Beer–Lambert Law - Beers Law

The Beerâ€"Lambert law, also known as Beer's law, the Lambertâ€"Beer law, or the Beerâ€"Lambertâ€"Bouguer law relates the attenuation of light to the properties of the material through which the light is traveling. The law is commonly applied to chemical analysis measurements and used in understanding attenuation in physical optics, for photons, neutrons or rarefied gases. In mathematical physics, this law arises as a solution of the BGK equation.

History

The law was discovered by Pierre Bouguer before 1729. It is often attributed to Johann Heinrich Lambert, who cited Bouguer's Essai d'optique sur la gradation de la lumière (Claude Jombert, Paris, 1729)â€"and even quoted from itâ€"in his Photometria in 1760. Lambert's law stated that absorbance of a material sample is directly proportional to its thickness (path length). Much later, August Beer discovered another attenuation relation in 1852. Beer's law stated that absorbance is proportional to the concentrations of the attenuating species in the material sample. The modern derivation of the Beerâ€"Lambert law combines the two laws and correlates the absorbance to both the concentrations of the attenuating species as well as the thickness of the material sample.

Beerâ€"Lambert law

By definition, the transmittance of material sample is related to its optical depth Ï„ and to its absorbance A as

${\displaystyle T={\frac {\Phi _{\mathrm {e} }^{\mathrm {t} }}{\Phi _{\mathrm {e} }^{\mathrm {i} }}}=e^{-\tau }=10^{-A},}$

where

• Φ_e^t is the radiant flux transmitted by that material sample;
• Φ_eⁱ is the radiant flux received by that material sample.

The Beerâ€"Lambert law states that, for N attenuating species in the material sample,

${\displaystyle T=e^{-\sum _{i=1}^{N}\sigma _{i}\int _{0}^{\ell }n_{i}(z)\mathrm {d} z}=10^{-\sum _{i=1}^{N}\varepsilon _{i}\int _{0}^{\ell }c_{i}(z)\mathrm {d} z},}$

or equivalently that

${\displaystyle \tau =\sum _{i=1}^{N}\tau _{i}=\sum _{i=1}^{N}\sigma _{i}\int _{0}^{\ell }n_{i}(z)\,\mathrm {d} z,}$

${\displaystyle A=\sum _{i=1}^{N}A_{i}=\sum _{i=1}^{N}\varepsilon _{i}\int _{0}^{\ell }c_{i}(z)\,\mathrm {d} z,}$

where

• σ_i is the attenuation cross section of the attenuating species i in the material sample;
• n_i is the number density of the attenuating species i in the material sample;
• ε_i is the molar attenuation coefficient of the attenuating species i in the material sample;
• c_i is the amount concentration of the attenuating species i in the material sample;
• â„" is the path length of the beam of light through the material sample.

Attenuation cross section and molar attenuation coefficient are related by

${\displaystyle \varepsilon _{i}={\frac {\mathrm {N_{A}} }{\ln {10}}}\,\sigma _{i},}$

and number density and amount concentration by

${\displaystyle c_{i}={\frac {n_{i}}{\mathrm {N_{A}} }},}$

where N_A is the Avogadro constant.

In case of uniform attenuation, these relations become

${\displaystyle T=e^{-\sum _{i=1}^{N}\sigma _{i}n_{i}\ell }=10^{-\sum _{i=1}^{N}\varepsilon _{i}c_{i}\ell },}$

or equivalently

${\displaystyle \tau =\sum _{i=1}^{N}\sigma _{i}n_{i}\ell ,}$

${\displaystyle A=\sum _{i=1}^{N}\varepsilon _{i}c_{i}\ell .}$

Cases of non-uniform attenuation occur in atmospheric science applications and radiation shielding theory for instance.

The law tends to break down at very high concentrations, especially if the material is highly scattering. If the radiation is especially intense, nonlinear optical processes can also cause variances. The main reason, however, is the following. At high concentrations, the molecules are closer to each other and begin to interact with each other. This interaction will change several properties of the molecule, and thus will change the attenuation.

