Beer–Lambert_law - Wikiparency

# Beer–Lambert law The **Beer–Lambert law**, also written as Beer's law or the Bouguer–Lambert–Beer law, states that the [[Absorbance|absorbance]] of a sample is proportional to the concentration of the absorbing species and to the length of the path the light travels through it: `A = ε·l·c`, where ε is the [[Molar_absorption_coefficient|molar absorption coefficient]] of the species at the wavelength used, *l* the path length and *c* the concentration.[^iupac-goldbook] Because absorbance is the negative decadic logarithm of the transmitted fraction, `A = −log10(I/I0)`, the law says that the transmitted intensity falls exponentially with concentration and with thickness while the absorbance rises linearly.[^openstax-ch6] The law carries three names because it was found three times: Pierre Bouguer measured the exponential dimming of light with thickness in 1729, Johann Heinrich Lambert restated it in 1760, and August Beer showed in 1852 that the same exponent depends on the concentration of a dissolved absorber.[^bouguer1729][^lambert1760][^beer1852] On the [[Chemistry]] flagship the law is the third section of Part XI, *Chemical laws*, under its own heading, following [[Graham's_law]] and preceding the analytical practice of Part XIII, where [[Mass_spectrometry]] and [[Spectrophotometry]] put it to work. Its spine neighbours are the absorbance and molar-absorption-coefficient pages and [[Ultraviolet–visible_spectroscopy]], the technique built on it. In the microsim below the reader sends a beam through a cuvette and slides two things: the concentration of the coloured solute and the path length, from 0.5 cm to 5 cm. The transmitted beam dims exponentially, following `I = I0·10^(−A)`, while the absorbance readout climbs in a straight line, following `A = ε·l·c`; the two displays side by side are the whole content of the law. A calibration panel plots three standards of known concentration, fits a line through them, and reads an unknown back from its absorbance, which is how the law is used every day in an [[Analytical_chemistry|analytical]] laboratory. ## History Bouguer's *Essai d'optique sur la gradation de la lumière*, published in Paris in 1729, reported photometric experiments in which he compared the brightness of light after passing through different thicknesses of absorbing media, [[Glass|glass]] plates and the air itself among them, and concluded that equal thicknesses absorb equal fractions, so that the intensity falls geometrically as the thickness grows arithmetically.[^bouguer1729] That is the exponential law of attenuation, and it is why astronomers still call the correction for atmospheric extinction Bouguer's law. Lambert's *Photometria*, published in Augsburg in 1760, restated the relation in the systematic form that gave photometry its vocabulary, and the dependence of absorbance on thickness has carried his name since.[^lambert1760] Neither man was concerned with solutions. In 1852 August Beer, working in Bonn, measured the absorption of red light by coloured solutions and found that the attenuation depends on the product of concentration and path length: doubling the concentration of the absorber has the same effect as doubling the thickness of the cell.[^beer1852] Beer's contribution turned a law of optics into a tool of chemistry, because it meant that a measurement of light could stand in for a measurement of amount. The combined statement in terms of a logarithmic absorbance that is linear in both variables is the modern form. The International Union of Pure and Applied Chemistry distinguishes the attenuance, which counts every loss from the beam including scattering, from the absorbance, which counts absorption alone; the two coincide for a clear solution, which is the case the law is usually applied to.[^iupac-goldbook] ## Mathematical formulations The law is written in two equivalent conventions. The decadic form used in chemistry defines the transmittance `T = I/I0` as the ratio of transmitted to incident intensity and the absorbance as `A = −log10 T`, and states `A = ε·l·c`, with ε in litres per mole per centimetre when *c* is in moles per litre and *l* in centimetres.