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{{short description|Logarithm of ratio of incident to transmitted radiant power through a sample}}
{{About|a quantitative expression|the process itself|Absorption (electromagnetic radiation)}}
{{Redirect|Optical density|other uses|Refractive index|and|Nucleic acid quantitation|and|Neutral-density filter}}

'''Absorbance''' is defined as "the logarithm of the ratio of incident to transmitted radiant power through a sample (excluding the effects on cell walls)".<ref name="GoldBook"/> Alternatively, for samples which scatter light, absorbance may be defined as "the negative logarithm of one minus absorptance, as measured on a uniform sample".<ref name=":0">{{GoldBookRef |title=decadic absorbance |file=D01536 }}</ref> The term is used in many technical areas to quantify the results of an experimental measurement. While the term has its origin in quantifying the absorption of light, it is often entangled with quantification of light which is “lost” to a detector system through other mechanisms. What these uses of the term tend to have in common is that they refer to a logarithm of the ratio of a quantity of light incident on a sample or material to that which is detected after the light has interacted with the sample. 

The term [[Absorption (electromagnetic radiation)|absorption]] refers to the physical process of absorbing light, while absorbance does not always measure only absorption; it may measure [[attenuation]] (of transmitted radiant power) caused by absorption, as well as reflection, scattering, and other physical processes. Sometimes the term "attenuance" or "experimental absorbance" is used to emphasize that radiation is lost from the beam by processes other than absorption, with the term "internal absorbance" used to emphasize that the necessary corrections have been made to eliminate the effects of phenomena other than absorption.<ref name=":1" />

== History and uses of the term absorbance ==

=== Beer-Lambert law ===
The roots of the term absorbance are in the [[Beer-Lambert law#Beer–Lambert law|Beer–Lambert law]]. As light moves through a medium, it will become dimmer as it is being "extinguished". Bouguer recognized that this extinction (now often called attenuation) was not linear with distance traveled through the medium, but related by what we now refer to as an exponential function. 

If <math>I_0</math> is the intensity of the light at the beginning of the travel and <math>I_d</math> is the intensity of the light detected after travel of a distance {{nowrap|<math>d</math>,}} the fraction transmitted, {{nowrap|<math>T</math>,}} is given by

<math display="block">T=\frac {I_d}{I_0} = \exp(-\mu d)\,,</math>

where <math>\mu</math> is called an [[Propagation constant#Attenuation constant|attenuation constant]] (a term used in various fields where a signal is transmitted though a medium) or coefficient. The amount of light transmitted is falling off exponentially with distance. Taking the natural logarithm in the above equation, we get

<math display="block">-\ln(T) = \ln \frac {I_0}{I_d} = \mu d\,.</math>

For scattering media, the constant is often divided into two parts,<ref>{{Cite book |last=Van de Hulst |first=H. C. |title=Light Scattering by Small Particles |publisher=John Wiley and Sons |year=1957 |isbn=9780486642284 |location=New York}}</ref> {{nowrap|<math>\mu = \mu_s + \mu_a </math>,}} separating it into a scattering coefficient <math>\mu _s</math> and an absorption coefficient {{nowrap|<math>\mu_a</math>,}} obtaining

<math display="block">-\ln(T) = \ln \frac {I_0}{I_s} = (\mu_s + \mu_a) d\,.</math>

If a size of a detector is very small compared to the distance traveled by the light, any light that is scattered by a particle, either in the forward or backward direction, will not strike the detector. (Bouguer was studying astronomical phenomena, so this condition was met.) In such case, a plot of <math>-\ln(T)</math> as a function of wavelength will yield a superposition of the effects of absorption and scatter. Because the absorption portion is more distinct and tends to ride on a background of the scatter portion, it is often used to identify and quantify the absorbing species. Consequently, this is often referred to as [[absorption spectroscopy]], and the plotted quantity is called "absorbance", symbolized as {{nowrap|<math>\Alpha</math>.}} Some disciplines by convention use decadic (base 10) absorbance rather than Napierian (natural) absorbance, resulting in: <math>\Alpha_{10} = \mu_{10}d </math> (with the subscript 10 usually not shown).

=== Absorbance for non-scattering samples ===
Within a homogeneous medium such as a solution, there is no scattering. For this case, researched extensively by [[August Beer]], the concentration of the absorbing species follows the same linear contribution to absorbance as the path-length. Additionally, the contributions of individual absorbing species are additive. This is a very favorable situation, and made absorbance an absorption metric far preferable to absorption fraction (absorptance). This is the case for which the term "absorbance" was first used.

