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 that relates the absorption of light to the properties of the material through which the light is travelling.

Equations


There are several ways in which the law can be expressed:


 * $$A=\alpha lc \,$$


 * $$ {I_{1}\over I_{0}} = 10^{-\alpha l c}$$


 * $$ A = -\log_{10}\frac{I_1}{I_0}$$


 * $$ \alpha = \frac{4 \pi k}{\lambda}$$

Here:
 * A is absorbance
 * 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
 * &alpha; is the absorption coefficient or the molar absorptivity of the absorber
 * &lambda; is the wavelength of the light
 * k is the extinction coefficient

In essence, the law states that there is a logarithmic 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 &alpha; are known, the concentration of a substance can be deduced from the amount of light transmitted by it.

The units of c and &alpha; 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 &alpha; 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 &alpha; is an absorption cross-section and has units of length squared (e.g. cm2). If concentration c is expressed in moles per unit volume, &alpha; is a molar absorptivity given in units of mol-1 cm-2 or sometimes L mol-1 cm-1.

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

In spectroscopy and spectrophotometry, the law is almost always defined in terms of common logarithms and powers of 10 as above. In general optics, the law is often defined in an alternate exponential form:


 * $$ {I_{1}\over I_{0}} = e^{-\alpha' l c}$$


 * $$ A' = - \alpha' l c = -\ln \frac{I_1}{I_0}$$

The values of &alpha;' and A' are approximately 2.3 (≈ln 10) times larger than the corresponding values of &alpha; and A defined in terms of base-10 functions. Therefore, care must be taken when interperting data that the correct form of the law is used.

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.

Beer-Lambert law in the atmosphere
This law is also applied to describe the attenuation of solar radiation as it travels through the atmosphere. In this case, there is scattering of radiation as well as absorption. The Beer-Lambert law for the atmosphere is usually written


 * $$I_n=(I_o/R^2)\,\exp(-(k_a+k_g+k_{NO2}+k_w+k_{O3}+k_r) m)$$ ,

where each $$k_x$$ is an extinction coefficient whose subscript identifies the source of the absorption or scattering it describes:


 * $$a$$ refers to aerosols (that absorb and scatter)
 * $$g$$ are uniformly mixed gases (mainly carbon dioxide ($$CO_2$$) and molecular oxygen ($$O_2$$) which only absorb)
 * $$NO_2$$ is nitrogen dioxide, mainly due to urban pollution (absorption only)
 * $$w$$ is water vapour absorption
 * $$O_3$$ is ozone (absorption only)
 * $$r$$ is Rayleigh scattering from molecular oxygen ($$O_2$$) and nitrogen ($$N_2$$) (responsible for the blue color of the sky).

$$m$$ is the optical mass, a term basically equal to $$1/\cos(\theta)$$ where $$\theta$$ is the solar azimuth (the solar angle with respect to a direction perpendicular to the Earth's surface at the observation site).

This equation can be used to retrieve $$k_a$$, the aerosol optical thickness, which is necessary for the correction of satellite images and also important in accounting for the role of aerosols in climate.

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.