6388:
5981:
36:
6383:{\displaystyle {\begin{aligned}T&=\exp \left(-\sum _{i=1}^{N}{\frac {\ln(10)}{\mathrm {N_{A}} }}\varepsilon _{i}\int _{0}^{\ell }n_{i}(z)\mathrm {d} z\right)\\&=\exp \left(-\sum _{i=1}^{N}\varepsilon _{i}\int _{0}^{\ell }{\frac {n_{i}(z)}{\mathrm {N_{A}} }}\mathrm {d} z\right)^{\ln(10)}\\&=10^{\;\!\wedge }\!\!\left(-\sum _{i=1}^{N}\varepsilon _{i}\int _{0}^{\ell }c_{i}(z)\mathrm {d} z\right).\end{aligned}}}
2253:
1982:
4834:
3681:
7005:
5587:
3852:
1987:
3016:
1747:
684:) of wavelengths commonly used in analytical spectroscopy (except upon entry and exit). The attenuation of a beam of light within a solution is assumed to be only due to absorption. In order to approximate the conditions required for the Beer Lambert law to hold, often the intensity of transmitted light through a reference sample
4611:
6446:
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
497:
relation. In the introduction to his classic paper, he wrote: "The absorption of light during the irradiation of a colored substance has often been the object of experiment; but attention has always been directed to the relative diminution of the various colors or, in the case of crystalline bodies,
1295:
The fundamental law of extinction states that the extinction process is linear in the intensity of radiation and amount of radiatively active matter, provided that the physical state is held constant. (Neither concentration or length are fundamental parameters.) There are two factors that determine
502:
through an absorbing medium, there is no information available." By studying absorption of red light in colored aqueous solutions of various salts, he concluded that "the transmittance of a concentrated solution can be derived from a measurement of the transmittance of a dilute solution". It is
3417:
6824:
3331:
are closer to each other interactions can set in. These interactions can be roughly divided into physical and chemical interactions. Physical interaction do not alter the polarizability of the molecules as long as the interaction is not so strong that light and molecular quantum state intermix
5398:
3986:
3326:
processes can also cause variances. The main reason, however, is that the concentration dependence is in general non-linear and Beer's law is valid only under certain conditions as shown by derivation below. For strong oscillators and at high concentrations the deviations are stronger. If the
556:
for double this thickness." Furthermore, Beer stated: "We shall take the absorption coefficient to be the coefficient giving the diminution in amplitude suffered by a light ray as it passes through a unit length of an absorbing material. We then have, according to theory, and as I have found
3301:
3159:
2761:
3686:
2839:
1608:, and extinction cross-sections are often used. The fraction of light extinguished by the sample may be described by the extinction cross section (fraction extinguished per particle). the number of particles in a unit distance and the distance in those units. For example:
5329:
5212:
2457:
4517:
2248:{\displaystyle {\begin{aligned}\tau &=\sum _{i=1}^{N}\tau _{i}=\sum _{i=1}^{N}\sigma _{i}\int _{0}^{\ell }n_{i}(z)\,\mathrm {d} z,\\A&=\sum _{i=1}^{N}A_{i}=\sum _{i=1}^{N}\varepsilon _{i}\int _{0}^{\ell }c_{i}(z)\,\mathrm {d} z,\end{aligned}}}
1977:{\displaystyle {\begin{aligned}T&=\exp \left(-\sum _{i=1}^{N}\sigma _{i}\int _{0}^{\ell }n_{i}(z)\mathrm {d} z\right)\\&=10^{\;\!\wedge }\!\!\left(-\sum _{i=1}^{N}\varepsilon _{i}\int _{0}^{\ell }c_{i}(z)\mathrm {d} z\right),\end{aligned}}}
5815:
1589:] has the area in both the numerator and denominator, the beam area cancels in the calculation of the absorbance. The units of the absorptivity must match the units in which the sample is described. For example, if the sample is described by
6777:
The
Bouguer–Lambert law may be 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 optical depth for a slant path is
6678:
3857:
4132:
3163:
3021:
2623:
4400:
2618:
5892:
4606:
4829:{\displaystyle {\frac {\mathrm {d} \Phi _{\mathrm {e} }(z)}{\mathrm {d} z}}\,\exp \left(\int _{0}^{z}\mu (z')\mathrm {d} z'\right)+\mu (z)\Phi _{\mathrm {e} }(z)\,\exp \left(\int _{0}^{z}\mu (z')\mathrm {d} z'\right)=0,}
6443:, or have at least a width that is narrower than that of the attenuating transition. Otherwise a spectrometer as detector for the power is needed instead of a photodiode which cannot discriminate between wavelengths.
6551:
5101:
3676:{\displaystyle {\begin{aligned}\mu (z)&=\sum _{i=1}^{N}\mu _{i}(z)=\sum _{i=1}^{N}\sigma _{i}n_{i}(z),\\\mu _{10}(z)&=\sum _{i=1}^{N}\mu _{10,i}(z)=\sum _{i=1}^{N}\varepsilon _{i}c_{i}(z)\end{aligned}}}
7000:{\displaystyle T=\exp {\big (}-m(\tau _{\mathrm {a} }+\tau _{\mathrm {g} }+\tau _{\mathrm {RS} }+\tau _{\mathrm {NO_{2}} }+\tau _{\mathrm {w} }+\tau _{\mathrm {O_{3}} }+\tau _{\mathrm {r} }+\cdots ){\bigr )},}
5217:
5986:
5582:{\displaystyle {\begin{aligned}T&=\exp \left(-\int _{0}^{\ell }\ln(10)\,\mu _{10}(z)\mathrm {d} z\right)\\&=10^{\;\!\wedge }\!\!\left(-\int _{0}^{\ell }\mu _{10}(z)\mathrm {d} z\right).\end{aligned}}}
5106:
5043:
2362:
233:
in 1760. Lambert expressed the law, which states that the loss of light intensity when it propagates in a medium is directly proportional to intensity and path length, in the mathematical form used today.
5687:
4972:
904:
Bouguer looked at astronomical phenomena where the size of a detector is very small compared to the distance traveled by the light. In this case, any light that is scattered by a particle, either in the
5976:
900:
168:
The fundamental law of extinction (the process is linear in the intensity of radiation and amount of radiatively active matter, provided that the physical state is held constant) is sometimes called the
5403:
4068:
3862:
3691:
3422:
3168:
3026:
2844:
2628:
1992:
1752:
821:
374:
2819:
1694:
1185:
4058:
1296:
the degree to which a medium containing particles will attenuate a light beam: the number of particles encountered by the light beam, and the degree to which each particle extinguishes the light.
1084:
4421:
3847:{\displaystyle {\begin{aligned}T&=\exp \left(-\int _{0}^{\ell }\mu (z)\mathrm {d} z\right)\\&=10^{\;\!\wedge }\!\!\left(-\int _{0}^{\ell }\mu _{10}(z)\mathrm {d} z\right),\end{aligned}}}
6726:
to determine the two amount concentrations from measurements made at more than two wavelengths. Mixtures containing more than two components can be analyzed in the same way, using a minimum of
5393:
287:
3412:
3011:{\displaystyle {\begin{aligned}T&=\exp \left(-\ell \sum _{i=1}^{N}\sigma _{i}n_{i}\right)\\&=10^{\;\!\wedge }\!\!\left(-\ell \sum _{i=1}^{N}\varepsilon _{i}c_{i}\right),\end{aligned}}}
3332:(strong coupling), but cause the attenuation cross sections to be non-additive via electromagnetic coupling. Chemical interactions in contrast change the polarizability and thus absorption.
1476:
5702:
661:
While the observations of
Bouguer and Beer have a similar form in the Beer–Lambert law, their areas of observation were very different. For both experimenters, the incident beam was well
7213:
6580:
1616:
A common and practical expression of the Beer–Lambert law relates the optical attenuation of a physical material containing a single attenuating species of uniform concentration to the
1554:
There must be a large number of particles that are uniformly distributed for this relationship to hold. In practice, the beam area is thought of as a constant, and since the fraction [
2537:
2494:
1231:
4212:
653:
of the attenuating species and the thickness of the material sample. An early, possibly the first, modern formulation was given by Robert Luther and
Andreas Nikolopulos in 1913.
433:
7866:
4063:
485:
588:
1505:
498:
the relation between the absorption and the direction of polarization. Concerning the absolute magnitude of the absorption that a particular ray of light suffers during its
554:
637:
Beer–Lambert law: The modern formulation of the Beer–Lambert law combines the observations of
Bouguer and Beer into the mathematical form of Lambert. It correlates the
4264:
527:
2546:
1317:
1285:
1258:
4232:
1587:
1549:
748:
715:
1397:
uses concentration and length in order to determine the number of particles the beam encounters. For a collimated beam (directed radiation) of cross-sectional area
1435:
973:
946:
4285:
3362:
1104:
913:
direction, will not strike the detector. The loss of intensity to the detector will be due to both absorption and scatter. Consequently, the total loss is called
608:
4154:
1529:
1415:
1013:
993:
628:
1395:
6491:
4314:
4533:
1110:
or coefficient. The amount of light transmitted is falling off exponentially with distance. Taking the natural logarithm in the above equation, we get:
7135:
the selection of the attenuators which have to be considered depends on the wavelength range and can include various other compounds. This can include
3981:{\displaystyle {\begin{aligned}\tau &=\int _{0}^{\ell }\mu (z)\,\mathrm {d} z,\\A&=\int _{0}^{\ell }\mu _{10}(z)\,\mathrm {d} z.\end{aligned}}}
5825:
7695:
4843:
4300:
as an axis parallel to the direction of the beam. Divide the material sample into thin slices, perpendicular to the beam of light, with thickness
3296:{\displaystyle {\begin{aligned}\tau &=\ell \sum _{i=1}^{N}\sigma _{i}n_{i},\\A&=\ell \sum _{i=1}^{N}\varepsilon _{i}c_{i}.\end{aligned}}}
3154:{\displaystyle {\begin{aligned}\tau &=\ell \sum _{i=1}^{N}\sigma _{i}n_{i},\\A&=\ell \sum _{i=1}^{N}\varepsilon _{i}c_{i}.\end{aligned}}}
2756:{\displaystyle {\begin{aligned}\tau &=\ell \sum _{i=1}^{N}\sigma _{i}n_{i},\\A&=\ell \sum _{i=1}^{N}\varepsilon _{i}c_{i}.\end{aligned}}}
921:). A single measurement cannot separate the two, but conceptually the contribution of each can be separated in the attenuation coefficient. If
634:
There is no evidence that Beer saw concentration and path length as symmetrical variables in an equation in the manner of the Beer–Lambert law.
1319:], which is defined as "the property of a body that determines the fraction of incident radiation absorbed by the body". The Beer–Lambert law
2765:
1627:
7225:(the angle measured from the direction perpendicular to the Earth's surface at the observation site). This equation can be used to retrieve
5695:
has the dimension of an area; it expresses the likelihood of interaction between the particles of the beam and the particles of the species
3995:
6430:
5048:
7313:
6682:
Therefore, measurements at two wavelengths yields two equations in two unknowns and will suffice to determine the amount concentrations
6484:
that is nearly unique for bilirubin and at a second wavelength in order to correct for possible interferences. The amount concentration
3322:. Absorbance within range of 0.2 to 0.5 is ideal to maintain linearity in the Beer–Lambert law. If the radiation is especially intense,
7726:
100:
4990:
72:
5611:
7238:, which is necessary for the correction of satellite images and also important in accounting for the role of aerosols in climate.