A common expression of the [[Beer's law]] relates the attenuation of light in a material as: {{nowrap|<math>\Alpha = \varepsilon\ell c </math>,}} where <math>\Alpha </math> is the '''absorbance;''' <math>\varepsilon </math> is the [[molar attenuation coefficient]] or [[Molar absorptivity|absorptivity]] of the attenuating species; <math>\ell </math> is the optical path length; and <math>c </math> is the concentration of the attenuating species.

=== Absorbance for scattering samples ===
For samples which scatter light, absorbance is defined as "the negative logarithm of one minus absorptance (absorption fraction: <math>\alpha</math>) as measured on a uniform sample".<ref name=":0" /> For decadic absorbance,<ref name=":1">{{cite book |doi=10.1002/0470027320.s8401 |chapter=Glossary of Terms used in Vibrational Spectroscopy |title=Handbook of Vibrational Spectroscopy |year=2006 |last1=Bertie |first1=John E. |isbn=0471988472 |editor1-first=Peter R |editor1-last=Griffiths }}</ref> this may be symbolized as {{nowrap|<math>\Alpha_{10}=-\log_{10}(1-\alpha)</math>.}} If a sample both transmits and [[Diffuse reflectance spectroscopy#Remission|remits light]], and is not luminescent, the fraction of light absorbed {{nowrap|(<math>\alpha</math>),}} remitted {{nowrap|(<math>R</math>),}} and transmitted {{nowrap|(<math>T</math>)}} add to 1: {{nowrap|<math>\alpha + R + T =1</math>.}} Note that {{nowrap|<math>1-\alpha = R+T </math>,}} and the formula may be written as {{nowrap|<math>\Alpha _{10}=-\log_{10}(R+T)</math>.}} For a sample which does not scatter, {{nowrap|<math>R=0 </math>,}} and {{nowrap|<math>1-\alpha = T</math>,}} yielding the formula for absorbance of a material discussed below. 

Even though this absorbance function is very useful with scattering samples, the function does not have the same desirable characteristics as it does for non-scattering samples. There is, however, a property called [[Representative layer theory#Absorbing Power: The Scatter Corrected Absorbance of a sample|absorbing power]] which may be estimated for these samples. The [[Representative layer theory#Absorbing Power: The Scatter Corrected Absorbance of a sample|absorbing power]] of a single unit thickness of material making up a scattering sample is the same as the absorbance of the same thickness of the material in the absence of scatter.<ref>{{cite book |doi=10.1255/978-1-901019-05-6 |title=Interpreting Diffuse Reflectance and Transmittance: A Theoretical Introduction to Absorption Spectroscopy of Scattering Materials |year=2007 |last1=Dahm |first1=Donald |last2=Dahm |first2=Kevin |isbn=9781901019056 }}</ref>

=== Optics ===
In [[optics]], '''absorbance''' or '''decadic absorbance''' is the ''[[common logarithm]]'' of the ratio of incident to {{em|transmitted}} [[radiant power]] through a material, and '''spectral absorbance''' or '''spectral decadic absorbance''' is the common logarithm of the ratio of incident to {{em|transmitted}} [[Radiant power|spectral radiant power]] through a material. Absorbance is [[Dimensionless quantity|dimensionless]], and in particular is not a length, though it is a monotonically increasing function of path length, and approaches zero as the path length approaches zero.

==Mathematical definitions==

===Absorbance of a material===
The '''absorbance''' of a material, denoted {{mvar|A}}, is given by<ref name="GoldBook">{{GoldBookRef|title=Absorbance|file=A00028|accessdate=2015-03-15}}</ref>

<math display="block">A = \log_{10} \frac{\Phi_\text{e}^\text{i}}{\Phi_\text{e}^\text{t}} = -\log_{10} T,</math>

where

* <math display="inline">\Phi_\text{e}^\text{t}</math> is the [[radiant flux]] {{em|transmitted}} by that material,
* <math display="inline">\Phi_\text{e}^\text{i}</math> is the [[radiant flux]] {{em|received}} by that material, and
* <math display="inline">T = \Phi_\text{e}^\text{t}/\Phi_\text{e}^\text{i}</math> is the [[transmittance]] of that material.

