural logarithm) I / I o = e – kc (making inverse on both sides) I = I o e -kc ….Equation (2) (equation of Beer’s law)

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Lambert’s law “The rate of decrease of intensity (monochromatic light) with the thickness of the medium is directly proportional to the intensity of incident light” i.e. –dI / dt α I This equation can be simplified similar to equation 2 to get the following equation (by replacing ‘c’ with ‘t’) I = I o e – kt ….. Equation (3) [equation of Lambert’s law]

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BEER – LAMBERT,S LAW Equations (2) and (3) can be combined to get I= I o e – kct I = I o 10 – kct (converting natural algorithm to base 10 & K = k * 0.4343) I / I o = 10 – kct (rearranging terms) I o / I = 10 kct (inverse on both side Log I o / I = kct (taking log on both sides) ….. Equation 4 It can be learnt that transmittance (T) = I / I o and Absorbance (A) = log 1 / T Hence A = log 1 / I/ I o A = log I o /I ……. Equation 5

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Using Equation 4 & 5 , Since A= log I o /I and log I o /I = Kct we can infer that A= Kct (instead of K, we can use ε) A= ε ct (Equation of beer – Lambert’s law) Where: A – Absorbance or optical density or extinction co- efficient. ε – Molecular extinction coefficient c – Concentration of the drug (mol/lit) t – Path length (normally 10mm or 1cm)

Deviations from Beer’s law:

Deviations from Beer’s law

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In beer’s law it states that if we plot absorbance A against concentration C a straight line passing through origin is obtained, but usually a deviation from a linear relationship between concentration and absorption and an apparent failure of beer’s law. There are two type of deviation : POSITIVE DEVIATION : When a small change in concentration produces a greater change in absorbance. NEGATIVE DEVIATION : When a large change in concentration produces a smaller change in absorbance.

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Beer’s law will holds good over a wide range of concentration provided the structure of the coloured ion or the coloured non electrolyte in the dissolved state does not change with concentration. If a coloured solution is having a foreign substance whose ions do not react chemically with the coloured components, its small concentration does not effect the light absorption, may affect light absorption and may also alter the value of the extinction coefficient .

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REASONS FOR DEVATION FROM BEER’S LAMBERT LAW Instrumental deviation Physicochemical change in solution Instrumental deviation: Factors like stray radiation, improper slit width, fluctuation in single beam and when monochromatic light is not used can influence the deviation. Physicochemical change in solution: Factors like association, dissociation, ionization (change in pH), faulty development of colour (incompletion of reaction) refractive index at high concentration, can influence such deviation.

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Example: Association: Methylene blue at a concentration 10 -5 M exists as monomer and has λmax of 660nm. But methylene blue at concentration above 10 -4 M exist as a dimer or trimer , but has a λmax of 600. Dissociation: Potassium dichromate at high concentration exist as a orange solution (λmax of 450nm).but on dilution, dichromate ions are dissociated into chromate ions which is yellow in colored (λmax of 410nm) Cr 2 O 7 2- + H 2 O → 2H + + 2CrO 4 ( orange ) (yellow)

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If coloured solute ionizes , dissociates in solution Benzyl alcohol in chloroform exists in a polymeric equilibrium; 4C6H5CH2OH ↔ (C6H5CH2OH)4. Dissociation of the polymer increases with dilution. The monomer absorbs at 2.750 to 2.765 µ. Where as the polymer absorbs at 3.0 µ. Hence absorption at 2.75 micron shows negative deviation whereas at 3.0 µ gives positive deviation.

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3)Deviation may also occur due to presence of impurities that fluoresce or absorb at the absorption wavelength the interference introduces an error in the measurement of absorption or radiation penetrating the sample . 4)If monochromatic light is not used deviation may occur .

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5)If width of slit is not proper and therefore it allows undesirable radiations to fall on the detector . These undesirable radiation might be absorbed by the impurities present in sample and there by change in absorbance.

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6)If the isolation species undergoes polymerization .Ex Benzyl alcohol in carbon tetrachloride in high concentration exits in polymeric form dissociation of this polymer increases with dilution which causes absorbance.

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7)In case of suspension Beer’s law can’t be applied but can be estimated colorimetrically with different known concentration reference curve.

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INSTRUMENTAL DEVIATION FROM BEER’S LAW Beer’s law requires monochromatic radiation .The possibility of error due to practical impossibility of obtaining a monochromatic radiation may be minimized by selection of spectral region, where the change in absorptivity with a change in wavelength is very small. This dictate that wavelength selection from a broad band rather than a sharply rising or sharply falling section of the absorption curve.

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Hence it is necessary to practice to prepare a calibrated curve for the absorbance concentration relationship at chosen wavelength.

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References Instrumental Methods of Chemical Analysis by Gurdeep R. Chatwal ( page no : 108 to 113 ) Instrumental Methods of Analysis by Scoog and West) http://en.wikipedia.org/wiki . Instrumental analysis by willard and merit ( page no : 160 to 163 )

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