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 (named after August Beer, Johann Heinrich Lambert, and Pierre Bouguer) relates the absorption of light to the properties of the material through which the light is travelling.

Equations
Diagram of Beer–Lambert absorption of a beam of light as it travels through a cuvette of width ℓ.

The law states that there is a logarithmic dependence between the transmission (or transmissivity), T, of light through a substance and the product of the absorption coefficient of the substance, α, and the distance the light travels through the material (i.e., the path length), ℓ. The absorption coefficient can, in turn, be written as a product of either a molar absorptivity (extinction coefficient) of the absorber, ε, and the molar concentration c of absorbing species in the material, or an absorption cross section, σ, and the (number) density N' of absorbers.

For liquids, these relations are usually written as:

\( T = {I\over I_{0}} = 10^{-\alpha\, \ell} = 10^{-\varepsilon\ell c} \)

whereas for gases, and in particular among physicists and for spectroscopy and spectrophotometry, they are normally written

\( T = {I\over I_{0}} = e^{-\alpha'\, \ell} = e^{-\sigma \ell N} \)

where I0 and I are the intensity (or power) of the incident light and the transmitted light, respectively; σ is cross section of light absorption by a single particle and N is the density (number per unit volume) of absorbing particles.

The base 10 and base e conventions must not be confused because they give different values for the absorption coefficient: \( \alpha\neq\alpha' \). However, it is easy to convert one to the other, using \( \alpha' = \alpha \ln(10)\approx 2.303\alpha \).

The transmission (or transmissivity) is expressed in terms of an absorbance which, for liquids, is defined as

\( A = -\log_{10} \left( \frac{I}{I_0} \right) \)

whereas, for gases, it is usually defined as

\( A' = -\ln \left( \frac{I}{I_0} \right). \)

This implies that the absorbance becomes linear with the concentration (or number density of absorbers) according to

A = \varepsilon \ell c = \alpha\ell \, \)

and

\( A' = \sigma \ell N = \alpha' \ell \, \)

for the two cases, respectively.

Thus, if the path length and the molar absorptivity (or the absorption cross section) are known and the absorbance is measured, the concentration of the substance (or the number density of absorbers) 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 absorption is proportional to the light path length, whereas the Beer law states that absorption is proportional to the concentration of absorbing species in the material.^[1]

If the concentration is expressed as a mole fraction i.e., a dimensionless fraction, the molar absorptivity (ε) takes the same dimension as the absorption coefficient, i.e., reciprocal length (e.g., m⁻¹). However, if the concentration is expressed in moles per unit volume, the molar absorptivity (ε) is used in L·mol⁻¹·cm⁻¹, or sometimes in converted SI units of m²·mol⁻¹.

The absorption coefficient α' is one of many ways to describe the absorption of electromagnetic waves. For the others, and their interrelationships, see the article: Mathematical descriptions of opacity. For example, α' can be expressed in terms of the imaginary part of the refractive index, κ, and the wavelength of the light (in free space), λ₀, according to

\( \alpha' = \frac{4 \pi \kappa}{\lambda_{0}}. \)

In molecular absorption spectrometry, the absorption cross section σ 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 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 absorbance (in base 10 or e) implies that the absorbance and the absorption coefficient for the cases with gases, A' and α', are ln 10 (approximately 2.3) times as large as the corresponding values for liquids, i.e., A and α, 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 light is especially intense, nonlinear optical processes can also cause variances.

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 (path length) of the sample.[2]

In concept, the derivation of the Beer–Lambert law is straightforward. Divide the absorbing 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 absorption are needed, a vast majority of the light entering the slice leaves without being absorbed. 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 absorption. The difference in intensity due to the slice of absorbing material dI is reduced; leaving the slice, it is a fraction \beta of the light entering the slice I. The thickness of the slice is dz, which scales the amount of absorption (thin slice does not absorb much light but a thick slice absorbs a lot). In symbols, \( dI = \beta I dz \), or \( dI/dz = \beta I \) . This conceptual overview uses \beta to describe how much light is absorbed. 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 absorption cross section (i.e., area), σ, 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 N.

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, σAN dz, divided by the area of the slab A, which yields σN 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 number of photons absorbed by the slab is given by

\( dI_z = - \sigma N\,I_z\,dz . \)

Note that because there are fewer photons which pass through the slab than are incident on it, dIz is actually negative (It is proportional in magnitude to the number of photons absorbed).