[^openstax-ch6][^iupac-goldbook] The Napierian form used in physics writes the intensity directly, `I = I0·exp(−μ·l)`, where μ is the linear attenuation coefficient of the medium in reciprocal length; the two are related by `μ = ln(10)·ε·c`, so a decadic absorbance of 1 corresponds to a Napierian optical depth of 2.303.[^iupac-goldbook] In either form the law describes a property of the beam, its fractional loss per unit length, that does not depend on how bright the beam is; the intensity along the path is an [[Exponential_decay|exponential decay]] in distance. That independence is what makes the law linear in concentration and is also what fails first when it fails. ### Formulation For a single absorber the working equation is `A = ε·l·c`, and for several absorbers that do not interact the absorbances add, `A = l·Σ εᵢ·cᵢ`, because each species removes its own fraction of whatever light reaches it.[^openstax-ch6] The microsim computes both lines of the display from these equations. With a molar absorption coefficient of 1.0 × 10⁴ L mol⁻¹ cm⁻¹, a value typical of a strongly coloured dye, a solution at 5.0 × 10⁻⁵ mol/L in a 1.00 cm cuvette has A = 0.50 and transmits 10^(−0.50) = 31.6 % of the light; in a 2.00 cm cell the absorbance is 1.00 and the transmittance 10 %; in a 5.00 cm cell the absorbance is 2.50 and only 0.32 % of the light gets through (ILLUSTRATIVE arithmetic, not a measured system). Doubling the concentration at fixed path length has exactly the effect of doubling the path length at fixed concentration, which is Beer's 1852 result restated.[^beer1852] The absorbance readout in the microsim therefore moves in a straight line as either slider moves, while the beam on screen dims by a constant factor for every equal step, and the reader can watch the last few percent of light vanish while the absorbance is still climbing steadily. ## Derivation The law follows from one assumption: a thin layer of the medium removes a fraction of the light that is proportional to the layer's thickness and independent of the intensity. Let a beam of [[Photon|photons]] with intensity *I* cross a slab of thickness d*z* containing absorbers at number density *n*, each presenting an absorption cross-section σ to the beam. The fraction of the beam intercepted is the fraction of the slab's area covered by cross-sections, `n·σ·dz`, so `dI = −n·σ·I·dz`. Dividing by *I* and integrating from the entrance face to a depth *l* gives `ln(I/I0) = −n·σ·l`, or `I = I0·exp(−n·σ·l)`; the product `n·σ` is the Napierian attenuation coefficient μ.[^hutchinson2002] Converting to base-ten logarithms and from number density to molar concentration gives the chemist's form, with ε proportional to σ times the [[Avogadro_constant|Avogadro constant]]. The derivation makes the assumptions explicit: the absorbers act independently of one another and of the beam, the light is monochromatic so that a single σ applies, the medium is homogeneous so that *n* is constant along the path, and nothing is scattered into or out of the beam or emitted along it. Each of these is a place the law can be broken, and the section on validity takes them in turn. The intensity itself is the energy flux of the [[Electromagnetic_radiation|electromagnetic wave]], proportional to the square of its field amplitude, which is why the law is stated for intensity and not for amplitude.[^uphys-ch16] ### Validity The law holds well for dilute solutions of a single absorber measured with nearly monochromatic light at moderate absorbance, and every departure from those conditions produces a curved calibration line. At concentrations above roughly 0.01 mol/L the absorbers begin to interact, the refractive index of the solution changes, and the effective ε drifts with concentration.[^skoog2018] Instrumental deviations arise when the light is not monochromatic, since the absorbers remove the strongly absorbed wavelengths first and the surviving light is one the sample absorbs less, and when stray light reaches the detector without passing through the sample, which puts a floor under the transmittance; both make high absorbances read low.[^skoog2018] Chemical deviations arise when the absorbing species takes part in an equilibrium, such as an acid–base or dimerization equilibrium, so that its concentration is not proportional to the amount added.