Equation Langmuir-Freundlich

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. 

In coal bed capacity estimation, the Langmuir equation provides a simple and efficient relation for single layer low-pressure conditions. In the case of high pressure and high temperature, Bi Langmuir, extended Langmuir, Sips, Langmuir-Freundlich, Toth, UNILAN, two-dimensional... 

This three-parameter equation behaves linearly in the Henry s law region and reduces to the Langmuir isotherm for m = 1. Other well-known isotherms include the Langmuir-Freundlich isotherm or Sips isotherm [Sips,/. Chem. Phys., 16, 490 (1948) Koble and Corrigan, Ind. Eng. Chem., 44,383 (1952)] or loading ratio correlation with prescribed temperature dependence [Yon and Turnock, AlChE Symp. Sen, 67(117), 75 (1971)]... 

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. 

It will be noted that the universal isotherm equation as written here has formal similarity to pressure explicit forms of Langmuir, Langmuir-Freundlich and LRC models. One key advantage of the universal form is that the heat of adsorption and the adsorption equilibrium are bound to be self-consistent. 

The problem of predicting multicomponent adsorption equilibria from single-component isotherm data has attracted considerable attention, and several more sophisticated approaches have been developed, including the ideal adsorbed solution theory and the vacancy solution theory. These theories provide useful quantitative correlations for a number of binary and ternary systems, although available experimental data are somewhat limited. A simpler but purely empirical approach is to use a modified form of isotherm expression based on Langmuir-Freundlich or loading ratio correlation equations ... 

Several different mathematical relationships (referred to as isotherms) have been developed to describe the relationship of fractional surface coverage with respect to the adsorbing species. Work by Langmuir, Freundlich, Tempkin, and others have attempted to describe the above mentioned pressure differential to fractional surface coverage. The most widely used and accepted isotherm is the BET equation, named after its orignators, Brunauer, Emmett, and Teller, and is as follows 29... 

At this point, it is feasible to correlate the liquid-phase adsorption equilibrium single component data, with the help of isotherm equations developed for gas-phase adsorption, since, in principle, it is feasible to extend these isotherms to liquid-phase adsorption by the simple replacement of adsorbate pressure by concentration [92], These equations are the Langmuir, Freundlich, Sips, Toth, and Dubinin-Radushkevich equations [91-93], Nevertheless, the Langmuir and Freudlich equations are the most extensively applied to correlate liquid-phase adsorption data. [2,87],... 

There have been numerous attempts to assign mathematical isothermal adsorption relations to various experimental data. Among the most frequently used isotherm equations are Langmuir, Freundlich, and BET. 

In the past, much attention was given to the study of dye and iodine adsorption by active carbons (Bmnauer, 1945 Orr and Dalla Valle, 1959). Many studies have been made with dye molecules of well-known size, shape and chemical properties, but the results have not been easy to interpret (Giles et al., 1970 McKay, 1982, 1984). In a systematic study of iodine adsorption (from aqueous solution) on a carbon black and four activated carbons (Femandez-Colinas etal., 1989b), it was found that the iodine isotherms could be analysed by the as-method. In this way it was possible to assess values of the available volume in pores of effective width of 0.5-1.5 nm. The adsorption of iodine was also featured in a recent study by Ziolkowska and Garbacz (1997), who applied the Langmuir, Freundlich and other isotherm equations. 

The fitting of the Langmuir, Freundlich, and R-P models to the data has been firstly applied to the photoreactivity results obtained from runs carried out at equal mass of cafalysf and lamp power. For the Langmuir model, the following procedure has been followed. In order to have an estimate of parameters values, the data at high initial concentration of benzyl alcohol have been fitted to Equation (A6) (see Appendix Al) and those at low initial concentration to Equation (A13). The parameters obtained by these fitting procedures have been used to determine Ng by means of Equation... 

In a kinetically controlled separation system using CMS or zeolite 4A as adsorbents, it is necessary to use more accurate rate model. Therefore, concentration dependent diffusivity model based on Darken equation combined with Langmuir-Freundlich isotherm was applied and each result was compared with the experimental data. 

A new mathematical model was developed to predict TPA behaviors of hydrocarbons in an adsorber system of honeycomb shape. It was incorporated with additional adsorption model of extended Langmuir-Freundlich equation (ELF). LDFA approximation and external mass transfer coefficient proposed by Ullah, et. al. were used. In addition, rate expression of power law model was employed. The parameters used in the power model were obtained directly from the conversion data of hydrocarbons in adsorber systems. To get numerical solutions for the proposed model, orthogonal collocation method and DVODE package were employed. 

For the special values of heterogeneity parameters m = m s (0,1) the isotherm (7) reduces to the known Langmuir-Freundlich (LF) equation. 

In the theories of gas adsorption the applicability of Freundlich equation was long ago associated with the energetic heterogeneity of the adsorption sites on the actual solid surfaces. It was also known, that Freundlich equation is a simplified form of a more general isotherm equation which is now commonly called Langmuir-Freundlich isotherm. Thus, in 1980 Sposito [13] suggested that it was high time it were used also in the case of ion adsorption at water/oxide interfaces. 

The Langmuir-Freundlich equation is similar in form to the Langmuir isotherm. 

To generate adsorption data, a known amount of adsorbate in aqueous solution is mixed with a known amount of adsorbent. At equilibrium, the amount of adsorbate removed from solution is assumed to be adsorbed. Secondary reactions (such as precipitation) must be eliminated or corrected for. Precipitation is indicated in some cases by a rapid increase in apparent adsorption (disappearance from solution) with a small change in solution concentration. Three equations are commonly used to describe adsorption the Langmuir, Freundlich, and Brunauer-Emmett-Teller (BET) equations. 

If data fail to conform to the Langmuir equation, the Freundlich equation often fits the data successfully ... 

Figure 2 shows the comparison of the fractal-layer (solid line a) and two-timescale (solid line b) models with the simulations in terms of effective diffusivity, eq. (13). Both the models furnish a satisfactory level of agreement with simulation data. We may therefore conclude that approximate models based on a Riemann-Liouville constitutive equation are able to furnish an accurate description of adsorption kinetics on fractal interfaces. These models can also be extended to nonlinear problems (e.g. in the presence of nonlinear isotherms, such as Langmuir, Freundlich, etc.). In order to extend the analysis to nonlinear cases, efficient numerical sJgorithms should be developed to solve partied differential schemes in the presence of Riemann-Liouville convolutional terms. 

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