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Beer-Lambert

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This page takes a brief look at the Beer-Lambert Law and explains the use of the terms absorbance and molar absorptivity relating to UV-visible absorption spectrometry. AbsorbanceMeasuring the absorbance of a solutionIf you have read the page about how an absorption spectrometer works, you will know that it passes a whole series of wavelengths of light through a solution of a substance (the sample cell) and also through an identical container (the reference cell) which only has solvent in it. | | Note: It isn't essential to read about how the spectrometerworks, but you could follow this link if you are interested or if it is on your syllabus. Everything you need from that page to understand the present topic is repeated below. | For each wavelength of light passing through the spectrometer, the intensity of the light passing through the reference cell is measured. This is usually referred to as Io - that's I for Intensity.The intensity of the light passing through the sample cell is also measured for that wavelength - given the symbol, I.If I is less than Io, then obviously the sample has absorbed some of the light. A simple bit of maths is then done in the computer to convert this into something called the absorbance of the sample - given the symbol, A.For reasons to do with the form of the Beer-Lambert Law (below), the relationship between A (the absorbance) and the two intensities is given by:On most of the diagrams you will come across, the absorbance ranges from 0 to 1, but it can go higher than that.An absorbance of 0 at some wavelength means that no light of that particular wavelength has been absorbed. The intensities of the sample and reference beam are both the same, so the ratio Io/I is 1. Log10 of 1 is zero.An absorbance of 1 happens when 90% of the light at that wavelength has been absorbed - which means that the intensity is 10% of what it would otherwise be.In that case, Io/I is 100/I0 (=10) and log10 of 10 is 1. | | Note: If you don't feel comfortable with logarithms, don't worry about it. Just accept that an absorbance scale often runs from zero to 1, but could go higher than that in extreme cases (in other words where more than 90% of a wavelength of light is absorbed). | Absorbance isn't very good for making comparisonsThe importance of concentrationThe proportion of the light absorbed will depend on how many molecules it interacts with. Suppose you have got a strongly coloured organic dye. If it is in a reasonably concentrated solution, it will have a very high absorbance because there are lots of molecules to interact with the light.However, in an incredibly dilute solution, it may be very difficult to see that it is coloured at all. The absorbance is going to be very low.Suppose then that you wanted to compare this dye with a different compound. Unless you took care to make allowance for the concentration, you couldn't make any sensible comparisons about which one absorbed the most light.The importance of the container shapeSuppose this time that you had a very dilute solution of the dye in a cube-shaped container so that the light travelled 1 cm through it. The absorbance isn't likely to be very high. On the other hand, suppose you passed the light through a tube 100 cm long containing the same solution. More light would be absorbed because it interacts with more molecules.Again, if you want to draw sensible comparisons between solutions, you have to allow for the length of the solution the light is passing through.Both concentration and solution length are allowed for in the Beer-Lambert Law. The Beer-Lambert LawWhat the Law looks likeYou will find that various different symbols are given for some of the terms in the equation - particularly for the concentration and the solution length. I'm going to use the obvious form where the concentration of the solution is "c" and the length is "l". | | Note: That's obviously "l" for length. The font I'm using won't distinguish between "l" for length and a capital letter "I" (for Intensity). That problem disappears in the equation below - where it is obvious which is which. | You should recognise the expression on the left of this equation as what we have just defined as the absorbance, A. You might also find the equation written in terms of A: That's obviously easier to remember than the first one, but you would still have to learn the equation for absorbance. It might be useful to learn it in the form: The Greek letter epsilon in these equations is called the molar absorptivity - or sometimes the molar absorption coefficient. Molar absorptivityIf you rearrange the simplest of the equations above to give an expression for epsilon (the molar absorptivity), you get: Remember that the absorbance of a solution will vary as the concentration or the size of the container varies. Molar absorptivity compensates for this by dividing by both the concentration and the length of the solution that the light passes through. Essentially, it works out a value for what the absorbance would be under a standard set of conditions - the light travelling 1 cm through a solution of 1 mol dm-3.That means that you can then make comparisons between one compound and another without having to worry about the concentration or solution length.Values for molar absorptivity can vary hugely. For example, ethanal has two absorption peaks in its UV-visible spectrum - both in the ultra-violet. One of these corresponds to an electron being promoted from a lone pair on the oxygen into a pi anti-bonding orbital; the other from a pi bonding orbital into a pi anti-bonding orbital. | | Note: These jumps are described in detail on the page explaining the theory of UV-visible spectrometry. Depending on your background knowledge, you may have to read another page first, but there is a link to that on the theory page.Use the BACK button on your browser to return to this page later if you choose to follow this link. | The table gives values for the molar absorptivity of a solution of ethanal in hexane. Notice that there are no units given for absorptivity. That's quite common. If you want them, and assuming the length is in cm and the concentration is mol dm-3, the units are mol-1 dm3 cm-1. electron jump | wavelength of maximum absorption (nm) | molar absorptivity | lone pair to pi anti-bonding orbital | 290 | 15 | pi bonding to pi anti-bonding orbital | 180 | 10000 | The ethanal obviously absorbs much more strongly at 180 nm than it does at 290 nm. (Although, in fact, the 180 nm absorption peak is outside the range of most spectrometers.)You may come across diagrams of absorption spectra plotting absorptivity on the vertical axis rather than absorbance. However, if you look at the figures above and the scales that are going to be involved, you aren't really going to be able to spot the absorption at 290 nm. It will be a tiny little peak compared to the one at 180 nm.To get around this, you may also come across diagrams in which the vertical axis is plotted as log10(molar absorptivity).If you take the logs of the two numbers in the table, 15 becomes 1.18, while 10000 becomes 4. That makes it possible to plot both values easily, but produces strangely squashed-looking spectra! |

