Partitioning processes, linear relationships

Roberts et al. criticized the attempts to predict permeabilities since permeability is the result of two processes, partitioning and diffusion [40], Therefore, instead of following the approach of Potts and Guy, Roberts et al. tried to find a predictive model for each of these processes separately. For the partitioning step they found a Collander-type linear relationship (Eq. 11) between the logarithms of the stratum corneum-water and the octanol-water partition coefficients with a high correlation coefficient (r2 = 0.839) ... 

This linear relationship is based on the change in the free energy in the partitioning process between the two phases. 

The LFER, which is also known as the linear solvation-energy relationship (LSER), was developed by Taft et al. (62) and established by Abraham and coworkers (63). The LFER has been used for characterization of two-phase partitioning processes of solutes such as octanol-water and chromatographic processes such as HPLC, GLC, and TLC. The general equation is expressed as follows ... 

Sorption of organic contaminants onto aquifer solids is frequently described as a partitioning process, where the hydrophobic organic compound partitions into natural organic material associated with the aquifer solids [8]. Sorption can be characterized as either an equilibrium or rate-limited phenomenon. Equilibrium sorption can be modeled as either a linear or non-linear process. Equilibrium sorption may be assumed when the flow of groundwater and other processes affecting contaminant transport are slow compared to the rate of sorption. In this event the sorption of the contaminant can be considered instantaneous. If we assume equilibrium sorption, the relationship between sorbed and aqueous contaminant concentrations may be described by a sorption isotherm. 

A linear free-energy relationship has been suggested 113 by Abraham to describe various partition processes of molecules. Eq. (12.14) shows the so-called linear solvation equation ... 

Equation (lb) establishes a parametric (or semi-parametric) non-linear relationship between p and c where a set of nw parameters W are involved. These parameters must be identified from measurements. Irrespective of the type of relationship defined in Eqs (lb), the goal of the identification procedure is to obtain the parameter vector W that minimises the deviation between the model and real process outputs. The real process reaction kinetics cannot be measured directly only the concentrations can be measured using adequate measuring devices. By definition, the reaction rates can be calculated using Eq. (la). In practice, only a partition of r equations is required... 

A linear solvation energy relationship (LSER) has been developed to predict the water-supercritical CO2 partition coefficients for a published collection of data. The independent variables in the model are empirically determined descriptors of the solute and solvent molecules. The LSER approach provides an average absolute relative deviation of 22% in the prediction of the water-supercritical CO2 partition coefficients for the six solutes considered. Results suggest that other types of equilibrium processes in supercritical fluids may be modeled using a LSER approach (Lagalante and Bruno, 1998). 

The linear correlation between BCF and Kow apparently breaks down for chemicals with a log Kow greater than approximately 6 (Figure 9.4), resulting in a "parabolic" or "bilinear" type relationship between the BCF and Kow (Bintein 1993, Meylan et al., 1999). For these superhydrophobic chemicals, the BCF appears to be much lower than expected from the chemical s octanol-water partition coefficient. A loss of linear correlation between the BCF and Kow can be caused by a number of experimental artifacts (described in section 9.4.3) and physiological processes, including metabolic transformation, fecal egestion, and growth. 

During the diffusion of solute molecules through the network of pores, some of the solute molecules are adsorbed onto the interior surface of the particle. This process of adsorption is normally very fast relative to the diffusion process, and so we can model it as local equilibrium between the solute in the pore fluid and the solute bound to the interior surface of the particle. This partitioning is often referred to as the adsorption isotherm, which implies constant temperature conditions. When the solute concentration in the pore is low enough, the relationship becomes linear (Henry s law) hence, mathematically, we can write... 

The free molecules in the pore space and the adsorbed molecules at any point within a particle are in equilibrium with each other even though their concentration gradients exist within the particle. This local equilibrium is feasible only when at any point within the particle the local adsorption kinetics is much faster than the diffusion process into the particle. This is usually the case in most practical solids. In this section, we will assume a linear partition between the two phases thus the relationship is known as the local linear adsorption isotherm. The term local is because that particular condition is only applicable to a given position as time approaches infinity this local adsorption isotherm will become the global adsorption isotherm (true equilibrium) as there is no gradient in concentration either in the pore space or on the surface phase at t = oo. The local linear isotherm takes the form ... 

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