Langmuir-Derived Model for Diffusion- and Reaction-Limited Adsorption of Organic Compounds on Fractal Aggregates

Abstract

Natural colloids, aerosols, and aggregates generated in wastewater treatments have fractal geometries, and so not all the surface sites of these aggregates possess the same collision probability. It is this property that modifies the classical Langmuir adsorption isotherm, since it describes the equilibrium of the rate of collisions that adhere with the rate of desorption. In this paper, the growth site probability distribution is computed for aggregates with different fractal dimensions in the two- and three-dimensional models. These distributions are then used to find the adsorption isotherms depending on a model parameter η. This parameter is related to the reactivity of the adsorbate, and its relationship to solubility and molecular size has tentatively been proposed and studied. The low concentration range of the adsorption isotherms can be fitted with the Freundlich isotherm for description purposes. When there is low affinity between adsorbate and matrix (reaction-limited adsorption), linear adsorption isotherms are obtained; conversely, isotherms are less linear when adsorbate reactivity increases. Furthermore, the increase of adsorbate solubility led to lower partition coefficients, in agreement with the results reported in the literature. Therefore, the isotherms obtained with the Langmuir-derived equation take into account the fractal geometry of aggregates and provide a good description of the linear and nonlinear isotherms of organic compounds in terms of their solubility, reactivity, and molecular size.

Adsorption and Partitioning of Organic Compounds onto/into Fractal Sorbents

“…The isotherms obtained from this process show a departure from the classical Langmuir isotherm and can be fitted by Freundlich-and Tempkin-like equations at low and medium surface coverages, respectively. 14,45 Thus, this Langmuir derived model is consistent with a wide range of experimental observations. 11,12 It is noteworthy that a unique adsorption mechanism can explain linear and nonlinear isotherms depending on the value of η, which is dependent on the physicochemical properties of the sorbate and the time of equilibration.…”

“…Both cases are consistent with very low values of the parameter η (η < 1) which lead to classical Langmuir isotherms and thus to linear sorption at low coverage. 14,45 Further evidence that partitioning is related to sorption into the inner sites of fractal macromolecules is given by the Gibbs free energy relationships obtained from close analyses of the adsorption isotherms for fractal surfaces obtained from eq 3. Indeed, the relationship between the partition coefficient (K OM ) for hydrophobic chemicals between organic matter and water to their solubility in the aqueous phase (C w sat ) has been derived from the application of the described model onto fractal surfaces: 14 where k SOL and const are the slope and constant, respectively.…”

“…22 Langmuir Derived Adsorption Model. The Langmuir derived adsorption model used in the present paper has been explained elsewhere, 14 and here only a summary of its characteristics will be provided. This is a monolayer adsorption model that takes into account the different adsorption probabilities of the surface active sites resulting from geometrical irregularities.…”

Applications of Fractals in the Study of Humic Materials

“…The real power of a fractal approach lies not in simply characterizing humic materials, but in applying it to modeling and predicting the chemical and geochemical behavior of these materials in the environment. For example, Dachs and coworkers [57,58] have developed a fractal approach that takes the 'geometric heterogeneity' of humic materials into account when modeling their interaction with organic contaminants. This approach [57] utilizes the different interaction probabilities of the active contaminant binding sites on fractal surfaces to accommodate linear and nonlinear sorption isotherms (i.e.…”

“…For example, Dachs and coworkers [57,58] have developed a fractal approach that takes the 'geometric heterogeneity' of humic materials into account when modeling their interaction with organic contaminants. This approach [57] utilizes the different interaction probabilities of the active contaminant binding sites on fractal surfaces to accommodate linear and nonlinear sorption isotherms (i.e. 'dual-mode' sorption) using coverage extent, physical and chemical properties of the adsorbate and the existence of equilibrium conditions.…”