Absorbance is a [[dimensionless]] quantity. Nevertheless, the '''absorbance unit''' or '''AU''' is commonly used in [[ultraviolet–visible spectroscopy]] and its [[high-performance liquid chromatography]] applications, often in derived units such as the milli-absorbance unit (mAU) or milli-absorbance unit-minutes (mAU×min), a unit of absorbance integrated over time.<ref>{{cite web |author= GE Health Care |title= ÄKTA Laboratory-Scale Chromatography Systems - Instrument Management Handbook |date= 2015 |publisher= GE Healthcare Bio-Sciences AB |location= Uppsala |url= https://cdn.gelifesciences.com/dmm3bwsv3/AssetStream.aspx?mediaformatid=10061&destinationid=10016&assetid=16189 |archive-url= https://web.archive.org/web/20200315013424/https://cdn.gelifesciences.com/dmm3bwsv3/AssetStream.aspx?mediaformatid=10061&destinationid=10016&assetid=16189 |archive-date= 2020-03-15 }}</ref>

Absorbance is related to [[optical depth]] by

<math display="block">A = \frac{\tau}{\ln 10} = \tau \log_{10} e \,,</math>

where {{mvar|τ}} is the optical depth.

===Spectral absorbance===
'''Spectral absorbance in frequency''' and '''spectral absorbance in wavelength''' of a material, denoted {{math|''A{{sub|ν}}''}} and {{math|''A{{sub|λ}}''}} respectively, are given by<ref name=GoldBook />

<math display="block">\begin{align}
A_\nu &= \log_{10} \frac{\Phi_{\text{e},\nu}^\text{i}}{\Phi_{\text{e},\nu}^\text{t}} = -\log_{10} T_\nu\,, \\
A_\lambda &= \log_{10} \frac{\Phi_{\text{e},\lambda}^\text{i}}{\Phi_{\text{e},\lambda}^\text{t}} = -\log_{10} T_\lambda\,,
\end{align}</math>

where

* <math display="inline">\Phi_{\mathrm{e},\nu}^t</math> is the [[Radiant flux|spectral radiant flux in frequency]] {{em|transmitted}} by that material;
* <math display="inline">\Phi_{\mathrm{e},\nu}^i</math> is the spectral radiant flux in frequency {{em|received}} by that material;
* <math display="inline">T_\nu</math> is the [[Transmittance|spectral transmittance in frequency]] of that material;
* <math display="inline">\Phi_{\mathrm{e},\lambda}^t</math> is the [[Radiant flux|spectral radiant flux in wavelength]] {{em|transmitted}} by that material;
* <math display="inline">\Phi_{\mathrm{e},\lambda}^i</math> is the spectral radiant flux in wavelength {{em|received}} by that material; and
* <math display="inline">T_\lambda</math> is the [[Transmittance|spectral transmittance in wavelength]] of that material.

Spectral absorbance is related to spectral optical depth by

<math display="block">\begin{align}
A_\nu &= \frac{\tau_\nu}{\ln 10} = \tau_\nu \log_{10} e \,, \\
A_\lambda &= \frac{\tau_\lambda}{\ln 10} = \tau_\lambda \log_{10} e \,,
\end{align}</math>

where
* {{mvar|τ{{sub|ν}}}} is the spectral optical depth in frequency, and
* {{mvar|τ{{sub|λ}}}} is the spectral optical depth in wavelength.

Although absorbance is properly unitless, it is sometimes reported in "absorbance units", or AU. Many people, including scientific researchers, wrongly state the results from absorbance measurement experiments in terms of these made-up units.<ref>{{cite journal |doi=10.1021/jz4006916 |title=How to Make Your Next Paper Scientifically Effective |journal=J. Phys. Chem. Lett. |date=2013 |volume=4 |pages=1578–1581 |issue=9|last1=Kamat |first1=Prashant |last2=Schatz |first2=George C. |pmid=26282316 |doi-access=free }}</ref>

== Relationship with attenuation ==

===Attenuance===
Absorbance is a number that measures the ''attenuation'' of the transmitted radiant power in a material. Attenuation can be caused by the physical process of "absorption", but also reflection, scattering, and other physical processes. Absorbance of a material is approximately equal to its attenuance{{clarify|reason=This term desperately needs a definition, otherwise the whole point of the fine distinctions in this section will be lost. Is attenuance a numerical measure of the physical process of attenuation, or are they also, like absorbance and absorption, unexpectedly distinct?|date=April 2015}} when both the absorbance is much less than 1 and the emittance of that material (not to be confused with [[radiant exitance]] or [[emissivity]]) is much less than the absorbance. Indeed,

