Langmuir equilibrium, local

Assume that typical time scales of both direct and reverse reactions in (3.3.2) are much shorter than any other time scale in the system. Then the reaction (3.3.2) yields the Langmuir s local equilibrium relation between Ci and 0 of the form... 

With such a surface, covered with two different kinds of sites that behave independently (and on each of which the two basic principles of the Langmuir model, local adsorption and lack of adsorbate-adsorbate interactions, apply), the equilibrium isotherm results from the addition of the two independent contributions of the two types of sites [35]. Since in most cases the Langmuir isotherm is appropriate to account for single-component adsorption on a homogeneous surface (in practice if not in principle, see earlier), we have the following isotherm (see example in Figure 3.12.)... 

Consider now desorption phenomena on surfaces which, in the equilibrium condition, obey the Langmuir isotherm locally. As the Langmuir isotherm can be obtained assuming desorption kinetics of the first order and adsorption kinetics of zero order (Langmuir arrived at his expression in this way), consistency reasons imply that R(t, q) is given by first-order kinetics... 

The Langmuir equation (Eq. 5.1), derived originally to describe the adsorption of gases on solids, assumes that the adsorbed entity is attached to the surface at specific, homogeneous, localized sites, forming a monolayer. It is also assumed that the heat of adsorption is constant over the entire monolayer, that there is no lateral interaction between adsorbed species, that equilibrium is reached, and that the energy of adsorption is independent of temperature ... 

In this section we address formation of concentration shocks in reactive ion-exchange as an asymptotic phenomenon. The prototypical case of local reaction equilibrium of Langmuir type will be treated in detail, following [1], [51], For a treatment of the effects of deviation from local equilibrium the reader is referred to [51]. The methodological point of this section consists of presentation of a somewhat unconventional asymptotic procedure well suited for singular perturbation problems with a nonlinear degeneration at higher-order derivatives. The essence of the method proposed is the use of Newton iterates for the construction of an asymptotic sequence. 

These equations are supplemented by the equilibrium relationship between the solid and local liquid phase concentrations. Langmuir isotherm was chosen for the adsorption isotherm of a target product or by product ... 

Figure 21-18. Total local derivatives of the isotherms belongiag to different equilibrium states. Experimental data (squares) and theoretical results (solid lines) using the modified competitive Langmuir isotherm equation (21-8). (From reference 124, with permission.)...

For the moment no transitional states between these extremes will be considered. For these two limiting modes of adsorption the establishment of equilibrium proceeds via different paths in localized adsorption, adsorbed molecules can move to other sites only by desorption and re-attachment, whereas for mobile adsorption the path of tangential transport is also open. Typical representatives of localized isotherms are those of Langmuir. Frumkln-Fowler-Guggen-helm (monolayer) and BET (multilayer). For mobile adsorbates examples are the Volmer and the Hill-De Boer (or two-dimensional Van der Waals) equations. 

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)... 

It is assumed that the instant local equilibrium described by Eq. (21) is established. The Langmuir equation or other sorption isotherms can be utilized to represent the equilibrium relationship. The initial and boundary conditions are... 

As the mass transfer resistance at the external surface of the silicalite crystal particle has negligible effects on the overall mass transfer in the present LC system (13), local equilibrium at the external surface thus can be assumed to simplify the mathematics involved in the numerical solution. Since the adsorption isotherm data of alcohols from dilute aqueous solution in silicalite, as reported by Milestone and Bibby (2), can be described by a Langmuir type equation, the following Langmuir type equation is used as the boundary condition at the external surface of the silicalite crystal particle ... 

Fig. 11-14. A schematic diagram of Fig. II-15. Change in the rates of adsorption, ua, localized adsorption according to and desorption, ud, as the dynamic equilibrium is Langmuir established...

The equilibrium relations that must be expected for a mixed sorbate, are relatively simple for localized sorbate particles on identical independent sites in a zeolite. The pure components (indices i, j) obey Langmuir isotherms, Eq. (11), and the sorption equilibrium of the mixture will be described by... 

Following Drazer and Zanette, we assume that, although the whole system is not at equilibrium, in different local areas of the column there is a local equilibrium. We invoke the homottatic patch approximation in heterogeneous catalysis [20], according to which the overall adsorption isotherm is made up of a contribution of local Langmuir adsorption isotherms ... 

The condition of local mass equilibrium can exist even when adsorption and chemical reaction are taking place (Whitaker, 1999, Problem 1-3). When local mass equilibrium is not valid, one must propose an interfacial flux constitutive equation. The classic linear form is given by (Langmuir, 1916, 1917)... 

The actual acid-based conditions are assumed in local chemical equilibrium. The reactions included and their equilibrium constants (Sigg and Stumm, 1989 Langmuir, 1997 Appelo and Postma, 1999) are given in Table 11.4. 

In nonlinear adsorption systems where parallel diffusion mechanisms hold, the mass balance equation given in eq. (9.2-lb) is still valid. The difference is in the functional relationship between the concentrations of the two phases, that is the local adsorption isotherm. In general, this relationship can take any form that can describe well equilibrium data. Adsorption isotherm such as Langmuir, Unilan, Toth, Sips can be used. In this section we present the mathematical model for a general isotherm and then perform simulations with a Langmuir isotherm as it is adequate to show the effect of isotherm nonlinearity on the dynamics behaviour. The adsorption isotherm takes the following functional form ... 

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 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. 

However, at still higher concentrations, an adsorption/desorption equilibrium tends to be approached leading to a Langmuir-type relationship. Eventually, in spite of incorporation, saturation of the surface is reached and the surface concentration is no longer sensitive to local changes in the diffusion flux of the additives. Hence, differences in inhibition vanish and leveling is lost [11]. 

Toi et al. applied the dual sorption mechanism to analysis of the time-lag diffusion (permeation) under the constraint that the penetrant fraction attributed as the Langmuir component is completely immobilized, but in local equilibrium with the Henry s law dissolution component [1]. They yielded a mathematical description of transient permeation, consisting of a nonlinear partial differential equation. This equation was then solved by a finite-difference technique for the case of permeation... 

It should be realized that fitting the Langmuir s model to the experimental data makes sense only if the conditions underlying the Langmuir s theory are fulfilled. These are adsorption equilibrium and identical and independent adsorption sites. It implies localized adsorption and no interaction between the adsorbed molecules. It may be obvious that in particular the last-mentioned condition is often not met in practice. A modification of the theory is discussed later, which takes interaction between adsorbed molecules into account. 

In this case, considering a Langmuir-type equilibrium for the local isotherm leads to the following equations ... 

---