Beer’s Law
Many compounds
absorb ultraviolet (UV) or visible (Vis.) light. The diagram below shows a beam
of monochromatic radiation of radiant power P_{0}, directed at a sample
solution. Absorption takes place and the beam of radiation leaving the sample has
radiant power P.

The relationship
between absorbance and transmittance is illustrated in the following diagram:
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 BeerLambert 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=ebc
Where A is
absorbance (no units, since A = log_{10} P_{0} / P )
e is the molar absorbtivity
with units of L mol^{1}
cm^{1
}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.^{
}
c is the concentration of the compound in solution, expressed in mol/L
Question : Why do we prefer to express
the BeerLambert law using absorbance as a measure of the absorption rather
than %T ?
Answer : To begin, let's think about
the equations...
A=ebc
%T = 100 P/P_{0} = e ^{}^{e}^{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:
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 
0 
0.3 
0.6 
0.9 
1.2 
1.5 


A = ebc tells us that absorbance
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. This deviation from the
Law is not dealt with here. 
The linear
relationship between concentration and absorbance is both simple and
straightforward, which is why we prefer to express the BeerLambert law using
absorbance as a measure of the absorption rather than %T.
Question : What is the significance of
the molar absorptivity, e ?
Answer : To begin we will rearrange
the equation A = ebc :
e = A / bc
In words, this
relationship can be stated as "e is a measure of the amount
of light absorbed per unit concentration".
Molar absorptivity
is a constant for a particular substance, so if the concentration of the
solution is halved so is the absorbance, which is exactly what you would
expect.
Let us take a
compound with a very high value of molar absorptivity, say 100,000 L mol^{1} cm^{1}, which is in a solution in
a 1 cm pathlength cuvette and gives an absorbance of 1.
e = 1 / 1 ´ c
Therefore, c = 1 /
100,000 = 1 ´ 10^{5}
mol L^{1}
Now let us take a
compound with a very low value of e, say 20 L mol^{1} cm^{1
} which is in solution in a 1
cm pathlength cuvette and gives an absorbance of 1.
e = 1 / 1 ´ c
Therefore, c = 1 /
20 = 0.05 mol L^{1}
The answer is now
obvious  a compound with a high molar absorptivity is very effective at
absorbing light (of the appropriate wavelength), and hence low concentrations
of a compound with a high molar absorptivity can be easily detected.
Question : What is the molar
absorptivity of Cu^{2+ } ions
in an aqueous solution of CuSO_{4} ? 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 color. However, the actual molar
absorptivity value is 20 L mol^{1} cm^{1} ! The bright blue color is seen because the
concentration of the solution is very high.
bcarotene is an organic
compound found in vegetables and is responsible for the color of carrots. It
is found at exceedingly low concentrations. You may not be surprised to learn
that the molar absorptivity of bcarotene is 100,000 L mol^{1} cm^{1} !