Freundlich equation assumption

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 Freundlich equation requires the assumption that sorption reactions are reversible. However, several studies (7) have recently shown that K and N depend on sorption direction, i.e., whether sorption or desorption occurred. In each case, N was less and K was greater for desorption than sorption. 

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 difference in thermodynamic equilibrium between crystals of different size is defined by the Gibbs-Thomson or Ostwald-Freundlich equation (Mullin 1993). The original Kelvin equation was written to describe droplets of liquid in a vapor in terms of vapor pressures around droplets of different size. However, in crystallization studies, the vapor pressure may be related, through the solution activity, to the equilibrium concentration in solution for crystals of different size, faking the assumption that concentration can be used directly fpr vapor pressure gives... 

Sorption and desorption are usually modeled as one fully reversible process, although hystersis is sometimes observed. Four types of equations are commonly used to describe sorption/desorption processes Langmuir, Freundlich, overall and ion or cation exchange. The Langmuir isotherm model was developed for single layer adsorption and is based on the assumption that maximum adsorption corresponds to a saturated monolayer of solute molecules on the adsorbent surface, that the energy of adsorption is constant, and that there is no transmigration of adsorbate on the surface phase. 

A number of attempts have been made to modify the IAS model (Eqs. 22-29) to improve its accuracy and reduce computational efforts. Using the IAS model, DiGiano et al. [80] derived a Simplified Competitive Equilibrium Adsorption Model (SCAM). This model, which is based on the Freundlich isotherm, assumes the single-solute isotherms of all the components are equal and it utilizes average isotherm constants when this assumption is not valid. The IAS model equations have been reduced to a single expression ... 

Eq. (4.140) is for liquid-film diffusion control and eq. (4.141) for solid diffusion control. The following equation is a solution of the fixed-bed model under the constant pattern and plug-flow assumption, for fluid-film diffusion control and the favorable Freundlich... 

Equation 1 is given in spherical coordinates, thus assuming a spherical shape for the carbon particle, an assumption which accords reasonably well with microscopic observations of the geometry of particles of the experimental carbon. In Equation 1, C represents the H30+ activity in solution t, time r, the radial distance from the particle center D, the diffusion coefficient and S, the H30+ concentration at the surface of the carbon. For the present experiments, the equilibrium relationship between S and C is described in terms of the Freundlich expression... 

Distribution functions (93) and (94) are substantiated by the fact that they lead to the most frequently observed adsorption isotherms for the region of medium surface coverages, viz., the Freundlich isotherm corresponds to distribution (93) with > 0 and T < as it has been demonstrated by Zel dovich (43) and the logarithmic isotherm corresponds to distribution (94) (40). Besides that, if we accept Assumption 1, then distribution (94) will give the equation of adsorption kinetics by Zel dovich and Roginskil and distribution (93) will result in the equation of adsorption kinetics by Bangham. Finally, if Assumption 1 is correct, then distribution (93), including distribution (94) as its particular case, follows from the kinetics of fractional order reactions (44). 

The treatment of Freundlich adsorption also becomes tractable only by making the assumption that the extents of adsorption are small. This leads to the kinetic equations... 

The isosteric heats of adsorption, AH so, were calculated as a function of nitrogen loading for each of the samples tested. For each sample, the AHiS0 was found to decrease logarithmically with increased loading and this functionality is consistent with the assumptions of the Freundlich adsorption equation. As before, the brevity of presentation, the heat of adsorption data were curve fitted to an equation of the form ... 

In obtaining the Langmuir isotherm equations, several aspects of the adsorption system were presupposed in the derivations. The most important of these, and the one that has been subject to the greatest doubt, is that a uniform surface is assumed, In other words, any active site has the same attraction for an impinging molecule as does any other site. Isotherms different from the Langmuir types, such as the Freundlich isotherm, may be derived based on various assumptions concerning the adsorption system, including different types of nonuniform surfaces. 

The two constants K and n (n a 1) are characteristic of a given adsorption system. The Freundlich isotherm is derivable from certain rather artificial assumptions [e.g.. Ref. (10)], but it is essentially an empirical fitting function. Equation (3-7) commonly provides a good match for experimental adsorption data over a limited range in sample concentration. Systems which exhibit an extended linear isotherm region (and these are today of greatest interest in adsorption chromatography) are less well described by the Freundlich isotherm. 

Sheindorf et al. (1981,1982) developed a multicomponent Freundlich-type equation to describe the adsorption of binary solute mixtures containing arsenate and phosphate or arsenate and molybdate. The derivation of the Sheindorf-Rebhun-Sheintuch (SRS) equation was based on the assumption that there is an exponential distribution of adsorption energies available for each solute. The SRS equation can be written for the solute i from a binary solute mixture as... 

The prediction of multicomponent equilibria based on the information derived from the analysis of single component adsorption data is an important issue particularly in the domain of liquid chromatography. To solve the general adsorption isotherm, Equation (27.2), Quinones et al. [156] have proposed an extension of the Jovanovic-Freundlich isotherm for each component of the mixture as local adsorption isotherms. They tested the model with experimental data on the system 2-phenylethanol and 3-phenylpropanol mixtures adsorbed on silica. The experimental data was published elsewhere [157]. The local isotherm employed to solve Equation (27.2) includes lateral interactions, which means a step forward with respect to, that is, Langmuir equation. The results obtained account better for competitive data. One drawback of the model concerns the computational time needed to invert Equation (27.2) nevertheless the authors proposed a method to minimize it. The success of this model compared to other resides in that it takes into account the two main sources of nonideal behavior surface heterogeneity and adsorbate-adsorbate interactions. The authors pointed out that there is some degree of thermodynamic inconsistency in this and other models based on similar -assumptions. These inconsistencies could arise from the simplihcations included in their derivation and the main one is related to the monolayer capacity of each component [156]. 

Originally postulated as an empirical equation, the Freundlich isotherm with two constants, k and , can be derived from thermodynamic or statistical considerations with the assumptions that AHads decreases exponentially with increasing surface coverage and that this decrease is due to surface heterogeneity ... 

Free gel was added to definite concentrations of the MB solutions (2-20 mg/1) at room temperature and was noted for its adsorption. It is clear from Figure 13.3 that the dye adsorption increases sharply with an increase in the initial dye concentration. When Cq was reached at 5 ppm and 10 ppm, the was reached at 10.04 and 20.81 respectively, which were much higher than reported Qe values of other adsorbents (Table 13.3). Equilibrium adsorption isotherm is an important criterion to determine the mechanism of dye adsorption on hydrogel. The Langmuir and Freundlich models are widely used to examine the adsorption isotherms. Freundlich isotherm models are based on the assumption that the surface of the adsorbent is not homogeneous. The experimental data in Figure 13.3 was also analyzed with the Freundlich isotherm model, which describes a heterogeneous system with multilayer adsorption. The linear form of Freundlich isotherm equation... 

Of the adsorption isotherms discussed above, the Langmuir equation is important because of its theoretical and conceptual contribution to the studies of adsorption and the kinetics of surface reactions. The Freundlich isotherm has found a wide use in correlating experimental data the Temkin isotherm is obeyed by certain systems [62-64], but it is expected to apply only in the intermediate range of coverage due to the simplifying assumptions made in its derivation. 

---