Absorbance

Mathematical definitions

Absorbance of a material

The absorbance of a material, denoted A, is given by

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

where

• ${\mathrm{\Phi }}_{\text{e}}^{\text{t}}$ is the radiant flux transmitted by that material,
• ${\mathrm{\Phi }}_{\text{e}}^{\text{i}}$ is the radiant flux received by that material, and
• $T={\mathrm{\Phi }}_{\text{e}}^{\text{t}}/{\mathrm{\Phi }}_{\text{e}}^{\text{i}}$ 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.

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

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

$${\displaystyle \begin{array}{cc}{A}_{\nu }& ={\log }_{10}\frac{{\mathrm{\Phi }}_{\text{e},\nu }^{\text{i}}}{{\mathrm{\Phi }}_{\text{e},\nu }^{\text{t}}}=-{\log }_{10}{T}_{\nu },\\ A_{\lambda }& ={\log }_{10}\frac{{\mathrm{\Phi }}_{\text{e},\lambda }^{\text{i}}}{{\mathrm{\Phi }}_{\text{e},\lambda }^{\text{t}}}=-{\log }_{10}{T}_{\lambda },\end{array}}$$

where

• ${\mathrm{\Phi }}_{\mathrm{e},\nu }^t$ is the spectral radiant flux in frequency transmitted by that material;
• ${\mathrm{\Phi }}_{\mathrm{e},\nu }^i$ is the spectral radiant flux in frequency received by that material;
• $T_{\nu }$ is the spectral transmittance in frequency of that material;
• ${\mathrm{\Phi }}_{\mathrm{e},\lambda }^t$ is the spectral radiant flux in wavelength transmitted by that material;
• ${\mathrm{\Phi }}_{\mathrm{e},\lambda }^i$ is the spectral radiant flux in wavelength received by that material; and
• $T_{\lambda }$ is the spectral transmittance in wavelength of that material.

Spectral absorbance is related to spectral optical depth by

$${\displaystyle \begin{array}{cc}{A}_{\nu }& =\frac{{\tau }_{\nu }}{\ln 10}={\tau }_{\nu }{\log }_{10}e,\\ A_{\lambda }& =\frac{{\tau }_{\lambda }}{\ln 10}={\tau }_{\lambda }{\log }_{10}e,\end{array}}$$

where

• τ_ν is the spectral optical depth in frequency, and
• τ_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.

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 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,

$${\displaystyle {\mathrm{\Phi }}_{\mathrm{e}}^{\mathrm{t}}+{\mathrm{\Phi }}_{\mathrm{e}}^{\mathrm{a}\mathrm{t}\mathrm{t}}={\mathrm{\Phi }}_{\mathrm{e}}^{\mathrm{i}}+{\mathrm{\Phi }}_{\mathrm{e}}^{\mathrm{e}},}$$

where

• ${\mathrm{\Phi }}_{\mathrm{e}}^{\mathrm{t}}$ is the radiant power transmitted by that material,
• ${\mathrm{\Phi }}_{\mathrm{e}}^{\mathrm{a}\mathrm{t}\mathrm{t}}$ is the radiant power attenuated by that material,
• ${\mathrm{\Phi }}_{\mathrm{e}}^{\mathrm{i}}$ is the radiant power received by that material, and
• ${\mathrm{\Phi }}_{\mathrm{e}}^{\mathrm{e}}$ is the radiant power emitted by that material.

This is equivalent to

$${\displaystyle T+\mathrm{A}\mathrm{T}\mathrm{T}=1+E,}$$

where

• $T={\mathrm{\Phi }}_{\mathrm{e}}^{\mathrm{t}}/{\mathrm{\Phi }}_{\mathrm{e}}^{\mathrm{i}}$ is the transmittance of that material,
• $\mathrm{A}\mathrm{T}\mathrm{T}={\mathrm{\Phi }}_{\mathrm{e}}^{\mathrm{a}\mathrm{t}\mathrm{t}}/{\mathrm{\Phi }}_{\mathrm{e}}^{\mathrm{i}}$ is the attenuance of that material,
• $E={\mathrm{\Phi }}_{\mathrm{e}}^{\mathrm{e}}/{\mathrm{\Phi }}_{\mathrm{e}}^{\mathrm{i}}$ is the emittance of that material.

According to the Beer's law, T = 10^−A, so

• ${\displaystyle \mathrm{A}\mathrm{T}\mathrm{T}=1-{10}^{-A}+E\approx A\ln 10+E,\text{if}A\ll 1,}$

and finally

• ${\displaystyle \mathrm{A}\mathrm{T}\mathrm{T}\approx A\ln 10,\text{if}E\ll A.}$

Attenuation coefficient

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, and
• 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:

$${\displaystyle A={\int }_0^l\epsilon c(z)\mathrm{d}z,}$$

where

• ε is the molar attenuation coefficient of that material, and
• c(z) is the molar concentration of that material at z.

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

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

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

Use in Analytical Chemistry

Absorbance is a widely used measurement in quantitative absorption spectroscopy. While the attenuation of a light beam can be also be described by transmittance (the ratio of transmitted incident light), the logarithmic formulation of absorbance is convenient for sample quantification: under conditions where the Beer's law is valid, absorbance will be linearly proportional to sample thickness and the concentration of the absorptive species.

For quantitative purposes, absorbance is often measured on a sample solution held in a cuvette, where the solution is sufficiently dilute that the linear relationship of the Beer's law holds. The cuvette provides a known and consistent path length for the light beam passing through the sample. Measuring first the absorbance of the cuvette and a "blank" solution containing no analyte, differences in absorbance between samples can be used to quantity the analyte. Spectrometers generally measure absorbance separately for a range of wavelengths: this data is then plotted as absorbance vs. wavelength.

Shade number

Some filters, notably welding glass, are rated by shade number (SN), which is 7/3 times the absorbance plus one:

$${\displaystyle \begin{array}{cc}\mathrm{S}\mathrm{N}& =\frac{7}{3}A+1\\ & =\frac{7}{3}(-{\log }_{10}T)+1.\end{array}}$$

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