Adsorption from ideal solutions

S is constant for ideal adsorption from ideal solutions [1-5], The constancy of the separation factor in many other cases may result fi om compensation effects. From the linear dependence of XiX2/n on xj, we determine the value of n g. 

The predicted adsorption isotherms From ideal solution theory (Equations 6—9) are also shown in Figures 3—5. Since it is diFFicult to see degree oF Fit on a log-log plot, the ability to describe the data is better illustrated in Figures 6-9, where the CACm is plotted For several adsorption levels as a Function oF monomer composition along with predictions From Equation 6. 

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. 

Assuming this standard state, the AC° value expresses a change in the Gibbs energy of adsorption of one molecule B upon being moved from the hypothetical ideal solution onto the electrode surface. This enables the particle-particle interactions on the surface to be separated from any other interactions and to be included in the term/(/i). 

When two similarly structured anionic surfactants adsorb on minerals, the mixed admicelle approximately obeys ideal solution theory (jUL - Below the CMC, the total adsorption at any total surfactant concentration is intermediate between the pure component adsorption levels. Adsorption of each surfactant component in these systems can be easily predicted from pure component adsorption isotherms by combining ideal solution theory with an empirical correspond ng states theory approach (Z3). ... 

Scamehorn et. al. (20) also presented a simple, semi—empirical method based on ideal solution theory and the concept of reduced adsorption isotherms to predict the mixed adsorption isotherm and admicellar composition from the pure component isotherms. In this work, we present a more general theory, based only on ideal solution theory, and present detailed mixed system data for a binary mixed surfactant system (two members of a homologous series) and use it to test this model. The thermodynamics of admicelle formation is also compared to that of micelle formation for this same system. 

Fig. 6a-d. Schematic view of adsorption from solution onto smooth, planar surfaces where the surface sites are considered to have the same area as the projected area of the solute of interest, a. Top, the ideal (Langmuir) case b. clustering of adsorbed solute due to attractive lateral interactions or positive cooperativity c. heterogeneous surface, i.e., two sets of binding sites d. patchwise heterogeneity or surface domains of different adsorptive properties... 

These assumptions are identical to those in the ideal theory of multiple equilibria used to analyze low molecular weight ligand binding to proteins 4-47 49). Figure 7 shows how these assumptions can be considered in more refined treatments of adsorption from solution 45). 

The surface excess amount, or Gibbs adsorption (see Section 6.2.3), of a component i, that is, /if, is defined as the excess of the quantity of this component actually present in the system, in excess of that present in an ideal reference system of the same volume as the real system, and in which the bulk concentrations in the two phases stay uniform up to the GDS. Nevertheless, the discussion of this topic is difficult on the other hand for the purposes of this book, it is enough to describe the practical methodology, in which the amount of solute adsorbed from the liquid phase is calculated by subtracting the remaining concentration after adsorption from the concentration at the beginning of the adsorption process. 

The adsorption of mixtures of surfactants has received comparatively little attention. The adsorption of mixtures of nonionic and anionic surfactants has been studied (10,25-27) and strong negative deviations from ideality were observed (10,27). Attempts to model the degree of non-ideality using regular solution theory failed (21). The adsorption of mixtures of anionic and cationic surfactants would be expected to exibit even larger deviations from ideality (28). 

The adsorption of binary mixtures of anionic surfactants in the bilayer region has also been modeled by using just the pure component adsorption isotherms and ideal solution theory to describe the formation of mixed admicelles (3 ). Positive deviation from ideality in the mixed admicelle phase was reported, and the non-ideality was attributed to the planar shape of the admicelle. However, a computational error was made in comparison of the ideal solution theory equations to the experimental data, even though the theoretical equations presented were correct. Thus, the positive deviation from ideal mixed admicelle formation was in error. 

The matter discussed in sec. 2.3 concerned the phenomenology of adsorption from solution. To make further progress, model assumptions have to be made to arrive at isotherm equations for the individual components. These assumptions are similar to those for gas adsorption secs. 1.4-1.7) and Include issues such as is the adsorption mono- or multlmolecular. localized or mobile is the surface homogeneous or heterogeneous, porous or non-porous is the adsorbate ideal or non-ideal and is the molecular cross-section constant over the entire composition range In addition to all of this the solution can be ideal or nonideal, the molecules may be monomers or oligomers and their interactions simple (as in liquid krypton) or strongly associative (as in water). 

These equations provide the means for calculationof activity coefficients from mixed-gas adsorptiondata. Alternatively, if y,- values can be predicted, tlrey allow calculationof adsorbate composition. In particnlar, if the mixed-gas adsorbate forms an ideal solution, tlren y, = 1, and the resulting equation is the adsorption analog of Raoult s law ... 

Predictions of adsorption equilibria by ideal-adsorbed-solution theory are usually satisfactory when the specific amount adsorbed is less than a third of the saturation value for mono-layer coverage. At higher adsorbed amounts, appreciable negative deviations from ideality are promoted by differences in size of the adsorbate molecules and by adsorbent heterogeneity. One must then have recourse to Eq. (14.123). The difficulty is in obtaining valnes of the activity coefficients, which are strong functions of both spreading pressnre and temperatnre. This is in contrast to activity coefficients for liquid phases, which for most applications are insensitive to pressure. This topic is treated by Talu et ai. ... 

The larger the selectivity, the easier the separation of component i from component j by adsorption. Zeolites with a selectivity as high as 10 for nitrogen relative to oxygen are used in pressure-swing adsorption processes" to produce oxygen from air. The specific amount of each component adsorbed for an ideal solution is given by... 

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