# Beer’s Law Equation and Example Beer’s law or the Beer-Lambert law states that the absorption of light by a sample is directly proportional to its path length through the sample and the solution concentration.

In spectroscopy, Beer’s law states that the absorption of light by a sample is directly proportional to the length of its path and its concentration. In other words, a solution absorbs more monochromatic light the further it passes through the sample or the more concentrated it is. The laser light in this solution of rhodamine dye is attenuated by the length of its path, illustrating the Beer-Lambert law. (Amirber, CC 4.0)

### History

Other names for Beer’s law are the Beer-Lambert law, the Lambert-Beer law, and the Beer–Lambert–Bouguer law. The law combines discoveries made by Bouger, Lambert, and Beer.

French scientist Pierre Bouger published the law in 1729 in Essai D’Optique Sur La Gradation De La Lumière. Johann Lambert often gets credit for the law, even though he quoted Bouger’s discovery in his Photometria in 1760. Lambert’s law says the absorbance of a sample is directly proportional to the path length of light. German scientist August Beer described a separate attenuation relation in 1852. Beer stated the transmittance of a solution is constant if the product of the path length and concentration are constant. The modern Beer-Lambert law correlates absorbance (the negative log of transmittance) to both sample thickness and species concentration.

### Beer’s Law Equation

The Beer’s law equation finds absorbance by relating the attenuation of light to the optical path length through a sample of uniform concentration:

A = εc

• A is the absorbance
• ε is the absorptivity or molar attenuation coefficient in M-1cm-1 (formerly called the extinction coefficient)
• is the optical path length in cm
• c is the concentration of the chemical species in mol/L or M

From this law, note:

1. Absorbance is directly proportional to path length. In spectroscopy, this is the width of a cuvette.
2. Absorbance is directly proportional to sample concentration.

### How to Use Beer’s Law

There is a linear relationship between the absorbance and concentration of a solution. Graphing a calibration curve using solutions of known concentration lets you find an unknown concentration. The graph only applies to dilute solutions.

### Beer’s Law Example Problem

Here is an example showing how to use Beer’s law.

A sample has a maximum absorbance of 275 nm and molar absorptivity of 8400 M-1cm-1. A spectrophotometer measures absorbance of 0.70 using a cuvette that is 1 cm wide. Find the solution concentration.

Start solving the problem by writing the formula for Beer’s law:

A = εc

Rearrange the equation and solve for concentration (c):

c = A/ε

Write down what you know:

• A = 0.70
• ε = 8400 M-1cm-1
• = 1 cm

Finally, plug in the values and obtain the answer:

c = (0.70) / (8400 M-1cm-1)(1 cm) = 8.33 x 10-5 mol/L = 8.33 x 10-5 M

### Limitations

The biggest limitation of Beer’s law is that it only works for relatively dilute homogeneous solutions. The law is not valid for concentrated solutions or turbid (cloudy or opaque) solutions. Deviations from the law also occur if there are interactions occurring within the solution.

The incident light must be monochromatic and consist of parallel rays. This is why the light source is a laser. The light must not influence the atoms or molecules within the sample.

### Importance of Beer’s Law

In addition to its usefulness in chemistry, Beer’s law applies to problems in physics, medicine, and meteorology. Remember, it applies to all forms of electromagnetic radiation, not just visible light.

In chemistry, Beer’s law finds solution concentration and helps assess oxidation and the rate of polymer degradation. In physics, the law describes the attenuation of particle beams, such as neutron beams passing through matter. Also, the Beer-Lambert law is a solution of the Bhatnagar-Gross-Krook (BKG) operator, which is in the Boltzmann equation for computational fluid dynamics. In medicine, the technicians apply the law to measure the amount of bilirubin in blood samples. Another application is finding the concentration of various chemicals in food and drugs. In meteorology, Beer’s law describes the attenuation of solar radiation in the Earth’s atmosphere.

### References

• Beer, August (1852). “”Bestimmung der Absorption des rothen Lichts in farbigen Flüssigkeiten” (Determination of the absorption of red light in colored liquids).” Annalen der Physik und Chemie. 162 (5): 78–88. doi:10.1002/andp.18521620505
• Bouguer, Pierre (1729). Essai d’optique sur la gradation de la lumière [Optics essay on the attenuation of light]. Paris, France: Claude Jombert.
• Ingle, J. D. J.; Crouch, S. R. (1988). Spectrochemical Analysis. New Jersey: Prentice Hall.
• Lambert, J.H. (1760). Photometria sive de mensura et gradibus luminis, colorum et umbrae [Photometry, or, On the measure and gradations of light intensity, colors, and shade]. Augsburg, Germany: Eberhardt Klett.
• Mayerhöfer, Thomas G.; Pahlow, Susanne; Popp, Jürgen (2020). “The Bouguer-Beer-Lambert Law: Shining Light on the Obscure”. ChemPhysChem. 21: 2031. doi:10.1002/cphc.202000464