# Lambert-Beer law

The Lambert-Beer law or Bouguer-Lambert-Beer law describes the attenuation of the intensity of a radiation in relation to its initial intensity when it passes through a medium with an absorbent substance, depending on the concentration of the absorbent substance and the layer thickness. The law forms the basis of modern photometry as an analytical method. It is a special case of the radiative transport equation without an emission term.

## history

The Bouguer-Lambert law was formulated by Pierre Bouguer before the year 1729 and describes the weakening of the radiation intensity with the path length when passing through an absorbent substance. It is also attributed to Johann Heinrich Lambert , sometimes even briefly referred to as Lambert's Law, although Lambert himself cites and even cites Bouguer's work " Essai d'optique sur la gradation de la lumière " in his "Photometria" (1760).

Lambert's law of cosines is also referred to as Lambert's law .

In 1852, August Beer extended Bouguer-Lambert's law by setting the concentration of the absorber in relation to the transmitted light. This relationship is referred to as Lambert-Beer's law or, more rarely, as Bouguer-Lambert-Beer's law.

## The law

The extinction (absorbance of the material for light of the wavelength ) is given by ${\ displaystyle E _ {\ lambda}}$${\ displaystyle \ lambda}$

${\ displaystyle E _ {\ lambda} = \ log _ {10} \ left ({\ frac {I_ {0}} {I_ {1}}} \ right) = \ varepsilon _ {\ lambda} \ cdot c \ cdot d}$

With

• ${\ displaystyle I_ {0}}$: Intensity of the incident (irradiated) light (unit: W m −2 )
• ${\ displaystyle I_ {1}}$: Intensity of the transmitted light (unit: W m −2 )
• ${\ displaystyle c}$: Molar concentration of the absorbing substance in the liquid (unit: mol m −3 )
• ${\ displaystyle \ varepsilon _ {\ lambda}}$: decadic extinction coefficient (often also referred to as spectral absorption coefficient ) at the wavelength . This is a specific variable for the absorbing substance and can depend , among other things, on the pH value or the solvent. If the concentration is given in mol, the decadic molar extinction coefficient is given, for example in the unit m 2 · mol −1${\ displaystyle \ lambda}$${\ displaystyle \ varepsilon _ {\ lambda}}$
• ${\ displaystyle d}$: Layer thickness of the irradiated body (unit: m)

## Derivation

The differential decrease in radiation intensity due to absorption is proportional to the intensity , the extinction coefficient , the molar concentration of the absorbing substance and its differential layer thickness : ${\ displaystyle \ mathrm {d} I_ {1}}$${\ displaystyle I_ {1}}$${\ displaystyle \ varepsilon ^ {*}}$${\ displaystyle c}$${\ displaystyle \ mathrm {d} d}$

${\ displaystyle \ mathrm {d} I_ {1} = - I_ {1} \, \ varepsilon ^ {*} \, c \, \ mathrm {d} d}$;

or

${\ displaystyle {\ frac {\ mathrm {d} I_ {1}} {I_ {1}}} = - \ varepsilon ^ {*} \, c \, \ mathrm {d} d}$

and after integration

${\ displaystyle \ ln \ left (I_ {1} \ right) = - \ varepsilon ^ {*} \ cdot c \, d + \ ln \ left (I_ {0} \ right)}$
${\ displaystyle \ ln \ left ({\ frac {I_ {1}} {I_ {0}}} \ right) = - \ varepsilon ^ {*} \, c \, d}$,

after the integration constant has been determined for too . This gives the falling exponential function , with which the decrease in light intensity when passing through a sample solution can be described with the concentration : ${\ displaystyle d = 0}$${\ displaystyle \ ln \ left (I_ {0} \ right)}$${\ displaystyle c}$

${\ displaystyle I_ {1} = I_ {0} \, e ^ {- \ varepsilon ^ {*} c \, d}}$

Transforming the equation gives:

${\ displaystyle - \ ln \ left ({\ frac {I_ {1}} {I_ {0}}} \ right) = \ varepsilon ^ {*} c \, d}$.

