# What is Beer-Lambert's Law?

#### Editorial Team - GoPhotonics

Nov 16, 2022

Beer-Lambert's law is a combination of Lambert's law (1730) and Beer's law (1850), which governs the light radiation absorption by molecules at low concentrations. Beer-Lambert law relates to the attenuation of light on passing through a physical material containing attenuating species of uniform concentration. The absorption of light is also directly proportional to the optical path length through the sample and the concentration of the species. This expression is give by:

Where A is the absorbance, ξ is a proportionality constant also called the molar attenuation / extinction coefficient or absorptivity of the attenuating species, l is the optical path length in cm, and C is the concentration of the attenuating species.

Derivation of Beer-Lambert's law

As a wavelength of light is passed through the spectrometer, the intensity of the light passing through the reference cell is measured. This is usually referred to as Io where I is Intensity.

The intensity of the light passing through the sample cell is also measured for that wavelength, represented with the symbol, I. If I is less than Io, then the sample has absorbed some of the light. Suppose A represents the absorbance of the sample given by,

The Absorbance depends on two external assumptions:

• Beer’s Law: The absorbance is directly proportional to the concentration (C) of the solution of the sample used in the experiment.

• Lambert’s Law: The absorbance is directly proportional to the length of the light path (l), which is equal to the width of the cuvette.

These proportionalities can be combined and converted into equality by including a proportionality constant (ξ).

The constant ξ is called molar absorptivity or molar extinction coefficient and is a measure of the probability of the electronic transition. Values for molar absorptivity can vary hugely. Eg, ethanal has two absorption peaks in its UV-visible spectrum - both in the ultra-violet. One of these corresponds to an electron being promoted from a lone pair on the oxygen into a pi anti-bonding orbital; the other from a  π bonding orbital into a  π   anti-bonding orbital.

Beer-Lambert Law Graph

The above graph illustrates the Beer-Lambert law. The x-axis will have units of concentration and the y-axis will be absorbance. This indicates that the other two variables in the equation, molar extinction coefficient, and path length, are held constant. As the concentration increases, the absorbance will also increase. This linear pattern is because as the concentration increases, there are more molecules present to absorb light, and this causes an increase in absorption.

The slope of the line will be the path length times the molar extinction coefficient. If you know the path length, the molar extinction coefficient can easily be determined. The molar extinction coefficient will be the slope of the line divided by the path length.

Beer-Lambert Law limitations:

1. At very high concentrations, especially if the material is highly scattered, the Beer–Lambart law is not followed. To maintain linearity in the Beer–Lambart law, the absorbance of the test material is within the range of 0.2 to 0.5. In case the concentration is higher, there will be a deviation and will follow a non-linear curve. When the concentration is high the molecules are closer to each other and thus interactions can set in. These interactions can be roughly divided into physical and chemical interactions. These interactions are so strong that light and molecular quantum state intermix which cause the attenuation of electromagnetic radiation completely.
2. If the radiation is intense, nonlinear optical processes can also cause variances. Chemical interactions in contrast change the polarizability and thus absorption.

However, limitations involve detection in a small concentration range, and deviations occur in high concentrations.

Beer-Lambert law in Chemical analysis

The Beer-Lambert law can be applied to the analysis of a mixture of chemical compounds by spectrophotometry. Spectrophotometry is a branch of electromagnetic spectroscopy that deals with the quantitative measurement of the reflection or transmission properties of a material as a function of wavelength. Eg. Let’s suppose we have a tablet and we don’t know which drug is present in it. Though we may know the drug, then the question arises about what its molar concentration is. In electromagnetic spectroscopy, we use electromagnetic radiation (we may take UV rays), which scans the tablet and determines the qualitative (drug present) and the quantitative (concentration) properties of the tablet.

We can determine the concentration of various substances in cell structures by measuring their absorbing spectra in the cell.

Other real-world applications involve unknown biomass concentration determination in an organic reactor, measuring the number of sulphonamides and peroxy-disulphate ions in drugs samples and industrial polymer production, affirming calcium ions and uranium concentration in natural waters, insight on molybdenum in low alloy steel, nonaqueous solvents anions, etc.