5911:
826:
753:
79:
6397:
Under certain conditions the Beer–Lambert law fails to maintain a linear relationship between attenuation and concentration of
4524:
918:
202:
157:
4311:
direction. The radiant flux of the light that emerges from a slice is reduced, compared to that of the light that entered, by
1113:
300:
630:
the length of the absorbing material traversed in the experiment." This is the relationship that might properly be called
53:
5324:{\displaystyle T=\mathrm {\frac {\Phi _{e}^{t}}{\Phi _{e}^{i}}} =\exp \left(-\int _{0}^{\ell }\mu (z)\mathrm {d} z\right).}
1018:
5207:{\displaystyle \mathrm {\Phi _{e}^{t}} =\mathrm {\Phi _{e}^{i}} \exp \left(-\int _{0}^{\ell }\mu (z)\mathrm {d} z\right),}
4159:
86:
5345:
4307:
sufficiently small that one particle in a slice cannot obscure another particle in the same slice when viewed along the
2452:{\displaystyle T={\frac {\Phi _{\mathrm {e} }^{\mathrm {t} }}{\Phi _{\mathrm {e} }^{\mathrm {i} }}}=e^{-\tau }=10^{-A},}
244:
7876:
7861:
7765:
7740:
7303:
6723:
3367:
6415:
There are at least six conditions that need to be fulfilled in order for the Beer–Lambert law to be valid. These are:
7542:
4238:
if the scattering centers are much smaller than the incident wavelength). Also note that for some systems we can put
119:
68:
1440:
17:
7308:
7178:
3683:
respectively, by definition of attenuation cross section and molar attenuation coefficient. Then the law becomes
1107:
7385:
7288:
7268:
7253:
6555:
For a more complicated example, consider a mixture in solution containing two species at amount concentrations
4415:
1590:
57:
6436:
The incident radiation must consist of parallel rays, each traversing the same length in the absorbing medium.
2508:
2465:
140:
measurements to determine the concentration of chemical species that absorb light. It is often referred to as
7283:
7263:
5820:
4512:{\displaystyle {\frac {\mathrm {d} \Phi _{\mathrm {e} }(z)}{\mathrm {d} z}}=-\mu (z)\Phi _{\mathrm {e} }(z).}
2300:
1713:
1190:
717:
consisting of pure solvent is measured, and compared to the intensity of light transmitted through a sample
4411:
1593:(g/L) and length (cm), then the units on the absorptivity would be , so that the absorbance has no units.
7222:
7014:
is the optical depth whose subscript identifies the source of the absorption or scattering it describes:
379:
7350:
4519:
The attenuation is caused by the photons that did not make it to the other side of the slice because of
3344:, but in this case is better called the Bouguer-Lambert's law. The (Napierian) attenuation coefficient
438:
6742:
6467:
in blood plasma samples. The spectrum of pure bilirubin is known, so the molar attenuation coefficient
560:
5810:{\displaystyle T=\exp \left(-\sum _{i=1}^{N}\sigma _{i}\int _{0}^{\ell }n_{i}(z)\mathrm {d} z\right).}
1481:
93:
293:
was often termed the "optical density" of the body. Integrating to find the intensity at a distance
7783:"Beer–Lambert law for optical tissue diagnostics: current state of the art and the main limitations"
6673:{\displaystyle \mu _{10}(\lambda )=\varepsilon _{1}(\lambda )c_{1}+\varepsilon _{2}(\lambda )c_{2}.}
6408:
Chemical—deviations observed due to specific chemical species of the sample which is being analyzed.
1605:
1601:
504:
6463:, without the need for extensive pre-processing of the sample. An example is the determination of
6451:
If any of these conditions are not fulfilled, there will be deviations from the Beer–Lambert law.
532:
529:
is the coefficient (fraction) of diminution, then this coefficient (fraction) will have the value
7846:
7732:
7258:
5606:
4407:
3341:
2266:
218:
164:
to the properties of the material through which the light is travelling. It had its first use in
46:
7560:"The Beer–Bouguer–Lambert law. Concepts of extinction (scattering plus absorption) and emission"
7365:
7340:
6765:
attenuation at about 6 micrometres can be detected quite easily, and degree of oxidation of the
4241:
510:
7273:
6821:
corresponding to the given path. The
Bouguer-Lambert law for the atmosphere is usually written
6738:
1597:
1302:
165:
7318:
3318:
The law tends to break down at very high concentrations, especially if the material is highly
1263:
1236:
7871:
7841:
7410:
4217:
1557:
1534:
720:
687:
672:
Beer specifically looked at solutions. Solutions are homogeneous and do not scatter light (
7641:"Absorption and Extinction Cross Sections and Photon Streamlines in the Optical Near-field"
7422:
7248:
5906:
4127:{\displaystyle {\begin{aligned}\tau &=\mu \ell ,\\A&=\mu _{10}\ell .\end{aligned}}}
2321:
1420:
951:
924:
499:
198:
7615:
4270:
3347:
1089:
593:
8:
7881:
7298:
7107:
6746:
4235:
3312:
3308:
1733:
1617:
7559:
7426:
7815:
7782:
7673:
7640:
7510:
7483:
4528:
4139:
2304:
1717:
1621:
1514:
1400:
998:
978:
823:. It is for this case that the common mathematical formulation (see below) applies:
613:
241:
of light traveling into an absorbing body would be given by the differential equation:
4395:{\displaystyle \mathrm {d\Phi _{e}} (z)=-\mu (z)\Phi _{\mathrm {e} }(z)\mathrm {d} z,}
1322:
7820:
7802:
7761:
7736:
7678:
7660:
7597:
7538:
7515:
6460:
5902:
3414:
of a material sample are related to its number densities and amount concentrations as
2829:
1508:
642:
137:
7705:
7577:
4156:, in which case one does not have to perform an integral and can express the law as:
7810:
7794:
7709:
7700:
7668:
7652:
7589:
7505:
7495:
7430:
7372:
Photometry, or, On the measure and gradations of light intensity, colors, and shade
7082:
7069:
7059:
6719:
6411:
Instrument—deviations which occur due to how the attenuation measurements are made.
3323:
1299:
For the case of absorption (Beer), this later quantity is called the absorptivity [
677:
2613:{\displaystyle \varepsilon _{i}={\frac {\mathrm {N_{A}} }{\ln(10)}}\,\sigma _{i},}
906:
662:
289:
which is compatible with
Bouguer's observations. The constant of proportionality
182:
5887:{\displaystyle \varepsilon _{i}={\tfrac {\mathrm {N_{A}} }{\ln 10}}\sigma _{i},}
7798:
7656:
7032:
6762:
5592:
2283:
214:
7855:
7806:
7704:, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "
7664:
7601:
7468:
7434:
7235:
6758:
6754:
6440:
2346:
2339:
666:
650:
646:
153:
7713:
6718:
are known at both wavelengths. This two system equation can be solved using
4527:. The solution to this differential equation is obtained by multiplying the
656:
7824:
7682:
7519:
7500:
7413:[Determination of the absorption of red light in colored liquids].
7144:
7079:
6818:
4837:
2497:
194:
2543:
Attenuation cross section and molar attenuation coefficient are related by
1290:
7448:
Pfieffer, Heinz; Liebhafshy, Herman (1951). "The
Origins of Beer's Law".
7136:
1187:. For scattering media, the constant is often divided into two parts,
914:
910:
673:
494:
490:
229:
6546:{\displaystyle c={\frac {\mu _{10}(\lambda )}{\varepsilon (\lambda )}}.}
7411:"Bestimmung der Absorption des rothen Lichts in farbigen Flüssigkeiten"
4520:
4214:
where the attenuation is usually an addition of absorption coefficient
3319:
2353:
1704:
638:
7593:
5905:, to describe the attenuation coefficient in a way independent of the
7293:
6750:
6700:
as long as the molar attenuation coefficients of the two components,
6464:
6426:
6422:
The attenuating medium must be homogeneous in the interaction volume.
6405:
Real—fundamental deviations due to the limitations of the law itself.
4601:{\displaystyle \exp \left(\int _{0}^{z}\mu (z')\mathrm {d} z'\right)}
6459:
The Beer–Lambert law can be applied to the analysis of a mixture by
5591:
To describe the attenuation coefficient in a way independent of the
5096:{\displaystyle \mathrm {\Phi _{e}^{t}} =\mathrm {\Phi _{e}} (\ell )}
2334:
is the path length of the beam of light through the material sample.
35:
7278:
7122:
7022:
3328:
681:
190:
7367:
Photometria sive de mensura et gradibus luminis, colorum et umbrae
975:
is the intensity of the light detected after travel of a distance
7164:
7148:
6800:
6766:
6398:
948:
is the intensity of the light at the beginning of the travel and
227:(Claude Jombert, Paris, 1729) – and even quoted from it – in his
186:
145:
7639:
Striebel, Moritz; Wrachtrup, Jӧrg; Gerhardt, Ilja (2017-11-13).
5038:{\displaystyle \mathrm {\Phi _{e}^{i}} =\mathrm {\Phi _{e}} (0)}
181:
The extinction law is also used in understanding attenuation in
7169:, a term approximately equal (for small and moderate values of
7111:
7040:
5605:
attenuating species of the material sample, one introduces the
213:
Bouguer–Lambert law: This law is based on observations made by
4136:
In many cases, the attenuation coefficient does not vary with
7482:
Mayerhöfer, Thomas G.; Pahlow, Susanne; Popp, Jürgen (2020).
7097:
5682:{\displaystyle \sigma _{i}={\tfrac {\mu _{i}(z)}{n_{i}(z)}}.}
4967:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}\left=0.}
4296:
Assume that a beam of light enters a material sample. Define
4234:(creation of electron-hole pairs) or scattering (for example
1740:
A more general form of the Beer–Lambert law states that, for
161:
7484:"The Bouguer–Beer–Lambert Law: Shining Light on the Obscure"
5971:{\displaystyle c_{i}(z)=n_{i}{\tfrac {z}{\mathrm {N_{A}} }}}
895:{\displaystyle \log _{10}(I_{R}/I_{S})=A=\varepsilon \ell c}
7578:"Speaking Theoretically ... ... Things Nobody Knows but Me"
7349:] (in French). Paris, France: Claude Jombert. pp.
7140:
6803:, and for a plane-parallel atmosphere it is determined as
6471:
is known. Measurements of decadic attenuation coefficient
3335:
1600:" (Bouguer), the sum of absorption and scatter, the terms
816:{\displaystyle {\log _{10}{{\bigl (}I_{R}/I_{S}{\big )}}}}
669:
which preferentially detected directly transmitted light.
7638:
6454:
6425:
The attenuating medium must not scatter the radiation—no
6401:. These deviations are classified into three categories:
5342:
is related to the (Napierian) attenuation coefficient by
657:
Differences between
Bouguer and Beer in application areas
369:{\textstyle \ln(I_{0}/I)=\int _{0}^{d}\mu \mathrm {d} x.}
6573:. The decadic attenuation coefficient at any wavelength
2814:{\displaystyle c_{i}={\frac {n_{i}}{\mathrm {N_{A}} }},}
1689:{\displaystyle \log _{10}(I_{0}/I)=A=\varepsilon \ell c}
1180:{\displaystyle -\ln(T)=\ln {\frac {I_{0}}{I_{d}}}=\mu d}
7374:] (in Latin). Augsburg, (Germany): Eberhardt Klett.
1291:
Absorptivity, cross-sections, and units of coefficients
489:
Beer's law: Much later, in 1852, the German scientist
7867:
Scattering, absorption and radiative transfer (optics)
7183:
6753:(also in biological tissue) as well as to measure the
5948:
5843:
5629:
5363:
4053:{\displaystyle T=e^{-\mu \ell }=10^{-\mu _{10}\ell },}
3385:
1417:, the number of particles encountered over a distance
303:
7181:
6827:
6583:
6494:
6419:
The attenuators must act independently of each other.