[^skoog2018] Scattering by turbid samples removes light without absorbing it, and fluorescence adds light at the detector. For these reasons analysts work in the middle of the range, at absorbances of a few tenths to about one, where the relative error in concentration is smallest; the microsim's 5 cm setting, which drives the transmittance below one percent, is there to show the top of the range where a real instrument stops being reliable. The law also assumes the sample does not change under the beam; photochemical bleaching breaks it in a different way, by making *c* itself a function of time. ## Applications The law is applied wherever light is used to count something. In chemistry it is the basis of [[Spectrophotometry|spectrophotometry]] and of the [[Absorption_spectroscopy|absorption spectroscopies]] across the [[Electromagnetic_spectrum|electromagnetic spectrum]] from the ultraviolet through the [[Infrared_spectroscopy|infrared]]; in physics it is the working equation of beam attenuation in gases and plasmas; in astronomy it is the correction that turns a measurement made through the atmosphere into one made above it. In each case the measured quantity is a transmitted intensity and the wanted quantity is a column density, the product of concentration and path length, and the law is the conversion between them. ### In plasma physics A probe beam sent through a [[Plasma_(physics)|plasma]] is attenuated by the same exponential, with the cross-section now standing for whatever process removes beam particles or photons: photoionization or line absorption for a light beam, ionization and charge exchange for a beam of neutral atoms. The optical depth `τ = ∫ n·σ·dz` along the line of sight measures how much of the beam survives, and a plasma is called optically thin when τ ≪ 1, so that a line of sight samples the whole path, and optically thick when τ ≫ 1, so that only the outer layer is seen.[^hutchinson2002] Beam attenuation is used in the other direction as a diagnostic: because the survival fraction depends on the integral of the density along the path, a measured attenuation gives the line-averaged density, and in a [[Magnetic_confinement_fusion|magnetically confined]] plasma the deposition profile of a heating beam is computed from exactly this integral.[^hutchinson2002] The [[Laser|laser]] absorption measurements used to follow atomic species in discharges are Beer's law applied with a cross-section rather than a molar coefficient. ### Chemical analysis by spectrophotometry The everyday use of the law is the calibration line. An analyst prepares standards of known concentration, measures the absorbance of each at a wavelength where the analyte absorbs strongly, usually the peak of its spectrum, and fits a straight line; the concentration of an unknown is then read from its absorbance on that line.[^openstax-ch6][^averill-ch13] The microsim's calibration panel does this with three standards. In its ILLUSTRATIVE preset, standards at 1.0, 2.0 and 3.0 × 10⁻⁵ mol/L read absorbances of 0.20, 0.40 and 0.60 in a 1.00 cm cell, so the fitted slope is ε·l = 2.0 × 10⁴ L mol⁻¹, and an unknown that reads 0.50 is at 2.5 × 10⁻⁵ mol/L. A real calibration carries an intercept from the blank, a [[Least_squares|least-squares]] fit rather than a line through the origin, and standards bracketing the unknown so that no extrapolation is needed. The same measurement, repeated in time, gives a [[Reaction_rate|reaction rate]]: if one species in a reaction absorbs and the others do not, its concentration can be followed continuously through its absorbance, and the Portal Book *General Chemistry* uses the law in that way when it introduces [[Chemical_kinetics|kinetics]].[^averill-ch13] A [[Molar_concentration|molar concentration]] read from an absorbance is only as good as the ε used, so the coefficient is measured on the same instrument rather than taken from a table when accuracy matters. ### In-atmosphere astronomy Light from a [[Star|star]] is attenuated by the [[Atmosphere_of_Earth|atmosphere]] according to Bouguer's original law, and the attenuation grows with the length of the atmospheric path, which for a star at zenith angle *z* is approximately sec *z* times the vertical path and is called the air mass *X*. In the magnitude system, which is logarithmic, the observed magnitude is `m = m0 + k·X`, where m0 is the magnitude the star would have above the atmosphere and *k* is the extinction coefficient in magnitudes per air mass at the wavelength observed.