Langmuir adsorption with lateral interactions

This equation is sometimes called the Frumkin-Fowler-Guggenheim (FFG) isotherm [374— 376], For j3 = nEP/RT 4 lateral interactions cause a steeper increase of the adsorption isotherm in the intermediate pressure range. Characteristic of all Langmuir isotherms is a saturation at high partial pressures P/Po — 1. 

A remarkable shape is calculated with Eq. (9.35) for (3 4. A region is obtained where the 0-versus-P curve has a negative slope (dotted curve in Fig. 9.7). This is physically nonsense The coverage is supposed to decrease with increasing pressure and for one pressure there are three possible values of 6. In reality this is a region of two-phase equilibrium. Single adsorbed molecules and clusters of adsorbed molecules coexist on the surface. The situation is reminiscent of the three-dimensional van der Waals equation of state which can be used to describe condensation. 

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The BET model has been generalized by using of various monolayer isotherms for heterogeneous surfaces Langmuir-Freundlich [94], Toth [95], generalized Freundheh (GF) [94], Dubinin-Radushkevich [67,95,96], and others [5]. These equations have been apphed to the interpretation of experimental data [5,6]. The above-discussed procedure has also extended to the adsorption with lateral interactions on randomly heterogeneous surfaces also [5]. 

From the asymmetrical concentration profile with front tailing (see Figure 2.4b), it can correctly be deduced that (1) the adsorbent layer is already overloaded by the analyte (i.e., the analysis is being run in the nonlinear range of the adsorption isotherm) and (2) the lateral interactions (i.e., those of the self-associative type) among the analyte molecules take place. The easiest way to approximate this type of concentration profile is by using the anti-Langmuir isotherm (which has no physicochemical explanation yet models the cases with lateral interactions in a fairly accurate manner). 

For an evaluation of the local model isotherm 6(p,T,Q) with constant interaction energy Q, the effects of multi-layer adsorption and lateral interactions between neighboring adsorbed molecules are considered by applying two modifications to the Langmuir isotherm (i) a multi-layer correction according to the well known BET-concept and (ii) a correction due to lateral interactions with neighboring gas molecules introduced by Fowler and Guggenheim (FG) [105] ... 

Quinones, I. Guiochon, G. Isotherm models for localized monolayers with lateral interactions. Application to singlecomponent and competitive adsorption data obtained in RP-HPLC. Langmuir 1996, 12, 5433-5443. 

The site energy distribution function f Q) can be calculated by using the experimentally observed overall isotherm (p,T) and a theoretical local isotherm function d(p,T,Q). Here a Langmuir type model equation 9(p,T,Q) with corrections for multilayer adsorption and lateral interactions between the adsorbed molecules is chosen [54—56]. Then the integral equation can be solved by an analytical iterative method based on numerical integration [57]. More details about this procedure are found in [22,53]. 

Various functional forms for / have been proposed either as a result of empirical observation or in terms of specific models. A particularly important example of the latter is that known as the Langmuir adsorption equation [2]. By analogy with the derivation for gas adsorption (see Section XVII-3), the Langmuir model assumes the surface to consist of adsorption sites, each having an area a. All adsorbed species interact only with a site and not with each other, and adsorption is thus limited to a monolayer. Related lattice models reduce to the Langmuir model under these assumptions [3,4]. In the case of adsorption from solution, however, it seems more plausible to consider an alternative phrasing of the model. Adsorption is still limited to a monolayer, but this layer is now regarded as an ideal two-dimensional solution of equal-size solute and solvent molecules of area a. Thus lateral interactions, absent in the site picture, cancel out in the ideal solution however, in the first version is a properly of the solid lattice, while in the second it is a properly of the adsorbed species. Both models attribute differences in adsorption behavior entirely to differences in adsorbate-solid interactions. Both present adsorption as a competition between solute and solvent. 

Data on the adsorption of caprylic acid on a hydrophobic (mercury) surface in terms of a double logarithmic plot of Eq. (4.13) Panel a) compares the experimental values with a theoretical Langmuir isotherm, using the same values for the adsorption constant B for both curves. Panel b) shows that the adsorption process can be described by introducing the parameter a, which accounts for lateral interaction in the adsorption layer. Eq. (4.13) postulates a linear relation between the ordinate [= log [0/ 1 - 0)] - 2a 0 / (In 10)] and the abscissa (log c). If the correct value for a is inserted, a straight line results. For caprylic acid at pH 4, a value of a = 1.5 gives the best fit. 

With this in mind, some important adsorption isotherms were introduced, and we found that each of them describes important characteristics of the adsorption process (Table 6.10). Thus, the Langmuir isotherm considers the basic step in the adsorption process the Frumkin isotherm was one of the first isotherms involving lateral interactions the Temkin is a surface heterogeneity isotherm and the Flory-Huggins-type isotherms include the substitution step of replacing adsorbed water molecules by the adsorbed entities (Fig. 6.98). 

The non-specific adsorption of surfactants is based on the interaction of the hydrophilic headgroup and the hydrophobic alkyl chain with the pigment and substrate surfaces as well as the solvent. For the adsorption of surfactants, different models have been developed which take into account different types of interactions. A simple model which excludes lateral interactions of the adsorbed molecules is the Langmuir equation ... 

The procedure is first illustrated in terms of its application to the analysis of an isotherm calculated from an arbitrary assumed site energy distribution, f(Q). For reasons discussed in more detail later, the Langmuir equation—i.e., localized adsorption with no lateral interaction—appears very satisfactory as the form to use for 0(P,T,Q) ... 

For multilayer adsorption, the BET model assumes about the same position as that of Langmuir, its monolayer counterpart adsorption is thought to be localized, the surface is homogeneous and non-porous, lateral interaction is disregarded. The extension is that on one site more than one molecule can adsorb with different affinities, one on top of the other. 

A Ithough the adsorption of polymers onto solid surfaces has been thor-oughly studied (I), relatively few studies can be found in the literature on the adsorption of proteins onto polymer surfaces. In 1905, Landsteiner and Uhliz (2) discussed the interaction of serum proteins with synthetic surfaces. Blitz and Steiner (3) showed that albumin adsorption onto solid surfaces increased with increasing albumin concentration and that adsorption was nearly irreversible. Hitchcock reported (4) that adsorption of egg albumin onto collodion membranes followed a Langmuir isotherm with maximum adsorption occurring near the isoelectric point. Later, Kemp and Rideal (5) reported that protein adsorption onto solids conforms with Langmuir adsorption. 

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 L-type, would follow the Langmuir model, which is site adsorption without any lateral interaction between the adsorbate molecules. The concavity of the curve, in normal scale, is always directed toward the concentration axis. The S-type would follow a more complex model in which lateral interactions between molecules are to be taken into account, using, e.g., the Bragg-Williams approximation [15]. A concavity of the adsorption isotherm directed toward the y-axis is a very strong indication of lateral interactions between molecules. If one looks at the lUPAC classification of gas adsorption isotherms [1], the same remark holds this type of concavity is related with phenomena involving interactions between adsorbate molecules capillary condensation, multilayer formation, 2-D phase changes, etc. 

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]. 

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