Information about Beer Lambert Law
In optics, the Beer-Lambert law, also known as Beer's law or the Lambert-Beer law or the Beer-Lambert-Bouguer law (in fact, most of the permutations of these three names appear somewhere in literature) is an empirical relationship that relates the absorption of light to the properties of the material through which the light is traveling.
There are several ways in which the law can be expressed,
where,
and 
are known, the concentration of a substance can be deduced from the amount of light transmitted by it.
The units of absorber concentration (
) and absorption coefficient () depend on the way that the concentration of the absorber is being expressed.
If the material is a liquid, it is usual to express the absorber concentration as a mole fraction i.e. a dimensionless fraction. The units of the absorption coefficient are thus reciprocal length (e.g. cm−1). If the concentration is expressed in moles per unit volume, is a molar absorptivity (usually given the symbol 
) in units of mol−1 cm−2 or sometimes L·mol−1·cm−1.
In the case of a gas, the concentration may be expressed as a number density (e.g. cm−3), in which case is an absorption cross-section and has units of area (e.g. cm2).
The value of the absorption coefficient varies between different absorbing materials and also with wavelength for a particular material. It is usually determined by experiments.
In spectroscopy and spectrophotometry, the law is almost always defined in terms of common logarithm. In optics, the law is often defined in an alternate exponential form,
The values of
and 
are approximately 2.3 (≈ln 10) times larger than the corresponding values of and 
defined in terms of common logarithm. Note though that when c is given as a number density and as an area it is often denoted by 
and represents the "true" cross section of the absorber (as can be seen from the derivation below). Therefore, care must be taken when interpreting data that the correct form of the law is used.
In molecular absorption spectrometry, the absorption coefficient α' is expressed in terms of a linestrength, S, and an (area-normalized) lineshape function, Φ. The frequency scale in molecular spectroscopy is often in cm-1, wherefore the lineshape function is expressed in units of 1/cm-1, which can look funny but is strictly correct. Since c is given as a number density in units of 1/cm3, the linestrength is often given in units of cm2cm-1/molecule. A typical linestrength in one of the vibrational overtone bands of smaller molecules, e.g. around 1.5 μm in CO or CO2, is around 10-23 cm2cm-1, although it can be larger for species with strong transitions, e.g. C2H2. The linestrengths of various transitions can be found in large databases, e.g. HITRAN. The lineshape function often takes a value around a few 1/cm-1, up to around 10/cm-1 under low pressure conditions, when the transition is Doppler broadened, and below this under atmospheric pressure conditions, when the transition is collision broadened. It has also become commonplace to express the linestrength in units of cm-2/atm since then the concentration is given in terms of a pressure in units of atm. A typical linestrength is then often in the order of 10-3 cm-2/atm. Under these conditions, the detectability of a given technique is often quoted in terms of ppm•m.
The law tends to break down at very high concentrations, especially if the material is highly scattering. If the light is especially intense, nonlinear optical processes can also cause variances.
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. We 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 concentration of particles in the slab is represented by c.
It follows that the fraction of photons absorbed when passing through this slab is equal to the total opaque area of the particles in the slab, αAc dz, divided by the area of the slab, or αc dz. Expressing the number of photons absorbed by the slab as dIz, and the total number of photons incident on the slab as Iz, the fraction of photons absorbed by the slab is given by
The solution to this simple differential equation is obtained by integrating both sides to obtain Iz as a function of z
For a slab of real thickness, ℓ, the difference in light intensity I0 at z = 0, and I1 at z = ℓ, is given by
or
It is instructive to consider the consequences of error in an assumption that is implicit in this derivation, namely that every absorbing particle behaves independently with respect to the light. Error is introduced when particles interact by lying along the same optical path such that some particles are in the shadow of others. The assumption approaches accuracy only in very dilute solutions, and it becomes increasingly inaccurate with increasingly concentrated solutions or long optical paths.
In practice, the accuracy of the assumption is better than the accuracy of most spectroscopic measurements up to an absorbance of 1 (or :
and to a good approximation, measurements of absorbance in this range are linearly related to the concentration of absorbing substances in solution. At higher absorbances, concentrations will be underestimated due to this shadow effect unless one employs a nonlinear relationship between absorbance and concentration.
where 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.
Equations
)
Diagram of Beer-Lambert absorption of a beam of light as it travels through a cuvette of size l.
There are several ways in which the law can be expressed,
where,
is absorbance
is the intensity of the incident light
is the intensity after passing through the material
- is the distance that the light travels through the material (the path length)
- is the concentration of absorbing species in the material
- is the absorption coefficient or the molar absorptivity of the absorber
is the wavelength of the light
is the extinction coefficient
is the transmittance
and are known, the concentration of a substance can be deduced from the amount of light transmitted by it.
The units of absorber concentration (
) and absorption coefficient () depend on the way that the concentration of the absorber is being expressed.
If the material is a liquid, it is usual to express the absorber concentration as a mole fraction i.e. a dimensionless fraction. The units of the absorption coefficient are thus reciprocal length (e.g. cm−1). If the concentration is expressed in moles per unit volume,
is a molar absorptivity (usually given the symbol ) in units of mol−1 cm−2 or sometimes L·mol−1·cm−1.
