Other Adsorption Models

The software package described elsewhere (Fainerman et al. 2001) was used to fit two isotherm models to the data the reorientation model of Eqs. (18) to (20) yields COi = 6.7 10 mVmol, cg = 2.56 10 m /mol, a = 1.18 while the Langmuir model gives CO = 2.9 10 mVmol. In the range given in Fig. 30 the quality of fitting for both models is the same, however, as one can easily see, at lower concentrations the experimental data could give a more accurate answer on the model. As discussed by other authors, for all oxethylated alcohols the reorientation model is superior to other adsorption models (Lee et al. 2003). 

The TMAB model combines the thermodynamic mass action and mass balance equations to produce adsorption equations which correctly predict adsorption at the S-MO interface and give correct pH dependence of adsorption. This model was used exclusively for all adsorption equations derived so far in this chapter. The TMAB model will be contrasted with the other adsorption models throughout the remainder of this section. 

The earliest derivations of solute-metal oxide adsorption equations using die TMAB adsorption model appeared in the literature in die mid-1990s [2—4], The details of the derivation of the TMAB adsorption equations have been explored in detail in previous sections of this chapter so no further review of the model is necessary. However, it is useful to compare the subtle but important differences between die TMAB adsorption models and all the other adsorption models discussed in diis section as shown in Table 20. 

Nevertheless, each of the more popular isotherm models have been found useful for modeling adsorption behavior in particular circumstances. The following outlines many of the isotherm models presently available. Detailed discussions of derivations, assumptions, strengths, and weaknesses of these and other isotherm models are given in references 4 and 7—16. 

Surface Area and Permeability or Porosity. Gas or solute adsorption is typicaUy used to evaluate surface area (74,75), and mercury porosimetry is used, ia coajuactioa with at least oae other particle-size analysis, eg, electron microscopy, to assess permeabUity (76). Experimental techniques and theoretical models have been developed to elucidate the nature and quantity of pores (74,77). These iaclude the kinetic approach to gas adsorptioa of Bmaauer, Emmett, and TeUer (78), known as the BET method and which is based on Langmuir s adsorption model (79), the potential theory of Polanyi (25,80) for gas adsorption, the experimental aspects of solute adsorption (25,81), and the principles of mercury porosimetry, based on the Young-Duprn expression (24,25). 

A pulse of a racemic mixture (5 g each enantiomer) was carried out to check the adsorption model and to predict the mass transfer coefficient. The other model parameters used in simulation were = 0.4 and Pe = 1000. The mass transfer coefficient used to fit experimental and model predictions in the pulse experiment was k = 0.4 s k Model and experimental results are compared in Figs. 9-16 and 9-17. 

FHH (Frenkel-Halsey-Hill) theory is valid for multi molecules adsorption model of the flat surfrtce material. When this model is applied for the surface fractal in the range of capillary condensation, in other words, in the state of interface which was controlled by the surface tension between liquid and gas, the modified FHH equation can be expressed as Eq. (3). 

It is of considerable interest to compare the present model with the predictions of the statistical theories. In particular, the results will be compared with the theories of Scheutjens and Fleer (9,10,13). These authors have pointed out that their results are in general agreement with other published models. Comparison of the adsorption isotherms shown in Figure 1 with those given by Scheutjens and Fleer (see, for example, Figure 5 of Reference 13) shows excellent qualitative agreement. In each case, the isotherms consist of three regions as described above. 

The linear equilibrium isotherm adsorption relationship (Eq. 11) requires a constant rate of adsorption, and is most often not physically valid because the ability of clay solid particles to absorb pollutants decreases as the adsorbed amount of pollutant increases, contrary to expectations from the liner model. If the rate of adsorption decreases rapidly as the concentration in the pore fluid increases, the simple Freundlich type model (Eqs. 8 and 9) must be extended to properly portray the adsorption relationship. Few models can faithfully portray the adsorption relationship for multicomponent COM-pollutant systems where some of the components are adsorbed and others are desorbed. It is therefore necessary to perform initial tests with the natural system to choose the adsorption model specific to the problem at hand. From leaching-column experimental data, using field materials (soil solids and COMs solutions), and model calibration, the following general function can be successfully applied [155] ... 

