Beer–Lambert law
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The Beer-Lambert law is commonly applied to
History
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 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 I of light traveling into an absorbing body would be given by the differential equation: which is compatible with Bouguer's observations. The constant of proportionality μ was often termed the "optical density" of the body. Integrating to find the intensity at a distance d into the body, one obtains: For a homogeneous medium, this reduces to: from which follows the exponential attenuation law: [3]
Beer's law: Much later, in 1852, the German scientist
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
Differences between Bouguer and Beer in application areas
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 consisting of pure solvent is measured, and compared to the intensity of light transmitted through a sample , with the absorbance of the sample taken as: . It is for this case that the common mathematical formulation (see below) applies:
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 is the intensity of the light at the beginning of the travel and is the intensity of the light detected after travel of a distance , the fraction transmitted, , is given by: , where is called an attenuation constant or coefficient. The amount of light transmitted is falling off exponentially with distance. Taking the natural logarithm in the above equation, we get: . For scattering media, the constant is often divided into two parts, , separating it into a scattering coefficient, , and an absorption coefficient, .[8]
Absorptivity, cross-sections, and units of coefficients
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 [], 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 uses concentration and length in order to determine the number of particles the beam encounters. If we know the area of a collimated beam (directed radiation), we can get the number of particles in a distance. The number of particles encountered can be calculated from Avogadro's number, the molar concentration, the cross-sectional area of the incident beam .
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 [] 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−1 cm−1], so that the absorbance has no units.
For the case of "
Mathematical formulations
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
- A is the absorbance
- ε is the absorptivityof the attenuating species
- ℓ is the optical path length
- c is the concentration of the attenuating species
A more general form of the Beer–Lambert law states that, for N attenuating species in the material sample,
- σi is the attenuation cross section of the attenuating species i in the material sample;
- ni is the number density of the attenuating species i in the material sample;
- εi is the absorptivityof the attenuating species i in the material sample;
- ci is the amount concentrationof the attenuating species i in the material sample;
- ℓ is the path length of the beam of light through the material sample.
In the above equations, the transmittance T of material sample is related to its optical depth τ and to its absorbance A by the following definition
- is the radiant flux transmitted by that material sample;
- is the radiant flux received by that material sample.
Attenuation cross section and molar attenuation coefficient are related by
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.
Expression with attenuation coefficient
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 and the decadic attenuation coefficient of a material sample are related to its number densities and amount concentrations as
In many cases, the attenuation coefficient does not vary with , in which case one does not have to perform an integral and can express the law as:
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 where μ is the (Napierian) attenuation coefficient, which yields the following first-order linear, ordinary differential equation:
Integrating both sides and solving for Φe for a material of real thickness ℓ, with the incident radiant flux upon the slice and the transmitted radiant flux gives
Since the decadic attenuation coefficient μ10 is related to the (Napierian) attenuation coefficient by we also have
To describe the attenuation coefficient in a way independent of the number densities ni of the N attenuating species of the material sample, one introduces the attenuation cross section σi has the dimension of an area; it expresses the likelihood of interaction between the particles of the beam and the particles of the species i in the material sample:
One can also use the
Validity
Under certain conditions the Beer–Lambert law fails to maintain a linear relationship between attenuation and concentration of analyte.[16] These deviations are classified into three categories:
- Real—fundamental deviations due to the limitations of the law itself.
- Chemical—deviations observed due to specific chemical species of the sample which is being analyzed.
- 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 the Beer–Lambert law to be valid. These are:
- The attenuators must act independently of each other.
- The attenuating medium must be homogeneous in the interaction volume.
- The attenuating medium must not scatter the radiation—no turbidity—unless this is accounted for as in DOAS.
- The incident radiation must consist of parallel rays, each traversing the same length in the absorbing medium.
- 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 cannot discriminate between wavelengths.
- 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 the Beer–Lambert law.
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 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
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
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 coefficients of the two components, ε1 and ε2 are known at both wavelengths. This two system equation can be solved using
The law is used widely in
Application for the atmosphere
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 τ′ = mτ, where τ refers to a vertical path, m is called the
- 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 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 HONO, formaldehyde, glyoxal, a series of halogen radicals and others.
m is the optical mass or
See also
- Applied spectroscopy
- Atomic absorption spectroscopy
- Absorption spectroscopy
- Cavity ring-down spectroscopy
- Clausius-Mossotti relation
- Infra-red spectroscopy
- Job plot
- Laser absorption spectrometry
- Lorentz-Lorenz relation
- Logarithm
- Polymer degradation
- Scientific laws named after people
- Quantification of nucleic acids
- Tunable diode laser absorption spectroscopy
- Transmittance#Beer–Lambert law
References
- ^ Bouguer, Pierre (1729). Essai d'optique sur la gradation de la lumière [Optics essay on the attenuation of light] (in French). Paris, France: Claude Jombert. pp. 16–22.
- ^ Lambert, J.H. (1760). Photometria sive de mensura et gradibus luminis, colorum et umbrae [Photometry, or, On the measure and gradations of light intensity, colors, and shade] (in Latin). Augsburg, (Germany): Eberhardt Klett.
- ^ "Bouguer-Lambert-Beer Absorption Law - Lumipedia". www.lumipedia.org. Retrieved 2023-04-25.
- .
- ^ Pfieffer, Heinz; Liebhafshy, Herman (1951). "The Origins of Beer's Law". Journal of Chemical Education (March, 1951): 123–125.
- ^ Ingle, J. D. J.; Crouch, S. R. (1988). Spectrochemical Analysis. New Jersey: Prentice Hall.
- PMID 32662939.
- ISBN 9780486642284.
- ^ Sokolik, Irina N. (2009). "The Beer-Bouguer-Lambert law. Concepts of extinction (scattering plus absorption) and emission" (PDF).
- ISSN 0960-3360.
- ^ "Definition of ABSORPTIVITY". www.merriam-webster.com. Retrieved 2023-05-17.
- PMID 29133925.
- ISBN 978-0199573370.
- ISBN 978-0198556862.
- PMID 34713647.