THANK's to the "Division of Chemistry" of the "Sheffield Hallam University" .  —   A very good job! Peter Forster.
(But I by myself will consequently use the Terms:  Absorption,  Absorptivity.)

SORRY! But with some green I made it PHYSICAL and ALGEBRAIC a little bit MORE CORRECT! Peter Forster.

Do you know:   The Beer-Lambert-Bouguer-law as usually written, is only the half of the true?
                         The more correct answer:  The:  GENERAL BEER LAMBERT LAW ! [9]
Peter Forster.



NEW       Have you also seen my: "Beer-Lambert Law  FAQ's / Spectroscopy  FAQ's"?       NEW
                              (FAQ's:  Frequently Asked Questions)

Introduction

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 amount of radiation absorbed may be measured in a number of ways:

Transmittance:         T = P / P0
% Transmittance: %T = 100 T

Absorbance:

A = log10 (P0 / P)
A = log10 (1 / T)     = -log10 (T)
A = log10 (100 / %T)
A = 2 - log10 (%T)

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.


The Beer-Lambert Law (Beer's Law)

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!

Where:

Almost used Unit:

   AU = Absorption Unit = 1.
mAU = milli AU.
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. 
b   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.
c   is the concentration of the compound in solution, expressed in mol L -1
         (Remember:   MolaRity = mol / L Solution;      MolaLity = mol / kg Solvent !)
  (used Synonyms: Molar absorptivity;
Molar absorption;
Molar extinctivity;
Molar extinction

Molar absorptivity coefficient;
Molar absorption coefficient;
Molar extinctivity coefficient;
Molar extinction coefficient;
(Molar-specific) absorptivity;   . . . . )
   "Molar" means[2, 3], all times the same number of 'molecules' inside the measuring beam (if b = const.)
   (Remember too: That contrary: "Specific Absorptivity Coefficient"
                                or short: "Specific Absorptivity" is expressed for the fixed Situation: 0.1g L-1 m-1   =>  1%, 1cm )[4]

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])

Let's have a look at a few questions... :

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

Answer :    To begin, let's think about the equations...

A(λ, env)= ε(λ, env) b c     [*]

%T(λ, env) = 100 P(λ, env)/P0(λ, env) = e (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)
Pierre Bouguer
                                                                                                                                    (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-1 which 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.

Review your learning

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.


And now, what's about:   The:  GENERAL BEER LAMBERT LAW ? [7] Peter Forster.


References References:
 
[*]  "Do you know, that this equation is still not correct and never correct enough?:" 

One big mistake is located in ε(λ, env)
Mother natur learned us, that we have to rewrite it at least as: 
Molar absorptivity:
  ε( λ, sw, [pH], solv, temp, μΘ, B,   etc.)
λ       : wavelength
sw     : Slit width in use.
pH    : equiv. Proton concentration
solv   : Solvent or Matrix or Mixture[8].
temp  : Temperature.
μΘ    : phys. & chem. Reaction State.
B      : Magnetic Field.
Now you must already be able to estimate by yourself of how much practical value all the published "molar" and not so "molar" absorptivity/extinction coefficient and their precision are!
-  Don't ask me, why nobody takes ever care of!  
-  Why also, is everybody so keen to tell you about:
TROUBLES & EXCEPTIONS & LIMITS & DEVIATIONS & 'NOT OBEYED' & 'PROBLEMS' of:
The Beer-Lambert-Bouguer-law?
-  But now you know, why I never used a ε-Value from the Literatur.
-  I'm sure, NOW you can tell me the value for all published "molar" absorptivity coefficients -
     for your copper sulfate and your β-carotene!

There is also a lot of Information on my:  Spectroscopy FAQ's / Beer-Lambert-Bouguer Law FAQ's Page for you.
(FAQ's:  Frequently Asked Questions)

There you will also be able to discover, why I wrote [pH] instead of pH:
          www.p-forster.com/english/themes/spectroscopy/Spectroscopy_FAQ.htm#FAQ_MA03

[1] "Spectroscopy: Monochromatic:
      Do you know, Who felt himself competent enough to 'introduce' the term "monochromatic"
      to The Beer-Lambert-Bouguer-law (BLBL) in such a .... kind?
      It's such an easy game to play, to show you the evidence of THE ABSOLUT CONTRARY !!:
"monochromatic" .IS NEVER a Requirement for The BLBL !      
      But that means that also Wikipedia and most of the other Wiki's must be completely wrong!
      www.p-forster.com/english/themes/spectroscopy/Spectroscopy_FAQ.htm#FAQ_US01
[2] "Molar:   —  is the Measure of Concentrations in the Unit: Mole per Litre”
      Note: Mole is a number of 'molecules' ("species") expressed as a weight!
      Mole: http://en.wikipedia.org/wiki/Mole (Unit)
[3] "Molar:”   —  is very frequently used on universities, colleges, and schools!
      Note: Molar is very rarely used in the practical field (quality control) and production!
      Molar makes only sense, if you are interested in comparing transition probabilities,
      oscillator strength? (force?), and other such parameters of different species.
      Molar:  1.0 molar  =  6.0221415 × 1023 'molecules' ('species') in 1.0 Litre SOLUTION!
      http://en.wikipedia.org/wiki/Concentration#Molarity
[4] "Pharmacopoeia (IP, EP, or USP):
      Specific Absorptivity Coefficient or Specific Absorptivity are otherwise then by xP's only rarely used.
      http://"The International Pharmacopoeia"
[5] "Why has direct proportionality from Concentration to Absorbance such Lacks?
      Or have you other such "Frequently Asked Questions" (FAQ's):
      UV/Vis- NNIR- NIR- Spectroscopy FAQ's / Beer-Lambert-Bouguer Law FAQ's
[6] "Spectroscopy: Lesson 2:
      6.  That's real linearity and how you have to test it !"    [Paragraph 5.), case a.)]
      www.p-forster.com/english/themes/spectroscopy
[7] "Spectroscopy: Lesson 1:
1.  .... (We were never able, to find a real case, when a solution was not following strictly the law of Beer - Lambert! ! ....)     —     What surprise!
www.p-forster.com/english/themes/spectroscopy
[8] "Linearity:   —  maybe a little more?”
      www.p-forster.com/english/themes/Spectroscopy/Linearity.htm
[9] The:  GENERAL BEER LAMBERT LAW !!!:
That's the TRUE as — mother nature, - Mr. A. Beer, - Mr. J. H. Lambert, and - Mr. P. Bouguer are telling you.
It is also still true, even when a Mixture (for that see: MCA),   more as only ONE Compound/Substance - or more precise: Species, is present!
Read: www.p-forster.com/english/themes/Spectroscopy/Spectroscopy.htm
[10] Limits of Concentration/Detection of the Beer-Lambert Law:
      What's the lower Limit of Concentration of the Beer-Lambert Law (Beer's Law)?
      What's the upper Limit of Concentration of the Beer-Lambert Law (Beer's Law)?



Beer's Law - Quiz
Corrected Beer's Law
UV-Vis. Absorption spectroscopy


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