In the case of a gas, the concentration may be expressed as a number density (e.g. cm−3), in which case
is an absorption cross-section and has units of area (e.g. cm2).
The value of the absorption coefficient
varies between different absorbing materials and also with wavelength for a particular material. It is usually determined by experiments.
In spectroscopy and spectrophotometry, the law is almost always defined in terms of common logarithm. In optics, the law is often defined in an alternate exponential form,
The values of
and are approximately 2.3 (≈ln 10) times larger than the corresponding values of and defined in terms of common logarithm. Note though that when c is given as a number density and as an area it is often denoted by and represents the "true" cross section of the absorber (as can be seen from the derivation below). Therefore, care must be taken when interpreting data that the correct form of the law is used.
In molecular absorption spectrometry, the absorption coefficient α' is expressed in terms of a linestrength, S, and an (area-normalized) lineshape function, Φ. The frequency scale in molecular spectroscopy is often in cm-1, wherefore the lineshape function is expressed in units of 1/cm-1, which can look funny but is strictly correct. Since c is given as a number density in units of 1/cm3, the linestrength is often given in units of cm2cm-1/molecule. A typical linestrength in one of the vibrational overtone bands of smaller molecules, e.g. around 1.5 μm in CO or CO2, is around 10-23 cm2cm-1, although it can be larger for species with strong transitions, e.g. C2H2. The linestrengths of various transitions can be found in large databases, e.g. HITRAN. The lineshape function often takes a value around a few 1/cm-1, up to around 10/cm-1 under low pressure conditions, when the transition is Doppler broadened, and below this under atmospheric pressure conditions, when the transition is collision broadened. It has also become commonplace to express the linestrength in units of cm-2/atm since then the concentration is given in terms of a pressure in units of atm. A typical linestrength is then often in the order of 10-3 cm-2/atm. Under these conditions, the detectability of a given technique is often quoted in terms of ppm•m.
The law tends to break down at very high concentrations, especially if the material is highly scattering. If the light is especially intense, nonlinear optical processes can also cause variances.
Derivation
Assume that particles may be described as having an area, α, perpendicular to the path of light through a solution, such that a photon of light is absorbed 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. We 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 concentration of particles in the slab is represented by c.
It follows that the fraction of photons absorbed when passing through this slab is equal to the total opaque area of the particles in the slab, αAc dz, divided by the area of the slab, or αc dz. Expressing the number of photons absorbed by the slab as dIz, and the total number of photons incident on the slab as Iz, the fraction of photons absorbed by the slab is given by
The solution to this simple differential equation is obtained by integrating both sides to obtain Iz as a function of z
For a slab of real thickness, ℓ, the difference in light intensity I0 at z = 0, and I1 at z = ℓ, is given by
or
It is instructive to consider the consequences of error in an assumption that is implicit in this derivation, namely that every absorbing particle behaves independently with respect to the light. Error is introduced when particles interact by lying along the same optical path such that some particles are in the shadow of others. The assumption approaches accuracy only in very dilute solutions, and it becomes increasingly inaccurate with increasingly concentrated solutions or long optical paths.
In practice, the accuracy of the assumption is better than the accuracy of most spectroscopic measurements up to an absorbance of 1 (or :
and to a good approximation, measurements of absorbance in this range are linearly related to the concentration of absorbing substances in solution. At higher absorbances, concentrations will be underestimated due to this shadow effect unless one employs a nonlinear relationship between absorbance and concentration.
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 Beer-Lambert law for the atmosphere is usually written
where each
is the optical depth whose subscript identifies the source of the absorption or scattering it describes:
refers to aerosols (that absorb and scatter)
are uniformly mixed gases (mainly carbon dioxide (
) and molecular oxygen (
) which only absorb)
is nitrogen dioxide, mainly due to urban pollution (absorption only)
is water vapour absorption
is ozone (absorption only)
is Rayleigh scattering from molecular oxygen () and nitrogen (
) (responsible for the blue color of the sky).
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.
History
The law was discovered by Pierre Bouguer before 1729. It is often mis-attributed to Johann Heinrich Lambert, who cited Bouguer's “Essai d'Optique sur la Gradation de la Lumiere” (Claude Jombert, Paris, 1729) — and even quoted from it — in his “Photometria” in 1760. Much later, August Beer extended the exponential absorption law in 1852 to include the concentration of solutions in the absorption coefficient.External links
See also
- Atomic absorption spectroscopy
- Logarithm
- Scientific laws named after people
- Quantification of nucleic acids
Optics (ὀπτική appearance or look in Ancient Greek) is a branch of physics that describes the behavior and properties of light and the interaction of light with matter.
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In science, an empirical relationship is one based solely on observation rather than theory. An empirical relationship requires only confirmatory data irrespective of theoretical basis.
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In physics, absorption is the process by which the energy of a photon is taken up by another entity, for example, by an atom whose valence electrons make transition between two electronic energy levels. The photon is destroyed in the process.
