Beyond Beer's Law: Why the Index of Refraction Depends (Almost) Linearly on Concentration

Abstract

Beer's empiric law states that absorbance is linearly proportional to the concentration. Based on electromagnetic theory, an approximately linear dependence can only be confirmed for comparably weak oscillators. For stronger oscillators the proportionality constant, the molar attenuation coefficient, is modulated by the inverse index of refraction, which is itself a function of concentration. For comparably weak oscillators, the index of refraction function depends, like absorbance, linearly on concentration. For stronger oscillators, this linearity is lost, except at wavenumbers considerably lower than the oscillator position. In these transparency regions, linearity between the change of the index of refraction and concentration is preserved to a high degree. This can be shown with help of the Kramers-Kronig relations which connect the integrated absorbance to the index of refraction change at lower wavenumbers than the corresponding band. This finding builds the foundation not only for refractive index sensing, but also for new interferometric approaches in IR spectroscopy, which allow measuring the complex index of refraction function.The exact origin of the form of Beer's law, as we employ it nowadays [1,2] AðṽÞ ¼ e * ðṽÞ � c � dwherein e * ðṽÞ is the molar attenuation coefficient, c the concentration and d the sample thickness, remains unclear. It seems that it was not employed in this form before about 1900, as it was not discussed nor provided in this form in Kayser's handbook of spectroscopy, [3] which was the reference work at this time. When Max Planck derived his particular kind of dispersion theory and employed it to compare the results with Beer's law in 1903, [4] he was most probably unaware of Equation (1), otherwise he probably would have derived it from dispersion theory. Recently, this derivation was carried out, first from dispersion theory [5] and, subsequently, from simple electromagnetic theory. [6] One of the main results of the derivation is that the molar attenuation coefficient is inversely proportional to the index of refraction, which is itself a function of concentration. Concerning the nature of this dependence, as was already stated in Ref.[5], a law comparable to Beer's law, but for the index of refraction function, can be derived. In ref. [5] the nature of this dependence was not investigated in detail. As we will show in the following, the use of the index of refraction variation instead of the absorbance to investigate concentration may have several advantages. To derive the concentration dependence of the complex index of refraction, it is possible to start from simple electromagnetic theory. Accordingly, the macroscopic polarizationP is related to the dipole momentp induced in an atom or a molecule by the following relation:Here, N is the number of dipole moments per unit volume (which is equal to the molar concentration multiplied by Avogadro's constant, N = N A ·c). Inherent to Equation (2) is the assumption that there are no interactions between the microscopic dipoles, i....

The Bouguer‐Beer‐Lambert Law: Shining Light on the Obscure

“…(17) one can conclude that for small variations the index of refraction is predicted to be also linearly depending on concentration. [81] This is in principle known for nearly as long as Beer's law. In contrast to the latter, the corresponding law was not formulated for the concentration, but for the density instead.…”

“…[82] We recently showed that a related law formulated for the concentration still works at resonance. [81] Interestingly, there is a law corresponding to eqn. (16), also dealing with the dependence from density instead of concentration, which is called the Newton-Laplace rule.…”

Electrochemistry and Optical Microscopy

“…. These different expressions allow predicting [8], from tabulated values, the index of refraction, 𝑛 ! "# , of a solution of a solute 𝑆 (e.g.…”

Beyond Beer's Law: Spectral Mixing Rules

“…4 This means that Beer's law would be purely empiric had it not recently been derived from dispersion and electromagnetic theory. 5–8 The derivation was to a high degree the same as the well-known Lorentz–Lorenz relation, or, equivalently, the Clausius–Mossotti relation. 9 Those relations are also known to provide mixing rules, and therefore it seems natural to see how they are related to Eq.…”

“…Again, it is often not realized in literature that this simplification also leads to a well-known mixing rule, which is the Gladstone–Dale rule, 12 also known as the Arago–Biot rule, 15 which we recently proved experimentally to be correct for small molar concentrations. 8…”