However, the extinction and the extinction coefficient are not defined using the natural logarithm . Since the decadic and natural logarithms are linearly related, the transition corresponds to a constant factor in the equation. This is simply included in the equation: becomes off . ${\ displaystyle \ varepsilon ^ {*}}$${\ displaystyle \ varepsilon = \ log _ {10} (e) \ varepsilon ^ {*} \ approx 0 {,} 434 \ varepsilon ^ {*}}$

${\ displaystyle - \ log _ {10} \ left ({\ frac {I_ {1}} {I_ {0}}} \ right) = \ varepsilon \, c \, d}$.

Here is the decadic molar extinction coefficient. ${\ displaystyle \ varepsilon}$

With the power rule of the logarithm, the usual notation results:

${\ displaystyle \ log _ {10} \ left ({\ frac {I_ {0}} {I_ {1}}} \ right) = \ varepsilon \, c \, d}$.

The extinction of a substance depends on the dispersion of the complex refractive index on the wavelength of the incident light. ${\ displaystyle \ lambda}$

The derivation of the concentration dependence is based on the electromagnetic theory. Accordingly, the macroscopic polarization of a medium results in the absence of interactions ${\ displaystyle P}$

${\ displaystyle P = N \ p}$

where is the dipole moment and the particle number density. On the other hand, the polarization is given by: ${\ displaystyle p}$${\ displaystyle N}$

${\ displaystyle P = (\ varepsilon _ {r} -1) \ cdot \ varepsilon _ {0} \ cdot E}$

Herein is the relative dielectric function, the electric field constant and the electric field. After equating and solving for the relative dielectric function, we get: ${\ displaystyle \ varepsilon _ {r}}$${\ displaystyle \ varepsilon _ {0}}$${\ displaystyle E}$

${\ displaystyle \ varepsilon _ {r} = 1 + {\ frac {P} {\ varepsilon _ {0} \ cdot E}}}$

If one takes into account that the polarizability is defined by and for the particle number density , it follows: ${\ displaystyle \ alpha}$${\ displaystyle p = \ alpha \ cdot E}$${\ displaystyle N = N _ {\ mathrm {A}} \ cdot c}$

${\ displaystyle \ varepsilon _ {r} = 1 + c {\ frac {N _ {\ mathrm {A}} \ cdot \ alpha} {\ varepsilon _ {0}}}}$

According to the electromagnetic wave equation resulting from the Maxwell equations and the definition of the refractive index , the relationship between the complex dielectric function and the complex refractive index applies to non-magnetic, isotropic and homogeneous media . So it follows: ${\ displaystyle \ varepsilon _ {r} = {\ hat {n}} ^ {2}}$

${\ displaystyle {\ hat {n}} = {\ sqrt {1 + c {\ frac {N _ {\ mathrm {A}} \ cdot \ alpha} {\ varepsilon _ {0}}}}}}$

The imaginary part of the complex refractive index is the absorption index . With the imaginary part of the polarizability and the approximation we get: ${\ displaystyle k}$${\ displaystyle \ alpha ''}$${\ displaystyle {\ sqrt {1 + x}} \ approx 1 + x / 2}$

${\ displaystyle k = c {\ frac {N _ {\ mathrm {A}} \ cdot \ alpha ''} {2 \ varepsilon _ {0}}}}$

If you consider the connection between and , it ultimately follows ${\ displaystyle k}$${\ displaystyle E _ {\ lambda}}$${\ displaystyle E _ {\ lambda} = 4 \ pi (\ log _ {10} e) k \ cdot c \ cdot d / \ lambda}$

${\ displaystyle E _ {\ lambda} = {\ frac {2 \ pi (\ log _ {10} e) N _ {\ mathrm {A}} \ alpha ''} {\ lambda \ cdot \ varepsilon _ {0}} } \ cdot c \ cdot d}$

It follows from this that the linear relationship between concentration and extinction is generally only given approximately, i.e. for small polarizabilities and thus weaker absorptions. If one does not introduce the approximation and instead uses the following relationship between the imaginary part of the relative dielectric function and the refractive and absorption index , one can see that the extinction coefficient depends on the refractive index (which in turn is also dependent on the concentration): ${\ displaystyle {\ sqrt {1 + x}} \ approx 1 + x / 2}$${\ displaystyle \ varepsilon _ {r} '' = 2nk}$