5984:
5914:
5828:
5705:
5614:
5401:
5348:
5220:
5109:
5051:
4993:
4846:
4614:
4536:
4424:
4317:
4273:
4244:
4220:
4162:
4142:
4066:
3998:
3860:
3689:
3420:
3370:
3350:
3166:
3024:
2842:
2768:
2626:
2549:
2539:
is the radiant flux received by that material sample.
2511:
2468:
2365:
1990:
1750:
1630:
1560:
1537:
1517:
1484:
1443:
1423:
1403:
1325:
1305:
1266:
1239:
1193:
1116:
1092:
1079:{\displaystyle T={\frac {I_{d}}{I_{0}}}=\exp(-\mu d)}
1021:
1001:
981:
954:
927:
829:
756:
723:
690:
616:
596:
563:
535:
513:
441:
435:
from which follows the exponential attenuation law:
382:
247:
7481:
60:. Unsourced material may be challenged and removed.
7447:
7207:
7062:, mainly due to urban pollution (absorption only);
6999:
6672:
6545:
6382:
5978:of the attenuating species of the material sample:
5970:
5886:
5809:
5681:
5581:
5388:{\displaystyle \mu _{10}={\tfrac {\mu }{\ln 10}},}
5387:
5323:
5206:
5095:
5037:
4966:
4828:
4600:
4511:
4394:
4279:
4258:
4226:
4206:
4148:
4126:
4052:
3980:
3846:
3675:
3406:
3356:
3295:
3153:
3010:
2813:
2755:
2612:
2531:
2488:
2451:
2247:
1976:
1688:
1581:
1543:
1523:
1499:
1470:
1429:
1409:
1389:
1311:
1279:
1252:
1225:
1179:
1098:
1078:
1007:
987:
967:
940:
894:
815:
742:
709:
622:
602:
582:
548:
521:
479:
427:
368:
282:{\displaystyle -\mathrm {d} I=\mu I\mathrm {d} x,}
281:
7386:"Bouguer-Lambert-Beer Absorption Law - Lumipedia"
6772:
6286:
6285:
6279:
5516:
5515:
5509:
3781:
3780:
3774:
3407:{\displaystyle \mu _{10}={\tfrac {\mu }{\ln 10}}}
2943:
2942:
2936:
1880:
1879:
1873:
7853:
4987:, with the incident radiant flux upon the slice
1233:, separating it into a scattering coefficient,
750:, with the absorbance of the sample taken as:
7780:
7342:Essai d'optique sur la gradation de la lumière
7132:) (responsible for the blue color of the sky);
4266:(1 over inelastic mean free path) in place of
2620:and number density and amount concentration by
1471:{\displaystyle N_{\mathrm {A} }c\sigma \ell S}
224:Essai d'optique sur la gradation de la lumière
6989:
6842:
806:
774:
237:Lambert began by assuming that the intensity
27:Scientific law describing absorption of light
7781:Oshina, Ilze; Spigulis, Janis (2021-10-28).
7755:
7532:
6439:The incident radiation should preferably be
1611:
376:For a homogeneous medium, this reduces to:
222:
7462:
7314:Tunable diode laser absorption spectroscopy
7208:{\displaystyle {\tfrac {1}{\cos \theta }},}
1744:attenuating species in the material sample,
7151:, a series of halogen radicals and others.
6278:
5508:
5333:Since the decadic attenuation coefficient
3773:
2935:
1872:
7814:
7672:
7509:
7499:
5460:
4755:
4657:
4410:, which yields the following first-order
3962:
3902:
2596:
2229:
2104:
120:Learn how and when to remove this message
7760:. Oxford Chemistry Primers. p. 26.
7689:
7537:. New York: John Wiley & Sons, Inc.
7347:Optics essay on the attenuation of light
3364:and the decadic attenuation coefficient
2532:{\displaystyle \mathrm {\Phi _{e}^{i}} }
2489:{\displaystyle \mathrm {\Phi _{e}^{t}} }
1531:the molar concentration (in mol/m), and
7557:
7463:Ingle, J. D. J.; Crouch, S. R. (1988).
7363:
7338:
4976:Integrating both sides and solving for
3336:Expression with attenuation coefficient
641:, most often expressed as the negative
217:before 1729. It is often attributed to
201:, this law arises as a solution of the
14:
7854:
6455:Chemical analysis by spectrophotometry
1226:{\displaystyle \mu =\mu _{s}+\mu _{a}}
6730:wavelengths for a mixture containing
3340:The law can be expressed in terms of
2345:of material sample is related to its
7847:Beer–Lambert Law Simpler Explanation
7756:Attard, Gary; Barnes, Colin (1998).
7575:
7408:
6429:—unless this is accounted for as in
4207:{\displaystyle I(z)=I_{0}e^{-\mu z}}
58:adding citations to reliable sources
29:
7724:
7535:Light Scattering by Small Particles
3992:attenuation, these relations become
2836:attenuation, these relations become
1624:of the species. This expression is:
428:{\displaystyle \ln(I_{0}/I)=\mu d,}
24:
7701:Compendium of Chemical Terminology
7304:Scientific laws named after people
7031:are uniformly mixed gases (mainly
6972:
6951:
6935:
6914:
6910:
6895:
6892:
6877:
6862:
6757:of various compounds in different
6722:. In practice it is better to use
6361:
6233:
6224:
6220:
6112:
6059:
6055:
5959:
5955:
5851:
5847:
5792:
5560:
5481:
5306:
5254:
5249:
5245:
5239:
5234:
5230:
5189:
5141:
5136:
5132:
5121:
5116:
5112:
5078:
5074:
5063:
5058:
5054:
5020:
5016:
5005:
5000:
4996:
4936:
4877:
4872:
4856:
4850:
4800:
4740:
4735:
4702:
4647:
4629:
4624:
4619:
4581:
4491:
4486:
4457:
4439:
4434:
4429:
4382:
4366:
4361:
4327:
4323:
4319:
3964:
3904:
3825:
3746:
2800:
2796:
2571:
2567:
2523:
2518:
2514:
2480:
2475:
2471:
2406:
2399:
2394:
2387:
2380:
2375:
2231:
2106:
1955:
1845:
1491:
1450:
610:is the absorption coefficient and
480:{\displaystyle I=I_{0}e^{-\mu d}.}
356:
269:
252:
25:
7893:
7835:
5045:and the transmitted radiant flux
4983:for a material of real thickness
1260:, and an absorption coefficient,
583:{\displaystyle \lambda =\mu ^{D}}
1500:{\displaystyle N_{\mathrm {A} }}
156:which relates the extinction or
34:
7774:
7749:
7718:
7632:
7608:
7569:
7309:Quantification of nucleic acids
45:needs additional citations for
7551:
7526:
7475:
7456:
7441:
7402:
7378:
7357:
7332:
7319:Transmittance#Beer–Lambert law
7254:Atomic absorption spectroscopy
6984:
6853:
6773:Application for the atmosphere
6654:
6648:
6622:
6616:
6600:
6594:
6534:
6528:
6520:
6514:
6357:
6351:
6258:
6252:
6214:
6208:
6108:
6102:
6049:
6043:
5931:
5925:
5821:molar attenuation coefficients
5788:
5782:
5669:
5663:
5648:
5642:
5556:
5550:
5477:
5471:
5457:
5451:
5302:
5296:
5185:
5179:
5090:
5084:
5032:
5026:
4932:
4921:
4889:
4883:
4796:
4785:
4752:
4746:
4731:
4725:
4698:
4687:
4641:
4635:
4577:
4566:
4503:
4497:
4482:
4476:
4451:
4445:
4416:ordinary differential equation
4378:
4372:
4357:
4351:
4339:
4333:
4172:
4166:
3959:
3953:
3899:
3893:
3821:
3815:
3742:
3736:
3666:
3660:
3613:
3607:
3560:
3554:
3534:
3528:
3481:
3475:
3434:
3428:
2590:
2584:
2226:
2220:
2101:
2095:
1951:
1945:
1841:
1835:
1665:
1644:
1384:
1363:
1342:
1326:
1132:
1126:
1073:
1061:
871:
843:
737:
724:
704:
691:
410:
389:
331:
310:
13:
1:
7450:Journal of Chemical Education
7415:Annalen der Physik und Chemie
7325:
7284:Laser absorption spectrometry
7264:Cavity ring-down spectroscopy
4291:
2301:molar attenuation coefficient
1714:molar attenuation coefficient
503:clear that he understood the
297:into the body, one obtains:
7787:Journal of Biomedical Optics
7728:Optical Properties of Solids
7616:"Definition of ABSORPTIVITY"
7533:Van de Hulst, H. C. (1957).
4836:which simplifies due to the
2338:In the above equations, the
1551:the particle cross section.
995:, the fraction transmitted,
549:{\displaystyle \lambda ^{2}}
7:
7842:Beer–Lambert Law Calculator
7241:
6795:refers to a vertical path,
6480:are made at one wavelength
6392:
2359:by the following definition
2324:of the attenuating species
2307:of the attenuating species
2286:of the attenuating species
2269:of the attenuating species
10:
7898:
7799:10.1117/1.JBO.26.10.100901
7657:10.1038/s41598-017-15528-w
7558:Sokolik, Irina N. (2009).
7269:Clausius–Mossotti relation
7025:(that absorb and scatter);
6743:near-infrared spectroscopy
6737:The law is used widely in
4259:{\displaystyle 1/\lambda }
1736:of the attenuating species
1726:is the optical path length
1720:of the attenuating species
522:{\displaystyle {\lambda }}
208:
7877:Electromagnetic radiation
7862:Eponymous laws of physics
7221:is the observed object's
5607:attenuation cross section
2267:attenuation cross section
1612:Mathematical formulations
1312:{\displaystyle \epsilon }
557:verified by experiment,
7576:Dahm, Donald J. (2010).
7465:Spectrochemical Analysis
7435:10.1002/andp.18521620505
7339:Bouguer, Pierre (1729).
2503:by that material sample;
1280:{\displaystyle \mu _{a}}
1253:{\displaystyle \mu _{s}}
505:exponential relationship
175:Bouguer–Beer–Lambert law
171:Beer–Bouguer–Lambert law
166:astronomical extinction.
7733:Oxford University Press
7714:10.1351/goldbook.B00626
7620:www.merriam-webster.com
7452:(March, 1951): 123–125.
7289:Lorentz–Lorenz relation
7259:Absorption spectroscopy
6447:to stimulated emission.
5699:in the material sample:
4408:attenuation coefficient
4227:{\displaystyle \alpha }
3342:attenuation coefficient
2328:in the material sample;
2311:in the material sample;
2290:in the material sample;
2273:in the material sample;
1620:through the sample and
1582:{\displaystyle I/I_{0}}
1544:{\displaystyle \sigma }
743:{\displaystyle (I_{S})}
710:{\displaystyle (I_{R})}
219:Johann Heinrich Lambert
136:is commonly applied to
7501:10.1002/cphc.202000464
7364:Lambert, J.H. (1760).
7274:Infra-red spectroscopy
7209:
7001:
6739:infra-red spectroscopy
6674:
6547:
6384:
6315:
6169:
6033:
5972:
5888:
5811:
5746:
5683:
5583:
5389:
5325:
5208:
5097:
5039:
4968:
4840:(applied backwards) to
4830:
4602:
4513:
4396:
4281:
4260:
4228:
4208:
4150:
4128:
4054:
3982:
3848:
3677:
3639:
3590:
3507:
3464:
3408:
3358:
3297:
3265:
3204:
3155:
3123:
3062:
3012:
2975:
2894:
2815:
2757:
2725:
2664:
2614:
2533:
2490:
2453:
2249:
2184:
2150:
2059:
2025:
1978:
1909:
1799:
1690:
1583:
1545:
1525:
1501:
1472:
1431:
1411:
1391:
1313:
1281:
1254:
1227:
1181:
1100:
1080:
1009:
989:
969:
942:
896:
817:
744:
711:
624:
604:
584:
550:
523:
481:
429:
370:
283:
223:
221:, who cited Bouguer's
7210:
7047:) which only absorb);
7002:
6675:
6548:
6385:
6295:
6149:
6013:
5973:
5907:amount concentrations
5889:
5819:One can also use the
5812:
5726:
5684:
5584:
5390:
5326:
5209:
5098:
5040:
4969:
4831:
4603:
4514:
4397:
4282:
4261:
4229:
4209:
4151:
4129:
4055:
3983:
3849:
3678:
3619:
3570:
3487:
3444:
3409:
3359:
3315:theory for instance.
