The Beer-Lambert Law states that the absorbance of a solution is directly proportional to the path length of the sample and the concentration of the absorbing species in the solution.
Consider a beam of electromagnetic radiation with a narrow range of wavelengths 
passing through a sample of molar concentration 
and length 
. Let 
be the number of photons striking on the sample per unit time, and 
be the number of photons absorbed by the sample. The probability of the number of photons absorbed by the sample 
is dependent on the length of sample 
and the concentration of the sample. Therefore, we have
where 
is a proportionality constant.
As spectroscopy involves the absorption of radiation intensity at different wavelengths, we need to express the above equation in terms of intensity.
The intensity of the radiation is proportional to the number of photons striking per unit area of the sample per unit time. Let the intensity of radiation that is incident on the sample and the outgoing intensity of the radiation be 
and 
respectively. It follows that, the greater the number of incident photons, the higher the outgoing intensity will be, i.e. 
. Furthermore, if 
is the change in intensity after the radiation passes through the sample, then 
, which implies that 
. So, we have
where 
is another proportionality constant.
Integrating the above equation throughout the length of the sample, we have 
, which gives 
, or equivalently,
where 
and 
.
Eq1 is the Beer-Lambert law. 
is the sample’s absorbance and 
is known as the molar absorption coefficient, which represents the nature of the sample. Conventionally, the units of and 
are 
and 
respectively. Moreover, is expressed in 
, and so, is unitless. The Beer-Lambert law is applicable to UV, visible and IR spectroscopy.