${\ displaystyle E _ {\ lambda} = {\ frac {2 \ pi (\ log _ {10} e) N _ {\ mathrm {A}} \ alpha ''} {n \ cdot \ lambda \ cdot \ varepsilon _ { 0}}} \ cdot c \ cdot d}$

For stronger oscillators and higher concentrations, the deviations from linearity can be significant. In such cases, the integrated absorbance can be used instead, because this is also linearly dependent on the concentration for higher concentrations and stronger oscillators, as long as the other conditions (see validity) are observed. It is no coincidence that the derivation of the concentration dependence of the absorbance follows the derivation of the Lorentz-Lorenz equation (or the Clausius-Mossotti equation ). In fact, it can be shown that the approximately linear concentration dependence also applies to changes in the refractive index of solutions, since both also follow from the Lorentz-Lorenz equation.

## validity

The law applies to:

• homogeneous distribution of the absorbent substance;
• negligible multiple scattering (especially for clear media);
• negligible variation of the absorption coefficient within the measured spectral range (for monochromatic radiation and in spectroscopy, the variation is not a problem).
• negligible self-emission (the transmitted radiation intensity must be considerably higher than the (especially thermal) natural radiation);
• Low-concentration solutions (usually less than 0.01 mol·l −1 ) (at high concentrations, interactions lead to greater deviations; in general, the extinction / absorbance is not directly proportional to the concentration, since the extinction coefficient depends in a complicated way on the concentration even without interactions Corresponding deviations are only significant for higher concentrations and greater oscillator strengths).
• In the event that the wave properties of the light may be neglected, for example no interference amplifications or attenuations occur

In general, it should be noted that the Lambert-Beer law is not compatible with Maxwell's equations . Strictly speaking, it does not describe the transmission through a medium, but only the propagation within it. Compatibility can, however, be established by relating the transmission of a solute to the transmission of the pure solvent, as is standardly done in spectrophotometry . However, this is not possible for pure media. There, the uncritical application of Lambert-Beer's law can lead to errors of the order of 100% and more. In this case the transfer matrix method must be used.

## Application in chemistry

The wavelength dependence of the absorption coefficient of a substance is determined by its molecular properties. Differences between substances cause their color and allow the quantitative analysis of substance mixtures through photometric measurements. Malachite green is one of the most intense dyes with a molar absorption coefficient of 8.07 · 10 4  l mol −1 cm −1 (622 nm, ethanol ).

The same law applies in general to the drop in the intensity of electromagnetic radiation propagating in damping substances. It describes the attenuation of optical radiation in optical waveguides (LWL) or in attenuating optical media or the attenuation of gamma or X-ray radiation in matter. Conversely, a thickness measurement can be made with this connection if both intensities are known.

The radiant power passing through a medium of length is: ${\ displaystyle d}$${\ displaystyle P (d)}$

${\ displaystyle P (d) = P_ {0} \ cdot e ^ {- \ varepsilon 'd}}$.

With

• ${\ displaystyle P_ {0}}$: incoming service
• ${\ displaystyle \ varepsilon '}$: Absorption coefficient in m −1
• ${\ displaystyle d}$: Material thickness or length in m.

It is often strongly dependent on the wavelength and the material. ${\ displaystyle \ varepsilon '}$ ${\ displaystyle \ lambda}$

### optical fiber

For the silicate glass used in long-distance optical waveguides , it decreases between the visible range by 0.6 µm to approx. 1.6 µm with the fourth power of the wavelength ; At this point there is a steep increase in attenuation due to material resonance in the glass. Another pole of attenuation is in the ultraviolet range. Hydroxide ions in the glass, which one tries to avoid through special manufacturing technologies, cause a selective increase in attenuation at around 1.4 µm ( infrared spectroscopy ). The attenuation values ​​for the plastic fibers used in short-distance fiber optic cables are higher and are also heavily dependent on the material and wavelength, and are lowest in the visible range for specific purposes. ${\ displaystyle \ varepsilon '(\ lambda)}$