3307:attenuation occur in
3298:
3245:
3184:
3156:
3103:
3042:
3013:
2955:
2874:
2816:
2758:
2705:
2644:
2615:
2534:
2491:
2454:
2250:
2164:
2130:
2039:
2005:
1979:
1889:
1779:
1691:
1584:
1546:
1526:
1502:
1473:
1432:
1430:{\displaystyle \ell }
1412:
1392:
1314:
1282:
1255:
1228:
1182:
1101:
1081:
1010:
990:
970:
968:{\displaystyle I_{d}}
943:
941:{\displaystyle I_{0}}
897:
818:
745:
712:
625:
605:
585:
551:
524:
482:
430:
371:
284:
7249:Applied spectroscopy
7179:
6825:
6724:linear least squares
6581:
6492:
5982:
5912:
5826:
5703:
5612:
5399:
5346:
5218:
5107:
5049:
4991:
4844:
4612:
4608:throughout to obtain
4534:
4422:
4315:
4280:{\displaystyle \mu }
4271:
4242:
4218:
4160:
4140:
4064:
3996:
3858:
3687:
3418:
3368:
3357:{\displaystyle \mu }
3348:
3164:
3022:
2840:
2766:
2624:
2547:
2509:
2466:
2363:
2322:amount concentration
1988:
1984:or equivalently that
1748:
1628:
1558:
1535:
1515:
1482:
1441:
1421:
1401:
1323:
1303:
1264:
1237:
1191:
1114:
1108:attenuation constant
1099:{\displaystyle \mu }
1090:
1019:
999:
979:
952:
925:
827:
754:
721:
688:
614:
603:{\displaystyle \mu }
594:
561:
533:
511:
439:
380:
301:
245:
199:mathematical physics
54:improve this article
7427:1852AnP...162...78B
7299:Polymer degradation
7108:Rayleigh scattering
7068:are effects due to
6747:polymer degradation
6340:
6194:
6091:
5771:
5539:
5444:
5292:
5258:
5243:
5175:
5145:
5125:
5067:
5009:
4917:
4781:
4683:
4562:
4406:is the (Napierian)
4236:Rayleigh scattering
3942:
3889:
3804:
3732:
3313:radiation shielding
3309:atmospheric science
2527:
2484:
2411:
2392:
2209:
2084:
1934:
1824:
1618:optical path length
507:, as he wrote: "If
351:
150:Bouguer–Lambert law
7725:Fox, Mark (2010).
7645:Scientific Reports
7205:
7200:
7100:(absorption only);
7072:in the atmosphere;
6997:
6670:
6543:
6380:
6378:
6326:
6180:
6077:
5968:
5966:
5884:
5869:
5807:
5757:
5679:
5674:
5579:
5577:
5525:
5430:
5385:
5380:
5321:
5278:
5244:
5229:
5204:
5161:
5131:
5111:
5093:
5053:
5035:
4995:
4964:
4903:
4826:
4767:
4669:
4598:
4548:
4529:integrating factor
4509:
4392:
4277:
4256:
4224:
4204:
4146:
4124:
4122:
4050:
3978:
3976:
3928:
3875:
3844:
3842:
3790:
3718:
3673:
3671:
3404:
3402:
3354:
3293:
3291:
3151:
3149:
3008:
3006:
2811:
2753:
2751:
2610:
2529:
2513:
2486:
2470:
2449:
2393:
2374:
2245:
2243:
2195:
2070:
1974:
1972:
1920:
1810:
1686:
1591:mass concentration
1579:
1541:
1521:
1497:
1468:
1427:
1407:
1387:
1309:
1277:
1250:
1223:
1177:
1096:
1076:
1005:
985:
965:
938:
892:
813:
740:
707:
620:
600:
580:
546:
519:
477:
425:
366:
337:
279:
69:"Beer–Lambert law"
7594:10.1255/nirn.1176
7390:www.lumipedia.org
7236:optical thickness
7199:
7039:) and molecular
6538:
6488:is then given by
6461:spectrophotometry
6230:
6065:
5965:
5903:Avogadro constant
5868:
5673:
5379:
5259:
4864:
4655:
4465:
4149:{\displaystyle z}
3401:
3324:nonlinear optical
3311:applications and
2830:Avogadro constant
2806:
2594:
2412:
1596:For the case of "
1524:{\displaystyle c}
1509:Avogadro constant
1410:{\displaystyle S}
1166:
1050:
1008:{\displaystyle T}
988:{\displaystyle d}
643:decadic logarithm
623:{\displaystyle D}
138:chemical analysis
130:
129:
122:
104:
16:(Redirected from
7889:
7829:
7828:
7818:
7778:
7772:
7771:
7753:
7747:
7746:
7722:
7716:
7706:Beer–Lambert law
7693:
7687:
7686:
7676:
7636:
7630:
7629:
7627:
7626:
7612:
7606:
7605:
7573:
7567:
7566:
7564:
7555:
7549:
7548:
7530:
7524:
7523:
7513:
7503:
7479:
7473:
7472:
7460:
7454:
7453:
7445:
7439:
7438:
7406:
7400:
7399:
7397:
7396:
7382:
7376:
7375:
7361:
7355:
7354:
7336:
7233:
7220:
7216:
7214:
7212:
7211:
7206:
7201:
7198:
7184:
7172:
7157:
7131:
7120:
7105:
7095:
7094:
7077:
7070:Raman scattering
7067:
7060:nitrogen dioxide
7057:
7056:
7030:
7020:
7013:
7006:
7004:
7003:
6998:
6993:
6992:
6977:
6976:
6975:
6962:
6961:
6960:
6959:
6958:
6940:
6939:
6938:
6925:
6924:
6923:
6922:
6921:
6900:
6899:
6898:
6882:
6881:
6880:
6867:
6866:
6865:
6846:
6845:
6816:
6812:
6801:relative airmass
6798:
6794:
6790:
6784:
6745:for analysis of
6733:
6729:
6717:
6708:
6699:
6690:
6679:
6677:
6676:
6671:
6666:
6665:
6647:
6646:
6634:
6633:
6615:
6614:
6593:
6592:
6576:
6572:
6563:
6552:
6550:
6549:
6544:
6539:
6537:
6523:
6513:
6512:
6502:
6487:
6483:
6479:
6470:
6389:
6387:
6386:
6381:
6379:
6372:
6368:
6364:
6350:
6349:
6339:
6334:
6325:
6324:
6314:
6309:
6284:
6283:
6266:
6262:
6261:
6244:
6240:
6236:
6231:
6229:
6228:
6227:
6217:
6207:
6206:
6196:
6193:
6188:
6179:
6178:
6168:
6163:
6127:
6123:
6119:
6115:
6101:
6100:
6090:
6085:
6076:
6075:
6066:
6064:
6063:
6062:
6052:
6035:
6032:
6027:
5977:
5975:
5974:
5969:
5967:
5964:
5963:
5962:
5949:
5946:
5945:
5924:
5923:
5900:
5893:
5891:
5890:
5885:
5880:
5879:
5870:
5867:
5856:
5855:
5854:
5844:
5838:
5837:
5816:
5814:
5813:
5808:
5803:
5799:
5795:
5781:
5780:
5770:
5765:
5756:
5755:
5745:
5740:
5698:
5694:
5688:
5686:
5685:
5680:
5675:
5672:
5662:
5661:
5651:
5641:
5640:
5630:
5624:
5623:
5604:
5600:
5593:number densities
5588:
5586:
5585:
5580:
5578:
5571:
5567:
5563:
5549:
5548:
5538:
5533:
5514:
5513:
5496:
5492:
5488:
5484:
5470:
5469:
5443:
5438:
5394:
5392:
5391:
5386:
5381:
5378:
5364:
5358:
5357:
5341:
5330:
5328:
5327:
5322:
5317:
5313:
5309:
5291:
5286:
5260:
5257:
5252:
5242:
5237:
5228:
5213:
5211:
5210:
5205:
5200:
5196:
5192:
5174:
5169:
5146:
5144:
5139:
5126:
5124:
5119:
5102:
5100:
5099:
5094:
5083:
5082:
5081:
5068:
5066:
5061:
5044:
5042:
5041:
5036:
5025:
5024:
5023:
5010:
5008:
5003:
4986:
4982:
4973:
4971:
4970:
4965:
4957:
4953:
4952:
4948:
4947:
4939:
4931:
4916:
4911:
4882:
4881:
4880:
4865:
4863:
4859:
4853:
4848:
4835:
4833:
4832:
4827:
4816:
4812:
4811:
4803:
4795:
4780:
4775:
4745:
4744:
4743:
4718:
4714:
4713:
4705:
4697:
4682:
4677:
4656:
4654:
4650:
4644:
4634:
4633:
4632:
4622:
4616:
4607:
4605:
4604:
4599:
4597:
4593:
4592:
4584:
4576:
4561:
4556:
4518:
4516:
4515:
4510:
4496:
4495:
4494:
4466:
4464:
4460:
4454:
4444:
4443:
4442:
4432:
4426:
4405:
4401:
4399:
4398:
4393:
4385:
4371:
4370:
4369:
4332:
4331:
4330:
4310:
4306:
4299:
4288:
4286:
4284:
4283:
4278:
4265:
4263:
4262:
4257:
4252:
4233:
4231:
4230:
4225:
4213:
4211:
4210:
4205:
4203:
4202:
4187:
4186:
4155:
4153:
4152:
4147:
4133:
4131:
4130:
4125:
4123:
4113:
4112:
4059:
4057:
4056:
4051:
4046:
4045:
4041:
4040:
4020:
4019:
3987:
3985:
3984:
3979:
3977:
3967:
3952:
3951:
3941:
3936:
3907:
3888:
3883:
3853:
3851:
3850:
3845:
3843:
3836:
3832:
3828:
3814:
3813:
3803:
3798:
3779:
3778:
3761:
3757:
3753:
3749:
3731:
3726:
3682:
3680:
3679:
3674:
3672:
3659:
3658:
3649:
3648:
3638:
3633:
3606:
3605:
3589:
3584:
3553:
3552:
3527:
3526:
3517:
3516:
3506:
3501:
3474:
3473:
3463:
3458:
3413:
3411:
3410:
3405:
3403:
3400:
3386:
3380:
3379:
3363:
3361:
3360:
3355:
3302:
3300:
3299:
3294:
3292:
3285:
3284:
3275:
3274:
3264:
3259:
3224:
3223:
3214:
3213:
3203:
3198:
3160:
3158:
3157:
3152:
3150:
3143:
3142:
3133:
3132:
3122:
3117:
3082:
3081:
3072:
3071:
3061:
3056:
3017:
3015:
3014:
3009:
3007:
3000:
2996:
2995:
2994:
2985:
2984:
2974:
2969:
2941:
2940:
2923:
2919:
2915:
2914:
2913:
2904:
2903:
2893:
2888:
2827:
2820:
2818:
2817:
2812:
2807:
2805:
2804:
2803:
2793:
2792:
2783:
2778:
2777:
2762:
2760:
2759:
2754:
2752:
2745:
2744:
2735:
2734:
2724:
2719:
2684:
2683:
2674:
2673:
2663:
2658:
2619:
2617:
2616:
2611:
2606:
2605:
2595:
2593:
2576:
2575:
2574:
2564:
2559:
2558:
2538:
2536:
2535:
2530:
2528:
2526:
2521:
2495:
2493:
2492:
2487:
2485:
2483:
2478:
2458:
2456:
2455:
2450:
2445:
2444:
2429:
2428:
2413:
2410:
2409:
2403:
2402:
2391:
2390:
2384:
2383:
2373:
2358:
2351:
2344:
2333:
2327:
2319:
2310:
2298:
2289:
2281:
2272:
2264:
2254:
2252:
2251:
2246:
2244:
2234:
2219:
2218:
2208:
2203:
2194:
2193:
2183:
2178:
2160:
2159:
2149:
2144:
2109:
