Editing Beer–Lambert law (section)

== Applications ==

===In plasma physics===
{{expand section|date=October 2024}}
The BBL extinction law also arises as a solution to the [[Bhatnagar–Gross–Krook operator|BGK equation]].

=== Chemical analysis by spectrophotometry ===
The Beer–Lambert law can be applied to the analysis of a mixture by [[spectrophotometry]], without the need for extensive pre-processing of the sample. An example is the determination of [[bilirubin]] in blood plasma samples. The spectrum of pure bilirubin is known, so the molar attenuation coefficient {{mvar|ε}} is known. Measurements of decadic attenuation coefficient {{math|''μ''<sub>10</sub>}} are made at one wavelength {{mvar|λ}} that is nearly unique for bilirubin and at a second wavelength in order to correct for possible interferences. The amount concentration {{mvar|c}} is then given by
<math display="block">c = \frac{\mu_{10}(\lambda)}{\varepsilon(\lambda)}.</math>

For a more complicated example, consider a mixture in solution containing two species at amount concentrations {{math|''c''<sub>1</sub>}} and {{math|''c''<sub>2</sub>}}. The decadic attenuation coefficient at any wavelength {{mvar|λ}} is, given by
<math display="block">\mu_{10}(\lambda) = \varepsilon_1(\lambda) c_1 + \varepsilon_2(\lambda) c_2.</math>

Therefore, measurements at two wavelengths yields two equations in two unknowns and will suffice to determine the amount concentrations {{math|''c''<sub>1</sub>}} and {{math|''c''<sub>2</sub>}} as long as the molar attenuation coefficients of the two components, {{math|''ε''<sub>1</sub>}} and {{math|''ε''<sub>2</sub>}} are known at both wavelengths. This two system equation can be solved using [[Cramer's rule]]. In practice it is better to use [[linear least squares (mathematics)|linear least squares]] to determine the two amount concentrations from measurements made at more than two wavelengths. 

Mixtures containing more than two components can be analyzed in the same way, using a minimum of {{mvar|m}} wavelengths for a mixture containing {{mvar|n}} components. So, in general: 

<math>A_{\lambda_i} = \sum_{j=1}^{n} \epsilon_{j, \lambda_i} c_j l
</math>

where <math>A_{\lambda_i}</math>is the absorbance at wavelength <math>\lambda_i</math>, <math>\epsilon_{j, \lambda_i}</math> is the molar absorptivity of component <math>j</math> at <math>\lambda_i</math>, <math>c_j</math> is the concentration of component <math>j</math>, and <math>l</math> is the path length.

The law is used widely in [[infra-red spectroscopy]] and [[near-infrared spectroscopy]] for analysis of [[polymer degradation]] and [[oxidation]] (also in biological tissue) as well as to measure the [[concentration]] of various compounds in different [[food sampling|food samples]]. The [[carbonyl group]] attenuation at about 6 micrometres can be detected quite easily, and degree of oxidation of the [[polymer]] calculated.

=== In-atmosphere astronomy ===
The Bouguer–Lambert law may be applied to describe the attenuation of solar or stellar radiation as it travels through the atmosphere. In this case, there is scattering of radiation as well as absorption. The optical depth for a slant path is {{math|1=''{{prime|τ}}'' = ''mτ''}}, where {{mvar|τ}} refers to a vertical path, {{mvar|m}} is called the [[airmass|relative airmass]], and for a plane-parallel atmosphere it is determined as {{math|1=''m'' = sec ''θ''}} where {{mvar|θ}} is the [[zenith angle]] corresponding to the given path. The Bouguer-Lambert law for the atmosphere is usually written
<math display="block">T = \exp \big( -m(\tau_\mathrm{a} + \tau_\mathrm{g} + \tau_\mathrm{RS} + \tau_\mathrm{NO_2} + \tau_\mathrm{w} + \tau_\mathrm{O_3} + \tau_\mathrm{r} + \cdots) \bigr),</math>
where each {{mvar|&tau;<sub>x</sub>}} is the optical depth whose subscript identifies the source of the absorption or scattering it describes:
* {{math|a}} refers to [[aerosols]] (that absorb and scatter);
* {{math|g}} are uniformly mixed gases (mainly [[carbon dioxide]] (CO<sub>2</sub>)  and molecular [[oxygen]] (O<sub>2</sub>) which only absorb);
* {{math|{{chem2|NO2}}}} is [[nitrogen dioxide]], mainly due to urban pollution (absorption only);
* {{math|RS}} are effects due to [[Raman scattering]] in the atmosphere;
* {{math|w}} is [[water vapour]] [[water absorption|absorption]];
* {{math|{{chem2|O3}}}} is [[ozone]] (absorption only);
* {{mvar|r}} is [[Rayleigh scattering]] from molecular [[oxygen]] ({{chem2|O2}}) and [[nitrogen]] ({{chem2|N2}}) (responsible for the blue color of the sky);
* the selection of the attenuators which have to be considered depends on the wavelength range and can include various other compounds. This can include [[tetraoxygen]], [[HONO]], [[formaldehyde]], [[glyoxal]], a series of halogen radicals and others.

{{mvar|m}} is the ''optical mass'' or ''[[airmass]] factor'', a term approximately equal (for small and moderate values of {{mvar|θ}}) to {{tmath|\tfrac{1}{\cos \theta},}} where {{mvar|θ}} is the observed object's [[celestial coordinate system|zenith angle]] (the angle measured from the direction perpendicular to the Earth's surface at the observation site). This equation can be used to retrieve {{math|''τ''<sub>a</sub>}}, the aerosol [[optical depth|optical thickness]], which is necessary for the correction of satellite images and also important in accounting for the role of aerosols in climate.