Lambert-Beer law

Lambert-Beer law
Lambert-Beer law

The Lambert-Beer law correlates the quantity of light absorbed by a substance with its concentration, with its chemical nature, and with the thickness of the medium crossed. When a substance is subjected to the action of electromagnetic radiation, the radiant energy contained in it can be absorbed by molecules to be transformed into other forms of energy. The following cases generally occur:

1) The absorbed energy hν increases the kinetic energy of translation of the molecules in space (½ mv 2 ); since the latter depends on the mass of the molecule, molecules having different masses absorb different amounts of energy due to their translational motions

2) The absorbed energy hν increases the kinetic energy of rotation of the molecules around their axis; this energy is related to the moment of inertia (½ I ω 2 ), so molecules with different moments of inertia will absorb different amounts of energy for their rotary motions

3) In multi-atomic molecules the absorbed energy hν can be transformed into vibration energy of the atoms, which depends on the atomic masses, their distance, the bond strength, etc. Therefore, the amount of energy absorbed is also different depending on the molecule

4) The energy hν can be such as to cause the quantum jump of an electron from one orbit to another of higher energy. Even for quantum leaps, the energy involved is different depending on the type of atom and the way in which the atom is linked to the molecule

Energy absorption

Energy absorption is therefore closely related to the molecular structure, and the amount of energy absorbed is characteristic of the molecule. But since the Planck equation E = hν correlates the energy and frequency of the radiation, it can be deduced that the absorbed frequencies are also characteristics of the molecules themselves. The energy absorbed for translational and rotational motions is very small, and the corresponding frequencies lie in the range of radio waves and far infrared, respectively. Vibratory motions require higher energies, which correspond to medium and near infrared frequencies. Electronic jumps, on the other hand, absorb even greater energies whose frequencies correspond to the visible and ultraviolet ranges.

When light radiation hits a transparent medium, it is partly reflected and partly refracted in the medium. The intensity of this fraction decreases as the radiation propagates, so it will be lower at the exit. In the figure:


o is the intensity of the incident light while I is the intensity of the escaping light. The I/I ratio is defined as the overall transmittance of the medium.


The decimal logarithm of the reciprocal of transmittance is called extinction or absorbance:

A = log 1/T = log I or /I.

The fundamental law of sorption analysis, known as the Lambert-Beer law, establishes a relationship between extinction and the concentration of the dissolved substance and is the basis of quantitative chemical analysis.

The Lambert-Beer law establishes a direct proportionality between the absorbance and the concentration of the absorbing species:

A = log I or /I = abc


  • a is a constant of proportionality called specific absorbance
  • b is the thickness of the solution crossed by the radiation expressed in centimeters
  • c the concentration of the absorbent substance in the solution.

If the latter is expressed in mol/L, the specific absorbance is called the molar extinction coefficient , ε and, consequently, A = ε bc

If linearity between absorbance and concentration can be demonstrated, a quantitative analysis method is available for ultraviolet and visible absorption. Linearity usually occurs in narrow concentration ranges. The linear proportionality relationship between A and c, in reality, no longer occurs as the concentration increases; in fact, deviations from the Lambert-Beer law can occur with poor reliability of the analytical data as can be seen in the figure:



Therefore this law is valid for diluted solutions: once the molar extinction coefficient and the thickness of the solution are known, once the absorbance has been determined instrumentally, the concentration of the solution can be obtained.
Usually we proceed with the construction of a calibration line. The absorbance of solutions of known titre is measured and these experimental points are plotted with the concentration on the abscissa and the absorbance on the ordinate. If the solution is diluted, these points align along a straight line which constitutes the calibration straight line, also known as the working straight line. The concentration of an unknown solution is determined by interpolation knowing its absorbance.


The device used to determine absorbance is the spectrophotometer


which, in its essential lines, consists of:

  • Light source. To cover the entire measurement range from 190 nm to 900 nm there are two lamps, one with a tungsten filament for the visible light, the other with a deuterium filament for the UV. The system decides which one to use depending on the wavelength which you work on. The selection occurs via a movable mirror with two positions, which directs the light to wards an exit slit
  • Monochromator. The light is then monochromatized through a diffraction grating and a slit, of selectable width. The narrower the slit, the greater the selectivity in λ but the smaller the amount of light that can be used in the measurement
  • Chopper. The chopper has three sectors: a mirror reflects the light, a black one absorbs it, and a perforated one transmits it. By keeping the chopper rotating, the light is sent alternately through the sample compartment, through the reference, or is suppressed to measure the dark current of the phototube used as a detector.
  • Compartment for samples and reference. They can each accommodate a 1 cm sided cuvette. The sample temperature is controllable using an external thermostated bath.
  • Detector: it is a photomultiplier. The photosensitive part is a cathode whose surface is covered with an alloy containing various alkali metals, to obtain an extended sensitivity range from 185 to 900 nm

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