Beer's Law - Beer Lambert Law - Beer Lambert Bouguer Law

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Introduction

The Beer-Lambert Law (Beer's Law)

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Many compounds absorb:
ultraviolet (UV: _˜190 nm - _˜370 nm) or visible (Vis.: _˜370 nm - _˜800 nm) or nnir (NNIR: _˜800 nm - _˜1050 nm) light.
The diagram below shows a beam of ~~monochromatic~~ ^[1] radiation of radiant power P₀, directed at a sample solution. Absorption and others takes place and the beam of radiation leaving the sample has radiant power P.
Monochromatic can never be a requirement, otherwise "Inverted Optics" would not be possible!

The last equation, A = 2 - log₁₀ (%T) , is worth remembering because it allows you to easily calculate absorbance from percentage transmittance data.

The relationship between absorbance and transmittance is illustrated in the following diagram:

So, if all the light passes through a solution without any absorption, then absorbance is zero, and percent transmittance is 100%. If all the light is absorbed, then percent transmittance is zero, and absorption is infinite.

Now let us look at the Beer-Lambert law and explore it's significance. This is important because people who use the law often don't understand it - even though the equation representing the law is so straightforward:

A_{(λ, env)}= ε_{(λ, env)}b c; ^[*] — but that's only true for 'one, sole' Species!

The reason why we prefer to express the law with this equation is because ~~absorbance~~ Absorption is directly proportional to the other parameters, as long as the law is obeyed. We are not going to deal with deviations from the law. (Why ??^[5])

Question : Why do we prefer to express the Beer-Lambert law using Absorbance as a measure of the Absorption rather than %T ?

%T_{(λ, env)} = 100 P_{(λ, env)}/P₀_{(λ, env)} = e ^{-ε_{(λ, env)}bc}

Now, suppose we have a solution of copper sulphate (which appears blue because it has an absorption maximum at 600 nm). We look at the way in which the intensity of the light (radiant power) changes as it passes through the solution in a 1 cm cuvette. We will look at the reduction every 0.2 cm as shown in the diagram below. The Law says that the fraction of the light absorbed by each layer of solution is the same. For our illustration, we will suppose that this fraction is 0.5 for each 0.2 cm "layer" and calculate the following data:

Total Path length [cm]	0	0.2	0.4	0.6	0.8	1.0
%T	100	50	25	12.5	6.25	3.125
Absorbance [AUs]	0	0.3	0.6	0.9	1.2	1.5


Original Work by Pierre Bouguer 1729: _{("Logarithms": Michael Stifel 1544, John Napiers: 1614)} (http://books.google.co.uk/books?id=JNkTAAAAQAAJ&pg=PA1)

A_{(λ, env)} = ε_{(λ, env)}bc tells us that ~~absorbance~~ Absorption depends on the total quantity of the absorbing compound in the light path through the cuvette. If we plot absorbance against concentration, we get a straight line passing through the origin (0,0).

Note that the Law is not obeyed at high concentrations.
( THAT's NOT TRUE !!! — Do YOU know why? )
If you don't have any idea, — then you have NOT UNDERSTOOD
Beer-Lambert-Bouguer! _{Hint: Try to think "GENERAL" !!} [6, 7]
and for Linearity see also: [8].

This deviation from the Law is not dealt with here.

The linear relationship between concentration and ~~absorbance~~ Absorption is both simple and straightforward, which is why we prefer to express the Beer-Lambert law (Please - Beer's Law, not Ber's Law) using Absorbance as a measure of the Absorption rather than %T.
(Be told: This simple and straightforward relationship between concentration and Absorbance is only for the "Special Beer-Lambert Law" and over a restricted concentration range true! More correct ... ... .^{[5, 9]})

Question : What is the significance of the molar absorbtivity, ε_{(λ, env)} ?

Answer : To begin we will rearrange the equation A_{(λ, env)} = ε_{(λ, env)}bc ^[*]:

ε_{(λ, env)} = A_{(λ, env)} / bc ^[*]

In words, this relationship can be stated as:
"ε_{(λ, env)} is a measure of the amount of light absorbed per unit concentration at the measuring wavelength _{(λ, env)}!".

Molar absorbtivity is a constant for a particular substance , more precise: Species!, - at the measuring wavelength _{(λ, env)}, so if the concentration of the solution is halved so is the ~~absorbance~~ Absorption, which is exactly what you would expect.

Let us take a compound with a very high value of molar absorbtivity, say 100,000 L mol ^-1 cm^-1, which is in a solution in a 1 cm pathlength cuvette and gives an ~~absorbance~~ Absorption of 1 AU.

ε_{(λ, env)} = 1_{(λ, env)} / 1 x c

Therefore, c = 1 / 100,000 = 1 x 10^-5 mol L^-1

Now let us take a compound with a very low value of ε_{(λ, env)}, say 20 L mol^-1 cm^-1which is in solution in a 1 cm pathlength cuvette and gives an ~~absorbance~~ Absorption of 1.

ε_{(λ, env)} = 1_{(λ, env)} / 1 ´ c

Therefore, c = 1 / 20 = 0.05 mol L^-1

The answer is now obvious - a compound with a high molar absorbtivity is very effective at absorbing light (of the appropriate wavelength), and hence low concentrations of a compound with a high molar absorbtivity can be easily detected.

Question : What is the molar absorbtivity of Cu²⁺ ions in an aqueous solution of CuSO₄ ? It is either ~20 or ~100,000 L mol^-1 cm^-1

Answer : I am guessing that you think the higher value is correct, because copper sulphate solutions you have seen are usually a beautiful bright blue colour. However, the actual molar absorbtivity value is ~20 L mol^-1 cm^-1 ! The bright blue colour is seen because the concentration of the solution is very high.

β-carotene (b-carotene) is an organic compound found in vegatables and is responsible for the colour of carrots. It is found at exceedingly low concentrations. You may not be surprised to learn that the molar absorbtivity of b-carotene is ~100,000 L mol^-1 cm^-1 !

The "CRIB": [german: "Die Eselsleiter"^[E] ] (for all, who likes so very much molar absorbtivity; molar absorptivity):

"Draw exactly" the picture the equation is telling you, and think of ε_{(λ, env.)} as of a concentration for your substance, in the manner of: Who many Litres of solvent you need for diluting ONE Mole of your compound to measure ONE Absorption in the ONE cm Cuvette at your measuring Wavelength.
"That means", if you need a lot of Litres, that you have a strong absorber, and if you need only a few Litres, that you have a weak absorber in your cuvette. — That's it. Now it's absolute easy to calculate all you like.

You should now have a good understanding of the Beer-Lambert Law; the different ways in which we can report absorption, and how they relate to each other. You should also understand the importance of molar absorbtivity, and how this affects the limit of detection^[10] of a particular compound.

ε_{(λ, env)}

is the molar absorbtivity with units of L mol ^-1 cm^-1;

but depends largely upon the wavelength of the measuring light.

is the path length of the sample - that is, the path length of the cuvette in which the sample is contained. We will express this measurement in centimetres. Some times You see also a d, or l instead in formulas.

is the concentration of the compound in solution, expressed in mol L ^-1
(Remember: MolaRity = mol / L Solution; MolaLity = mol / kg Solvent !)


				Almost used Unit:	AU = Absorption Unit = 1.
					mAU = milli AU.