Beer's Law Definition and Equation

Beer's Law is an equation that relates the attenuation of light to properties of a material. The law states that the concentration of a chemical is directly proportional to the absorbance of a solution. The relation may be used to determine the concentration of a chemical species in a solution using a colorimeter or spectrophotometer. The relation is most often used in UV-visible absorption spectroscopy. Note that Beer's Law is not valid at high solution concentrations.

Other Names for Beer's Law

Beer's Law is also known as the Beer-Lambert Law, the Lambert-Beer Law, and the Beer–Lambert–Bouguer Law. The reason there are so many names is because more than one law is involved. Basically, Pierre Bouger discovered the law in 1729 and published it in Essai D'Optique Sur La Gradation De La Lumière. Johann Lambert quoted Bouger's discovery in his Photometria in 1760, saying the absorbance of a sample is directly proportional to the path length of light.

Even though Lambert didn't claim discovery, he was often credited with it. August Beer discovered a related law in 1852. Beer's Law stated that the absorbance is proportional to the concentration of the sample. Technically, Beer's Law relates only to concentration, while the Beer-Lambert Law relates absorbance to both concentration and sample thickness.

Equation for Beer's Law

Beer's Law may be written simply as:

A = εbc

where A is absorbance (no units)
ε is the molar absorptivity with units of L mol^-1 cm^-1 (formerly called the extinction coefficient)
b is the path length of the sample, usually expressed in cm
c is the concentration of the compound in solution, expressed in mol L^-1

Calculating the absorbance of a sample using the equation depends on two assumptions:

1. The absorbance is directly proportional to the path length of the sample (the width of the cuvette).
2. The absorbance is directly proportional to the concentration of the sample.

How to Use Beer's Law

While many modern instruments perform Beer's Law calculations by simply comparing a blank cuvette with a sample, it's easy to prepare a graph using standard solutions to determine the concentration of a specimen. The graphing method assumes a straight-line relationship between absorbance and concentration, which is valid for dilute solutions. 

Beer's Law Example Calculation

A sample is known to have a maximum absorbance value of 275 nm. Its molar absorptivity is 8400 M^-1cm^-1. The width of the cuvette is 1 cm. A spectrophotometer finds A = 0.70. What is the concentration of the sample?

To solve the problem, use Beer's Law:

A = εbc

0.70 = (8400 M^-1cm^-1)(1 cm)(c)

Divide both sides of the equation by [(8400 M^-1 cm^-1)(1 cm)]

c = 8.33 x 10^-5 mol/L

Importance of Beer's Law

Beer's Law is especially important in the fields of chemistry, physics, and meteorology. Beer's Law is used in chemistry to measure the concentration of chemical solutions, to analyze oxidation, and to measure polymer degradation. The law also describes the attenuation of radiation through the Earth's atmosphere. While normally applied to light, the law also helps scientists understand the attenuation of particle beams, such as neutrons. In theoretical physics, the Beer-Lambert Law is a solution to the Bhatnagar-Gross-Krook (BKG) operator, which is used in the Boltzmann equation for computational fluid dynamics.

Sources

• Beer, August. ""Bestimmung der Absorption des rothen Lichts in farbigen Flüssigkeiten" (Determination of the absorption of red light in colored liquids)." Annalen der Physik und Chemie, vol. 86, 1852, pp. 78–88.
• Bouguer, Pierre. Essai d'optique sur la gradation de la lumière. Claude Jombert, 1729 pp. 16–22.
• Ingle, J. D. J., and S. R. Crouch. Spectrochemical Analysis. Prentice Hall, 1988.
• Lambert, J. H. Photometria sive de mensura et gradibus luminis, colorum et umbrae [Photometry, or, On the measure and gradations of light, colors, and shade]. Augsburg ("Augusta Vindelicorum"). Eberhardt Klett, 1760.
• Mayerhöfer, Thomas Günter, and Jürgen Popp. "Beer's law - why absorbance depends (almost) linearly on concentration." Chemphyschem, vol. 20, no. 4, December 2018. doi:10.1002/cphc.201801073