Extended Langmuir equation

For multicomponent systems, the local adsorption isotherm for a given micropore is assumed to follow the extended Langmuir equation. 

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)... 

The BET equation has been applied to ion adsorption from soil solutions, although the extended Langmuir equation (Eq. 9.11) would seem to apply as well. The BET equation has also been used to study the adsorption of pesticides having relatively high vapor pressures. 

For liquid mixtures that are dilute in two or more solutes, the multicomponent adsorption may be estimated from a concentration form of the extended Langmuir equation (9-38). 

Chapters 2 to 4 deal with pure component adsorption equilibria. Chapter 5 will deal with multicomponent adsorption equilibria. Like Chapter 2 for pure component systems, we start this chapter with the now classical theory of Langmuir for multicomponent systems. This extended Langmuir equation applies only to ideal solids, and therefore in general fails to describe experimental data. To account for this deficiency, the Ideal Adsorption Solution Theory (lAST) put forward by Myers and Prausnitz is one of the practical approaches, and is presented in some details in Chapter 5. Because of the reasonable success of the IAS, various versions have been proposed, such as the FastlAS theory and the Real Adsorption Solution Theory (RAST), the latter of which accounts for the non-ideality of the adsorbed phase. Application of the RAST is still very limited because of the uncertainty in the calculation of activity coefficients of the adsorbed phase. There are other factors such as the geometrical heterogeneity other than the adsorbed phase nonideality that cause the deviation of the IAS theory from experimental data. This is the area which requires more research. 

Further, if Uj = 1 for all i, eq. (2.4-12) reduces to the extended Langmuir equation. We shall further discuss multicomponent systems in details in Chapter 5. 

Although the extended Langmuir equations can be derived from the kinetic approach presented in the last section, it can be derived from thermodynamics, which should be the right tool to study the equilibria (Helfferich, 1992). The two approaches, however, yield identical form of the equilibrium equation. What to follow is the analysis due to Helfferich. 

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. 

We have shown in the last few sections the LAS theory as well as its computation implementation to obtain multicomponent adsorption isotherm. Since this theory is based on solution thermodynamics it can be applied to prove the thermodynamic consistency of the extended Langmuir equation. 

Now to calculate the binary adsorption equilibria for this system with the partial pressures of the two components being Pj and P2, we apply the IAS theory to the adsorbed phase 1. It has been shown in the Section 5.3.6 that if the saturation capacities of the adsorbates are the same, the adsorption of these adsorbates in the binary mixtures can be described by the extended Langmuir equation, that is ... 

To see the explicit form of the diffusion matrix in terms of concentration, we take an example of the extended Langmuir equation ... 

If the adsorption isotherm takes the form of the extended Langmuir equation (11.6-1), the local flux written in terms of the gradient of gas phase concentrations... 

When the system consists of a gas mixture containing N components, the overall adsorption equilibrium is described by the general integral equation, Eq. (16). Kapoor et al. applied the extended Langmuir equation as a local isotherm and a uniform energy distribution. The uniform distribution function is... 

The MPSD model can be conveniently extended to describe the adsorption equilibria of a gas mixture with N components. The generalized MPSD model simply takes the extended Langmuir equation as the local isotherm while using the MPSD information obtained from pure component data ... 

The equilibrium interaction between different adsorbates is confined within the same pore where they are accommodated i.e., adsorbates inaccessible to a pore do not participate in the competition and hence are not accounted for in the local extended Langmuir equation for that pore (M < AO- This effect of size exclusion is reflected in the lower integration limit. 

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... 

As in the extended Langmuir equation for mixtures, the hybrid Lang-muir-Freundlich equation (Eq. 3.3) can also be extended to an -component mixture (Yon and Tumock, 1971) ... 

For systems following the Langmuir isotherm, the IAS theory is identical to the extended Langmuir equation for mixtures, if the saturated amounts are equal... 

Eq. 3.1, is followed by botb components. For a binary mixture, the extended Langmuir equation, Eq. 3.4., is... 

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