Freundlich expression

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... 

Two-Stage Crosscurrent Optimization. When fresh adsorbent is used at each stage, qo 0 and using an isotherm expression, to describe equilibrium, the least total amount of adsorbent can be calculated analytically. In this example, the Freundlich expression, Eq. (15.5), is used. For stage 1,... 

Application of the Freundlich equation Trial-and-error calculation for the adsorbent/solvent ratio can be eliminated if the equilibrium curve can be conveniently described algebraically. If the equilibrium curve is linear, the Kremser equations (5.50) to (5.53) and Fig. 5.16 apply. More frequently the Freundlich expression (11.5) is useful, and fresh adsorbent (Xj + i 0) is used in the last stage [86]. For a typical two-stage cascade (Fig. " 11.22) a solute material balance for the entire plant is... 

Now, if the solutes interact with themselves more strongly than they do with the stationary phase, then their presence will increase the interaction of further solute with the stationary phase mixture. This gives an isotherm having the shape shown in Figure 10. This type of isotherm is called a Freundlich isotherm, the expression for... 

In the simplest case, a linear relationship may exist between solutes in soil solution and those on the solid pha.se, such that C, = hCr, where C, is the. solute concentration on the solid phase and h defines the buffer power. In perhaps more typical situations, the relationship might be defined by a more complicated, concentration-dependent relationship such as a Freundlich or Langmuir expression here the instantaneous buffer power may be defined as the tangent to the exchange isotherm i.e., h = dCJdCi. Some care is needed when defining this buffer power (7). 

For different acceptor particle adsorption isotherms expressions (1.85) - (1.89) provide various dependencies of equilibrium values of <7s for a partial pressure P (ranging from power indexes up to exponential). Thus, in case when the logarithmic isotherm Nt InP is valid the expression (1.85 ) leads to dependence <75 P" often observed in experiments [20, 83, 155]. In case of the Freundlich isotherm we arrive to the same type of dependence of - P" observed in the limit case described by expression (1.87). 

The Freundlich sorptive isotherm is an empirical model expressed by ... 

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). 

Geochemical models, as with the approach, are commonly formulated with a variant of the Freundlich isotherm based on a chemical reaction, like Reaction 9.1. In this approach, known as the reaction Freundlich model or the activity Freundlich model, the extent of sorption by the reaction can be expressed,... 

Predominantly, Freundlich s fitted adsorption isotherms computed by means of simple linear regression were proposed for the mathematical description of the process studied. Unlike the Langmuir equation, the Freundlich model did not reduce to a linear adsorption expression at very low nor very high solute concentrations, as above resulted. 

The Freundlich model, which was first used to describe gas phase adsorption and solute adsorption, is an empirical adsorption model that has been widely used in environmental chemistry [138]. It can be expressed as... 

Sips [38] modified the Langmuir adsorption model by introducing a power law expression of the Freundlich equation ... 

The Sips [38] model (Eq. 10) can easily be extended to binary or multicomponent systems [34, 74]. The resulting expression for the multicomponent Langmuir-Freundlich adsorption model is ... 

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 ... 

The Freundlich equation was derived empirically, based on the logarithmic decrease in adsorption energy with increasing coverage of the adsorbent snrface. Freundlich fonnd that adsorption data for many dilnte solntions conld be fit by the expression... 

Many attempts have been made to express the curves relating the amount adsorbed per gm. of charcoal to the gas pressure in simple mathematical form, notably by Freundlich, Arrhenius, Williams and Langmuir. 

Freundlich isotherm the bulk concentrations Ci, (7a, necessary to produce precipitation will evidently follow no regular order, but be dependent on the variables a and n of the isotherm expression. Thus on the addition of aluminium chloride to arsenious sulphide... 

The dSp/dC term can be found from the equilibrium relationship of Freundlich isotherms, expressed as... 

Note that Freundlich s adsorption non-linearity is not differentiable since in most applications 0lacking smoothness of [Pg.17]

Consistent with experience with adsorbents like activated carbons, the fitted Klhc values are greater for sorbates with larger hydrophobicities (Fig. 9.136). Note that when using Freundlich isotherms, the KiF value depends nonlinearly on the units in which the concentration in the aqueous phase is expressed (see Problem 9.5). 

Note When using Freundlich isotherms, be aware that the numerical value of Kif depends nonlinearly on the unit in which the concentration in the aqueous phase is expressed. Hence for solving this problem, you may first convert pg - LH to mg-mL 1 or you may express the Freundlich equation using, for example, pg-kg l and pg-L respectively ... 

If the equilibrium isotherm can be expressed by the Freundlich equation and fresh adsorbent is used in each stage (Y0 = 0), the total amount of adsorbent used for a two-stage crosscurrent adsorption unit (Figure 10.9) is... 

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... 

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