The Beer–Lambert law is commonly applied to chemical analysis measurements to determine the concentration of chemical species that absorb light. It is often referred to as Beer's law. In physics, the Bouguer–Lambert law is an empirical law which relates the extinction or attenuation of light to the properties of the material through which the light is travelling. It had its first use in astronomical extinction. The fundamental law of extinction (the process is linear in the intensity of radiation and amount of radiatively active matter, provided that the physical state is held constant) is sometimes called the Beer–Bouguer–Lambert law or the Bouguer–Beer–Lambert law or merely the extinction law. The extinction law is also 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.
Bouguer–Lambert law: This law is based on observations made by Pierre Bouguer before 1729.[1] It is often attributed to Johann Heinrich Lambert, who cited Bouguer's French: 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 expressed the law, which states that the loss of light intensity when it propagates in a medium is directly proportional to intensity and path length, in the mathematical form used today.
Lambert began by assuming that the intensity of light traveling into an absorbing body would be given by the differential equation:
-dI=\muIdx,
ln(I0/I)=\mud,
I=I0e-\mu.
Beer's law: Much later, in 1852, the German scientist August Beer studied another attenuation relation. In the introduction to his classic paper,[4] he wrote: "The absorption of light during the irradiation of a colored substance has often been the object of experiment; but attention has always been directed to the relative diminution of the various colors or, in the case of crystalline bodies, the relation between the absorption and the direction of polarization. Concerning the absolute magnitude of the absorption that a particular ray of light suffers during its propagation through an absorbing medium, there is no information available." By studying absorption of red light in colored aqueous solutions of various salts, he concluded that "the transmittance of a concentrated solution can be derived from a measurement of the transmittance of a dilute solution". It is clear that he understood the exponential relationship, as he wrote: "If
{λ}
λ2
λ=\muD
\mu
D
Beer–Lambert law: The modern formulation of the Beer–Lambert law combines the observations of Bouguer and Beer into the mathematical form of Lambert. It correlates the absorbance, most often expressed as the negative decadic logarithm of the transmittance, to both the concentrations of the attenuating species and the thickness of the material sample.[6] An early, possibly the first, modern formulation was given by Robert Luther and Andreas Nikolopulos in 1913.[7]
While the observations of Bouguer and Beer have a similar form in the Beer–Lambert law, their areas of observation were very different. For both experimenters, the incident beam was well collimated, with a light sensor which preferentially detected directly transmitted light.
Beer specifically looked at solutions. Solutions are homogeneous and do not scatter light (Ultraviolet, visible, Infrared) of wavelengths commonly used in analytical spectroscopy (except upon entry and exit). The attenuation of a beam of light within a solution is assumed to be only due to absorption. In order to approximate the conditions required for the Beer Lambert law to hold, often the intensity of transmitted light through a reference sample
(IR)
(IS)
{log10{l(IR/IS)}}
log10(IR/IS)=A=\varepsilon\ellc
Bouguer looked at astronomical phenomena where the size of a detector is very small compared to the distance traveled by the light. In this case, any light that is scattered by a particle, either in the forward or backward direction, will not strike the detector. The loss of intensity to the detector will be due to both absorption and scatter. Consequently, the total loss is called attenuation (rather than absorption). A single measurement cannot separate the two, but conceptually the contribution of each can be separated in the attenuation coefficient. If
I0
Id
d
T
T= | Id |
I0 |
=\exp(-\mud)
\mu
-ln(T)=ln
I0 | |
Id |
=\mud
\mu=\mus+\mua
\mus
\mua
The fundamental law of extinction states[9] that the extinction process is linear in the intensity of radiation and amount of radiatively active matter, provided that the physical state is held constant. (Neither concentration or length are fundamental parameters.) There are two factors that determine the degree to which a medium containing particles will attenuate a light beam: the number of particles encountered by the light beam, and the degree to which each particle extinguishes the light.[10]
For the case of absorption (Beer), this later quantity is called the absorptivity [<math>\epsilon</math>], which is defined as "the property of a body that determines the fraction of incident radiation absorbed by the body".[11] The Beer–Lambert law
[log10(I0/I)=A=\epsilon\ellc]
S
\ell
NAc\sigma\ellS
NA
c
\sigma
There must be a large number of particles that are uniformly distributed for this relationship to hold. In practice, the beam area is thought of as a constant, and since the fraction [<math>I/I_0</math>] has the area in both the numerator and denominator, the beam area cancels in the calculation of the absorbance. The units of the absorptivity must match the units in which the sample is described. For example, if the sample is described by mass concentration (g/L) and length (cm), then the units on the absorptivity would be [L g<sup>−1</sup> cm<sup>−1</sup>], so that the absorbance has no units.