<math display="block">\Phi_\mathrm{e}^\mathrm{t} + \Phi_\mathrm{e}^\mathrm{att} = \Phi_\mathrm{e}^\mathrm{i} + \Phi_\mathrm{e}^\mathrm{e}\,,</math>

where
* <math display="inline">\Phi_\mathrm{e}^\mathrm{t}</math> is the radiant power transmitted by that material,
* <math display="inline">\Phi_\mathrm{e}^\mathrm{att}</math> is the radiant power attenuated by that material,
* <math display="inline">\Phi_\mathrm{e}^\mathrm{i}</math> is the radiant power received by that material, and
* <math display="inline">\Phi_\mathrm{e}^\mathrm{e}</math> is the radiant power emitted by that material.

This is equivalent to

<math display="block">T + \mathrm{ATT} = 1 + E\,,</math>

where
* <math display="inline">T = \Phi_\mathrm{e}^\mathrm{t} / \Phi_\mathrm{e}^\mathrm{i}</math> is the transmittance of that material,
* <math display="inline">\mathrm{ATT} = \Phi_\mathrm{e}^\mathrm{att} / \Phi_\mathrm{e}^\mathrm{i}</math> is the {{em|attenuance}} of that material,
* <math display="inline">E = \Phi_\mathrm{e}^\mathrm{e} / \Phi_\mathrm{e}^\mathrm{i}</math> is the emittance of that material.

According to the [[Beer–Lambert law]], {{math|1=''T'' = 10<sup>-''A''</sup>}}, so
* <math>\mathrm{ATT} = 1 - 10^{-A} + E \approx A \ln 10 + E, \quad \text{if}\ A \ll 1,</math>
and finally
* <math>\mathrm{ATT} \approx A \ln 10, \quad \text{if}\ E \ll A.</math>

===Attenuation coefficient===
Absorbance of a material is also related to its ''[[Attenuation coefficient|decadic attenuation coefficient]]'' by

<math display="block">A = \int_0^l a(z)\, \mathrm{d}z\,,</math>

where
* {{mvar|l}} is the thickness of that material through which the light travels, and
* {{math|''a''(''z'')}} is the ''decadic attenuation coefficient'' of that material at {{mvar|z}}.

If ''a''(''z'') is uniform along the path, the attenuation is said to be a ''linear attenuation'', and the relation becomes
<math display="block">A = al.</math>

Sometimes the relation is given using the ''[[molar attenuation coefficient]]'' of the material, that is its attenuation coefficient divided by its [[molar concentration]]:

<math display="block">A = \int_0^l \varepsilon c(z)\, \mathrm{d}z\,,</math>

where
* {{mvar|ε}} is the ''molar attenuation coefficient'' of that material, and
* {{math|''c''(''z'')}} is the molar concentration of that material at {{mvar|z}}.

If {{math|''c''(''z'')}} is uniform along the path, the relation becomes

<math display="block">A = \varepsilon cl\,.</math>

The use of the term "molar absorptivity" for molar attenuation coefficient is discouraged.<ref name=GoldBook />

==Measurements==

===Logarithmic vs. directly proportional measurements===
The amount of light transmitted through a material diminishes [[Exponential function|exponentially]] as it travels through the material, according to the Beer–Lambert law ({{math|1=''A'' = (''ε'')(''l'')}}). Since the absorbance of a sample is measured as a logarithm, it is directly proportional to the thickness of the sample and to the concentration of the absorbing material in the sample. Some other measures related to absorption, such as transmittance, are measured as a simple ratio so they vary exponentially with the thickness and concentration of the material.

