Absorptance

The absorptance measurement capacity of a medium to absorb the light which crosses it. It is also called optical Densité or extinction. It is about a size without unit given by the relation:

$A\_\; \backslash \; lambda\; =\; \backslash \; log\_\; \{10\}\; \backslash \; left\; (\backslash \; frac\; \{I\_0\}\; \{I\}\; \backslash \; right).$

The absorptance is defined by the ratio between the incidental light $I\_0$ which crosses the medium to a Wavelength λ, and the transmitted light $I$ expressed in Logarithme basic 10.

The absorptance differs according to nature from the element and the wavelength under which he is studied.

Principles of absorption of the light

That is to say a Monochromatic beam of light which crosses a layer of transparent matter (like a piece of ice for example). The majority of radiant energy will pass through this substance, but a small portion will be considered or absorptive. The total sum of incidental energy will be preserved, and expressed by the relation:

$P\_0\; =\; P\_a\; +\; P\_t\; +\; P\_r.\; \backslash ;$

$P\_0$ is incidental energy, $P\_a$ is absorptive energy, $P\_t$ is transmitted energy and $P\_r$ is considered energy. If one can compensate for the energy lost by the reflection, with the help of a development technical, one will interest in the relation between incidental energy and transmitted energy. This relation was formulated by Pierre Bouguer in 1729. The law of Bouguer (known also as the law of Lambert) consists of two parts; the first defines the Transmittance, the second the variation of the absorptance according to the thickness of the layer of substance crossed by the light.

Law of Bouguer

Like agreed previously, the energy transmitted by a homogeneous medium is proportional to the energy applied to this one. Thus, the energy transmitted by this one will be always part of total energy applied. This east report/ratio defines as the transmittance , T, which can be expressed by

$T=\backslash frac\{P\_t\}\{P\_0\}\; .\; \backslash ;$

For a given substance, with a thickness and a defined wavelength, T is a constant. How does transmittance according to the thickness of the medium vary then? To answer this question one will take the following example: it is supposed that a substance, 1cm thickness, allows 50% energy received to pass through it. Otherwise, it has a transmittance of 0,5. If this light obtained passes by a second of the same layer thickness, another time, 50% of that if will succeed in passing through. In the successive passing by the two layers, only 25% of the energy of the incidental light were transmitted by 2cm of this substance (0,50 X 0,50 = 0,50²). In the same way, 3cm will transmit 12,5% of the received total (0,50 X 0,50 X 0,50 = 0,5³). It is thus there about a geometric progression, for each unit added thickness, transmitted energy will be attenuated with the half. This attenuation or extinction evolves/moves in an exponential way . Transmittance thus does not decrease in a linear way but exponential (see figure). On the other hand, the Logarithme of T (logT) decreases linearly according to the thickness.

The change undergone by radiant energy $P$ according to the length of the optical way crossed $l$, is defined by the relation:

$\backslash \; frac\; \{dP\}\; \{DLL\}\; =\; -\; kP.\; \backslash ;$

where $k$ is a constant of proportionality. By integrating this equation, one obtains:

$\backslash \; int\_\; \{P\_0\}\; ^P\; \backslash \; frac\; \{dP\}\; \{P\}\; =\; -\; K\; \backslash ,\; \backslash \; int\_0^l\; \backslash ,\; DLL.\; \backslash ;$

from where:

$\backslash \; ln\; \{\backslash \; frac\; \{P\}\; \{P\_0\}\}\; =\; -\; kl.\; \backslash ;$

For the chemists, one substitutes the Napierian logarithm by the Decimal logarithm by dividing the factor K by 2,303 incorporating it in a new noted constant has ( α for the physicists):

$\backslash \; log\_\; \{10\}\; \backslash \; frac\; \{P\}\; \{P\_0\}\; =\; -\; Al.\; \backslash ;$

Like $T=\; \backslash \; frac\; \{P\_t\}\; \{P\_0\}.\; \backslash ;$, one can write:

$-\; \backslash \; log\; T\; =\; Al.\; \backslash ;$

One now definite that the absorptance is:

$has\; =\; -\; \backslash \; log\; T\; =\; Al.\; \backslash ;$

has is the coefficient absorption or absorptivity of the medium expressed in m^-1 or cm^-1. In a solution, one can divide the absorption coefficient by many moles N of the entities contained in the volume crossed by a beam of light:

$\backslash \; varepsilon\; =\; \backslash \; frac\; \{has\}\; \{N\}\; =\; -\; \backslash \; frac\; \{1\}\; \{L\; \backslash \; cdot\; N\}\; \backslash \; log\; \backslash \; frac\; \{P\}\; \{P\_0\}\; =\; \backslash \; frac\; \{has\}\; \{L\; \backslash \; cdot\; N\}\; \backslash ;$

ε, expressed in M^-1.cm^-1, is the molar Absorptivité or molar coefficient of extinction of the entity in solution. It depends on the nature of the absorbing body, selected wavelength and temperature.

Law of Beer-Lambert

For more details, to see the article Law of Beer-Lambert

Using the same reasoning as that of the law of Bouguer, August Beer proposes in 1852 an equation connecting the absorptance and transmittance to the concentration of a substance in solution . The law is stated in the following way:

$A\_\; \backslash \; lambda\; =\; -\; \backslash \; log\; T\; =\; has\; \backslash \; cdot\; C.\; \backslash ,$

The concentration C is expressed in mol. L^-1 or in mol.m^-3. The absorptivity has can be substituted in the equation by the molar coefficient of extinction ε like definite previously. Then, by the combination of the two equations, one obtains the law of Beer-Bouguer better known like the law of Beer-Lambert :

$A\_\; \backslash \; lambda\; =\; \backslash \; varepsilon\_\; \backslash \; lambda\; \backslash \; cdot\; L\; \backslash \; cdot\; C.\; \backslash ;$

The measurement of the absorptance is done thanks to a Spectrophotomètre. L is with the length of the optical way crossed by the light in the solution in cm. In practice, this length corresponds to the thickness of the tank of measurement (generally taken 1cm).

This law makes it possible to the chemists to determine the unknown concentration of one or more elements in a given solution. However, this proportionality between the concentration and the absorptance would not be applicable any more for C > 0,01 mol. L^-1 (it is where the phenomenon of reflection becomes considerable).

Colorimetry

If an element does not absorb enough the light to take correct measurements, one makes it react with another element so that the product of the reaction posts a quite visible color. The intensity of coloring obtained is proportional to the real concentration.

Turbidimetry

The Turbidimétrie bases on an optical system of detection which measures the Turbidité, i.e. the concentration of very small suspended particles in a solution (Mg. L^-1). The light transmitted through a turbide medium depends on the concentration in diffusing objects and their cross sections of extinction, therefore their sizes, their forms, their indexes of refraction, and wavelength considered.
For weak concentrations, the transmitted intensity can be given by the law of Beer-Lambert. The measurement of the transmitted intensity allows thus, to go back with the distribution of size and the concentration of the absorbing particles.

Example of application
traditional study, by Spectrophotometry, of the bacterial Growth in a liquid culture medium agitated.

References

James HENKEL, Essentials off drug product quality (p 130,133). 1978, The Mosby Company.

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