Editing Absorbance (section)

== 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.