Transmittance

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Diagram of Beer-Lambert Law of transmittance of a beam of light as it travels through a cuvette of width l.
Earth's atmospheric transmittance. Because of the natural radiation of the hot atmosphere, the intensity of radiation is different from the transmitted part.
Transmittance of ruby in optical and near-IR spectra. Note the two broad blue and green absorption bands and one narrow absorption band on the wavelength of 694 nm, which is the wavelength of the ruby laser.

In optics and spectroscopy, transmittance is the fraction of incident light (electromagnetic radiation) at a specified wavelength that passes through a sample.[1][2] The terms visible transmittance (VT) and visible absorptance (VA), which are the respective fractions for the spectrum of light visible radiation, are also used. The natural radiation of the cuvette corresponding to the temperature of the cuvette remains ignored - see radiative transfer equation.

A related terms is absorbance,[3] or absorption factor,[4] which is the fraction of radiation absorbed by a sample at a specified wavelength.

Beer–Lambert law[edit]

In equation form,

\mathcal{T}_\lambda = {I\over I_{0}} \qquad \mathcal{A}_\lambda = \frac{I_0-I}{I_0}

where I_0 is the intensity of the incident radiation and I is the intensity of the radiation coming out of the sample and \mathcal{T}_\lambda and \mathcal{A}_\lambda are transmittance and absorptance respectively. In these equations, scattering and reflection are considered to be close to zero or otherwise accounted for. The transmittance of a sample is sometimes given as a percentage.

For liquids, transmittance is related to absorbance A (not to be confused with absorptance) as

A = - \log_{10}\mathcal{T}\ = - \log_{10}\left({I\over I_{0}}\right)

In the case of gases it is customary to use natural logarithms instead, making absorbance A for gases

A = - \ln\mathcal{T}\ = - \ln\left({I\over I_{0}}\right)

From the above equation and the Beer-Lambert law, the transmittance for gases is thus given by

\mathcal{T} = e^{-\alpha \, x},

where \alpha is the attenuation coefficient and x is the path length. For liquids e is replaced by 10.

Note that the term "transmission" refers to the physical process of radiation passing through a sample, whereas transmittance refers to the mathematical quantity.

Transmittance in the atmosphere[edit]

For atmospheric science applications, it is useful to introduce the quantity of optical depth \tau. Given a coefficient of extinction within an atmospheric column, \beta_e, optical depth is defined to be

\tau = \int_{z_1}^{z_2} \beta_e(z) \mathrm{d}z.

With this, transmittance can be given as

\mathcal{T} = e^{-\tau},

or

\mathcal{T} = e^{-\tau / \mu }

where, when the plane parallel assumption is invoked, \mu=|\cos(\theta)| with \theta the angle of propagation of the ray from the normal of the surface.

See also[edit]

Another equation that can be useful in solving for \tau is the following: A = 2 - \log(\tau) Where A is a measure of absorbance. By manipulating the equation you can generate the more direct form of the equation: \tau = 10^{2-A}

References[edit]

  1. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "Transmittance".
  2. ^ Verhoeven, J. W. (1996). "Glossary of terms used in photochemistry (IUPAC Recommendations 1996)". Pure and Applied Chemistry 68 (12): 2223–2286. doi:10.1351/pac199668122223. ISSN 0033-4545. 
  3. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "Absorbance".
  4. ^ "CRC Dictionary of pure and applied physics, CRC Press, Editor: Dipak Basu (2001)".