To that end, an important idea contributed by Robertson and Michaels was that oxygen reduction on Pt could potentially be co-limited by adsorption and diffusion rather than by just one or the other. In modeling the system, they noted that it is not possible for adsorbed oxygen to be in chemical equilibrium with the gas at the gas-exposed Pt surface while at the same time being in electrochemical equilibrium with the applied potential at the three-phase boundary. To resolve this singularity, prior (and several subsequent) models for diffusion introduce an artificial fixed diffusion length governing transport from the gas-equilibrated surface to the TPb.56,57,59,64,65,70 coutrast, Robertsou and Michaels... 

Adsorption at liquid surfaces can be monitored using the Gibbs adsorption isotherm since the surface energy, y, of a solution can be readily measured. However, for solid substrates, this is not the case, and the adsorption density has to be measured in some other manner. In the present case, the concentration of adsorbate in solution will be monitored. In place of the Gibbs equation, we can use a simple adsorption model based on the mass action approach. 

Fig. 3 Comparison of the surface tension for nonionic surfactant CnEg as measured at T = 298.15 K, data points [45], with improved models considering orientational states of surfactant molecules at the surface. The data shown are obtained by regression analysis minimizing the revised chi-square The calculation with fi = 0 represents the best fit of the improved Szyszkowski-Langmuir model described by Eqs. 21 and 22. The other calculated curve with =- 3.921 shows the best fit of the improved Frumkin adsorption model described by Eqs. 23 and 24...

Additional information on adsorption mechanisms and models is in Stollenwerk (2003), 93-99 and Prasad (1994). Foster (2003) also discusses in considerable detail how As(III) and As(V) may adsorb and coordinate on the surfaces of various iron, aluminum, and manganese (oxy)(hydr)oxides. In adsorption studies, relevant laboratory parameters include arsenic and adsorbent concentrations, adsorbent chemistry and surface area, surface site densities, and the equilibrium constants of the relevant reactions (Stollenwerk, 2003), 95. Once laboratory data are available, MINTEQA2 (Allison, Brown and Novo-Gradac, 1991), PHREEQC (Parkhurst and Appelo, 1999), and other geochemical computer programs may be used to derive the adsorption models. 

Illustrative calculations appear below for estimating the particulate fraction (c()) of p,p -DDT in urban air at 20°C using the Junge-Pankow adsorption model (Equation 3), the Mackay adsorption model (Equation 15), and the octanol-air partition coefficient model (Equation 25). Table 10.3 lists values of P3 and Koafor p,p -DDT and other POPs of different chemical classes. All model calculations are for an urban air TSP = 80 pg/m3 (Shah et al., 1986). 

Theoretical models based on first principles, such as Langmuir s adsorption model, help us understand what is happening at the catalyst surface. However, there is (still) no substitute for empirical evidence, and most of the papers published on heterogeneous catalysis include a characterization of surfaces and surface-bound species. Chemists are faced with a plethora of characterization methods, from micrometer-scale particle size measurement, all the way to angstrom-scale atomic force microscopy [77]. Some methods require UHV conditions and room temperature, while others work at 200 bar and 750 °C. Some methods use real industrial catalysts, while others require very clean single-crystal model catalysts. In this book, I will focus on four main areas classic surface characterization methods, temperature-programmed techniques, spectroscopy and microscopy, and analysis of macroscopic properties. For more details on the specific methods see the references in each section, as well as the books by Niemantsverdriet [78] and Thomas [79]. 

At this stage of developments, most fixed-bed adsorption models assume that film mass transfer resistance is small compared with the other transport resistances in the system and that equilibrium is reached instantaneously between the solute in the pore liquid and at the surface of the sorbent. Even if it assumed that a homogeneous adsorptive HPLC sorbent is used, it can be readily shown, however, that both film and pore diffusion mass transfer resistances cannot be ignored393,394 and that the dynamic behavior of the adsorption stage is greatly dependent on the rate of the polypeptide- or protein-ligate interaction.8,357,395 Breakthrough of solute(s) may thus occur... 

A frequently used adsorption model that allows for adsorption in multilayers has been introduced by Brunauer, Emmett and Teller [10] and is known as the BET equation. With the exception of the assumption that the adsorption process terminates at monolayer coverage, these authors have retained all the other assumptions made in deriving the Langmuir adsorption isotherm. Hence all objections to the application of the Langmuir equation apply here, too. 

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