In physical optics and transport theory, 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 one substance like light, neutrons or host rarefied
gases is traveling. It is strongly related to BGK model.
Equations The law states that there is a logarithmic
dependence between the transmission, T, of light through a substance and the product
of the attenuation coefficient of the substance, Σ, and the distance the light travels through
the material, ℓ. The attenuation coefficient can, in turn, be written as a product of either
an absorptivity of the attenuator, ε, and the concentration c of attenuating species
in the material, or a total cross section, σ, and the density N' of attenuators. In
some chemistry applications for liquids these relations are usually written with the notation:
whereas in biology and physics, they are normally written as:
where and are the intensity of the incident radiation and the transmitted radiation, respectively;
σ is attenuation cross section and N is the concentration of attenuating medium. The base
10 and base e conventions must not be confused because they have different numerical values
for the attenuation coefficient: . However, it is easy to convert one to the other, using
and to express the decimal logarithm quantity with the decibel unit of measure.
The transmissivity is expressed in terms of an absorbance which is defined as
whereas it can be expressed in decibels as:
This implies that the absorbance becomes linear with the concentration according to
and
for the two cases, respectively. Thus, if the path length and the attenuation coefficient
are known and the absorbance is measured, the concentration of the substance can be
deduced. Although several of the expressions above often are used as Beer–Lambert law,
the name should strictly speaking only be associated with the latter two. The reason
is that historically, the Lambert law states that attenuation is proportional to the light
path length, whereas the Beer law states that attenuation is proportional to the concentration
of attenuating species in the material. If the concentration is expressed as a mole fraction
i.e., a dimensionless fraction, the attenuation coefficient takes the same dimension as the
attenuation coefficient, i.e., reciprocal length. However, if the concentration is expressed
in moles per unit volume, the attenuation coefficient is used in L·mol−1·cm−1,
or sometimes in converted SI units of m2·mol−1. The attenuation coefficient Σ' is one of
many ways to describe the atenuation of electromagnetic waves. For the others, and their interrelationships,
see the article: Mathematical descriptions of opacity. For example, Σ' can be expressed
in terms of the imaginary part of the refractive index, κ, and the wavelength of the radiation(in
free space), λ0, according to
In molecular attenuation spectrometry, the attenuation cross section σ is expressed
in terms of a linestrength, S, and an lineshape function, Φ. The frequency scale in molecular
spectroscopy is often in cm−1, where the lineshape function is expressed in units of
1/cm−1. Since N 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 fact that there are two commensurate definitions of attenuation implies that the
absorbance and the attenuation coefficient for the cases with gases, A' and Σ', are
ln 10 times as large as the corresponding values for liquids, i.e., A and Σ, respectively.
Therefore, care must be taken when interpreting data that the correct form of the law is used.
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. If the attenuation
is different at higher concentrations than at lower ones, then the plot of the attenuation
coefficient will not be linear, as is suggested by the equation, so you can only use it when
all the concentrations you are working with are low enough that the absorbtivity is the
same for all of them. Derivation
Classically, the Beer–Lambert law was first devised independently where Lambert's law
stated that absorbance is directly proportional to the thickness of the sample, and Beer's
law stated that absorbance is proportional to the concentration of the sample. 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. 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—intensity
before and after light passes through the slice—we can easily write an ordinary differential
equation model for attenuation. The difference in intensity due to the slice of attenuating
material is reduced; leaving the slice, it is a fraction of the light entering the slice
. The thickness of the slice is , which scales the amount of attenuation. In symbols, , or
. This conceptual overview uses 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, 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 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 N. It follows
that the fraction of photons attenuated when passing through this slab is equal to the
total opaque area of the particles in the slab, σAN dz, divided by the area of the
slab A, which yields σN dz. Expressing the number of photons attenuated by the slab as
dIz, and the total number of photons incident on the slab as Iz, the number of photons attenuated
by the slab is given by
Note that because there are fewer photons which pass through the slab than are incident
on it, dIz is actually negative. The solution to this simple differential equation is obtained
by integrating both sides to obtain Iz as a function of z
The difference of intensity for a slab of real thickness ℓ is I0 at z = 0, and Il
at z = ℓ. Using the previous equation, the difference in intensity can be written
as,
rearranging and exponentiating yields,
This implies that
and
The quantity Σ is called the total macroscopic cross section or attenuation coefficient,
depending on the topic. The derivation assumes that every attenuating particle behaves independently
with respect to the light and is not affected by other particles. While it is commonly thought
that error is introduced when particles are lying along the same optical path such that
some particles are in the shadow of others, this is actually a key part of the derivation
and why integration is used. When the path taken is long enough to make
the medium attenuation coefficient not uniform, the original equation must be modified as
follows:
where z is the distance along the path through the medium, all other symbols are as defined
above. This is taken into account in each in the atmospheric equation above.
Deviations from Beer–Lambert law Under certain conditions Beer–Lambert law
fails to maintain a linear relationship between attenuation and concentration of analyte.
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. Prerequisites
There are at least six conditions that need to be fulfilled in order for Beer’s 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 e.g. 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 intensity is needed instead of a photodiode
which has not a selective wavelength dependence; and
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’s law. Chemical analysis
Beer's 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 are made at one wavelength that is nearly unique for
bilirubin and at a second wavelength in order to correct for possible interferences.The
concentration is given by c = Acorrected / ε. For a more complicated example, consider a
mixture in solution containing two components at concentrations c1 and c2. The absorbance
at any wavelength, λ is, for unit path length, given by
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 absorbances 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.
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 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
are uniformly mixed gases and molecular oxygen which only absorb)
is nitrogen dioxide, mainly due to urban pollution are effects due to Raman scattering in the
atmosphere is water vapour absorption
is ozone is Rayleigh scattering from molecular oxygen
and nitrogen. 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. is the optical mass or airmass factor, a term
approximately equal to , where is the observed object's zenith angle. 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 attributed to Johann Heinrich Lambert, who cited Bouguer's Essai d'Optique
sur la Gradation de la Lumiere — and even quoted from it — in his Photometria in 1760.
Much later, August Beer extended the exponential attenuation law in 1852 to include the concentration
of solutions in the attenuation coefficient. See also
References
External links Beer–Lambert Law Calculator
Beer–Lambert Law Simpler Explanation