[^hardie1962] Observing a standard star at several zenith angles through the night gives *k* as the slope of *m* against *X*, and the intercept at X = 0 is the extra-atmospheric magnitude; the procedure is the astronomer's calibration line, with air mass playing the part of path length.[^hardie1962] The extinction has a molecular part from Rayleigh scattering, which rises steeply toward the blue, an aerosol part that varies with the weather, and an absorption part from [[Ozone|ozone]] and water vapour, and it is why the same star is redder and fainter near the horizon than overhead. [[Sunlight|Sunlight]] at sunset has crossed many times the zenith air mass, and its colour is Beer's law seen with the naked eye. ## See also - [[Absorbance]] - [[Spectrophotometry]] - [[Ultraviolet–visible_spectroscopy]] - [[Molar_absorption_coefficient]] - [[Absorption_spectroscopy]] - [[Exponential_decay]] - [[Graham's_law]] - [[Mass_spectrometry]] ## References [^iupac-goldbook]: International Union of Pure and Applied Chemistry. *Compendium of Chemical Terminology* (the "Gold Book"), online edition; entries "Beer–Lambert law", "absorbance" and "attenuation". https://goldbook.iupac.org/ [^openstax-ch6]: Flowers, P.; Neth, E.; Robinson, W.; et al. (2019). *Chemistry: Atoms First*, 2nd ed. OpenStax. Chapter 6, "Composition of Substances and Solutions", pp. 283–312 (the spectrophotometry feature and Beer's law; page to pin). https://open.umn.edu/opentextbooks/textbooks/chemistry-atoms-first [^averill-ch13]: Averill, B.; Eldredge, P. (2011). *General Chemistry: Principles, Patterns, and Applications*. Index chapter 13 (pp. 1171–2366 as indexed; the kinetics discussion that uses Beer's law to follow concentration; page to pin). https://open.umn.edu/opentextbooks/textbooks/general-chemistry-principles-patterns-and-applications [^uphys-ch16]: Sanny, J.; Ling, S. (2016). *University Physics Volume 2*. OpenStax. Chapter 16, "Electromagnetic Waves", pp. 669–710 (intensity of an electromagnetic wave; page to pin). https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-2 [^bouguer1729]: Bouguer, P. (1729). *Essai d'optique sur la gradation de la lumière*. Paris: Claude Jombert. [^lambert1760]: Lambert, J. H. (1760). *Photometria sive de mensura et gradibus luminis, colorum et umbrae*. Augsburg: Eberhardt Klett. [^beer1852]: Beer, A. (1852). "Bestimmung der Absorption des rothen Lichts in farbigen Flüssigkeiten." *Annalen der Physik und Chemie*, vol. 86. [^skoog2018]: Skoog, D. A.; Holler, F. J.; Crouch, S. R. (2018). *Principles of Instrumental Analysis*, 7th ed. Boston: Cengage. Chapter on the introduction to ultraviolet–visible molecular absorption spectrometry (limits to Beer's law; chapter and page to pin). [^hutchinson2002]: Hutchinson, I. H. (2002). *Principles of Plasma Diagnostics*, 2nd ed. Cambridge: Cambridge University Press (optical depth, beam attenuation and line-integrated density; chapter and page to pin). [^hardie1962]: Hardie, R. H. (1962). "Photoelectric reductions." In Hiltner, W. A. (ed.), *Astronomical Techniques* (Stars and Stellar Systems, vol. II). Chicago: University of Chicago Press. ## External links - [IUPAC Gold Book](https://goldbook.iupac.org/), the source of the recommended definitions of absorbance and attenuation - [Open Textbook Library record for *Chemistry: Atoms First*](https://open.umn.edu/opentextbooks/textbooks/chemistry-atoms-first) - The Wikipedia pair's external links section lists the remaining calculators and teaching resources.  **Microsim — three.js (Wikitube framework), pending deploy:** *Beer–Lambert law* will play here once `https://wikitube-3d-microsims.netlify.app/matter/Beer–Lambert_law.html` is live.   ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Beer–Lambert_law) : [Wikitube](https://en.wikitube.io/wiki/Beer–Lambert_law) · pinned revision [1372078887](https://en.wikipedia.org/w/index.php?oldid=1372078887) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Chemistry]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Chemistry row K53 · sim pending (matter/Beer–Lambert_law).*