2094:
2093:
2083:
2078:
2069:
2068:
2058:
2053:
2035:
2034:
2024:
2019:
1983:
1981:
1980:
1975:
1973:
1966:
1962:
1958:
1944:
1943:
1933:
1928:
1919:
1918:
1908:
1903:
1878:
1877:
1860:
1856:
1852:
1848:
1834:
1833:
1823:
1818:
1809:
1808:
1798:
1793:
1743:
1731:
1725:
1711:
1702:
1695:
1693:
1692:
1687:
1661:
1656:
1655:
1640:
1639:
1588:
1586:
1585:
1580:
1578:
1577:
1568:
1550:
1548:
1547:
1542:
1530:
1528:
1527:
1522:
1506:
1504:
1503:
1498:
1496:
1495:
1494:
1477:
1475:
1474:
1469:
1455:
1454:
1453:
1436:
1434:
1433:
1428:
1416:
1414:
1413:
1408:
1396:
1394:
1393:
1390:{\displaystyle }
1388:
1359:
1354:
1353:
1338:
1337:
1318:
1316:
1315:
1310:
1286:
1284:
1283:
1278:
1276:
1275:
1259:
1257:
1256:
1251:
1249:
1248:
1232:
1230:
1229:
1224:
1222:
1221:
1209:
1208:
1186:
1184:
1183:
1178:
1167:
1165:
1164:
1155:
1154:
1145:
1105:
1103:
1102:
1097:
1085:
1083:
1082:
1077:
1051:
1049:
1048:
1039:
1038:
1029:
1014:
1012:
1011:
1006:
994:
992:
991:
986:
974:
972:
971:
966:
964:
963:
947:
945:
944:
939:
937:
936:
901:
899:
898:
893:
870:
869:
860:
855:
854:
839:
838:
822:
820:
819:
814:
812:
811:
810:
809:
803:
802:
793:
788:
787:
778:
777:
767:
766:
749:
747:
746:
741:
736:
735:
716:
714:
713:
708:
703:
702:
629:
627:
626:
621:
609:
607:
606:
601:
589:
587:
586:
581:
579:
578:
555:
553:
552:
547:
545:
544:
528:
526:
525:
520:
518:
493:studied another
486:
484:
483:
478:
473:
472:
457:
456:
434:
432:
431:
426:
406:
401:
400:
375:
373:
372:
367:
359:
350:
345:
327:
322:
321:
296:
292:
288:
286:
285:
280:
272:
255:
240:
226:
134:Beer–Lambert law
125:
118:
114:
111:
105:
103:
62:
38:
30:
21:
18:Beer-Lambert law
7897:
7896:
7892:
7891:
7890:
7888:
7887:
7886:
7852:
7851:
7838:
7833:
7832:
7779:
7775:
7768:
7754:
7750:
7743:
7723:
7719:
7694:
7690:
7637:
7633:
7624:
7622:
7614:
7613:
7609:
7574:
7570:
7562:
7556:
7552:
7545:
7531:
7527:
7480:
7476:
7461:
7457:
7446:
7442:
7407:
7403:
7394:
7392:
7384:
7383:
7379:
7362:
7358:
7337:
7333:
7328:
7323:
7244:
7232:
7226:
7218:
7188:
7182:
7180:
7177:
7176:
7174:
7170:
7155:
7130:
7126:
7119:
7115:
7110:from molecular
7103:
7093:
7089:
7088:
7075:
7065:
7055:
7051:
7050:
7046:
7038:
7028:
7018:
7012:
7008:
6988:
6987:
6971:
6970:
6966:
6954:
6950:
6949:
6948:
6944:
6934:
6933:
6929:
6917:
6913:
6909:
6908:
6904:
6891:
6890:
6886:
6876:
6875:
6871:
6861:
6860:
6856:
6841:
6840:
6826:
6823:
6822:
6814:
6804:
6796:
6792:
6782:
6779:
6775:
6731:
6727:
6716:
6710:
6707:
6701:
6698:
6692:
6689:
6683:
6661:
6657:
6642:
6638:
6629:
6625:
6610:
6606:
6588:
6584:
6582:
6579:
6578:
6574:
6571:
6565:
6562:
6556:
6524:
6508:
6504:
6503:
6501:
6493:
6490:
6489:
6485:
6481:
6478:
6472:
6468:
6457:
6395:
6377:
6376:
6360:
6345:
6341:
6335:
6330:
6320:
6316:
6310:
6299:
6291:
6287:
6277:
6273:
6264:
6263:
6245:
6232:
6223:
6219:
6218:
6202:
6198:
6197:
6195:
6189:
6184:
6174:
6170:
6164:
6153:
6145:
6141:
6140:
6125:
6124:
6111:
6096:
6092:
6086:
6081:
6071:
6067:
6058:
6054:
6053:
6036:
6034:
6028:
6017:
6009:
6005:
5992:
5985:
5983:
5980:
5979:
5958:
5954:
5953:
5947:
5941:
5937:
5919:
5915:
5913:
5910:
5909:
5899:
5895:
5875:
5871:
5857:
5850:
5846:
5845:
5842:
5833:
5829:
5827:
5824:
5823:
5791:
5776:
5772:
5766:
5761:
5751:
5747:
5741:
5730:
5722:
5718:
5704:
5701:
5700:
5696:
5693:
5689:
5657:
5653:
5652:
5636:
5632:
5631:
5628:
5619:
5615:
5613:
5610:
5609:
5602:
5599:
5595:
5576:
5575:
5559:
5544:
5540:
5534:
5529:
5521:
5517:
5507:
5503:
5494:
5493:
5480:
5465:
5461:
5439:
5434:
5426:
5422:
5409:
5402:
5400:
5397:
5396:
5368:
5362:
5353:
5349:
5347:
5344:
5343:
5340:
5334:
5305:
5287:
5282:
5274:
5270:
5253:
5248:
5238:
5233:
5227:
5219:
5216:
5215:
5188:
5170:
5165:
5157:
5153:
5140:
5135:
5130:
5120:
5115:
5110:
5108:
5105:
5104:
5077:
5073:
5072:
5062:
5057:
5052:
5050:
5047:
5046:
5019:
5015:
5014:
5004:
4999:
4994:
4992:
4989:
4988:
4984:
4981:
4977:
4940:
4935:
4924:
4912:
4907:
4902:
4898:
4876:
4875:
4871:
4870:
4866:
4855:
4854:
4849:
4847:
4845:
4842:
4841:
4804:
4799:
4788:
4776:
4771:
4766:
4762:
4739:
4738:
4734:
4706:
4701:
4690:
4678:
4673:
4668:
4664:
4646:
4645:
4628:
4627:
4623:
4618:
4617:
4615:
4613:
4610:
4609:
4585:
4580:
4569:
4557:
4552:
4547:
4543:
4535:
4532:
4531:
4490:
4489:
4485:
4456:
4455:
4438:
4437:
4433:
4428:
4427:
4425:
4423:
4420:
4419:
4403:
4381:
4365:
4364:
4360:
4326:
4322:
4318:
4316:
4313:
4312:
4308:
4301:
4297:
4294:
4272:
4269:
4268:
4267:
4248:
4243:
4240:
4239:
4219:
4216:
4215:
4192:
4188:
4182:
4178:
4161:
4158:
4157:
4141:
4138:
4137:
4121:
4120:
4108:
4104:
4097:
4091:
4090:
4074:
4067:
4065:
4062:
4061:
4060:or equivalently
4036:
4032:
4028:
4024:
4009:
4005:
3997:
3994:
3993:
3975:
3974:
3963:
3947:
3943:
3937:
3932:
3921:
3915:
3914:
3903:
3884:
3879:
3868:
3861:
3859:
3856:
3855:
3841:
3840:
3824:
3809:
3805:
3799:
3794:
3786:
3782:
3772:
3768:
3759:
3758:
3745:
3727:
3722:
3714:
3710:
3697:
3690:
3688:
3685:
3684:
3670:
3669:
3654:
3650:
3644:
3640:
3634:
3623:
3595:
3591:
3585:
3574:
3563:
3548:
3544:
3541:
3540:
3522:
3518:
3512:
3508:
3502:
3491:
3469:
3465:
3459:
3448:
3437:
3421:
3419:
3416:
3415:
3390:
3384:
3375:
3371:
3369:
3366:
3365:
3349:
3346:
3345:
3338:
3290:
3289:
3280:
3276:
3270:
3266:
3260:
3249:
3235:
3229:
3228:
3219:
3215:
3209:
3205:
3199:
3188:
3174:
3167:
3165:
3162:
3161:
3148:
3147:
3138:
3134:
3128:
3124:
3118:
3107:
3093:
3087:
3086:
3077:
3073:
3067:
3063:
3057:
3046:
3032:
3025:
3023:
3020:
3019:
3018:or equivalently
3005:
3004:
2990:
2986:
2980:
2976:
2970:
2959:
2948:
2944:
2934:
2930:
2921:
2920:
2909:
2905:
2899:
2895:
2889:
2878:
2867:
2863:
2850:
2843:
2841:
2838:
2837:
2826:
2822:
2799:
2795:
2794:
2788:
2784:
2782:
2773:
2769:
2767:
2764:
2763:
2750:
2749:
2740:
2736:
2730:
2726:
2720:
2709:
2695:
2689:
2688:
2679:
2675:
2669:
2665:
2659:
2648:
2634:
2627:
2625:
2622:
2621:
2601:
2597:
2577:
2570:
2566:
2565:
2563:
2554:
2550:
2548:
2545:
2544:
2522:
2517:
2512:
2510:
2507:
2506:
2479:
2474:
2469:
2467:
2464:
2463:
2437:
2433:
2421:
2417:
2405:
2404:
2398:
2397:
2386:
2385:
2379:
2378:
2372:
2364:
2361:
2360:
2356:
2349:
2342:
2331:
2325:
2318:
2314:
2308:
2297:
2293:
2287:
2280:
2276:
2270:
2263:
2259:
2242:
2241:
2230:
2214:
2210:
2204:
2199:
2189:
2185:
2179:
2168:
2155:
2151:
2145:
2134:
2123:
2117:
2116:
2105:
2089:
2085:
2079:
2074:
2064:
2060:
2054:
2043:
2030:
2026:
2020:
2009:
1998:
1991:
1989:
1986:
1985:
1971:
1970:
1954:
1939:
1935:
1929:
1924:
1914:
1910:
1904:
1893:
1885:
1881:
1871:
1867:
1858:
1857:
1844:
1829:
1825:
1819:
1814:
1804:
1800:
1794:
1783:
1775:
1771:
1758:
1751:
1749:
1746:
1745:
1741:
1729:
1723:
1709:
1700:
1657:
1651:
1647:
1635:
1631:
1629:
1626:
1625:
1614:
1573:
1569:
1564:
1559:
1556:
1555:
1536:
1533:
1532:
1516:
1513:
1512:
1490:
1489:
1485:
1483:
1480:
1479:
1449:
1448:
1444:
1442:
1439:
1438:
1422:
1419:
1418:
1402:
1399:
1398:
1355:
1349:
1345:
1333:
1329:
1324:
1321:
1320:
1304:
1301:
1300:
1293:
1271:
1267:
1265:
1262:
1261:
1244:
1240:
1238:
1235:
1234:
1217:
1213:
1204:
1200:
1192:
1189:
1188:
1160:
1156:
1150:
1146:
1144:
1115:
1112:
1111:
1091:
1088:
1087:
1044:
1040:
1034:
1030:
1028:
1020:
1017:
1016:
1015:, is given by:
1000:
997:
996:
980:
977:
976:
959:
955:
953:
950:
949:
932:
928:
926:
923:
922:
865:
861:
856:
850:
846:
834:
830:
828:
825:
824:
805:
804:
798:
794:
789:
783:
779:
773:
772:
771:
762:
758:
757:
755:
752:
751:
731:
727:
722:
719:
718:
698:
694:
689:
686:
685:
659:
615:
612:
611:
595:
592:
591:
574:
570:
562:
559:
558:
540:
536:
534:
531:
530:
514:
512:
509:
508:
462:
458:
452:
448:
440:
437:
436:
402:
396:
392:
381:
378:
377:
355:
346:
341:
323:
317:
313:
302:
299:
298:
294:
290:
268:
251:
246:
243:
242:
238:
211:
183:physical optics
179:extinction law.
