Freundlich sorption equation

Equilibrium sorption and desorption behavior in natural soil systems are often nonlinear, and conunonly modeled well by the Freundlich sorption equation ... 

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... 

This equation has been successfully applied to many sorption and desorption reactions of dissolved metals and organic compounds. In the case of irreversible sorption (hysteresis), sorption and desorption isotherms are not identical. However, both sorption and desorption Freundlich isotherm equations can be substituted into the transport equation(2) ... 

Bacterial sorption of some metals can be described by the linearized Freundlich adsorption equation log S = log K+n log C, where S is the amount of metal absorbed in pmol g, C is the equilibrium solution concentration in pmol L, and K and n are the Freundlich constants. 

The capacity for sorption at various phosgene concentrations, with dry carbon and dry air, follows Freundlich s equation. 

The simplest mathematical models to describe sorption from solution are the so-called isotherm equations. The simplest of these is the Freundlich isotherm equation, which may be written ... 

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... 

Figure 5. Schematic representation and equations defining the 2-box (top) and 3-box (bottom) kinetic models. X = dissolved metal Y, Yi and Y2 = reversibly sorbed metal on Freundlich sorption sites f = fraction of Freundlich sorption sites reaching equilibrium instantaneously K and n are the Freundlich-isotherm constants r and k are the reversible and irreversible rate constant, respectively Z = irreversibly sorbed metal. The subscript 0 in the mass balance equations denotes concentrations at time zero, and Cp = particle concentration. (Adapted from ref. 15)...

Three Domain Model. Weber and Huang (16) devised a temporal phase-distribution relationship (PDR) approach for measuring sorption under non-equilibrium conditions. They observed that sorption rate data obtained at a given time, t, for systems initiated at different solution-phase concentrations can be fitted using a Freundlich-like equation having the form... 

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. 

Results of adsorption experiments for butylate, alachlor, and metolachlor in Keeton soil at 10, 19, and 30°C were plotted using the Freundlich equation. A summary of the coefficients obtained from the Freundlich equation for these experiments is presented in TABLE IV. Excellent correlation using the Freundlich equation over the concentration ranges studied (four orders of magnitude) is indicated by the r values of 0.99. The n exponent from the Freundlich equation indicates the extent of linearity of the adsorption isotherm in the concentration range studied. If n = 1 then adsorption is constant at all concentrations studied (the adsorption isotherm is linear) and K is equivalent to the distribution coefficient between the soil and water (Kd), which is the ratio of the soil concentration (mole/kg) to the solution concentration (mole/L). A value of n > 1 indicates that as the solution concentration increases the sorption sites become saturated, resulting in a disproportionate amount of chemical being dissolved. Since n is nearly equal to 1 in these studies, the adsorption isotherms are nearly linear and the values for Kd (shown in TABLE IV) correspond closely to K. These Kd values were used to calculate heats of adsorption (AH). 

The Langmuir and Freundlich equations have often been employed to model the sorption of metal ions by bacteria. Mullen et al. (1989) used the Freundlich isotherm to describe the sorption of Cd and Cu by B. cereus, B. subtilis, E. coli and P. aeruginosa over the concentration range of 0.001-lmM. The respective values of the Freundlich constant (Kf) indicated that E. coli was most efficient at sorbing Cd2+ and Cu2+. 

Hall et al. (2001) measured the biosorption of copper by P. syringae, fitting the experimental data to the Freundlich, Brunauer-Emmett-Teller (BET), and Langmuir equations. Meaningful maximum sorption capacities... 

Loukidou et al. (2005) fitted the data for the equilibrium sorption of Cd from aqueous solutions by Aeromonas caviae to the Langmuir and Freundlich isotherms. They also conducted, a detailed analysis of sorption rates to validate several kinetic models. A suitable kinetic equation was derived, assuming that biosorption is chemically controlled. The so-called pseudo second-order rate expression could satisfactorily describe the experimental data. The adsorption data of Zn on soil bacterium Pseudomonas putida were fit with the van Bemmelen-Freundlich model (Toner et al. 2005). 

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. 

Eq. (4.56) is typically observed for absorption reactions, for example for the (ab)sorption of nonpolar organic substances in a solid matrix containing organic (humus-like) material. Eq. (4.56) corresponds also to the initial linear portion of a Langmuir isotherm or to a Freundlich equation S = Kp cb where b = 1. 

