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The Beer-Lambert Law By Rebecca Hylton 1

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The Beer-Lambert Law is a mathematical equation which relates the absorption of light to the properties of a substance light source enters a cell of know length and hits a detector at the end. Light intensity decreases between entering the cell and leaving the cell 2

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Figure 1: Experimental set up for visible spectrometry (M. Hippler) Light intensity decreases according to the Beer-Lambert law: I=I 0 exp(-σCl) 3

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More commonly written in terms of Absorbance, A: A = ln(I o /I) = σCl Where: A = Absorbance (no units) I 0 = intensity of light incident upon sample cell I = intensity of light leaving sample cell C = concentration of molecules, number of molecules per unit volume (m -3 ) l = length of sample cell (m) σ = molar absorption cross section (m -2 ) 4

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There are other formulations of the Beer-Lambert Law. Instead of using C and σ, it is possible to use c and σ mol. The Beer-Lamber t Law then becomes: A = ln(I o /I) = σ mol cl Where: σ mol = σ x Avagadros number = molar absorption cross section (m 2 mol -1 ) c = concentration of molecules, number of moles per volume (mol -3 ) 5

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There is an alternate form of the Beer-Lambert Law, using the decadic logarithm: A 10 = log(I 0 /I) =εcl Where: A 10 = decadic absorbance I o = intensity of light incident upon sample cell I = intensity of light leaving sample cell ε = molar extinction coefficient (m 2 mol -1 ) l = length of sample cell c = concentration of molecules, number of moles per unit volume (mol m -3 ) 6

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The natural and decadic forms of the Beer Lambert law can be easily converted between using the following relations: ε = σ mol /ln(10) = (σN A )/ln(10) σ mol = ln(10)ε σ = (ln(10)ε)/N A If the light intensity is measured before and after the cell, and we know ε, σ, or σ mol and we know the path length then it is possible to find the concentration of the sample 7

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The greater the number of molecules able to absorb light the greater the extent of the absorbance More effectively a molecule absorbs light the greater the absorbance For σ the greater the cross-sectional area of the molecule the greater its ability to block the incidental radiation passing through it ε and σ are distinct for each molecule Values for ε and σ can be found in the literature 8

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From the way the Beer-Lambert Law is constructed, it is necessary for dilute samples to have a sufficiently long pathway for the substance to have an absorbance For liquids, the concentration of molecules is relatively high and so the path length can be short – typically 1 cm 9

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10 For gasses the concentration of molecules is relatively very small. When studying atmospheric gasses the path length is 1000s of meters. This is sufficiently long enough for absorbance to occur for even very small concentrations.

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References: D. L. Pavia, G. M. Lampman, G. S. Kriz and J. R. Vyvyan, Introduction to Spectroscopy, Brooks/Cole, Belmont, USA, 2009, 4 th edition, p383-384 J. M. Hollas, Modern Spectroscopy, John Wiley & Sons Ltd, West Sussex, England, 2007, 4 th edition, p32-33 P. Atkins and J. De Paula, Atkins Physical Chemistry, OUP, Oxford, 2008, 8 th Edition, p432-433 M. Hippler, Level 1 Chemical Reaction Kinetics. 11

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