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Light is electromagnetic radiation of a wavelength that is visible to the eye (visible light). In a scientific context, the word "light" is sometimes used to refer to the entire electromagnetic spectrum.
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In spectroscopy, the absorbance A is defined as
where is the intensity of light at a specified wavelength λ that has passed through a sample (transmitted light intensity) and is the intensity of the light before it enters the sample
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- ,
where is the intensity of light at a specified wavelength λ that has passed through a sample (transmitted light intensity) and is the intensity of the light before it enters the sample
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In physics, intensity is a measure of the time-averaged energy flux. To find the intensity, take the energy density (that is, the energy per unit volume) and multiply it by the velocity at which the energy is moving. The resulting vector has the units of power divided by area (i.e.
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In chemistry, the path length is defined as the distance that light (UV/VIS) travels through a sample in an analytical cell. Typically, a sample cell is made of quartz, glass, or a plastic rhombic cuvette with a volume typically ranging from 0.
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In chemistry, concentration is the measure of how much of a given substance there is mixed with another substance. This can apply to any sort of chemical mixture, but most frequently the concept is limited to homogeneous solutions, where it refers to the amount of
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The absorption coefficient α is a property of a material. It defines the extent to which a material absorbs energy, for example that of sound waves or electromagnetic radiation.
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The molar extinction coefficient, also known as molar absorptivity, is a measure of how strongly a chemical species at a given wavelength absorbs light at that wavelength.
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In physics, wavelength is the distance between repeating units of a propagating wave of a given frequency. It is commonly designated by the Greek letter lambda (λ). Examples of wave-like phenonomena are light, water waves, and sound waves.
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In optics and spectroscopy, transmittance is the fraction of incident light at a specified wavelength that passes through a sample.
where is the intensity of the light and I is the intensity of the light coming out of the sample.
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where is the intensity of the light and I is the intensity of the light coming out of the sample.
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In chemistry, the mole fraction of a component in a mixture is the relative proportion of molecules belonging to the component to those in the mixture, by number of molecules. It is one way of measuring concentration.
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The mole (symbol: mol) is the SI base unit that measures an amount of substance. One mole contains Avogadro's number (approximately 6.0221023) entities.
A mole is much like "a dozen" in that both are absolute numbers (having no units) and can describe any type of
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A mole is much like "a dozen" in that both are absolute numbers (having no units) and can describe any type of
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The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in the three-dimensional space.
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The molar extinction coefficient, also known as molar absorptivity, is a measure of how strongly a chemical species at a given wavelength absorbs light at that wavelength.
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The litre or liter (see spelling differences) is a unit of volume. There are two official symbols, namely the Latin letter L both in lower and upper case: l and L.
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Spectroscopy is the study of the interaction between radiation (electromagnetic radiation, or light, as well as particle radiation) and matter. Spectrometry is the measurement of these interactions and an instrument which performs such measurements is a spectrometer or
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In physics, spectrophotometry is the quantitative study of electromagnetic spectra. It is more specific than the general term electromagnetic spectroscopy in that spectrophotometry deals with visible light, near-ultraviolet, and near-infrared.
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In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm, named after its base. It is indicated by log10(x), or sometimes Log(x) with a capital L
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Optics (ὀπτική appearance or look in Ancient Greek) is a branch of physics that describes the behavior and properties of light and the interaction of light with matter.
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Scattering is a general physical process whereby some forms of radiation, such as light, sound or moving particles, for example, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which it passes.
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Nonlinear optics (NLO) is the branch of optics that describes the behaviour of light in nonlinear media, that is, media in which the polarization P responds nonlinearly to the electric field E of the light.
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Optical depth is a measure of transparency, and is defined as the fraction of radiation (or light) that is scattered or absorbed on a path. One way of visualizing optical depth is to think of a fog.
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Particulates, alternatively referred to as particulate matter (PM), aerosols or fine particles, are tiny particles of solid or liquid suspended in a gas. They range in size from less than 10 nanometres to more than 100 micrometres in diameter.
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Carbon dioxide is a chemical compound composed of two oxygen atoms covalently bonded to a single carbon atom. It is a gas at standard temperature and pressure and exists in Earth's atmosphere in this state.
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2, −1
(neutral oxide)
Electronegativity 3.44 (Pauling scale)
Ionization energies
(more) 1st: 1313.9 kJmol−1
2nd: 3388.3 kJmol−1
3rd: 5300.5 kJmol−1
Atomic radius 60 pm
Atomic radius (calc.
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(neutral oxide)
Electronegativity 3.44 (Pauling scale)
Ionization energies
(more) 1st: 1313.9 kJmol−1
2nd: 3388.3 kJmol−1
3rd: 5300.5 kJmol−1
Atomic radius 60 pm
Atomic radius (calc.
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Nitrogen dioxide is the chemical compound NO2. It is one of the several nitrogen oxides. This reddish-brown gas has a characteristic sharp, biting odor. NO2 is one of the most prominent air pollutants and a poison by inhalation.
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Water vapor or water vapour (see spelling differences), also aqueous vapor, is the gas phase of water. Water vapor is one state of the water cycle within the hydrosphere.
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