**Beer-Lambert law: When light is passed through a medium
and comes out at the other side, its intensity will be reduced, because part of
the light will be absorbed by the medium. The actual amount of light absorbed
by the medium is governed by the Beer-Lambert law. The derivation of the law is
as follows:**** **

*Lambert’s law*: (This
law was actually observed by Pierre Bouguer in 1729. Its origin is wrongly
attributed to Johann Heinrich Lambert, who only quoted from Bouguer’s work in
1760). This law says that the decrease in intensity 'dI' when light passes
through a medium of thickness ‘dx’ is directly proportional to (a) the
intensity of the incident radiation ‘I’ and (b) the thickness of the medium.
This may be mathematically expressed as:** **

*-dI a I dx
or -dI =
k*_{1 }I dx

**where ‘k’ is the proportionality
constant. The minus sign indicates that the intensity decreases as light passes
through the medium.**

*Beers Law*: (August Beer, 1852): This law says
that the decrease in intensity (dI) when light passes through a solution of
concentration ‘c’ is directly proportional to the concentration. This may be
mathematically expressed as:

*-dI a c or -dI =
k*_{2 }c

** ****Combining the two relationships, we get:**

*-dI
a**
I c dx* or *-dI
= k I c dx* or

**
Samples in spectroscopy are usually in the form of
solutions. The solution of molar
concentration ‘c’ is taken in a sample cell. When light passes through the
solution in the cell, it has to travel a distance ‘***l***’ through the solution.
Therefore this is called the path length. If ‘I**_{0}’ is the intensity
of incident light, and it decreases to ‘I’ after passing through the solution,
then the intensity is I_{0} when x=0 and I when x=*l***. Therefore,
integrating the above equation between these limits,**
** or or **

**or **** or **

** If is represented as ‘A’
and the constant as ‘e’,
then we can write**

**A = e c ***l*** (Beer-Lambert law)**

**where ‘A’ is known as the ‘absorbance’
and ‘e’
is a constant for a given substance and called its ‘molar absorptivity
coefficient’ (earlier known as ‘molar extinction coefficient’). When
the path length is 1 cm and the concentration of the solution is 1 molar, then
A = e.
Therefore the molar absorptivity coefficient ‘e’ can be defined as the
absorbance of a 1 molar solution for a path length of 1 cm. (Therefore most
sample cells commercially available have a path length of 1 cm). The unit of ‘e’ is
mol**^{-1}
cm^{-1}.
‘A’ has no unit since it is a ratio.

**If a substance has a very high
value for ‘e’,
then very low concentrations can be detected and very dilute solutions can be
used for spectroscopic analysis. If the value of ‘e’ for the substance is
low, then concentrated solutions have to be used for spectroscopic analysis.**