Absorbance

Absorbance of a material, denoted A, is given by[1]

${\displaystyle A=\log _{10}{\frac {\Phi _{\text{e}}^{\text{i}}}{\Phi _{\text{e}}^{\text{t}}}}=-\log _{10}T,}$

where

${\displaystyle \Phi _{\text{e}}^{\text{t}}}$ is the radiant flux transmitted by that material,

${\displaystyle \Phi _{\text{e}}^{\text{i}}}$ is the radiant flux received by that material,

${\displaystyle T=\Phi _{\text{e}}^{\text{t}}/\Phi _{\text{e}}^{\text{i}}}$ 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.[2]

Absorbance is related to optical depth by

${\displaystyle A={\frac {\tau }{\ln 10}}=\tau \log _{10}e\,,}$

where τ is the optical depth.

Spectral absorbance in frequency and spectral absorbance in wavelength of a material, denoted A_ν and A_λ respectively, are given by[1]

${\displaystyle A_{\nu }=\log _{10}{\frac {\Phi _{{\text{e}},\nu }^{\text{i}}}{\Phi _{{\text{e}},\nu }^{\text{t}}}}=-\log _{10}T_{\nu },}$

${\displaystyle A_{\lambda }=\log _{10}{\frac {\Phi _{{\text{e}},\lambda }^{\text{i}}}{\Phi _{{\text{e}},\lambda }^{\text{t}}}}=-\log _{10}T_{\lambda },}$

where

Φ_e,ν^t is the spectral radiant flux in frequency transmitted by that material,

Φ_e,νⁱ is the spectral radiant flux in frequency received by that material,

T_ν is the spectral transmittance in frequency of that material,

Φ_e,λ^t is the spectral radiant flux in wavelength transmitted by that material,

Φ_e,λⁱ is the spectral radiant flux in wavelength received by that material,

T_λ is the spectral transmittance in wavelength of that material.

Spectral absorbance is related to spectral optical depth by

${\displaystyle A_{\nu }={\frac {\tau _{\nu }}{\ln 10}}=\tau _{\nu }\log _{10}e\,,}$

${\displaystyle A_{\lambda }={\frac {\tau _{\lambda }}{\ln 10}}=\tau _{\lambda }\log _{10}e\,,}$

where

τ_ν is the spectral optical depth in frequency,

τ_λ is the spectral optical depth in wavelength.

Although absorbance is properly unitless, it is sometimes reported in "arbitrary units", or AU. Many people, including scientific researchers, wrongly state the results from absorbance measurement experiments in terms of these made-up units.[3]

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 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,

${\displaystyle \Phi _{\text{e}}^{\text{t}}+\Phi _{\text{e}}^{\text{att}}=\Phi _{\text{e}}^{\text{i}}+\Phi _{\text{e}}^{\text{e}},}$

where

Φ_e^t is the radiant power transmitted by that material,

Φ_e^att is the radiant power attenuated by that material,

Φ_eⁱ is the radiant power received by that material,

Φ_e^e is the radiant power emitted by that material,

that is equivalent to

${\displaystyle T+{\text{ATT}}=1+E,}$

where

T = Φ_e^t/Φ_eⁱ is the transmittance of that material,

ATT = Φ_e^att/Φ_eⁱ is the attenuance of that material,

E = Φ_e^e/Φ_eⁱ is the emittance of that material,

and according to Beer–Lambert law, T = 10^−A, so

${\displaystyle {\text{ATT}}=1-10^{-A}+E\approx A\ln 10+E,\quad {\text{if}}\ A\ll 1,}$

and finally

${\displaystyle {\text{ATT}}\approx A\ln 10,\quad {\text{if}}\ E\ll A.}$

Absorbance of a material is also related to its decadic attenuation coefficient by

${\displaystyle A=\int _{0}^{l}a(z)\,\mathrm {d} z,}$

where

l is the thickness of that material through which the light travels,

a(z) is the decadic attenuation coefficient of that material at z.

If a(z) is uniform along the path, the attenuation is said to be a linear attenuation, and the relation becomes

${\displaystyle A=al.}$

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

${\displaystyle A=\int _{0}^{l}\varepsilon c(z)\,\mathrm {d} z,}$

where

ε is the molar attenuation coefficient of that material,

c(z) is the molar concentration of that material at z.

If c(z) is uniform along the path, the relation becomes

${\displaystyle A=\varepsilon cl.}$

The use of the term "molar absorptivity" for molar attenuation coefficient is discouraged.[1]

The amount of light transmitted through a material diminishes exponentially as it travels through the material, according to the Beer–Lambert law (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.

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 10⁻⁴) with commercially available instruments for chemical analysis. In such cases, 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 (detections have been demonstrated all the way down to 5 × 10⁻¹³). 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.

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.[4] 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

${\displaystyle A_{\lambda }=\log _{10}\!\left({\frac {\Phi _{\mathrm {e} ,\lambda }^{\mathrm {i} }}{\Phi _{\mathrm {e} ,\lambda }^{\mathrm {t} }}}\right)\!.}$

The absorbance spectrum is plotted on a graph of absorbance vs. wavelength.[5]

A UV-Vis 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.