Beer–Lambert law

Lambert-Beer law or Bouguer - Lambert-Beer 's law describes the attenuation of the intensity of radiation in passing through a medium with an absorbent substance, depending on the concentration of the absorbing substance, and the layer thickness. The law forms the basis of modern photometry as an analytical method. It is a special case of the radiation transport equation without emission term.

History

The Bouguer - Lambertian law was formulated by Pierre Bouguer before 1729 and describes the attenuation of the radiation intensity with path length as it passes through an absorbing substance. It is also Johann Heinrich Lambert attributed, sometimes even referred to briefly as lambert 's law, although Lambert himself Bouguers work " Essai d' optique sur la gradation de la lumière " in his " Photometria " (1760 ) cites and even quotes from it.

As lambert 's law and the Lambertian cosine law is referred to.

In 1852 August Beer extended the Bouguer - Lambertian law by asking the concentration of Absorbanten depending on the transmitted light. This relationship is called the lambert -beer 's law or rarer than Bouguer - Lambert - Beer 's law.

The law

The absorbance ( the absorbance of the material for the light of the wavelength ) is given by

With

  • : Intensity of transmitted light (unit: W · m-2)
  • : Intensity of the incident ( incident ) light (unit: W · m-2)
  • : Molar concentration of the absorbing substance in the liquid ( unit: mol · l -1)
  • : Decadic extinction coefficient (often referred to as a spectral absorption coefficient) at the wavelength. This is a specific size for the absorbing substance, and can recognize the pH value or the solvent dependent. At a concentration specified in moles is expressed as a decadic molar extinction coefficient, for example, in the unit m2 · mol -1
  • : Thickness of the irradiated body ( unit: m)

Derivation

The differential loss due to absorption of the radiation intensity is proportional to the intensity, the extinction coefficient, of the molar concentration of the absorbing substance and its differential layer thickness:

After the integration constant was determined for the. This yields the falling exponential function with which the decrease of the light intensity when traversing a sample solution can be described by the concentration of:

Rearranging the equation gives:

The absorbance and the extinction coefficient are, however, not defined by the natural logarithm. Since the decadal and the natural logarithm linearly related, corresponds to the transition to a constant factor in the equation. This is simply added to the equation: Off is.

Here is the decadic molar extinction coefficient.

With the power rule of logarithms results in the usual notation:

The absorbance (as the refractive index of a substance ) of the wavelength of the incident light depending (see dispersion) in the formulation of the wavelength dependence of the extinction coefficient k ( imaginary part of complex refractive index ):

Validity

The Act applies to:

  • Homogeneous distribution of the absorbing substance;
  • Negligible multiple scattering (especially for clear media);
  • Negligible variation of the absorption coefficient in the measured spectral range (especially for monochromatic radiation );
  • Negligible intrinsic emission ( the transmitted radiation intensity must be significantly higher than the (especially thermal ) Self- radiation);
  • Low concentrated solutions (usually less than 0.01 mol · l -1) ( at high concentrations interactions lead to deviations ).

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 by photometric measurements. Malachite Green is one of the most intense dyes with a molar absorption coefficient of 8.07 x 104 L mol -1cm -1 ( 622 nm, ethanol).

Radiation damping generally

The same law applies generally to the decrease in the intensity of in -damping materials propagating electromagnetic radiation. It describes the attenuation of optical radiation in fiber-optic cables (FO ) or in absorbing optical media or the attenuation of gamma or X- radiation in matter. Conversely, with this context, with knowledge of both intensities thickness measurement carried out.

The light passing through a medium to the radiant power length is:

With

  • : Incoming power
  • : Absorption coefficient in m-1
  • : Material thickness or length in meters.

It is often highly dependent on the wavelength and the material.

Fiber Optic

For the silicate glass used in long-haul optical fibers decreases between the visible range of 0.6 microns to about 1.6 microns with the fourth power of the wavelength; Then, at this point, a sharp increase in attenuation due to a resonance of the glass material. Another attenuation lies in the ultraviolet range. Hydroxide ions in the glass, which can indeed be avoided by special production technologies studied selective damping cause increase at about 1.4 microns ( infrared spectroscopy). The attenuation values ​​for the plastic fibers used in fiber-optic short distances are higher and are also strong material and wavelength dependent, earmarked in the visible range at its lowest.

In place of the above-mentioned notation in the signal transmission technique, the representation

Used ( the attenuation in dB / km and the length of the fiber-optic cable in km ), because, in communications throughout the ratio of ( electrical as well as optical ) services in decimal logarithmic measure dB ( decibels ) is given:

Remote Sensing / atmosphere

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

Where stands for the atmospheric mass and the optical thicknesses of the material contained. In the example stands for:

  • : Aerosols ( absorbing and scattering )
  • : Homogeneous gases such as carbon dioxide and molecular oxygen (excluding absorbing)
  • : Nitrogen dioxide ( absorbing)
  • : Water vapor ( absorbing)
  • : Ozone ( absorbing)
  • : Rayleigh scattering by molecular oxygen and nitrogen ( blue sky )

The determination of need for the correction of satellite imagery and example of interest in climate observation.

Computed tomography

In computed the attenuation of the x-ray radiation by the Lambert-Beer law will be described. The extinction coefficient (absorption coefficient) is a function of position, i.e., varied within the object ( the patient ), and takes, for example, bone has a larger value than in the lungs. Therefore, the measured intensity of the X-ray results from the following integral:

Wherein the light emitted from the X-ray tube and radiation intensity is the integral of the parameterized path of the beam. The computed tomographic image then provides the function is (see Hounsfield scale ) as a grayscale image. The object of the reconstruction, it is therefore to be determined from the measured intensities of the absorption coefficient, i.e., to solve the related inverse problem.

Radiative transfer Optical depth Medical imaging Gerd Wedler
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