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Lambert-Beer's law
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Last updated 3 months ago
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The law was first developed by Pierre Bouguer before 1729. It was later attributed to Johann Heinrich Lambert who cited Bouguer’s findings. The law included path length as a variable that affected absorbance. Later, Beer extended in 1852 the law to include the concentration of solutions, thus giving the law its name Beer-Lambert Law.
Contents
[hide] * 1 Definition & Equation * 2 Derivation of Law * 3 Deviations to the law * 4 Applications * 5 Links * 5.1 Related articles * 5.2 External links * 5.3 Bibliography
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Definition & Equation[ edit | edit source] * The Beer-Lambert law states that the quantity of light absorbed by a substance dissolved in a fully transmitting solvent is directly proportional to the concentration of the substance and the path length of the light through the solution. * Because Beer's law states this, it means we can both calculate the concentration of a solution by using the absorbancies, or plot a graph of various concentrations, align them to their correct absorbencies, and use a colorimeter to find the concentration of an unknown solution * The law states that:
A(λ) = e(λ) l c.
The proportionality constant e (λ) is called the absorptivity of the substance at the wavelength λ. e (λ) is called the molar absorptivity if the concentration is measured in moles/liter. * The absorbance is inversely proportional to the transmittence of the solution
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Derivation of Law[ edit | edit source]
A spectrophotometer is an apparatus that measures the intensity, energy carried by the radiation per unit area per unit time, of the light entering a sample solution and the light going out of a sample solution. The two intensities can be expressed as transmittance: the ratio of the intensity of the exiting light to the entering light or percent transmittance (%T). Different substances absorb different wavelengths of light. Therefore, the wavelength of maximum absorption by a substance is one of the characteristic properties of that material. A completely transparent substance will have It = I0 and its percent transmittance will be 100. Similarly, a substance which allows no radiation of a particular wavelength to pass through it will have It = 0, and a corresponding percent transmittance of 0.
Transmittance
T = It / I0
% Transmittance: %T = 100 T
Absorbance
A = log10 (I0/It)
A = log10 (1/T) = -log10 (T)
A = log10 (100/%T)
A = 2 - log10 (%T)
Transmittance for liquids is usually written as: T = I/I0=10-αl =10Σlc''
Transmittance for gases is written as T = I/I0 = 10-αl = e-σlN
I0 and I are the intensity (or power) of the incident light and the transmitted light, respectively.
Absorbance for liquids is written as A = -log10 = (I/I0)
Absorbance for gases it is written as A´ = -ln(I/I0)
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Deviations to the law[ edit | edit source]
The Beer-Lambert law maintains linearity under specific conditions only. The law will make inaccurate measurements at high concentrations because the molecules of the analyte exhibit stronger intermolecular and electrostatics interactions which is due to the lesser amount of space between molecules. This can change the molar absorptivity of the analyte. Not only does high concentrations change molar absorptivity, but it also changes the refractive index of the solution causing departures from the Beer-Lambert law.
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Applications[ edit | edit source]
Beer-Lamberts law is applied to the analysis of a mixture by spectrophotometry, without the need for extensive pre-processing of the sample. Examples include the determination of bilirubin in blood plasma samples. The spectrum of pure bilirubin is known thus the molar absorbance is known. Measurements are made at one specific wavelength almost unique for bilirubin and another measurement at a second wavelength so interferences or deviations can be eliminated or corrected. Generally, it can be used to determine concentrations of a particular substance, or determine the molar absorptivity of a substance.

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