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Absorption

Measurement of Absorption

Page updated: Aug 18, 2017
Principal author: Collin Roesler
 

From Theory to Reality

Consider a scenario where the goal is to measure the absorption spectrum of a thin layer of material (Figure 1A). The incident radiant power is given by $ \Phi_o$ , in the form of a collimated beam. The radiant power transmitted through the layer,$ \Phi_t$ , is detected. If $ \Phi_t = \Phi_o$ , there is no loss of radiant power and therefore no attenuation. If however the medium absorbs some quantity of radiant power, $ \Phi_a$ , then $ \Phi_t < \Phi_o$ , and $ \Phi_o = \Phi_t + \Phi_a$ (Figure 1B). In the case of material that absorbs and scatters, the scattered radiant power is given by $ \Phi_b$ , and $ \Phi_o = \Phi_t + \Phi_a + \Phi_b$ .

Figure: Figure 1. Diagrammatic representation of theoretical attenuation by a thin layer of A. non-attenuating, B. absorbing, and C. absorbing and scattering material. The thickness of the layer is given by $ \Delta x$ .
Image 223f828acf858245309d563cf5f83df1

To quantify the absorbed radiant power only, it is necessary to measure both the transmitted and scattered radiant power. This is a requirement for an absorption meter. Consider first a nonscattering material. The measured dimensionless transmittance, $ T$ , is the fraction of incident power transmitted through the layer:

$\displaystyle T = \frac{\Phi_t}{\Phi_o} \,.$    

The absorptance, $ A$ , is the fraction of incident radiant power that is absorbed ($ 1 - T$ ):

$\displaystyle A = \frac{\Phi_a}{\Phi_o} = \frac{\Phi_o - \Phi_t}{\Phi_o} \,.$    

The absorption coefficient, $ a~(\unit{m^{-1}})$ , is the absorptance per unit distance

$\displaystyle a = \frac{A}{\Delta x}$    

which, for an infinitesimally thin layer can be expressed as:

$\displaystyle a = \frac{ \frac{\Delta \Phi}{\Phi} }{\Delta x} = \frac{\Delta \Phi}{\Phi \Delta x} ,.$    

Rearranging this expression and taking the limit as $ \Delta x \to 0$ yields:

$\displaystyle a \Delta x = \lim_{\Delta x \to 0} \left( \frac {\Delta \Phi}{\Phi} \right )$    

Integrating the function over the layer:

$\displaystyle \int_0^x a \,d x =$ $\displaystyle ~ -\int_{\Phi_o}^{\Phi_t} \frac{d \Phi}{(\Phi}$    
$\displaystyle ax \vert _0^x =$ $\displaystyle ~ -\ln \Phi \vert _{\Phi_o}^{\Phi_t}$    
$\displaystyle ax =$ $\displaystyle ~ -[\ln(\Phi_t ) - \ln(\Phi_o ) ] = -\ln \left ( \frac{\Phi_t}{\Phi_o} \right )$    
$\displaystyle a =$ $\displaystyle ~ - \frac{1}{x} \ln \left ( \frac{\Phi_t}{\Phi_o} \right )$    

This equation provides a guide toward designing instruments to accurately measure absorption. The Level 2 pages beginning at Benchtop Spectrometry of Solutions give the specifics on techniques to measure absorption by dissolved and particulate constituents in seawater.