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• Author: Frank Baumann
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## Abstract

The aim of this article is to generalize the diffusion based adsorption model to a fractional diffusion and fractional adsorption model. The models are formulated as nonlinear fractional boundary value problems equivalent to a singular Hammerstein integral equation. The novelty is that not only the diffusion component of the model is fractionalized but also the adsorption part. The singular Hammerstein integral equation is solved by Sinc approximations. Specific numerical schemes are presented. Based on these solutions we are able to identify different regimes of adsorption diffusion processes controlled by fractional derivatives verified by experimental data. These regimes allow to classify experiments if examined with respect to their scaling behavior.

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## Abstract

Fractional integrals, fractional derivatives, fractional integral equations, and fractional differential equations are numerically solved by Sinc methods. Sinc methods are able to deal with singularities of the weakly singular integral equations of Riemann-Liouville and Caputo type. The convergence of the numerical method is numerically examined and shows exponential behavior. Different examples are used to demonstrate the effective derivation of numerical solutions for different types of fractional differential and integral equations, linear and non-linear ones. Equations of mixed ordinary and fractional derivatives, integro-differential equations are solved using Sinc methods. We demonstrate that the numerical calculation needed in fractional calculus can be gained with high accuracy using Sinc methods.

A simple liquid solution model is proposed to describe the effect of solvent-solute interactions on the solubility of nonpolar and slightly polar substances in supercritical solvents. Treating the system as an ideal solution, the effect of pressure on the solubility is zero or nearly zero, as it is governed by the difference in molar volume of the pure supercooled liquid solute and the pure solid solute, and this may be nearly zero. Deviations from ideal behavior are given by activity coefficients of the Margules type with the interaction parameter w interpreted as interchange energy as in the lattice theory. The hypothesis is put forward that the interchange energy is of the same form as a function proposed by Liptay and others to describe the effect of the solvent on the wavelength of the absorption maximum of the solute dissolved in the solvent. The function consists of a radius of interaction a and a function g(ε ) of the dielectric constant ε of the solvent, treated as a continuum. The function g depends on pressure through the pressure dependence of the dielectric constant ε (P). The attractive feature of this formalism, introduced by Baumann et al. and here justified thermodynamically, is that plots of the logarithm of solubility vs. g are linear, except for polar solutes near the solvent’s critical point. Changes in slope then admit interpretation as changes in the radius of interaction a with possible clues about the mechanism of solvation of these molecules.

The first UV-absorption band of the polar molecules trans-4-dimethylamino-4'-nitrostilbene (DMANS), 5-dimethylamino-5'-nitro-2,2'-bithiophene (DMANBT), and 4-nitroanisole (NA), and of the slightly polar pesticide diclofop-methyl in the nonpolar supercritical solvent carbon dioxide (CO2) and in the slightly polar supercritical solvent trifluoromethane (CHF3) was measured in the pressure range from 1.5 to 30 MPa and the temperature range from 298 to 353 K with the purpose of studying the solvent-solute interactions in these molecules. The theory of Liptay for the effect of the solvent on the wave number of the electronic absorption of the solute molecule was applied. In this theory the solvent is represented by an isotropic and homogeneous dielectric continuum characterized by a pressure and temperature dependent dielectric constant ε (P,T), and an optical refraction index n. For the isotherms which approach the critical point in both solvents there is a change in the slope of the plot of the wave number maximum against the solvent parameter g=(ε−1)/(2ε+1) which is reminiscent of complex formation. A possible mechanism for this phenomenon is the arrangement of the solvent molecules around the dilute solute in the intermediate region at the (continuous) transition of the solvent from the dense vapour to the supercritical fluid

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