Generalized Freundlich isotherm

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. 

Equation 4.27 has the form of the generalized Freundlich isotherm equation.11 If the product A[C]cb < < 1, it reduces to the van Bemmelen-Freundlich isotherm equation and, ifb 1, it becomes the Langmuir isotherm equation (cf. Eq. 3.50b).M Thus Eq. 4.18 and a gamma distribution of the two rate coefficients it contains are sufficient to generate three very common adsorption... 

Though the suggestions of Sips were received well, by Sips himself who proposed a generalized Freundlich isotherm, by Misra who considered a kind of generalized DR equation, by Rudzinski and co-workers who extended the method to multilayer adsorption, and by Toth et who... 

The data of Loukidou et al. (2004) for the equilibrium biosorption of chromium (VI) by Aeromonas caviae particles were well described by the Langmuir and Freundlich isotherms. Sorption rates estimated from pseudo second-order kinetics were in satisfactory agreement with experimental data. The results of XAFS study on the sorption of Cd by B. subtilis were generally in accord with existing surface complexation models (Boyanov et al. 2003). Intrinsic metal sorption constants were obtained by correcting the apparent sorption constants by the Boltzmann factor. A 1 2 metal-ligand stoichiometry provides the best fit to the experimental data with log K values of 6.0 0.2 for Sr(II) and 6.2 0.2 for Ba(II). 

Nevertheless, surfactant sorption isotherms on natural surfaces (sediments and biota) are generally non-linear, even at very low concentrations. Their behaviour may be explained by a Freundlich isotherm, which is adequate for anionic [3,8,14,20,30], cationic [7] and non-ionic surfactants [2,4,15,17] sorbed onto solids with heterogeneous surfaces. Recently, the virial-electrostatic isotherm has been proposed to explain anionic surfactant sorption this is of special interest since it can be interpreted on a mechanistic basis [20]. The virial equation is similar to a linear isotherm with an exponential factor, i.e. with a correction for the deviation caused by the heterogeneity of the surface or the energy of sorption. 

The early experimental data of Freundlich Eapillarchemie) indicated that the adsorption of a solute from a solution followed the general laws governing the adsorption of vapours by solids, i.e. the Freundlich isotherm... 

The Freundlich isotherm, where the dye in fiber D, is directly proportional to (D,)- and a plot of log D against log D, gives a straight line, is generally found with cellulosic and other ionic hydrophobic fibers. 

Sorption is measured by recording sorption isotherms, which themselves are a way to express the amount of surfactant sorbed as function of the concentration of the compound in the solution. The Freundlich isotherm (Equation II) is a general sorption isotherm which describes sorption behavior and often is used in studies of surfactant sorption. KF is the Freundlich sorption coefficient which expresses the affinity of a surfactant for a given solid... 

TABLE 4.6. Freundlich Adsorption Constants (1In and K) and Distribution Coefficients (Kd) for Adsorption of Alachlor (Generally Linear Isotherms), Imazethapyr (Nonlinear Freundlich Isotherms), and Rimsulfuron (Langmuir Isotherms) on Humic Acids (HAs) Isolated from Two Sewage Sludges (SSI and SS2), a Soil Amended with 10tha 1yr 1 of SSI for 2 Years (SOI + SSI), and a Soil Amended with 40(ha 1yr 1 of SS2 for 2 Years (S02 + SS2), with the Corresponding Unamended Soils (SOI and S02, Respectively) (from Senesi et al., 2001)... 

Generalize the derivation of Eq. 4.27 to include the dissociable species QzQ(aq) in Eq. 4.3. Show that the result is the generalized competitive Freundlich isotherm equation ... 

Between the simplicity of the model and the complexity of the TLM, there are several other sorption models. These include various forms of isotherm equations (e.g., Langmuir and Freundlich isotherms) and models that include kinetic effects. The generalized two-layer model (Dzombak and Morel, 1990) (also referred to as the DLM) recently has been used to model radionuclide sorption by several research groups (Langmuir, 1997a Jenne, 1998 Davis, 2001). Constants used in this model are dependent upon the concentration of background electrolytes and... 

A plot of log(A /m) against log c should be linear, with an intercept of log a and slope of Xjn-, it is generally assumed that, for systems that obey this equation, adsorption results in the formation of multilayers rather than a single monolayer. Figure 6.17 shows Freundlich isotherms for the adsorption of local anaesthetics on activated carbon the method of calculating the constants a and jn from these plots is given in Example 6.5. 

Natural surfaces generally are too complex to be characterized as having uniform solute interaction energies, however. It is therefore not unexpected that sorption equilibrium data for natural soils and sediments are rarely described adequately by the Langmuir model. Such data are commonly described more satisfactorily by the empirical Freundlich isotherm model, which has the form... 

To decide if the experimental sorption behavior of a substance obeys the Freundlich isotherm or K, model, the approach generally used is to first linearize the isotherm by taking the logarithm of expression (10.2), which yields... 

In the theories of gas adsorption the applicability of Freundlich equation was long ago associated with the energetic heterogeneity of the adsorption sites on the actual solid surfaces. It was also known, that Freundlich equation is a simplified form of a more general isotherm equation which is now commonly called Langmuir-Freundlich isotherm. Thus, in 1980 Sposito [13] suggested that it was high time it were used also in the case of ion adsorption at water/oxide interfaces. 

Of the equilibrium adsorption isotherms observed on real surfaces, three hold particular positions. They are the Temkin isotherm, the Freundlich isotherm, and the Dubinin-Ra-dushkevich isotherm, and are reviewed in Refs. [22, 23]. The analysis carried out in Ref. [24] of several previously reported, often quoted, adsorption systems has shown that the Freundlich and the Dubinin-Radushkevich isotherms are the high-pressure and low-pressure limits, respectively, of a unique isotherm referred to as FDR. In turn, the FDR isotherm is a special case of a more general isotherm proposed by Jajoniec [25] on the basis of a previous idea of Heer [26],... 

Equation 2.66 shows that the velocity of a chemical solute concentration Vq is slower than the solution (water) velocity v by a retardation factor Rq. The retardation is caused by the adsorption of the chemical on the solid. Adsorption can be defined using the Freundlich isotherm or Langmuir isotherm. The general form of the Freundlich isotherm is... 

It may be considered a general empirical form encompassing the other more specific types. Second, a single isotherm of the Langmuir or Temkin type cannot be expected to fit data over the entire range of 6. Similarly, seldom does the Freundlich isotherm, with a single set of values of c and n, fit extensive data. 

The Langmuir isotherm, in the form of Eq. (18.46), is generally more successful in interpreting the data than is the Freundlich isotherm if only a monolayer is formed. A plot of 6 versus p is shown in Fig. 18.16. At low pressures, Kp 1 and 6 = Kp, so that 9 increases linearly with pressure. At high pressures, Kp > 1, so that 0 1. The surface is nearly covered with a monomolecular layer at high pressures, so that change in pressure produces little change in the amount adsorbed. 

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