Freundlich Equilibrium Approach

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 

Since SOM is most often the main component responsible for sorption of organic chemicals, e.g., contaminants (Fig. 4.19), researchers use octanol (an eight-carbon alcohol) to simulate organic matter-hydrophobic contaminant sorption phenomena. 

They carry out this process by extrapolating the octanol-water partition coefficient (jBTow) of a particular organic contaminant to the SOM-water partition coefficient (K0 and finally to the soil-water partition coefficient (KD) as follows. Consider that 

The magnitude of Koc for a particular contaminant can be obtained from the octanol-contaminant partition coefficient K0VI by the following relationship (Means et al., 

The Langmuir equilibrium approach was developed in 1918 by Langmuir to describe vapor adsorption on a homogeneous surface. It incorporates several assumptions when employed to model adsorption of chemical species in soil-solution suspensions. These assumptions are that 

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Figure 12.4 shows the use of a linear equilibrium approach (equation 12.8) for several Kd values (ranging from 0.5 to 3 cm3/g) in order to describe the BTC from the Bs-I column (C0 = 0.005 M). It is obvious that a linear approach failed to describe the shape of the BTC results. We therefore attempted to use the Freundlich approach of equation 12.1 in the transport equation where the dimensionless parameter n was allowed to vary from 0.5 to 4. As shown in Figure 12.5, the use of n < 1, which is... 

Sorption. Capture of neutral organics by non-living particulates depends on the organic carbon content of the solids (9). Equilibrium sorption of such "hydrophobic" compounds can be described by a carbon-normalized partition coefficient on both a whole-sediment basis and by particle size classes. The success of the whole-sediment approach derives from the fact that most natural sediment organic matter falls in the "silt" or "fine" particle size fractions. So long as dissolved concentrations do not exceed 0.01 mM, linear isotherms (partition coefficients) can be used. At higher concentrations, the sorptive capacity of the solid can be exceeded, and a nonlinear Freundlich or Langmuir isotherm must be invoked. 

On the other hand, in a sufficient number of cases a definite equilibrium is undoubtedly reached in a short time, and if we confine ourselves to these, it becomes possible to approach the second question we have put, that referring to the connection between concentration and amount adsorbed. Among the investigators who have treated this problem both mathematically and experimentally Freundlich deserves to be mentioned particularly. 

Fig. 4.18 shows the result of Cd2+ adsorption on illite in presence of Ca2+ (Comans, 1987). The data are fitted by Freundlich isotherms after an equilibration time of 54 days. It was shown in the experiments leading to these isotherms that adsorption approaches equilibrium faster than desorption. Comans has also used 109Cd to assess the isotope exchange he showed that at equilibrium (7-8 weeks equilibration time) the isotopic exchangeabilities are approximately 100 % i.e., all adsorbed Cd2+ is apparently in kinetic equilibrium with the solution. The available data do not allow a definite conclusion on the specific sorption mechanism. 

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. 

Equilibrium between solution and adsorbed or sorbed phases is a condition commonly used to evaluate adsorption or sorption processes in soils or soil-clay minerals. As previously stated, equilibrium is defined as the point at which the rate of the forward reaction equals the rate of the reverse reaction. Two major techniques commonly used to model soil adsorption or sorption equilibrium processes are (1) the Freundlich approach and (2) the Langmuir approach. Both involve adsorption or sorption isotherms. A sorption isotherm describes the relationship between the dissolved concentration of a given chemical species (adsorbate) in units of micrograms per liter (pg L 1), milligrams per liter (mg L-1), microequivalents per liter (pequiv L-1), or millimoles per liter (mmol L-1), and the sorbed quantity of the same species by the solid phase (adsorbent) in units of adsorbate per unit mass of adsorbent (solid) (e.g., pg kg-1, mg kg-1, peq kg-1, or mmol kg 1) at equilibrium under constant pressure and temperature. Sorption isotherms have been classified into four types, depending on their general shape (Fig. 4.13) ... 

Since the use of equilibrium (Freundlich) type with n > 1 is uncommon, we also attempted the kinetic reversible approach given by equation 12.2 to describe the effluent results from the Bs-I column. The use of equation 12.2 alone represents a fully reversible S04 sorption of the n-th order reaction where kj to k2 are the associated rates coefficients (Ir1). Again, a linear form of the kinetic equation is derived if m = 1. As shown in Figure 12.7, we obtained a good fit of the Bs-I effluent data for the linear kinetic curve with r2 = 0.967. The values of the reaction coefficients kj to k2, which provided the best fit of the effluent data, were 3.42 and 1.43 h with standard errors of 0.328 and 0.339 h 1, respectively (see Table 12.3). Efforts to achieve improved predictions using nonlinear (m different from 1) kinetics was not successful (figures not shown). We also attempted to incorporate irreversible (or slowly reversible) reaction as a sink term (see equation 12.5) concurrently with first-order kinetics. A value of kIIT = 0.0456 h 1 was our best estimate, which did not yield improved predictions of the effluent results as shown in Figure 12.7. 

Figure 13.2. Evidence for reversibility for sorption on illite. Adsorption-desorption equilibrium for Cd(II) on illite after 54 days of equilibration. The solution contains HCO, 2 X 10 M Ca, and has a pH = 7.8. Freundlich isotherms are based on separate adsorption ( ) and desorption (o). Adsorption approaches equilibrium faster than desorption. The data do not allow a conclusion on the specific absorption mechanism. (Data are from Comans, 1987.)...

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. 

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... 

Theoretical breakthrough curves for nonlinear systems may be calculated by numerical solution of the model equations using standard finite difference or collocation methods. Such solutions have been obtained by many authors and a brief summary is given in Table 8.4. In all cases plug flow was assumed and the equilibrium relationship was taken to be of cither Langmuir or Freundlich form. As linearity is approached ( ->1.0) the linearized rate models approach the Anzelius model (Table 8.1, model la) while the diffusion models approach the Rosen model (Table 8.1, model la). The conformity of the numerical solution to the exact analytic solution in the linear limit was confirmed by Garg and Ruthven. ... 

A third approach for CDI modeling is to quantify experimental data for salt adsorption in the electrodes as function of salt concentration in the external bath (recycle volume) using one of several adsorption isotherms, such as those based on the Langmuir or Freundlich equation. From the fitted parameters such as equilibrium constant K, useful information can be extracted on the interaction energy between ion and substrate. The fitted isotherms can also be used to predict adsorption at other values of the reservoir ionic strength. 

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