Langmuir-Freundlich isotherm methods

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. 

A common method of extracting f K) from Eq. 3.82 is to assume a form of the distribution function by differentiation of a smooth fimction describing the data. The function obtained by this method is called the affinity spectrum (AS) and the method, the AS method [71]. The most general approach uses a cubic spline to approximate the data. However, a simpler procedure uses a Langmuir-Freundlich (LF) isotherm model and the AS distribution is derived from the best parameters of a fit of the experimental isotherm data to the LF model [71]. This approach yields a unimodal distribution of binding affinity with a central peak, if the range... 

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]. 

The following table 5.3-3 shows the various formula for the spreading pressure and the pure component hypothetical pressure for various commonly used isotherms. Some isotherms such as Langmuir, Freundlich, LRC have analytical expressions for the spreading pressure as well as the pure component hypothetical pressure. Other isotherms, such as O Brien Myers, Ruthven, Toth and Nitta have analytical expression for the spreading pressure, but the pure component hypothetical pressure expressed in terms of the reduced pressure must be determined from a numerical method. For other general isotherms, such as Unilan, Aranovich, Dubinin-Radushkevich, Dubinin-Astakhov, Dubinin-Stoeckli, Dubinin-Jaroniec, one must resort to a numerical method to obtain the spreading pressure as well as the pure component hypothetical pressure. 

Carbonization seems to be an effective method to adjust the pore size of PAF-1 to increase the gas selectivity. PAF-1-450 (PAF-1 carbonized at 450 °C), with a narrow micropore distribution of 0.8 nm, shows obvious increased CO2 sorption. Besides, on the basis of single component isotherm data, the dual-site Langmuir-Freundlich adsorption model-based lAST prediction indicates that the CO2/N2 adsorption selectivity may be as high as 209 at a 15 85 CO2 N2 ratio. Also, the CO2/CH4 adsorption selectivity should be in the range of 7.8-9.8 at a 15 85 C02 CH4 ratio at 0adsorption selectivity could be about 392 at 273 K and 1 bar for the 20 80 CO2 H2 mixture (Figure 10.2). ... 

Beyond these relatively simple systems and for all other non-linear isotherms, it is necessary to obtain solutions for the breakthrough curves by applying numerical approximation techniques to the model equations. Standard finite difference or collocation methods are commonly used. Table 6.3 provides a brief source list to solutions for plug flow and axially dispersed models with Langmuir, Freundlich or more general isotherms. 

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. 

Many empirical equations have been proposed to describe sorption behavior of aqueous species on solid surfaces. The most conventional methods of plotting sorption data are the Freundlich and Langmuir isotherms (7). Both isotherms were developed for the sorption of gases by solid surfaces. However, they often, but not always, describe the sorption of nonelectrolytes and other species from liquid solution onto solid... 

This method consists to find the thermodynamic model (Henry, Langmuir, Fowler, Freundlich...) which gives the best fit with adsorption isotherms measured at various temperatures [7], The parameter of the model Kmodei is temperature dependent. It can be related to the dimensionless equilibrium constant of adsorption K(T) by the... 

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