The Beer-Lambert law

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.


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