Filters

2019 - 20203

More filters

20213
Reset filters

Settings

Author: Popp, Jürgen × Subject: Beer's law ×

Search Results

Now showing 1 - 3 of 3
  • Thumbnail Image
    Item
    Beyond Beer's Law: Revisiting the Lorentz-Lorenz Equation
    (Weinheim : Wiley-VCH Verl., 2020) Mayerhöfer, Thomas G.; Popp, Jürgen
    In this contribution we show how the Lorentz-Lorenz and the Clausius-Mosotti equations are related to Beer's law. Accordingly, the linear concentration dependence of absorbance is a consequence of neglecting the difference between the local and the applied electric field. Additionally, it is necessary to assume that the absorption index and the related refractive index change is small. By connecting the Lorentz-Lorenz equations with dispersion theory, it becomes obvious that the oscillators are coupled via the local field. We investigate this coupling with numerical examples and show that, as a consequence, the integrated absorbance of a single band is in general no longer linearly depending on the concentration. In practice, the deviations from Beer's law usually do not set in before the density reaches about one tenth of that of condensed matter. For solutions, the Lorentz-Lorenz equations predict a strong coupling also between the oscillators of solute and solvent. In particular, in the infrared spectral region, the absorption coefficients are prognosticated to be much higher due to this coupling compared to those in the gas phase. © 2020 The Authors. Published by Wiley-VCH Verlag GmbH & Co. KGaA.
  • Thumbnail Image
    Item
    Beer's Law-Why Integrated Absorbance Depends Linearly on Concentration
    (Weinheim : Wiley-VCH Verl., 2019) Mayerhöfer, Thomas G.; Pipa, Andrei V.; Popp, Jürgen
    As derived by Max Planck in 1903 from dispersion theory, Beer's law has a fundamental limitation. The concentration dependence of absorbance can deviate from linearity, even in the absence of any interactions or instrumental nonlinearities. Integrated absorbance, not peak absorbance, depends linearly on concentration. The numerical integration of the absorbance leads to maximum deviations from linearity of less than 0.1 %. This deviation is a consequence of a sum rule that was derived from the Kramers-Kronig relations at a time when the fundamental limitation of Beer's law was no longer mentioned in the literature. This sum rule also links concentration to (classical) oscillator strengths and thereby enables the use of dispersion analysis to determine the concentration directly from transmittance and reflectance measurements. Thus, concentration analysis of complex samples, such as layered and/or anisotropic materials, in which Beer's law cannot be applied, can be achieved using dispersion analysis. ©2019 The Authors. Published by Wiley-VCH Verlag GmbH & Co. KGaA.
  • Thumbnail Image
    Item
    Beyond Beer's Law: Why the Index of Refraction Depends (Almost) Linearly on Concentration
    (Weinheim : Wiley-VCH Verl., 2020) Mayerhöfer, Thomas G.; Dabrowska, Alicja; Schwaighofer, Andreas; Lendl, Bernhard; Popp, Jürgen
    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. © 2020 The Authors. Published by Wiley-VCH Verlag GmbH & Co. KGaA.