The solution to this simple differential equation is obtained by integrating both sides to obtain Iz as a function of z

\( \ln(I_z) = - \sigma N z + C . \, \)

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,

\( \ln(I_l) - \ln(I_0) = (- \sigma \ell N + C) - ( - \sigma 0 N + C) = - \sigma \ell N \, \)

rearranging and exponentiating yields,

\( \ T = \frac{I_l}{I_0} = e ^ {- \sigma \ell N} = e ^ {- \alpha'\ell} . \)

This implies that

\( A' = - \ln\left( \frac{I_l}{I_0} \right) = \alpha' \ell = \sigma\ell N \, \)

and

\( A = - \log_{10}\left( \frac{I_l}{I_0} \right) = \frac{\alpha'\ell}{2.303} = \alpha \ell = \varepsilon \ell c. \, \)

The derivation assumes that every absorbing particle behaves independently with respect to the light and is not affected by other particles. Error is introduced when particles are lying along the same optical path such that some particles are in the shadow of others. This occurs in highly concentrated solutions. In practice, when large absorption values are measured, dilution is required to achieve accurate results. Measurements of absorption in the range of \( {I_1} / {I_0} = 0.1 \) to 1 are less affected by shadowing than other sources of random error. In this range, the ODE model developed above is a good approximation; measurements of absorption in this range are linearly related to concentration. At higher absorbances, concentrations will be underestimated due to this shadow effect unless one employs a more sophisticated model that describes the non-linear relationship between absorption and concentration.

Prerequisites

There are at least six conditions that need to be fulfilled in order for Beer’s law to be valid. These are:

1. The absorbers must act independently of each other;
2. The absorbing medium must be homogeneous in the interaction volume
3. The absorbing medium must not scatter the radiation – no turbidity;
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 absorbing transition; and
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’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 absorbance 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 = A^corrected / 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

\( A(\lambda)=c_1\ \varepsilon_1(\lambda)+c_2\ \varepsilon_2(\lambda). \)

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 analysed 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 absorption 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

\( I = I_0\,\exp(-m(\tau_a+\tau_g+\tau_{\rm NO_2}+\tau_w+\tau_{\rm O_3}+\tau_r)), \)

where each \tau_{x} is the optical depth whose subscript identifies the source of the absorption or scattering it describes:

a refers to aerosols (that absorb and scatter)
g are uniformly mixed gases (mainly carbon dioxide \( (\mathrm{CO}_2) and molecular oxygen \( (\mathrm{O}_2) \) which only absorb)
\( \mathrm{NO}_2 \) is nitrogen dioxide, mainly due to urban pollution (absorption only)
w is water vapour absorption
\( \mathrm{O}_3 \) is ozone (absorption only)
r is Rayleigh scattering from molecular oxygen \( (\mathrm{O}_2) \) and nitrogen \( (\mathrm{N}_2) \) (responsible for the blue color of the sky).

m is the optical mass or airmass factor, a term approximately equal (for small and moderate values of \( \theta) to \( 1/\cos(\theta) \) , where \( \theta \) 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 \( \tau_{a}, 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.

When the path taken by the light is through the atmosphere, the density of the absorbing gas is not constant, so the original equation must be modified as follows:

\( T = {I_{1}\over I_{0}} = e^{-\int\alpha'\, dz} = e^{-\sigma\int N dz} \)

where z is the distance along the path through the atmosphere, all other symbols are as defined above.[3] This is taken into account in each \( \tau_{x} \) in the atmospheric equation above.
History

The law was discovered by Pierre Bouguer before 1729.[4] It is often 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.[5] Much later, August Beer extended the exponential absorption law in 1852 to include the concentration of solutions in the absorption coefficient.[6]
See also

Applied spectroscopy
Atomic absorption spectroscopy
Absorption spectroscopy
Cavity ring-down spectroscopy
Infra-red spectroscopy
Job plot
Laser absorption spectrometry
Logarithm
Polymer degradation
Scientific laws named after people
Quantification of nucleic acids
Tunable diode laser absorption spectroscopy

References

^ J. D. J. Ingle and S. R. Crouch, Spectrochemical Analysis, Prentice Hall, New Jersey (1988)
^ Derivation of Beer-Lambert Law
^ ISBN 0521339561 Houghton, J.T. The Physics of Atmospheres 2nd ed. Chapter 2
^ Pierre Bouguer, Essai d'Optique, sur la gradation de la lumiere (Paris, France: Claude Jombert, 1729) pp. 16–22.
^ J.H. Lambert, Photometria sive de mensura et gradibus luminis, colorum et umbrae [Photometry, or, On the measure and gradations of light, colors, and shade] (Augsburg ("Augusta Vindelicorum"), Germany: Eberhardt Klett, 1760). See especially p. 391.
^ Beer (1852) Bestimmung der Absorption des rothen Lichts in farbigen Flüssigkeiten (Determination of the absorption of red light in colored liquids), Annalen der Physik und Chemie, vol. 86, pp. 78–88.

External links

Beer–Lambert Law Calculator
Beer-Lambert Law Simpler Explanation
Reasons for Deviations from Beer-Lambert Law

Physics Encyclopedia