126:
115:
109:
106:
63:
61:
51:
39:
28:
23:
22:
15:
12:
11:
5:
7895:
7885:
7884:
7879:
7874:
7869:
7864:
7850:
7849:
7844:
7837:
7836:External links
7834:
7831:
7830:
7793:(10): 100901.
7773:
7767:978-0198556862
7766:
7748:
7742:978-0199573370
7741:
7731:(2 ed.).
7717:
7688:
7631:
7607:
7568:
7550:
7543:
7525:
7474:
7467:. New Jersey:
7455:
7440:
7401:
7377:
7356:
7330:
7329:
7327:
7324:
7322:
7321:
7316:
7311:
7306:
7301:
7296:
7291:
7286:
7281:
7276:
7271:
7266:
7261:
7256:
7251:
7245:
7243:
7240:
7234:, the aerosol
7230:
7204:
7197:
7194:
7191:
7187:
7153:
7152:
7133:
7128:
7117:
7101:
7091:
7086:
7073:
7063:
7053:
7048:
7044:
7036:
7033:carbon dioxide
7026:
7010:
6996:
6991:
6986:
6983:
6980:
6974:
6969:
6965:
6957:
6953:
6947:
6943:
6937:
6932:
6928:
6920:
6916:
6912:
6907:
6903:
6897:
6894:
6889:
6885:
6879:
6874:
6870:
6864:
6859:
6855:
6852:
6849:
6844:
6839:
6836:
6833:
6830:
6799:is called the
6774:
6771:
6763:carbonyl group
6714:
6705:
6696:
6687:
6669:
6664:
6660:
6656:
6653:
6650:
6645:
6641:
6637:
6632:
6628:
6624:
6621:
6618:
6613:
6609:
6605:
6602:
6599:
6596:
6591:
6587:
6569:
6560:
6542:
6536:
6533:
6530:
6527:
6522:
6519:
6516:
6511:
6507:
6500:
6497:
6476:
6456:
6453:
6449:
6448:
6444:
6437:
6434:
6423:
6420:
6413:
6412:
6409:
6406:
6394:
6391:
6375:
6371:
6367:
6363:
6359:
6356:
6353:
6348:
6344:
6338:
6333:
6329:
6323:
6319:
6313:
6308:
6305:
6302:
6298:
6294:
6290:
6282:
6276:
6272:
6269:
6267:
6265:
6260:
6257:
6254:
6251:
6248:
6243:
6239:
6235:
6226:
6222:
6216:
6213:
6210:
6205:
6201:
6192:
6187:
6183:
6177:
6173:
6167:
6162:
6159:
6156:
6152:
6148:
6144:
6139:
6136:
6133:
6130:
6128:
6126:
6122:
6118:
6114:
6110:
6107:
6104:
6099:
6095:
6089:
6084:
6080:
6074:
6070:
6061:
6057:
6051:
6048:
6045:
6042:
6039:
6031:
6026:
6023:
6020:
6016:
6012:
6008:
6004:
6001:
5998:
5995:
5993:
5991:
5988:
5987:
5961:
5957:
5952:
5944:
5940:
5936:
5933:
5930:
5927:
5922:
5918:
5897:
5883:
5878:
5874:
5866:
5863:
5860:
5853:
5849:
5841:
5836:
5832:
5806:
5802:
5798:
5794:
5790:
5787:
5784:
5779:
5775:
5769:
5764:
5760:
5754:
5750:
5744:
5739:
5736:
5733:
5729:
5725:
5721:
5717:
5714:
5711:
5708:
5691:
5678:
5671:
5668:
5665:
5660:
5656:
5650:
5647:
5644:
5639:
5635:
5627:
5622:
5618:
5597:
5574:
5570:
5566:
5562:
5558:
5555:
5552:
5547:
5543:
5537:
5532:
5528:
5524:
5520:
5512:
5506:
5502:
5499:
5497:
5495:
5491:
5487:
5483:
5479:
5476:
5473:
5468:
5464:
5459:
5456:
5453:
5450:
5447:
5442:
5437:
5433:
5429:
5425:
5421:
5418:
5415:
5412:
5410:
5408:
5405:
5404:
5384:
5377:
5374:
5371:
5367:
5361:
5356:
5352:
5338:
5320:
5316:
5312:
5308:
5304:
5301:
5298:
5295:
5290:
5285:
5281:
5277:
5273:
5269:
5266:
5263:
5256:
5251:
5247:
5241:
5236:
5232:
5226:
5223:
5203:
5199:
5195:
5191:
5187:
5184:
5181:
5178:
5173:
5168:
5164:
5160:
5156:
5152:
5149:
5143:
5138:
5134:
5129:
5123:
5118:
5114:
5092:
5089:
5086:
5080:
5076:
5071:
5065:
5060:
5056:
5034:
5031:
5028:
5022:
5018:
5013:
5007:
5002:
4998:
4979:
4963:
4960:
4956:
4951:
4946:
4943:
4938:
4934:
4930:
4927:
4923:
4920:
4915:
4910:
4906:
4901:
4897:
4894:
4891:
4888:
4885:
4879:
4874:
4869:
4862:
4858:
4852:
4825:
4822:
4819:
4815:
4810:
4807:
4802:
4798:
4794:
4791:
4787:
4784:
4779:
4774:
4770:
4765:
4761:
4758:
4754:
4751:
4748:
4742:
4737:
4733:
4730:
4727:
4724:
4721:
4717:
4712:
4709:
4704:
4700:
4696:
4693:
4689:
4686:
4681:
4676:
4672:
4667:
4663:
4660:
4653:
4649:
4643:
4640:
4637:
4631:
4626:
4621:
4596:
4591:
4588:
4583:
4579:
4575:
4572:
4568:
4565:
4560:
4555:
4551:
4546:
4542:
4539:
4508:
4505:
4502:
4499:
4493:
4488:
4484:
4481:
4478:
4475:
4472:
4469:
4463:
4459:
4453:
4450:
4447:
4441:
4436:
4431:
4391:
4388:
4384:
4380:
4377:
4374:
4368:
4363:
4359:
4356:
4353:
4350:
4347:
4344:
4341:
4338:
4335:
4329:
4325:
4321:
4293:
4290:
4276:
4255:
4251:
4247:
4223:
4201:
4198:
4195:
4191:
4185:
4181:
4177:
4174:
4171:
4168:
4165:
4145:
4119:
4116:
4111:
4107:
4103:
4100:
4098:
4096:
4093:
4092:
4089:
4086:
4083:
4080:
4077:
4075:
4073:
4070:
4069:
4049:
4044:
4039:
4035:
4031:
4027:
4023:
4018:
4015:
4012:
4008:
4004:
4001:
3973:
3970:
3966:
3961:
3958:
3955:
3950:
3946:
3940:
3935:
3931:
3927:
3924:
3922:
3920:
3917:
3916:
3913:
3910:
3906:
3901:
3898:
3895:
3892:
3887:
3882:
3878:
3874:
3871:
3869:
3867:
3864:
3863:
3839:
3835:
3831:
3827:
3823:
3820:
3817:
3812:
3808:
3802:
3797:
3793:
3789:
3785:
3777:
3771:
3767:
3764:
3762:
3760:
3756:
3752:
3748:
3744:
3741:
3738:
3735:
3730:
3725:
3721:
3717:
3713:
3709:
3706:
3703:
3700:
3698:
3696:
3693:
3692:
3668:
3665:
3662:
3657:
3653:
3647:
3643:
3637:
3632:
3629:
3626:
3622:
3618:
3615:
3612:
3609:
3604:
3601:
3598:
3594:
3588:
3583:
3580:
3577:
3573:
3569:
3566:
3564:
3562:
3559:
3556:
3551:
3547:
3543:
3542:
3539:
3536:
3533:
3530:
3525:
3521:
3515:
3511:
3505:
3500:
3497:
3494:
3490:
3486:
3483:
3480:
3477:
3472:
3468:
3462:
3457:
3454:
3451:
3447:
3443:
3440:
3438:
3436:
3433:
3430:
3427:
3424:
3423:
3399:
3396:
3393:
3389:
3383:
3378:
3374:
3353:
3337:
3334:
3288:
3283:
3279:
3273:
3269:
3263:
3258:
3255:
3252:
3248:
3244:
3241:
3238:
3236:
3234:
3231:
3230:
3227:
3222:
3218:
3212:
3208:
3202:
3197:
3194:
3191:
3187:
3183:
3180:
3177:
3175:
3173:
3170:
3169:
3146:
3141:
3137:
3131:
3127:
3121:
3116:
3113:
3110:
3106:
3102:
3099:
3096:
3094:
3092:
3089:
3088:
3085:
3080:
3076:
3070:
3066:
3060:
3055:
3052:
3049:
3045:
3041:
3038:
3035:
3033:
3031:
3028:
3027:
3003:
2999:
2993:
2989:
2983:
2979:
2973:
2968:
2965:
2962:
2958:
2954:
2951:
2947:
2939:
2933:
2929:
2926:
2924:
2922:
2918:
2912:
2908:
2902:
2898:
2892:
2887:
2884:
2881:
2877:
2873:
2870:
2866:
2862:
2859:
2856:
2853:
2851:
2849:
2846:
2845:
2824:
2810:
2802:
2798:
2791:
2787:
2781:
2776:
2772:
2748:
2743:
2739:
2733:
2729:
2723:
2718:
2715:
2712:
2708:
2704:
2701:
2698:
2696:
2694:
2691:
2690:
2687:
2682:
2678:
2672:
2668:
2662:
2657:
2654:
2651:
2647:
2643:
2640:
2637:
2635:
2633:
2630:
2629:
2609:
2604:
2600:
2592:
2589:
2586:
2583:
2580:
2573:
2569:
2562:
2557:
2553:
2541:
2540:
2525:
2520:
2516:
2504:
2482:
2477:
2473:
2448:
2443:
2440:
2436:
2432:
2427:
2424:
2420:
2416:
2408:
2401:
2396:
2389:
2382:
2377:
2371:
2368:
2336:
2335:
2329:
2316:
2312:
2295:
2291:
2284:number density
2278:
2274:
2261:
2240:
2237:
2233:
2228:
2225:
2222:
2217:
2213:
2207:
2202:
2198:
2192:
2188:
2182:
2177:
2174:
2171:
2167:
2163:
2158:
2154:
2148:
2143:
2140:
2137:
2133:
2129:
2126:
2124:
2122:
2119:
2118:
2115:
2112:
2108:
2103:
2100:
2097:
2092:
2088:
2082:
2077:
2073:
2067:
2063:
2057:
2052:
2049:
2046:
2042:
2038:
2033:
2029:
2023:
2018:
2015:
2012:
2008:
2004:
2001:
1999:
1997:
1994:
1993:
1969:
1965:
1961:
1957:
1953:
1950:
1947:
1942:
1938:
1932:
1927:
1923:
1917:
1913:
1907:
1902:
1899:
1896:
1892:
1888:
1884:
1876:
1870:
1866:
1863:
1861:
1859:
1855:
1851:
1847:
1843:
1840:
1837:
1832:
1828:
1822:
1817:
1813:
1807:
1803:
1797:
1792:
1789:
1786:
1782:
1778:
1774:
1770:
1767:
1764:
1761:
1759:
1757:
1754:
1753:
1738:
1737:
1727:
1721:
1707:
1685:
1682:
1679:
1676:
1673:
1670:
1667:
1664:
1660:
1654:
1650:
1646:
1643:
1638:
1634:
1613:
1610:
1576:
1572:
1567:
1563:
1540:
1520:
1493:
1488:
1467:
1464:
1461:
1458:
1452:
1447:
1426:
1406:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1365:
1362:
1358:
1352:
1348:
1344:
1341:
1336:
1332:
1328:
1308:
1292:
1289:
1274:
1270:
1247:
1243:
1220:
1216:
1212:
1207:
1203:
1199:
1196:
1176:
1173:
1170:
1163:
1159:
1153:
1149:
1143:
1140:
1137:
1134:
1131:
1128:
1125:
1122:
1119:
1095:
1075:
1072:
1069:
1066:
1063:
1060:
1057:
1054:
1047:
1043:
1037:
1033:
1027:
1024:
1004:
984:
962:
958:
935:
931:
891:
888:
885:
882:
879:
876:
873:
868:
864:
859:
853:
849:
845:
842:
837:
833:
808:
801:
797:
792:
786:
782:
776:
770:
765:
761:
739:
734:
730:
726:
706:
701:
697:
693:
658:
655:
651:concentrations
649:, to both the
619:
599:
577:
573:
569:
566:
543:
539:
517:
476:
471:
468:
465:
461:
455:
451:
447:
444:
424:
421:
418:
415:
412:
409:
405:
399:
395:
391:
388:
385:
365:
362:
358:
354:
349:
344:
340:
336:
333:
330:
326:
320:
316:
312:
309:
306:
278:
275:
271:
267:
264:
261:
258:
254:
250:
215:Pierre Bouguer
210:
207:
195:rarefied gases
177:or merely the
128:
127:
42:
40:
33:
26:
9:
6:
4:
3:
2:
7894:
7883:
7880:
7878:
7875:
7873:
7870:
7868:
7865:
7863:
7860:
7859:
7857:
7848:
7845:
7843:
7840:
7839:
7826:
7822:
7817:
7812:
7808:
7804:
7800:
7796:
7792:
7788:
7784:
7777:
7769:
7763:
7759:
7752:
7744:
7738:
7735:. p. 3.
