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Categories: Optics | Exponentials | Eponymous laws

Beer-Lambert law

In optics, the Beer-Lambert law, also known as Beer's law or the Beer-Lambert-Bouguer law is an empirical relationship in relating the absorption of light to the properties of the material the light is travelling through.

Basically, the law states that absorbance is proportional to the concentration of light-absorbing molecules in the sample.

Equations

$A=\epsilon lc \,$

${I_{1}\over I_{0}} = e^{-\alpha l c}\; ,$

$A = log\frac{I_1}{I_0}$

$\alpha = \frac{4 \pi k}{\lambda}. \,$

A is absorbance
ε is molar absorptivity
I₀ is the intensity of the incident light
I₁ is the intensity after passing through the material
l is the distance that the light travels through the material (the path length)
c is the concentration of absorbing species in the material
α is the absorption coefficient of the absorber.
λ is the wavelength of the light
k is the extinction coefficient

             | . . . . . . |
    I₀  ==>  |. . c,α . . .| ==> I₁
             | . . . . . . |
             
             <- - - l - - ->

In essence, the law states that there is an exponential dependence between the transmission of light through a substance and the concentration of the substance, and also between the transmission and the length of material that the light travels through. Thus if l and α are known, the concentration of a substance can be deduced from the amount of light transmitted by it.

The units of c and α depend on the way that the concentration of the absorber is being expressed. If the material is a liquid, it is usual to express the absorber concentration c as a mole fraction i.e. a dimensionless fraction. The units of α are thus reciprocal length (e.g. cm^-1). In the case of a gas, c may be expressed as a density (units of reciprocal length cubed, e.g. cm^-3), in which case α is an absorption cross-section and has units of length squared (e.g. cm²).

The value of the absorption coefficient α varies between different absorbing materials and also with wavelength for a particular material. It is usually determined by experiment.

The law tends to break down at very high concentrations, especially if the material is highly scattering . If the light is especially intense, nonlinear optical processes can also cause variances.

The law's link between concentration and light absorption is the basis behind the use of spectroscopy to identify substances.

History

Beer's law was independently discovered (in various forms) by Pierre Bouguer in 1729, Johann Heinrich Lambert in 1760 and August Beer in 1852.

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