Extended Langmuir Model

The derivations given in Sections 2.4 and 3.3 may be easily extended to binary or multicomponent systems. The resulting expression for the isotherm is 

Kemball, Ridcal, and Guggenheim and independently Broughton have shown that thermodynamic consistency requires = q 2- physical adsorption of molecules of widely different size such an assumption is unrealistic. If the extended Langmuir equation is regarded as an analytical description rather than a physical model, the use of different values of q for each component becomes permis.sible although the equations cannot then be expected to apply over the entire concentration range and extrapolation on such 

The separation factor for a system described by Eq. (4.11) is given simply by the ratio of the adsorption equilibrium constants for the two components  

This assumption of a constant separation factor leads to considerable simplifi- 

FIGURE 4.15. Icsl of the Lewis correlation [Eq. (4.14)] for sorption of hydrocarbons on silica gel and two activated carbons. 

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In the study, Langmuir model was assumed to fit the adsorption equilibrium relationship of fluoxetine on the column packed by the new CSP and the following approximated extended Langmuir model was used to describe the adsorption behavior... 

Adsorbate Extended Langmuir model Dual model at 293 K ... 

The extended Langmuir model (ELM) is the most common approach used to represent multicomponent gas adsorption in coal. This is based on direct extension of the single component isotherm to a multicomponent system giving the analogous expression... 

This equation can be readily obtained from the extended Langmuir model, Eq. 3.4, for the binary system. Thus, when the saturated amounts for the pure gases are the same, the IAS is identical to the extended Langmuir model. 

Markham andBenton. This model (34) is known as the extended Langmuir isotherm equation for two components, i and j. 

The Langmuir model was extended to include interaction between the adsorbed atoms/molecules by Fowler and Guggenheim [31], The model now becomes... 

We extend the model of the previous section. Instead of one regulatory site we now have two sites, say R and / 2, which are identical and bind the effector R (Fig. 8.9). We still have one active site A, hence in the absence of the effector the binding isotherm 0 (Q, Cg = 0) is a simple Langmuir curve. 

With respect to the kinetics of aromatic oxidations, (extended) redox models are suitable, and often provide an adequate fit of the data. Not all authors agree on this point, and Langmuir—Hinshelwood models are proposed as well, particularly to describe inhibition effects. It may be noted once more that extended redox models also account for certain inhibition effects, for mixtures of components that are oxidized with different velocities. The steady state degree of reduction (surface coverage with oxygen) is mainly determined by the component that reacts the fastest. This component therefore inhibits the reaction of a slower one, which, on its own, would be in contact with surface richer in oxygen (see also the introduction to Sect. 2). 

In Langmuir model, the maximal adsorption is that of a monolayer. Langmuir adsorption isotherms all saturate at high vapor pressures. This is unrealistic for many cases. In order to consider the adsorption of multilayers, Brunauer, Emmett, and Teller extended the Langmuir theory and derived the so-called BET adsorption isotherm [378], The basic idea in the BET theory was to assume a Langmuir adsorption for each of the layers (Fig. 9.8). 

Brunauer, Emmett, and Teller (BET) proposed a model that extends the Langmuir model to multilayer adsorption and which has several AHads components. The BET equation treats the extra adsorption layers by analogy with evaporation and condensation processes, and takes the form ... 

Hiester and Vermeulin [39] extended the model of Thomas by deriving a breakthrough curve for a slow diffusion rate-limiting step with a nonlinear Langmuir equilibrium isotherm. The model was further adapted by Arnold and Blanch [40] to describe affinity chromatographic experiments ... 

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. 

The pure component adsorption equilibrium of ethane and propane are measured on Norit AC at three temperatures (30, 60 and 90 °C). All experimental data of two species at three temperatures are employed simultaneously to fit the isotherm equation to extract the isothermal parameters. Since an extended Langmuir equation is used to describe the local multicomponent isotherm, the maximum adsorbed capacity is forced to be the same for ethane and propane in order to satisfy the thermodynamic consistency. The saturation capacity was assumed to be temperature dependent while the other parameters, bo and u], are temperature independent but species dependent. The derived isotherm parameters for ethane and propane are tabulated in Table 1. The experimental data (symbols) and the model fittings (solid lines)... 