Expression with attenuation coefficient

The Beerâ€"Lambert law can be expressed in terms of attenuation coefficient, but in this case is better called Lambert's law since amount concentration, from Beer's law, is hidden inside the attenuation coefficient. The (Napierian) attenuation coefficient μ and the decadic attenuation coefficient μ₁₀ = μ/ln 10 of a material sample are related to its number densities and amount concentrations as

${\displaystyle \mu (z)=\sum _{i=1}^{N}\mu _{i}(z)=\sum _{i=1}^{N}\sigma _{i}n_{i}(z),}$

${\displaystyle \mu _{10}(z)=\sum _{i=1}^{N}\mu _{10,i}(z)=\sum _{i=1}^{N}\varepsilon _{i}c_{i}(z)}$

respectively, by definition of attenuation cross section and molar attenuation coefficient. Then the Beerâ€"Lambert law becomes

${\displaystyle T=e^{-\int _{0}^{\ell }\mu (z)\mathrm {d} z}=10^{-\int _{0}^{\ell }\mu _{10}(z)\mathrm {d} z},}$

and

${\displaystyle \tau =\int _{0}^{\ell }\mu (z)\,\mathrm {d} z,}$

${\displaystyle A=\int _{0}^{\ell }\mu _{10}(z)\,\mathrm {d} z.}$

In case of uniform attenuation, these relations become

${\displaystyle T=e^{-\mu \ell }=10^{-\mu _{10}\ell },}$

or equivalently

${\displaystyle \tau =\mu \ell ,}$

${\displaystyle A=\mu _{10}\ell .}$

Derivation

Assume that a beam of light enters a material sample. Define z as an axis parallel to the direction of the beam. Divide the material sample into thin slices, perpendicular to the beam of light, with thickness dz sufficiently small that one particle in a slice cannot obscure another particle in the same slice when viewed along the z direction. The radiant flux of the light that emerges from a slice is reduced, compared to that of the light that entered, by dΦ_e(z) = âˆ'μ(z)Φ_e(z) dz, where μ is the (Napierian) attenuation coefficient, which yields the following first-order linear ODE:

${\displaystyle {\frac {\mathrm {d} \Phi _{\mathrm {e} }}{\mathrm {d} z}}(z)=-\mu (z)\Phi _{\mathrm {e} }(z).}$

The attenuation is caused by the photons that did not make it to the other side of the slice because of scattering or absorption. The solution to this differential equation is obtained by multiplying the integrating factor

${\displaystyle e^{\int _{0}^{z}\mu (z')\mathrm {d} z'}}$

throughout to obtain

${\displaystyle {\frac {\mathrm {d} \Phi _{\mathrm {e} }}{\mathrm {d} z}}(z)\,e^{\int _{0}^{z}\mu (z')\mathrm {d} z'}+\mu (z)\Phi _{\mathrm {e} }(z)\,e^{\int _{0}^{z}\mu (z')\mathrm {d} z'}=0,}$

which simplifies due to the product rule (applied backwards) to

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}{\bigl (}\Phi _{\mathrm {e} }(z)\,e^{\int _{0}^{z}\mu (z')\mathrm {d} z'}{\bigr )}=0.}$

Integrating both sides and solving for Φ_e for a material of real thickness â„", with the incident radiant flux upon the slice Φ_eⁱ = Φ_e(0) and the transmitted radiant flux Φ_e^t = Φ_e(â„" ) gives

${\displaystyle \Phi _{\mathrm {e} }^{\mathrm {t} }=\Phi _{\mathrm {e} }^{\mathrm {i} }\,e^{-\int _{0}^{\ell }\mu (z)\mathrm {d} z},}$

and finally

${\displaystyle T={\frac {\Phi _{\mathrm {e} }^{\mathrm {t} }}{\Phi _{\mathrm {e} }^{\mathrm {i} }}}=e^{-\int _{0}^{\ell }\mu (z)\mathrm {d} z}.}$

Since the decadic attenuation coefficient μ₁₀ is related to the (Napierian) attenuation coefficient by μ₁₀ = μ/ln 10, one also have

${\displaystyle T=e^{-\int _{0}^{\ell }\ln {10}\,\mu _{10}(z)\mathrm {d} z}={\bigl (}e^{-\int _{0}^{\ell }\mu _{10}(z)\mathrm {d} z}{\bigr )}^{\ln {10}}=10^{-\int _{0}^{\ell }\mu _{10}(z)\mathrm {d} z}.}$

To describe the attenuation coefficient in a way independent of the number densities n_i of the N attenuating species of the material sample, one introduces the attenuation cross section σ_i = μ_i(z)/n_i(z). σ_i has the dimension of an area; it expresses the likelihood of interaction between the particles of the beam and the particles of the specie i in the material sample:

${\displaystyle T=e^{-\sum _{i=1}^{N}\sigma _{i}\int _{0}^{\ell }n_{i}(z)\mathrm {d} z}.}$

One can also use the molar attenuation coefficients ε_i = (N_A/ln 10)σ_i, where N_A is the Avogadro constant, to describe the attenuation coefficient in a way independent of the amount concentrations c_i(z) = n_i(z)/N_A of the attenuating species of the material sample:

${\displaystyle {\begin{aligned}T=e^{-\sum _{i=1}^{N}{\frac {\ln {10}}{\mathrm {N_{A}} }}\varepsilon _{i}\int _{0}^{\ell }n_{i}(z)\mathrm {d} z}=\\{\Bigl (}e^{-\sum _{i=1}^{N}\varepsilon _{i}\int _{0}^{\ell }{\frac {n_{i}(z)}{\mathrm {N_{A}} }}\mathrm {d} z}{\Bigr )}^{\ln {10}}=10^{-\sum _{i=1}^{N}\varepsilon _{i}\int _{0}^{\ell }c_{i}(z)\mathrm {d} z}.\end{aligned}}}$

Validity

Under certain conditions Beerâ€"Lambert law fails to maintain a linear relationship between attenuation and concentration of analyte. These deviations are classified into three categories:

1. Realâ€"fundamental deviations due to the limitations of the law itself.
2. Chemicalâ€"deviations observed due to specific chemical species of the sample which is being analyzed.
3. Instrumentâ€"deviations which occur due to how the attenuation measurements are made.

There are at least six conditions that need to be fulfilled in order for Beerâ€"Lambert law to be valid. These are:

1. The attenuators must act independently of each other.
2. The attenuating medium must be homogeneous in the interaction volume.
3. The attenuating medium must not scatter the radiationâ€"no turbidityâ€"unless this is accounted for as in DOAS.
4. The incident radiation must consist of parallel rays, each traversing the same length in the absorbing medium.
5. The incident radiation should preferably be monochromatic, or have at least a width that is narrower than that of the attenuating transition. Otherwise a spectrometer as detector for the power is needed instead of a photodiode which has not a selective wavelength dependence.
6. The incident flux must not influence the atoms or molecules; it should only act as a non-invasive probe of the species under study. In particular, this implies that the light should not cause optical saturation or optical pumping, since such effects will deplete the lower level and possibly give rise to stimulated emission.

If any of these conditions are not fulfilled, there will be deviations from Beerâ€"Lambert law.

The Beerâ€"Lambert law is not compatible with Maxwell's equations. Being strict, the law does not describe the transmittance through a medium, but the propagation within that medium. It can be made compatible with Maxwell's equations if the transmittance of a sample with solute is ratioed against the transmittance of the pure solvent which explains why it works so well in spectrophotometry. As this is not possible for pure media, the uncritical employment of the Beerâ€"Lambert law can easily generate errors of the order of 100% or more. In such cases it is necessary to apply the Transfer-matrix method.

Chemical analysis by spectrophotometry

Beerâ€"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 ε is known. Measurements of decadic attenuation coefficient μ₁₀ are made at one wavelength λ that is nearly unique for bilirubin and at a second wavelength in order to correct for possible interferences. The amount concentration c is then given by

${\displaystyle c={\frac {\mu _{10}(\lambda )}{\varepsilon (\lambda )}}.}$

For a more complicated example, consider a mixture in solution containing two species at amount concentrations c₁ and c₂. The decadic attenuation coefficient at any wavelength λ is, given by

${\displaystyle \mu _{10}(\lambda )=\varepsilon _{1}(\lambda )c_{1}+\varepsilon _{2}(\lambda )c_{2}.}$

Therefore, measurements at two wavelengths yields two equations in two unknowns and will suffice to determine the amount concentrations c₁ and c₂ as long as the molar attenuation coefficient of the two components, ε₁ and ε₂ 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 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 N wavelengths for a mixture containing N components.

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 samples. The carbonyl group attenuation at about 6 micrometres can be detected quite easily, and degree of oxidation of the polymer calculated.

Beerâ€"Lambert law in the atmosphere

This law is also 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 τ′ = mÏ„, where Ï„ refers to a vertical path, m is called the relative airmass, and for a plane-parallel atmosphere it is determined as m = sec θ where θ is the zenith angle corresponding to the given path. The Beerâ€"Lambert law for the atmosphere is usually written

${\displaystyle T=e^{-m(\tau _{\mathrm {a} }+\tau _{\mathrm {g} }+\tau _{\mathrm {RS} }+\tau _{\mathrm {NO_{2}} }+\tau _{\mathrm {w} }+\tau _{\mathrm {O_{3}} }+\tau _{\mathrm {r} }+\ldots )},}$

where each Ï„_x is the optical depth 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 (CO₂) and molecular oxygen (O₂) which only absorb);
• NO₂ is nitrogen dioxide, mainly due to urban pollution (absorption only);
• RS are effects due to Raman scattering in the atmosphere;
• w is water vapour absorption;
• O₃ is ozone (absorption only);
• r is Rayleigh scattering from molecular oxygen (O₂) and nitrogen (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.

m is the optical mass or airmass factor, a term approximately equal (for small and moderate values of θ) to 1/cos θ, where θ is the observed object's 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 τ_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.