For the case of "extinction" (Bouguer), the sum of absorption and scatter, the terms absorption, scattering, and extinction cross-sections are often used.[12] The fraction of light extinguished by the sample may be described by the extinction cross section (fraction extinguished per particle). the number of particles in a unit distance and the distance in those units. For example: [(fraction extinguished / particle) (# particles / meter) (# meters / sample) = fraction extinguished / sample ]
A common and practical expression of the Beer–Lambert law relates the optical attenuation of a physical material containing a single attenuating species of uniform concentration to the optical path length through the sample and absorptivity of the species. This expression is:where
A more general form of the Beer–Lambert law states that, for attenuating species in the material sample,or equivalently thatwhere
In the above equations, the transmittance of material sample is related to its optical depth and to its absorbance by the following definitionwhere
|
|
Attenuation cross section and molar attenuation coefficient are related byand number density and amount concentration bywhere is the Avogadro constant.In case of uniform attenuation, these relations becomeor equivalentlyCases 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. Absorbance within range of 0.2 to 0.5 is ideal to maintain linearity in the Beer–Lambert law. If the radiation is especially intense, nonlinear optical processes can also cause variances. The main reason, however, is that the concentration dependence is in general non-linear and Beer's law is valid only under certain conditions as shown by derivation below. For strong oscillators and at high concentrations the deviations are stronger. If the molecules are closer to each other interactions can set in. These interactions can be roughly divided into physical and chemical interactions. Physical interaction do not alter the polarizability of the molecules as long as the interaction is not so strong that light and molecular quantum state intermix (strong coupling), but cause the attenuation cross sections to be non-additive via electromagnetic coupling. Chemical interactions in contrast change the polarizability and thus absorption.
The law can be expressed in terms of attenuation coefficient, but in this case is better called the Bouguer-Lambert's law. The (Napierian) attenuation coefficient
\mu
\mu10=\tfrac{\mu}{ln10}
In many cases, the attenuation coefficient does not vary with
z
\alpha
1/λ
Assume that a beam of light enters a material sample. Define 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 sufficiently small that one particle in a slice cannot obscure another particle in the same slice when viewed along the direction. The radiant flux of the light that emerges from a slice is reduced, compared to that of the light that entered, by
d\Phie(z) |
=-\mu(z)\Phie(z)dz,
Integrating both sides and solving for for a material of real thickness, with the incident radiant flux upon the slice
|
=
\Phie(0) |
|
=
\Phie(\ell) |
Since the decadic attenuation coefficient is related to the (Napierian) attenuation coefficient by
\mu10=\tfrac{\mu}{ln10},
\sigmai=\tfrac{\mui(z)}{ni(z)}.
One can also use the molar attenuation coefficients
\varepsiloni=
\tfrac{NA |
ci(z)=ni
\tfrac{z}{NA |
Under certain conditions the Beer–Lambert law fails to maintain a linear relationship between attenuation and concentration of analyte.[15] These deviations are classified into three categories:
There are at least six conditions that need to be fulfilled in order for the Beer–Lambert law to be valid. These are:
If any of these conditions are not fulfilled, there will be deviations from the Beer–Lambert law.
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 is known. Measurements of decadic attenuation coefficient are made at one wavelength that is nearly unique for bilirubin and at a second wavelength in order to correct for possible interferences. The amount concentration is then given by
For a more complicated example, consider a mixture in solution containing two species at amount concentrations and . The decadic attenuation coefficient at any wavelength is, given by
Therefore, measurements at two wavelengths yields two equations in two unknowns and will suffice to determine the amount concentrations and as long as the molar attenuation coefficients of the two components, and 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 wavelengths for a mixture containing 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.
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, where refers to a vertical path, is called the relative airmass, and for a plane-parallel atmosphere it is determined as where is the zenith angle corresponding to the given path. The Bouguer-Lambert law for the atmosphere is usually writtenwhere each is the optical depth whose subscript identifies the source of the absorption or scattering it describes:
is the optical mass or airmass factor, a term approximately equal (for small and moderate values of) to where 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, 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.