Instead of the notation given above, in signal transmission technology the representation

${\ displaystyle P (d) = P_ {0} \ cdot 10 ^ {- (\ varepsilon d) / 10}}$

( the attenuation is in dB / km and the length of the fiber-optic cable is in km), because in communications engineering the ratio of (electrical as well as optical) power is given in decimal-logarithmic dB ( decibel ): ${\ displaystyle \ varepsilon}$${\ displaystyle d}$

${\ displaystyle \ varepsilon d = 10 \ cdot \ log {\ frac {P_ {0}} {P (d)}}}$

### Remote sensing / atmosphere

For the atmosphere, the Lambert-Beer law is usually formulated as follows:

${\ displaystyle I = I_ {0} \, e ^ {- m \ left (\ tau _ {\ mathrm {a}} + \ tau _ {\ mathrm {g}} + \ tau _ {\ mathrm {NO_ { 2}}} + \ tau _ {\ mathrm {w}} + \ tau _ {\ mathrm {O_ {3}}} + \ tau _ {\ mathrm {r}} \ right)}}$

where stands for the atmospheric mass and for the optical thickness of the substance contained . The example stands for: ${\ displaystyle m}$${\ displaystyle \ tau _ {x}}$${\ displaystyle x}$${\ displaystyle x}$

• ${\ displaystyle \ mathrm {a}}$: Aerosols (absorbing and scattering)
• ${\ displaystyle \ mathrm {g}}$: homogeneous gases such as carbon dioxide and molecular oxygen (only absorbing)${\ displaystyle \ mathrm {CO} _ {2}}$ ${\ displaystyle \ mathrm {O} _ {2}}$
• ${\ displaystyle \ mathrm {NO} _ {2}}$: Nitrogen dioxide (absorbing)
• ${\ displaystyle \ mathrm {w}}$: Water vapor (absorbing)
• ${\ displaystyle \ mathrm {O} _ {3}}$: Ozone (absorbing)
• ${\ displaystyle \ mathrm {r}}$: Rayleigh scattering by molecular oxygen and nitrogen (sky blue)${\ displaystyle \ mathrm {O} _ {2}}$ ${\ displaystyle \ mathrm {N} _ {2}}$

The determination of is necessary for the correction of satellite images and is of interest, for example, for climate observation . ${\ displaystyle \ tau _ {a}}$

### Computed Tomography

In computed tomography , the attenuation of the X-rays is described by the Lambert-Beer law. The attenuation coefficient ( absorption coefficient ) is a function of the location, i.e. i.e., varies within the object ( patient ) and takes on a greater value in bones than in the lungs , for example . The measured intensity of the X-ray radiation results from the following integral: ${\ displaystyle \ mu}$${\ displaystyle \ mu}$ ${\ displaystyle I}$

${\ displaystyle I = I_ {0} \ exp \ left (- \ int \ mu (x) \ operatorname {d} x \ right)}$

where represents the radiation intensity emitted by the X-ray tube and parameterizes the integral over the path of the beam. The computed tomographic image then represents the function as a grayscale image (see Hounsfield scale ). The task of the reconstruction is therefore to determine the absorption coefficients from the measured intensities , that is to say to solve the associated inverse problem . ${\ displaystyle I_ {0}}$${\ displaystyle x}$${\ displaystyle \ mu (x)}$${\ displaystyle I}$${\ displaystyle \ mu}$

