Equation generalized Freundlich

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 Generalized Freundlich (GF) equation supposes that the heterogeneity and other properties of the surface and the adsorbed layer can take into account similarly to Toth s idea The coverage present in the Langmuir equation should be raised to a power n. Thus, we have... 

For certain values of the heterogeneity parameters (m, q), the Langmuir-Freundlich, general Freundlich (GF), and Toth equations for gas mixtures have been derived and verified experimentally by munerous authors [175,176,182,186,187]. 

The special forms of the MJ isotherm (61) have been frequently used to study adsorption from solution generalized Freundlich equation [199], Langmuir-Freundlich isotherm [170,200-205],... 

The following important semiconsistent relationship is the generalized Freundlich (GF) equation, which has flie following form ... 

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. 

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 main limitation of the Freundhch equation is that it does not predict a maximum adsorption capacity, because linear adsorption generally occurs at very low solute concentration and low loading of the sorbent. However, in spite of this limitation, the Freundlich equation is used widely for describing contaminant adsorption on geosorbents. 

Equation 10.27 is generally known as Freundlich equation. Equation 10.27 with concentration replaced by pressure was also used to describe the adsorption isotherms of gases on solids, suggesting the incorrect idea that adsorption from solution by a solid could be paralleled with gas or vapor adsorption on the same adsorbents. Whereas in some cases the restriction to dilute solutions was imposed by the solubility of solids (e.g., benzoic acid in water or stearic acid in benzene) it was not imposed on the investigation of mixtures of completely miscible liquids, e.g., acetic acid in water. 

Moreover, the adsorbability of an ion is generally greater the greater the valency. Matsuno 9 used the precipitating values of cobalt-ammines to determine the valency of the complex ions, employing the equation, deduced from Freundlich s adsorption hypothesis, Sif = S/1V4, where Sy is the equivalent concentration of an A-valent ion, N being the valency of the complex ion, and S the precipitating value of a univalent ion. The results confirmed those obtained by spectroscopic and conductivity methods. 

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... 

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. 

Kuo and Lotse (1973) presumed that the slope of a Freundlich plot was time independent. The physical meaning of the concentration term exponent in the Freundlich equation is unclear, but has generally been <1 and related to the characteristics of the sorbent. However, the exponent is time-independent, whereas the intercept is time-dependent. 

Tin and americium were so extensively sorbed under all conditions that isotherm data could not be obtained. These elements are not significantly mobile in the Mabton Interbed aquifer. Values of Freundlich constants for technetium, radium, uranium, neptunium, and plutonium are given in Table IV. The Freundlich equation did not fit the selenium sorption data very well probably because of slow sorption kinetics or precipitation. Precipitation was also observed for technetium at 23°C for concentrations above 10 7M. This is about the same solubility observed for technetium in the sandstone isotherm measurements. Linear isotherms were observed only in the case of radium sorption. In general, sorption on the Mabton Interbed was greater than on the Rattlesnake Ridge sandstone. This is probably due to the greater clay content of the Mabton standard. 

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 ... 

The variation of adsorption with pressure at a given constant temperature is generally expressed graphically as shown in the figure given above. Each curve is known as adsorption isotherm for a particular temperature. The relation ship between the magnitude of adsorption and pressure can be expressed mathematically by a empirically equation commonly known as Freundlich adsorption isotherm, viz. 

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... 

The study of adsorption from solution is experimentally straightforward. A known mass of the adsorbent material is shaken with a solution of known concentration at a fixed temperature. The concentration of the supernatant solution is determined by either physical or chemical means and the experiment continued until no further change in the concentration of the supernatant is observed, that is, until equilibrium conditions have been established. Equations originally derived for the adsorption of gases on solids are generally used in the interpretation of the data, the Langmuir and Freundlich equations being the most commonly used. 

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. 

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