{| class="wikitable sortable" style="text-align: center;"
|+ Aborbances and equivalent transmittances
|-
! scope="col" | Absorbance: <math display="inline">-\log_{10}\left(\Phi_\mathrm{e}^\mathrm{t}/\Phi_\mathrm{e}^\mathrm{i}\right)</math>
! scope="col" | Transmittance: <math display="inline">\Phi_\mathrm{e}^\mathrm{t}/\Phi_\mathrm{e}^\mathrm{i}</math>
|-
| 0
| 1
|-
| 0.1
| 0.79
|-
| 0.25
| 0.56
|-
| 0.5
| 0.32
|-
| 0.75
| 0.18
|-
| 0.9
| 0.13
|-
| 1
| 0.1
|-
| 2
| 0.01
|-
| 3
| 0.001
|}

===Instrument measurement range===
Any real measuring instrument has a limited range over which it can accurately measure absorbance. An instrument must be calibrated and checked against known standards if the readings are to be trusted. Many instruments will become non-linear (fail to follow the Beer–Lambert law) starting at approximately 2 AU (~1% transmission). It is also difficult to accurately measure very small absorbance values (below {{val|e=-4}}) with commercially available instruments for chemical analysis. In such cases, [[Laser absorption spectrometry|laser-based absorption techniques]] can be used, since they have demonstrated detection limits that supersede those obtained by conventional non-laser-based instruments by many orders of magnitude (detection has been demonstrated all the way down to {{val|5e-13}}). The theoretical best accuracy for most commercially available non-laser-based instruments is attained in the range near 1 AU. The path length or concentration should then, when possible, be adjusted to achieve readings near this range.

===Method of measurement===
Typically, absorbance of a dissolved substance is measured using [[absorption spectroscopy]]. This involves shining a light through a solution and recording how much light and what wavelengths were transmitted onto a detector. Using this information, the wavelengths that were absorbed can be determined.<ref>{{cite web|last1=Reusch|first1=William|title=Visible and Ultraviolet Spectroscopy|url=https://www2.chemistry.msu.edu/faculty/reusch/virttxtjml/Spectrpy/UV-Vis/spectrum.htm|access-date=2014-10-29}}</ref> First, measurements on a "blank" are taken using just the solvent for reference purposes. This is so that the absorbance of the solvent is known, and then any change in absorbance when measuring the whole solution is made by just the solute of interest. Then measurements of the solution are taken. The transmitted spectral radiant flux that makes it through the solution sample is measured and compared to the incident spectral radiant flux. As stated above, the spectral absorbance at a given wavelength is

<math display="block">A_\lambda = \log_{10}\!\left(\frac{\Phi_{\mathrm{e},\lambda}^\mathrm{i}}{\Phi_{\mathrm{e},\lambda}^\mathrm{t}}\right)\!.</math>

The absorbance spectrum is plotted on a graph of absorbance vs. wavelength.<ref>{{cite web|last1=Reusch|first1=William|title=Empirical Rules for Absorption Wavelengths of Conjugated Systems|url=https://www2.chemistry.msu.edu/faculty/reusch/virttxtjml/Spectrpy/UV-Vis/uvspec.htm#uv1|access-date=2014-10-29}}</ref>

An [[Ultraviolet-visible spectroscopy#Ultraviolet–visible spectrophotometer]] will do all this automatically. To use this machine, solutions are placed in a small [[cuvette]] and inserted into the holder. The machine is controlled through a computer and, once it has been "blanked", automatically displays the absorbance plotted against wavelength. Getting the absorbance spectrum of a solution is useful for determining the concentration of that solution using the Beer–Lambert law and is used in [[HPLC]].

==Shade number==
Some filters, notably [[welding]] glass, are rated by shade number (SN), which is 7/3 times the absorbance plus one:<ref>{{cite web |url=http://www.unc.edu/~rowlett/units/dictS.html |author=Russ Rowlett |title=How Many? A Dictionary of Units of Measurement |publisher=Unc.edu |date=2004-09-01 |access-date=2010-09-20 |archive-date=1998-12-03 |archive-url=https://web.archive.org/web/19981203072555/http://www.unc.edu/~rowlett/units/dictS.html |url-status=dead }}</ref>

<math display="block">\begin{align}
\mathrm{SN} &= \frac{7}{3} A + 1 \\
            &= \frac{7}{3}(-\log_{10} T) + 1\,.
\end{align}</math>

For example, if the filter has 0.1% transmittance (0.001 transmittance, which is 3 absorbance units), its shade number would be 8.

==See also==
*[[Absorptance]]
*[[Tunable Diode Laser Absorption Spectroscopy]] (TDLAS)
*[[Densitometry]]
*[[Neutral density filter]]
*[[Mathematical descriptions of opacity]]

==References==
{{Reflist}}

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[[Category:Spectroscopy]]
[[Category:Optical filters]]
[[Category:Logarithmic scales of measurement]]
[[Category:Physical quantities]]