Freundlich equation empirical model

Empirical Models vs. Mechanistic Models. Experimental data on interactions at the oxide-electrolyte interface can be represented mathematically through two different approaches (i) empirical models and (ii) mechanistic models. An empirical model is defined simply as a mathematical description of the experimental data, without any particular theoretical basis. For example, the general Freundlich isotherm is considered an empirical model by this definition. Mechanistic models refer to models based on thermodynamic concepts such as reactions described by mass action laws and material balance equations. The various surface complexation models discussed in this paper are considered mechanistic models. 

In general, there is an array of equilibrium-based mathematical models which have been used to describe adsorption on solid surfaces. These include the widely used Freundlich equation, a purely empirical model, and the Langmuir equation as discussed in the following sections. More detailed modeling approaches of sorption mechanisms are discussed in more detail in Chap. 3 of this volume. 

The Freundlich form is often employed along with the Langmuir form and is referred to as the Langmuir—FreundUch equation. That model is the basis of a great many useful modifications including many empirical forms that serve to describe adsorption data quite accurately. 

The above surface complexation models enable adsorption to be related to such parameters as the number of reactive sites available on the oxide surface, the intrinsic, ionization constants for each type of surface site (see Chap. 10), the capacitance and the binding constants for the adsorbed species. They, therefore, produce adsorption isotherms with a sounder physical basis than do empirical equations such as the Freundlich equation. However, owing to differences in the choice of adjustable... 

Sorption is most commonly quantified using distribution coefficients (Kd), which simplistically model the sorption process as a partitioning of the chemical between homogeneous solid and solution phases. Sorption is also commonly quantified using sorption isotherms, which allow variation in sorption intensity with triazine concentration in solution. Sorption isotherms are generally modeled using the empirical Freundlich equation, S = K CUn, in which S is the sorbed concentration after equilibration, C is the solution concentration after equilibration, and Kt and 1 In are empirical constants. Kd and K are used to compare sorption of different chemicals on one soil or sorbent, or of one chemical on several sorbents. Kd and K are also commonly used in solute leaching models to predict triazine interactions with soils under various environmental conditions. 

Geochemical models of sorption and desorption must be developed from this work and incorporated into transport models that predict radionuclide migration. A frequently used, simple sorption (or desorption) model is the empirical distribution coefficient, Kj. This quantity is simply the equilibrium concentration of sorbed radionuclide divided by the equilibrium concentration of radionuclide in solution. Values of Kd can be used to calculate a retardation factor, R, which is used in solute transport equations to predict radionuclide migration in groundwater. The calculations assume instantaneous sorption, a linear sorption isotherm, and single-valued adsorption-desorption isotherms. These assumptions have been shown to be erroneous for solute sorption in several groundwater-soil systems (1-2). A more accurate description of radionuclide sorption is an isothermal equation such as the Freundlich equation ... 

The modeling of the obtained isotherms was carried out by applying the Freundlich equation. This is an empirical relationship based on the assumption of a logarithmic decrease in adsorption heat with adsorption surface coverage. The Freundlich equation is commonly used in the form ... 

The frequent good fit of adsorption data to the Freundlich equation is influenced by the insensitivity of log-log plots and by the flexibility afforded curve fitting by the two empirical constants K and n. This flexibility does not guarantee accuracy, however, if the data are extrapolated beyond the experimental range. The Freundlich equation has the further limitation that it does not predict a maximum adsorption capacity, however mythical the adsorption maximum may be. Despite its shortcomings, the Freundlich equation is a common adsorption equation and is included in several models for predicting pesticide behavior in soil. 

The mathematical models that have been applied to the physical adsorption from liquid solutions are generally extensions of the theories that have been developed to describe the sorption of gases on solid surfaces with modifications to account for the competition between the solute and solvent for the adsorption sites. Two of these models have been applied to the adsorption isotherms of nonelectrolytes from solution they are the Langmuir model and the Brunauer, Emmett, and Teller (BET) model in addition the Freundlich empirical equation has also been used. In the Langmuir model it is assumed that the adsorbed species forms a monolayer on the surface of the adsorbent, that the adsorbed molecules... 

Most of the research on metal sorption at the mineral/water interface has dealt with equilibrium aspects. Numerous studies have used macroscopic approaches such as adsorption isotherms, empirical and semi-empirical equations (e.g., Freundlich, Langmuir), and surface complexation models (e.g., constant capacitance, triple layer) to describe adsorption, usually based on a 24 hour reaction time. 

The Freundlich isotherm model is usually applied to the heterogeneous surfaces. It is an empirical relationship, implying the binding energies of the adsorbate related to the adjacent sites. This model has the following form, Equation (11.4) [5] ... 

Adsorption at the solid-liquid interface is generally similar to the adsorption at the solid phase-gaseous phase interface. Theoretical modelling of the adsorption process is more difficult because, in addition to adsorbing dissolved substances, solvent is present (e.g. water), the molecules of which can also adsorb, and interactions of adsorbed molecules with molecules of the solvent may occur. Molecular adsorptions, when molecules of a substance are adsorbed, and ion adsorptions, in which ions of a substance are adsorbed, can similarly take place. Solid phase-liquid phase interfaces are usually described by empirical equations and theoretically derived equations of the Freundlich and Langmuir type. 

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