These assumptions are justifiable as the heat of adsorption of the small inert sorbate (e.g., N2 or Ar) is rather low and, hence, differences between sorption sites at the surface will be very small. Similarly, the interaction between the first and the following layers will be close to the heat of condensation, as the effect of polarization by the surface will be small beyond the first layer (screening of the long-range van der Waals forces). From its conception, the BET theory extends the Langmuir model to multilayer adsorption. It postulates that under dynamic equilibrium conditions the rate of adsorption in each layer is equal to the rate of desorption from that layer. Molecules in the first layer are located on sites of constant interaction strength and the molecules in that layer serve as sorption sites for the second layer and so forth. The surface is, therefore, composed of stacks of sorbed molecules. Lateral interactions are assumed to be absent. With these simplifications one arrives at the BET equation... 

The most conspicuous difference between the experimental and calculated band profiles is the lower maximum concentration peak (center figures) and the lower plateau concentration (far right figures) measured for resorcinol and catechol than calculated. Note, however, that the isotherm was determined in a concentration range which does not extend to the displacer concentration used. Hence, there is a possible error in the exact position of the operating line. In this concentration range, deviations of the adsorption behavior from the Langmuir model is probable, as this model assumes the solution to be ideal. 

The simplest model for the electrical double layer is the Helmholtz condenser. A distribution of counterions in the bulk phase described by a Boltzmann distribution agree with the Gouy-Chapman theory. On the basis of a Langmuir isotherm Stem (1924) derived a generalisation of the double layer models given by Helmholtz and Gouy. Grahame (1955) extended this model with the possibility of adsorption of hydrated and dehydrated ions. This leads to a built-up of an inner and an outer Helmholtz double layer. Fig. 2.14. shows schematically the model of specific adsorption of ions and dipoles. 

Other flexible isotherm models extending the Langmuir model are based on the theory of heterogeneous surfaces (Jaroniec and Madey, 1988) and on concepts provided by statistical thermodynamics (Hill, 1960). The latter approach allows deriving the following second-order isotherm model that is capable to describe inflection points in the isotherm courses ... 

The full range of adsorption isotherms under the lUPAC system is shown in Figure 7.46. Brunauer, Emmett and Teller were able to extend Langmuir s theory of monolayer adsorption to obtain an isotherm (the BET equation) which models Type II behaviour, for meso- and macroporous systems. Briefly, in the theory, molecules in one layer act as adsorption sites for molecules in the next layer, so that the adsorbed layer is not of uniform thickness, but rather is made up of a random stack of molecules. The theory has limitations, such as the assumption of liquid-like behaviour in all adsorbed layers but the first however, it has become a... 

The last three chapters deal with the fundamental and empirical approaches of adsorption isotherm for pure components. They provide the foundation for the investigation of adsorption systems. Most, if not all, adsorption systems usually involve more than one component, and therefore adsorption equilibria involving competition between molecules of different type is needed for the understanding of the system as well as for the design purposes. In this chapter, we will discuss adsorption equilibria for multicomponent system, and we start with the simplest theory for describing multicomponent equilibria, the extended Langmuir isotherm equation. This is then followed by a very popularly used IAS theory. Since this theory is based on the solution thermodynamics, it is independent of the actual model of adsorption. Various versions of the IAS theory are presented, starting with the Myers and Prausnitz theory, followed by the LeVan and Vermeulen approach for binary systems, and then other versions, such as the Fast IAS theory which is developed to speed up the computation. Other multicomponent equilibria theories, such as the Real Adsorption Solution Theory (RAST), the Nitta et al. s theory, the potential theory, etc. are also discussed in this chapter. 

Recognizing the deficiency of the extended Langmuir equation, despite its sound theoretical footing on basic thermodynamics and kinetics theories, and the empiricism of the loading ratio correlation, other approaches such as the ideal adsorbed solution theory of Myers and Prausnitz, the real adsorption solution theory, the vacancy solution theory and the potential theory have been proposed. In this section we will discuss the ideal adsorbed solution theory and we first develop some useful thermodynamic equations which will be used later to derive the ideal adsorbed solution model. 

AffyILM Relies on inputs from hybridization thermodynamics and uses an extended Langmuir isotherm model to compute transcript concentrations Berger and Carlon (2011)... 

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