# Beer–Lambert law

An example of Beer–Lambert law: green laser light in a solution of Rhodamine 6B. The beam power becomes weaker as it passes through solution

The Beer–Lambert law, also known as Beer's law, the Lambert–Beer law, or the Beer–Lambert–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.[1] 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)—and even quoted from it—in his Photometria in 1760.[2] Lambert's law stated that absorbance of a 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 concentration of the sample in 1852.[3] The modern derivation of the Beer–Lambert law combines the two laws and correlate the absorbance to both, the concentration as well as the thickness of the sample.[4]

## Law

By definition, transmittance is related to optical depth τ and to absorbance A as

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

where

• Φet is the radiant flux transmitted by that surface;

The Beer–Lambert law states that

$T = e^{-\sigma \int_0^\ell N(z)\mathrm{d}z} = 10^{-\varepsilon \int_0^\ell c(z)\mathrm{d}z},$

or equivalently that

$\tau = \sigma \int_0^\ell N(z)\,\mathrm{d}z,$
$A = \varepsilon \int_0^\ell c(z)\,\mathrm{d}z,$

where

In case of uniform attenuation, these relations become[5]

$T = e^{-\sigma N\ell} = 10^{-\varepsilon c\ell},$

or equivalently

$\tau = \sigma N\ell,$
$A = \varepsilon c\ell.$

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.

## Attenuation coefficent

The Beer–Lambert law can be expressed in terms of the attenuation coefficient, but in this case is better called Lambert's law since molar concentration, from Beer's law, is hidden inside the attenuation coefficient. As the (Napierian) attenuation coefficient α and the decadic attenuation coefficient α10 are defined as

$\alpha(z) = \sigma N(z),$
$\alpha_{10}(z) = \varepsilon c(z),$

the Beer–Lambert law gives

$T = e^{-\int_0^\ell \alpha(z)\mathrm{d}z} = 10^{-\int_0^\ell \alpha_{10}(z)\mathrm{d}z},$

and

$\tau = \int_0^\ell \alpha(z)\,\mathrm{d}z,$
$A = \int_0^\ell \alpha_{10}(z)\,\mathrm{d}z.$

In case of uniform attenuation, these relations become

$T = e^{-\alpha\ell} = 10^{-\alpha_{10}\ell},$

or equivalently

$\tau = \alpha\ell,$
$A = \alpha_{10}\ell.$

## Derivation of the law

In concept, the derivation of the Beer–Lambert law is straightforward. Divide the attenuating sample into thin slices that are perpendicular to the beam of light. The light that emerges from a slice is slightly less intense than the light that entered because some of the photons have run into molecules in the sample and did not make it to the other side. For most cases where measurements of attenuation are needed, a vast majority of the light entering the slice leaves without being attenuated. Because the physical description of the problem is in terms of differences—power before and after light passes through the slice—we can easily write an ordinary differential equation model for attenuation. The difference in power due to the slice of attenuating material dΦet is reduced; leaving the slice, it is a fraction k of the light entering the slice Φei. The thickness of the slice is dz, which scales the amount of attenuation (thin slice does not attenuates much light but a thick slice attenuates a lot). In symbols, dΦet = kΦet dz, or dΦet/dz = kΦet. This conceptual overview uses k to describe how much light is attenuated. All we can say about the value of this constant is that it will be different for each material. Also, its values should be constrained between −1 and 0. The following paragraphs cover the meaning of this constant and the whole derivation in much greater detail.

Assume that particles may be described as having an attenuation cross section (i.e., area) σ, perpendicular to the path of light through a solution, such that a photon of light is attenuated if it strikes the particle, and is transmitted if it does not. Define z as an axis parallel to the direction that photons of light are moving, and A and dz as the area and thickness (along the z axis) of a 3-dimensional slab of space through which light is passing. Assume that dz is sufficiently small that one particle in the slab cannot obscure another particle in the slab when viewed along the z direction. The number density of particles in the slab is represented by N(z). It follows that the fraction of photons attenuated (absorbed and scattered away) when passing through this slab is equal to the total opaque area of the particles in the slab, σAN(z) dz, divided by the area of the slab A, which yields σN(z) dz. Expressing the number of photons attenuated by the slab as dΦet, and the total number of photons incident on the slab as Φet, gives the following first-order linear ODE:

$\frac{\mathrm{d}\Phi_\mathrm{e}^\mathrm{t}}{\mathrm{d}z} = -\sigma N(z)\Phi_\mathrm{e}^\mathrm{t}.$

Note that because there are fewer photons which pass through the slab than are incident on it, Φet is actually negative (it is proportional in magnitude to the number of photons attenuated). The solution to this simple differential equation is obtained by multiplying the integrating factor

$e^{\sigma \int_0^z N(z')\mathrm{d}z'}$

throughout to obtain

$\frac{\mathrm{d}\Phi_\mathrm{e}^\mathrm{t}}{\mathrm{d}z}\,e^{\sigma \int_0^z N(z')\mathrm{d}z'} = -\sigma N(z)\Phi_\mathrm{e}^\mathrm{t}\,e^{\sigma \int_0^z N(z')\mathrm{d}z'},$

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

$\frac{\mathrm{d}(\Phi_\mathrm{e}^\mathrm{t}\,e^{\sigma \int_0^z N(z')\mathrm{d}z'})}{\mathrm{d}z} = 0,$

which, on integrating both sides and solving for Φet for a slab of real thickness , with Φet = Φei at z = 0, gives

$\Phi_\mathrm{e}^\mathrm{t} = \Phi_\mathrm{e}^\mathrm{i}\,e^{-\sigma \int_0^\ell N(z)\mathrm{d}z},$

and finally

$T = \frac{\Phi_\mathrm{e}^\mathrm{t}}{\Phi_\mathrm{e}^\mathrm{i}} = e^{-\sigma \int_0^\ell N(z)\mathrm{d}z}.$

## Validity

Under certain conditions Beer–Lambert law fails to maintain a linear relationship between attenuation and concentration of analyte.[6] These deviations are classified into three categories:

1. Real—fundamental deviations due to the limitations of the law itself.
2. Chemical—deviations observed due to specific chemical species of the sample which is being analyzed.
3. Instrument—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 Beer–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 radiation—no turbidity—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 Beer–Lambert law.

## Chemical analysis

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 ε is known. Measurements of decadic attenuation coefficient α10 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 molar concentration c is then given by

$c = \frac{\alpha_{10}(\lambda)}{\varepsilon(\lambda)}.$

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

$\alpha_{10}(\lambda) = \varepsilon_1(\lambda) c_1 + \varepsilon_2(\lambda) c_2.$

Therefore, measurements at two wavelengths yields two equations in two unknowns and will suffice to determine the 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 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). The carbonyl group attenuation at about 6 micrometres can be detected quite easily, and degree of oxidation of the polymer calculated.

## Beer–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′ = , where τ 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 Beer–Lambert law for the atmosphere is usually written

$T = e^{-m(\tau_\mathrm{a} + \tau_\mathrm{g} + \tau_\mathrm{RS} + \tau_\mathrm{NO_2} + \tau_\mathrm{w} + \tau_\mathrm{O_3} + \tau_\mathrm{r} + \ldots)},$

where each τx is the optical depth whose subscript identifies the source of the absorption or scattering it describes:

m is the optical mass or airmass factor, a term approximately equal (for small and moderate values of θ) to 1/cosθ, 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 τa, 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.