## Individual evidence

1. ^ Pierre Bouguer: Essai d'optique, Sur la gradation de la lumière . Claude Jombert, Paris 1729, p. 164 ff .
2. ^ Johann Heinrich Lambert: Photometria, sive de mensura et gradibus luminis, colorum et umbrae . Sumptibus Vidae Eberhardi Klett, Augsburg 1760 ( docnum.u-strasbg.fr [accessed on August 12, 2010] digitized version available from the University of Strasbourg).
3. August Beer: Determination of the absorption of red light in colored liquids . In: Annals of Physics and Chemistry . tape 86 , first piece, 1852, p. 78-88 ( digitized on Gallica ).
4. Lothar Matter (Ed.): Food and environmental analysis with spectrometry. Tips, tricks and examples for practice . VCH, Weinheim et al. 1995, ISBN 3-527-28751-5 .
5. Thomas G. Mayerhöfer, Jürgen Popp: Beer's law derived from electromagnetic theory . In: Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy . tape 215 , May 15, 2019, ISSN  1386-1425 , p. 345–347 , doi : 10.1016 / j.saa.2019.02.103 .
6. Thomas G. Mayerhöfer, Jürgen Popp: Beer's Law - Why Absorbance Depends (Almost) Linearly on Concentration . In: ChemPhysChem . tape 20 , no. 4 , February 18, 2019, p. 511-515 , doi : 10.1002 / cphc.201801073 .
7. Thomas Günter Mayerhöfer, Andrei Pipa, Jürgen Popp: Beer's law - why integrated absorbance depends linearly on concentration . In: ChemPhysChem . September 23, 2019, ISSN  1439-4235 , p. 2748-2753 , doi : 10.1002 / cphc.201900787 .
8. Thomas G. Mayerhöfer, Alicja Dabrowska, Andreas Schwaighofer, Bernhard Lendl, Jürgen Popp: Beyond Beer's Law: Why the Index of Refraction Depends (Almost) Linearly on Concentration . In: ChemPhysChem . tape 21 , no. 8 , April 20, 2020, ISSN  1439-4235 , p. 707-711 , doi : 10.1002 / cphc.202000018 .
9. ^ Thomas Günter Mayerhöfer, Jürgen Popp: Beyond Beer's law: Revisiting the Lorentz-Lorenz equation . In: ChemPhysChem . n / a, n / a, May 12, 2020, ISSN  1439-4235 , doi : 10.1002 / cphc.202000301 .
10. Jürgen Popp, Sonja Höfer, Thomas G. Mayerhöfer: Deviations from Beer's law on the microscale - nonadditivity of absorption cross sections . In: Physical Chemistry Chemical Physics . tape 21 , no. 19 , May 15, 2019, ISSN  1463-9084 , p. 9793-9801 , doi : 10.1039 / C9CP01987A .
11. ^ Thomas Günter Mayerhöfer, Jürgen Popp: Beer's law - why absorbance depends (almost) linearly on concentration . In: ChemPhysChem . tape 0 , yes, ISSN  1439-7641 , doi : 10.1002 / cphc.201801073 .
12. Thomas G. Mayerhöfer, Jürgen Popp: The electric field standing wave effect in infrared transflection spectroscopy . In: Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy . tape 191 , February 15, 2018, ISSN  1386-1425 , p. 283-289 , doi : 10.1016 / j.saa.2017.10.033 .
13. ^ Thomas G. Mayerhöfer, Harald Mutschke, Jürgen Popp: The Electric Field Standing Wave Effect in Infrared Transmission Spectroscopy . In: ChemPhysChem . tape 18 , no. 20 , 2017, ISSN  1439-7641 , p. 2916-2923 , doi : 10.1002 / cphc.201700688 .
14. ^ A b Thomas G. Mayerhöfer, Harald Mutschke, Jürgen Popp: Employing Theories Far beyond Their Limits — The Case of the (Boguer) Beer-Lambert Law . In: ChemPhysChem . tape 17 , no. 13 , ISSN  1439-7641 , p. 1948–1955 , doi : 10.1002 / cphc.201600114 .
15. A detailed discussion of the incompatibility of the Lambert-Beer law with Maxwell's equations is offered in the review: Thomas Günter Mayerhöfer, Susanne Pahlow, Jürgen Popp: The Bouguer-Beer-Lambert law: Shining light on the obscure . In: ChemPhysChem . n / a, n / a, July 14, 2020, ISSN  1439-4235 , doi : 10.1002 / cphc.202000464 .
16. Avinash C. Kak, Malcolm Slaney: Principles of Computerized Tomographic Imaging