The Beer-Lambert Law for absorption of electromagnetic radiation is
A = abc
where A is the absorbance
A = - log T
and T is the transmittance,
T = I/I0
defined as the fraction of the intensity of the light beam that is transmitted through the sample, I/I0. a is the absorptivity, b is the distance the beam passes through the sample, and c is the concentration. The molar extinction coefficient, often denoted by epsilon, is the special case of a where the concentration is expressed in moles/liter, with b usually in cm.
This law applies in the limit of infinite dilution, because the derivation assumes that all solute molecules behave independently. The experimentalist has control over b and c. a depends on the nature of the absorbing species, on the wavelength (and on the polarization of the light, if the sample is oriented or chiral), and on the environment (solvent, temperature, etc.). Real systems often exhibit deviations from this limiting law when A is as large as 1. The deviations may arise from physical effects in the sample, such as interactions between pairs of absorbing molecules as c increases. They may also arise for instrumental reasons, such as the inefficiency with which the particular apparatus rejects stray light (light with other than the desired frequency).
Consider an infinitesimally small volume element along the light path through the sample, with the light path through this volume element being dl. The intensity of the light beam is I as it enters this volume element, and it change to I+dI when it leaves. We assume dI = 0 if c = 0 (and therefore the pure solvent is our reference). Otherwise dI is negative and proportional to the probability that a photon will encounter an absorber. This probability is proportional to the product of the intensity of photons (I) and the number of absorbers encountered during traversal across the volume element (cdl), with proportionality constant a.
dI = -acIdl
We now integrate over the entire light path through the sample, using limits of
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