7734:
7730:
7729:
7721:
7715:
7711:
7707:
7703:
7702:
7697:
7692:
7684:
7680:
7675:
7670:
7666:
7662:
7658:
7654:
7650:
7646:
7642:
7635:
7621:
7617:
7611:
7603:
7599:
7595:
7591:
7587:
7583:
7579:
7572:
7561:
7554:
7546:
7544:9780486642284
7540:
7536:
7529:
7521:
7517:
7512:
7507:
7502:
7497:
7493:
7489:
7485:
7478:
7470:
7469:Prentice Hall
7466:
7459:
7451:
7444:
7436:
7432:
7428:
7424:
7420:
7417:(in German).
7416:
7412:
7409:Beer (1852).
7405:
7391:
7387:
7381:
7373:
7369:
7368:
7360:
7352:
7348:
7344:
7343:
7335:
7331:
7320:
7317:
7315:
7312:
7310:
7307:
7305:
7302:
7300:
7297:
7295:
7292:
7290:
7287:
7285:
7282:
7280:
7277:
7275:
7272:
7270:
7267:
7265:
7262:
7260:
7257:
7255:
7252:
7250:
7247:
7246:
7239:
7237:
7229:
7224:
7202:
7195:
7192:
7189:
7185:
7168:
7166:
7161:
7150:
7146:
7142:
7138:
7134:
7124:
7113:
7109:
7102:
7099:
7087:
7084:
7081:
7074:
7071:
7064:
7061:
7049:
7042:
7034:
7027:
7024:
7017:
7016:
7015:
6994:
6981:
6978:
6967:
6963:
6955:
6945:
6941:
6930:
6926:
6918:
6905:
6901:
6887:
6883:
6872:
6868:
6857:
6850:
6847:
6837:
6834:
6831:
6828:
6820:
6811:
6807:
6802:
6789:
6785:
6770:
6768:
6764:
6760:
6756:
6755:concentration
6752:
6748:
6744:
6740:
6735:
6725:
6721:
6720:Cramer's rule
6713:
6704:
6695:
6686:
6680:
6667:
6662:
6658:
6651:
6643:
6639:
6635:
6630:
6626:
6619:
6611:
6607:
6603:
6597:
6589:
6585:
6577:is, given by
6568:
6559:
6553:
6540:
6531:
6525:
6517:
6509:
6505:
6498:
6495:
6475:
6466:
6462:
6452:
6445:
6442:
6441:monochromatic
6438:
6435:
6432:
6428:
6424:
6421:
6418:
6417:
6416:
6410:
6407:
6404:
6403:
6402:
6400:
6390:
6373:
6369:
6365:
6354:
6346:
6342:
6336:
6331:
6327:
6321:
6317:
6311:
6306:
6303:
6300:
6296:
6292:
6288:
6280:
6274:
6270:
6268:
6255:
6249:
6246:
6241:
6237:
6211:
6203:
6199:
6190:
6185:
6181:
6175:
6171:
6165:
6160:
6157:
6154:
6150:
6146:
6142:
6137:
6134:
6131:
6129:
6120:
6116:
6105:
6097:
6093:
6087:
6082:
6078:
6072:
6068:
6046:
6040:
6037:
6029:
6024:
6021:
6018:
6014:
6010:
6006:
6002:
5999:
5996:
5994:
5989:
5950:
5942:
5938:
5934:
5928:
5920:
5916:
5908:
5904:
5881:
5876:
5872:
5864:
5861:
5858:
5839:
5834:
5830:
5822:
5817:
5804:
5800:
5796:
5785:
5777:
5773:
5767:
5762:
5758:
5752:
5748:
5742:
5737:
5734:
5731:
5727:
5723:
5719:
5715:
5712:
5709:
5706:
5676:
5666:
5658:
5654:
5645:
5637:
5633:
5625:
5620:
5616:
5608:
5594:
5589:
5572:
5568:
5564:
5553:
5545:
5541:
5535:
5530:
5526:
5522:
5518:
5510:
5504:
5500:
5498:
5489:
5485:
5474:
5466:
5462:
5454:
5448:
5445:
5440:
5435:
5431:
5427:
5423:
5419:
5416:
5413:
5411:
5406:
5382:
5375:
5372:
5369:
5365:
5359:
5354:
5350:
5337:
5331:
5318:
5314:
5310:
5299:
5293:
5288:
5283:
5279:
5275:
5271:
5267:
5264:
5261:
5224:
5221:
5201:
5197:
5193:
5182:
5176:
5171:
5166:
5162:
5158:
5154:
5150:
5147:
5127:
5087:
5069:
5029:
5011:
4974:
4961:
4958:
4954:
4949:
4944:
4941:
4928:
4925:
4918:
4913:
4908:
4904:
4899:
4895:
4892:
4886:
4867:
4860:
4839:
4823:
4820:
4817:
4813:
4808:
4805:
4792:
4789:
4782:
4777:
4772:
4768:
4763:
4759:
4756:
4749:
4728:
4722:
4719:
4715:
4710:
4707:
4694:
4691:
4684:
4679:
4674:
4670:
4665:
4661:
4658:
4651:
4638:
4594:
4589:
4586:
4573:
4570:
4563:
4558:
4553:
4549:
4544:
4540:
4537:
4530:
4526:
4522:
4506:
4500:
4479:
4473:
4470:
4467:
4461:
4448:
4417:
4413:
4409:
4389:
4386:
4375:
4354:
4348:
4345:
4342:
4336:
4305:
4289:
4274:
4253:
4249:
4245:
4237:
4221:
4199:
4196:
4193:
4189:
4183:
4179:
4175:
4169:
4163:
4143:
4134:
4117:
4114:
4109:
4105:
4101:
4099:
4094:
4087:
4084:
4081:
4078:
4076:
4071:
4047:
4042:
4037:
4033:
4029:
4025:
4021:
4016:
4013:
4010:
4006:
4002:
3999:
3991:
3971:
3968:
3956:
3948:
3944:
3938:
3933:
3929:
3925:
3923:
3918:
3911:
3908:
3896:
3890:
3885:
3880:
3876:
3872:
3870:
3865:
3837:
3833:
3829:
3818:
3810:
3806:
3800:
3795:
3791:
3787:
3783:
3775:
3769:
3765:
3763:
3754:
3750:
3739:
3733:
3728:
3723:
3719:
3715:
3711:
3707:
3704:
3701:
3699:
3694:
3663:
3655:
3651:
3645:
3641:
3635:
3630:
3627:
3624:
3620:
3616:
3610:
3602:
3599:
3596:
3592:
3586:
3581:
3578:
3575:
3571:
3567:
3565:
3557:
3549:
3545:
3537:
3531:
3523:
3519:
3513:
3509:
3503:
3498:
3495:
3492:
3488:
3484:
3478:
3470:
3466:
3460:
3455:
3452:
3449:
3445:
3441:
3439:
3431:
3425:
3397:
3394:
3391:
3387:
3381:
3376:
3372:
3351:
3343:
3333:
3330:
3325:
3321:
3316:
3314:
3310:
3306:
3286:
3281:
3277:
3271:
3267:
3261:
3256:
3253:
3250:
3246:
3242:
3239:
3237:
3232:
3225:
3220:
3216:
3210:
3206:
3200:
3195:
3192:
3189:
3185:
3181:
3178:
3176:
3171:
3144:
3139:
3135:
3129:
3125:
3119:
3114:
3111:
3108:
3104:
3100:
3097:
3095:
3090:
3083:
3078:
3074:
3068:
3064:
3058:
3053:
3050:
3047:
3043:
3039:
3036:
3034:
3029:
3001:
2997:
2991:
2987:
2981:
2977:
2971:
2966:
2963:
2960:
2956:
2952:
2949:
2945:
2937:
2931:
2927:
2925:
2916:
2910:
2906:
2900:
2896:
2890:
2885:
2882:
2879:
2875:
2871:
2868:
2864:
2860:
2857:
2854:
2852:
2847:
2835:
2832:. In case of
2831:
2808:
2789:
2785:
2779:
2774:
2770:
2746:
2741:
2737:
2731:
2727:
2721:
2716:
2713:
2710:
2706:
2702:
2699:
2697:
2692:
2685:
2680:
2676:
2670:
2666:
2660:
2655:
2652:
2649:
2645:
2641:
2638:
2636:
2631:
2607:
2602:
2598:
2587:
2581:
2578:
2560:
2555:
2551:
2505:
2502:
2499:
2462:
2461:
2460:
2446:
2441:
2438:
2434:
2430:
2425:
2422:
2418:
2414:
2369:
2366:
2355:
2348:
2347:optical depth
2341:
2340:transmittance
2330:
2323:
2313:
2306:
2302:
2292:
2285:
2275:
2268:
2258:
2257:
2256:
2238:
2235:
2223:
2215:
2211:
2205:
2200:
2196:
2190:
2186:
2180:
2175:
2172:
2169:
2165:
2161:
2156:
2152:
2146:
2141:
2138:
2135:
2131:
2127:
2125:
2120:
2113:
2110:
2098:
2090:
2086:
2080:
2075:
2071:
2065:
2061:
2055:
2050:
2047:
2044:
2040:
2036:
2031:
2027:
2021:
2016:
2013:
2010:
2006:
2002:
2000:
1995:
1967:
1963:
1959:
1948:
1940:
1936:
1930:
1925:
1921:
1915:
1911:
1905:
1900:
1897:
1894:
1890:
1886:
1882:
1874:
1868:
1864:
1862:
1853:
1849:
1838:
1830:
1826:
1820:
1815:
1811:
1805:
1801:
1795:
1790:
1787:
1784:
1780:
1776:
1772:
1768:
1765:
1762:
1760:
1755:
1735:
1734:concentration
1728:
1722:
1719:
1715:
1708:
1706:
1699:
1698:
1697:
1683:
1680:
1677:
1674:
1671:
1668:
1662:
1658:
1652:
1648:
1641:
1636:
1632:
1623:
1619:
1609:
1607:
1603:
1599:
1594:
1592:
1574:
1570:
1565:
1561:
1552:
1538:
1518:
1510:
1486:
1465:
1462:
1459:
1456:
1445:
1424:
1404:
1381:
1378:
1375:
1372:
1369:
1366:
1360:
1356:
1350:
1346:
1339:
1334:
1330:
1306:
1297:
1288:
1272:
1268:
1245:
1241:
1218:
1214:
1210:
1205:
1201:
1197:
1194:
1174:
1171:
1168:
1161:
1157:
1151:
1147:
1141:
1138:
1135:
1129:
1123:
1120:
1117:
1109:
1106:is called an
1093:
1070:
1067:
1064:
1058:
1055:
1052:
1045:
1041:
1035:
1031:
1025:
1022:
1002:
982:
960:
956:
933:
929:
920:
917:(rather than
916:
912:
908:
902:
889:
886:
883:
880:
877:
874:
866:
862:
857:
851:
847:
840:
835:
831:
799:
795:
790:
784:
780:
768:
763:
759:
732:
728:
699:
695:
683:
679:
675:
670:
668:
664:
654:
652:
648:
647:transmittance
644:
640:
635:
633:
617:
597:
575:
571:
567:
564:
541:
537:
515:
506:
501:
496:
492:
487:
474:
469:
466:
463:
459:
453:
449:
445:
442:
422:
419:
416:
413:
407:
403:
397:
393:
386:
383:
363:
360:
352:
347:
342:
338:
334:
328:
324:
318:
314:
307:
304:
276:
273:
265:
262:
259:
256:
248:
235:
232:
231:
225:
220:
216:
206:
204:
200:
196:
192:
188:
184:
180:
176:
172:
167:
163:
159:
155:
154:empirical law
151:
147:
143:
139:
135:
124:
121:
113:
110:February 2008
102:
99:
95:
92:
88:
85:
81:
78:
74:
71: –
70:
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
7872:Spectroscopy
7790:
7786:
7776:
7757:
7751:
7727:
7720:
7699:
7691:
7651:(1): 15420.