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. 

It is also recognized that the sorption of hydrophobic organic compounds onto dissolved organic matter can significantly increase the solubility of the compounds under aqueous conditions. The importance of solubility in determining the fate of PPCPs will be discussed in Section 3.3.2. Where available, the sorption data can be htted to the Freundlich equation [Eq. (3.17)] to derive sorption parameters for the matrix ... 

Note that this Kid value is significantly smaller than the Kjd obtained in the linear part of the isotherm (i.e., at low 1,4-DNB concentrations). Furthermore, as can be seen from Fig. 2, the Freundlich equation overestimates C(S (and thus Kid) at both the low and the high end of the concentration range considered. In fact, inspection of Fig. 2 reveals that at very high concentrations, the K+-illite surface seems to become saturated with 1,4-DNB, which is not surprising considering that only limited adsorption sites are available. In such a case, the sorption isotherm can also be approximated by a Langmuir equation (Eq. 9-3). 

Logarithmic plots of the Freundlich equation, Q = kpn, where Q is the amount of methane adsorbed at a pressure p, and k and n are constants, for methane adsorption at 0°, 30° and 50°C. in Figures 6 and 7 indicate that the equation is valid up to at least 1000 torr. Equilibrium sorption points obtained on different samples in a manostatic adsorption apparatus are shown as solid points in Figures 6 and 7. The exponent n varied from 0.72 at 0° to 0.87 at 50°C. for the Pocahontas coal and from 0.78 at 0° to 0.94 at 50°C. for Pittsburgh coal (Table III). 

Figure 7. Plots of Freundlich equation for methane on Pittsburgh coal solid circles were obtained using mano-static sorption apparatus...

If solute-solute interactions occur (e.g., because of the presence of a limited number of sorption sites), then sorption can be modeled by fitting the experimentally derived isotherm to theoretical equations, the Freundlich isotherm... 

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) used a two-constant rate equation, derived below, which is adapted from the Freundlich equation to study the kinetics of P04 sorption and desorption on hematite and gibbsite. 

Sorption and desorption isotherms were obtained for sorption of radionuclides under oxidizing and reducing conditions. The Freundlich equation accurately describes most of these isotherms. Most radionuclides are apparently irreversibly sorbed on each of the geologic solids since the slopes of sorption and desorption isotherms for a given radionuclide are different. This hysteresis effect is very large and will cause a significant delay in radionuclide transport. It, therefore, should be included in modeling radionuclide transport to accurately assess the isolation capabilities of a repository in basalt. 

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 ... 

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. 

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. 

Rate-limited sorption can also be modeled assuming a kinetic rate expression coupled with a nonlinear equilibrium expression. If we assume a Freundlich isotherm and a first-order rate expression, we can use the following equation to model sorption kinetics [21] ... 

Equations (6)-(8) may be substituted into Eq. (18) to describe first-order rate-limited sorption for linear, Freundlich, and Langmuir isotherms, respectively. If equilibrium sorption can be assumed, Eqs. (3)-(5) may be used to define S in Eq. (18) in terms of C for linear, Freundlich, and Langmuir isotherms, respectively [21]. 

Numerical methods allow one to obtain a solution to the groundwater contaminant transport equation even when the equation is nonlinear (for example, when sorption is described by either a Freundlich or Langmuir isotherm). These numerical methods also permit solution of the governing partial differential equation when parameter values and initial and boundary conditions vary in space and time. 

Soils are multicomponent systems consisting of solid, liquid, and gaseous phases. These three phases are constantly in a dynamic state, trying to maintain equilibrium. Any type of perturbation in one phase influences the other two phases so that a new equilibrium state is approached. An equilibrium process that has been extensively investigated in soil systems employing the Freundlich equation involves sorption. Consider the reaction... 

Many studies indicated that in the presence of DOM, the metal sorption capacity decreased markedly for most soils, and the effect on the calcareous soil was greater than on the acidic sandy loam. Figure 10.4 shows the metal sorption equilibrium isotherms onto soils with or without the addition of 400 mg C/l of DOM. The equilibrium isotherms could be better depicted according to the linear Freundlich equation with the high value for the correlation coefficient of determination (r2) ... 

Parameters of Freundlich Equation for Metal Sorption by Soils in Presence and Absence of 400 mg C/l of DOM... 

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