7648:
7644:
7634:
7623:. Retrieved
7619:
7610:
7588:(2): 14–16.
7585:
7581:
7571:
7553:
7534:
7528:
7494:(18): 2031.
7491:
7488:ChemPhysChem
7487:
7477:
7464:
7458:
7449:
7443:
7421:(5): 78–88.
7418:
7414:
7404:
7393:. Retrieved
7389:
7380:
7371:
7366:
7359:
7346:
7341:
7334:
7227:
7223:zenith angle
7163:
7160:optical mass
7159:
7154:
7145:formaldehyde
7080:water vapour
6819:zenith angle
6809:
6805:
6787:
6780:
6776:
6769:calculated.
6759:food samples
6736:
6734:components.
6711:
6702:
6693:
6684:
6681:
6566:
6557:
6554:
6473:
6458:
6450:
6414:
6396:
5818:
5590:
5395:we also have
5335:
5332:
4975:
4838:product rule
4303:
4295:
4135:
3989:
3339:
3317:
3304:
2833:
2542:
2500:
2498:radiant flux
2337:
2305:absorptivity
1739:
1718:absorptivity
1622:absorptivity
1615:
1595:
1553:
1298:
1294:
903:
671:
667:light sensor
660:
636:
631:
488:
236:
228:
212:
203:BGK equation
178:
174:
170:
149:
141:
133:
131:
116:
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
7137:tetraoxygen
7007:where each
5214:and finally
3988:In case of
3305:non-uniform
2501:transmitted
2352:and to its
915:attenuation
674:Ultraviolet
632:Beer's law.
500:propagation
495:attenuation
491:August Beer
230:Photometria
158:attenuation
7882:Visibility
7856:Categories
7625:2023-05-17
7395:2023-04-25
7326:References
7083:absorption
7021:refers to
4525:absorption
4521:scattering
4292:Derivation
3320:scattering
2354:absorbance
1705:absorbance
1606:scattering
1602:absorption
1598:extinction
919:absorption
663:collimated
639:absorbance
142:Beer's law
80:newspapers
7807:1083-3668
7665:2045-2322
7602:0960-3360
7294:Logarithm
7196:θ
7193:
6982:⋯
6968:τ
6946:τ
6931:τ
6906:τ
6888:τ
6873:τ
6858:τ
6848:−
6838:
6751:oxidation
6652:λ
6640:ε
6620:λ
6608:ε
6598:λ
6586:μ
6532:λ
6526:ε
6518:λ
6506:μ
6465:bilirubin
6427:turbidity
6337:ℓ
6328:∫
6318:ε
6297:∑
6293:−
6281:∧
6250:
6191:ℓ
6182:∫
6172:ε
6151:∑
6147:−
6138:
6088:ℓ
6079:∫
6069:ε
6041:
6015:∑
6011:−
6003:
5873:σ
5862:
5831:ε
5768:ℓ
5759:∫
5749:σ
5728:∑
5724:−
5716:
5634:μ
5617:σ
5542:μ
5536:ℓ
5527:∫
5523:−
5511:∧
5463:μ
5449:
5441:ℓ
5432:∫
5428:−
5420:
5373:
5366:μ
5351:μ
5294:μ
5289:ℓ
5280:∫
5276:−
5268:
5246:Φ
5231:Φ
5177:μ
5172:ℓ
5163:∫
5159:−
5151:
5133:Φ
5113:Φ
5088:ℓ
5075:Φ
5055:Φ
5017:Φ
4997:Φ
4919:μ
4905:∫
4896:
4873:Φ
4783:μ
4769:∫
4760:
4736:Φ
4723:μ
4685:μ
4671:∫
4662:
4625:Φ
4564:μ
4550:∫
4541:
4487:Φ
4474:μ
4471:−
4435:Φ
4362:Φ
4349:μ
4346:−
4324:Φ
4275:μ
4254:λ
4222:α
4197:μ
4194:−
4115:ℓ
4106:μ
4085:ℓ
4082:μ
4072:τ
4043:ℓ
4034:μ
4030:−
4017:ℓ
4014:μ
4011:−
3945:μ
3939:ℓ
3930:∫
3891:μ
3886:ℓ
3877:∫
3866:τ
3807:μ
3801:ℓ
3792:∫
3788:−
3776:∧
3734:μ
3729:ℓ
3720:∫
3716:−
3708:
3642:ε
3621:∑
3593:μ
3572:∑
3546:μ
3510:σ
3489:∑
3467:μ
3446:∑
3426:μ
3395:
3388:μ
3373:μ
3352:μ
3329:molecules
3303:Cases of
3268:ε
3247:∑
3243:ℓ
3207:σ
3186:∑
3182:ℓ
3172:τ
3126:ε
3105:∑
3101:ℓ
3065:σ
3044:∑
3040:ℓ
3030:τ
2978:ε
2957:∑
2953:ℓ
2950:−
2938:∧
2897:σ
2876:∑
2872:ℓ
2869:−
2861:
2728:ε
2707:∑
2703:ℓ
2667:σ
2646:∑
2642:ℓ
2632:τ
2599:σ
2582:
2552:ε
2515:Φ
2472:Φ
2439:−
2426:τ
2423:−
2395:Φ
2376:Φ
2206:ℓ
2197:∫
2187:ε
2166:∑
2132:∑
2081:ℓ
2072:∫
2062:σ
2041:∑
2028:τ
2007:∑
1996:τ
1931:ℓ
1922:∫
1912:ε
1891:∑
1887:−
1875:∧
1821:ℓ
1812:∫
1802:σ
1781:∑
1777:−
1769:
1681:ℓ
1678:ε
1642:
1539:σ
1463:ℓ
1460:σ
1425:ℓ
1379:ℓ
1376:ϵ
1340:
1307:ϵ
1269:μ
1242:μ
1215:μ
1202:μ
1195:μ
1172:μ
1142:
1124:
1118:−
1094:μ
1068:μ
1065:−
1059:
887:ℓ
884:ε
841:
769:
665:, with a
598:μ
572:μ
565:λ
538:λ
516:λ
467:μ
464:−
417:μ
387:
353:μ
339:∫
308:
263:μ
249:−
7825:34713647
7758:Surfaces
7683:29133925
7582:NIR News
7520:32662939
7279:Job plot
7242:See also
7123:nitrogen
7023:aerosols
6791:, where
6393:Validity
4945:′
4929:′
4809:′
4793:′
4711:′
4695:′
4590:′
4574:′
1478:, where
1086:, where
911:backward
682:Infrared
191:neutrons
7816:8553265
7674:5684246
7511:7540309
7423:Bibcode
7215:
7175:
7165:airmass
7158:is the
7149:glyoxal
6817:is the
6767:polymer
6399:analyte
5901:is the
5601:of the
3990:uniform
2834:uniform
2828:is the
2496:is the
2320:is the
2299:is the
2282:is the
2265:is the
1732:is the
1712:is the
1703:is the
1507:is the
907:forward
678:visible
645:of the
209:History
187:photons
173:or the
146:physics
94:scholar
7823:
7813:
7805:
7764:
7739:
7681:
7671:
7663:
7600:
7541:
7518:
7508:
7217:where
7167:factor
7121:) and
7112:oxygen
7041:oxygen
7009:τ
6813:where
6808:= sec
6761:. The
5894:where
4412:linear
4402:where
2821:where
2459:where
2350:τ
2260:σ
2255:where
1696:where
590:where
291:μ
185:, for
152:is an
148:, the
144:. In
96:
89:
82:
75:
67:
7696:IUPAC
7563:(PDF)
7370:[
7345:[
7173:) to
7098:ozone
5103:gives
197:. In
193:, or
162:light
101:JSTOR
87:books
7821:PMID
7803:ISSN
7762:ISBN
7737:ISBN
7679:PMID
7661:ISSN
7598:ISSN
7539:ISBN
7516:PMID
7353:–22.
7141:HONO
6749:and
6741:and
6709:and
6691:and
6564:and
6431:DOAS
132:The
73:news
7811:PMC
7795:doi
7710:doi
7708:".
7669:PMC
7653:doi
7590:doi
7506:PMC
7496:doi
7431:doi
7419:162
7190:cos
7162:or
7106:is
7096:is
7078:is
7058:is
7035:(CO
6835:exp
6135:exp
6000:exp
5713:exp
5417:exp
5265:exp
5148:exp
4893:exp
4757:exp
4659:exp
4538:exp
4523:or
3854:and
3705:exp
2858:exp
2303:or
1766:exp
1716:or
1633:log
1437:is
1331:log
1056:exp
909:or
832:log
760:log
160:of
56:by
7858::
7819:.
7809:.
7801:.
7791:26
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