Langmuirs Adsorption Theory

To evaluate the time-dependent function, X(t), a simple model of diffusion is proposed. Starting from Langmuir adsorption theory, we consider that liquid molecules having diffused into the elastomer are localized on discrete sites (which might be free volume domains). In these conditions, we can deduce the rate of occupation of these sites by TCP with time. Only the filhng of the first layer of the sites situated below the liquid/solid interface at a distance of the order of the length of intermolecular interaction, i.e., a few nanometers, needs to be considered to estimate X(t). 

The deviations from the Szyszkowski-Langmuir adsorption theory have led to the proposal of a munber of models for the equihbrium adsorption of surfactants at the gas-Uquid interface. The aim of this paper is to critically analyze the theories and assess their applicabihty to the adsorption of both ionic and nonionic surfactants at the gas-hquid interface. The thermodynamic approach of Butler [14] and the Lucassen-Reynders dividing surface [15] will be used to describe the adsorption layer state and adsorption isotherm as a function of partial molecular area for adsorbed nonionic surfactants. The traditional approach with the Gibbs dividing surface and Gibbs adsorption isotherm, and the Gouy-Chapman electrical double layer electrostatics will be used to describe the adsorption of ionic surfactants and ionic-nonionic surfactant mixtures. The fimdamental modeling of the adsorption processes and the molecular interactions in the adsorption layers will be developed to predict the parameters of the proposed models and improve the adsorption models for ionic surfactants. Finally, experimental data for surface tension will be used to validate the proposed adsorption models. 

Rate equations for simple reversible reactions are often developed from mechanistic models on the assumption that the kinetics of elementary steps can be described in terms of rate constants and surface concentrations of intermediates. An application of the Langmuir adsorption theory for such development was described in the classic text by Hougen and Watson (/ ), and was used for constructing rate equations for a number of heterogeneous catalytic reactions. In their treatment it was assumed that one step would be rate-controlling for a unique mechanism with the other steps at equilibrium. 

Derylo-Marczewska and Jaroniec [28] have reviewed the adsorption of organic solutes from dilute solutions and have provided a useful compilation of published experimental data for both single- and multisolute adsorption isotherms on carbonaceous adsorbents. They also presented a survey of theoretical approaches used to describe the solute adsorption equilibria, including the Polanyi adsorption model, the solvophobic interaction model, the Langmuir adsorption theory, the vacancy solution model, as well as considerations based on the energetic heterogeneity of the adsorbent. In particular, these authors emphasize the... 

The mechanism of clathrate hydrate formation shows similarities to adsorption of molecules at sites on a surface. The assumptions made for the mechanism of Langmuir adsorption are also applicable for hydrate formation [2]. The occupancy of the sites on a surface in the Langmuir adsorption theory is given by a so-called Langmuir isotherm, which can also be developed for the occupancy of the cavities in clathrate hydrates. 

This gives rise to direct application of Eq. (45). Such a Langmuir adsorption theory on heterogeneous surfaces has been applied in order to demonstrate the effect of negative deviation from ideal mixing, which has been observed for the most adsorbed binary mixtures. 

Langmuir adsorption isotherm A theoretical equation, derived from the kinetic theory of gases, which relates the amount of gas adsorbed at a plane solid surface to the pressure of gas in equilibrium with the surface. In the derivation it is assumed that the adsorption is restricted to a monolayer at the surface, which is considered to be energetically uniform. It is also assumed that there is no interaction between the adsorbed species. The equation shows that at a gas pressure, p, the fraction, 0, of the surface covered by the adsorbate is given by ... 

For nonlinear systems, however, the evaluation of the flow rates is not straightforward. Morbidelli and co-workers developed a complete design of the binary separation by SMB chromatography in the frame of Equilibrium Theory for various adsorption equilibrium isotherms the constant selectivity stoichiometric model [21, 22], the constant selectivity Langmuir adsorption isotherm [23], the variable selectivity modified Langmuir isotherm [24], and the bi-Langmuir isotherm [25]. The region for complete separation was defined in terms of the flow rate ratios in the four sections of the equivalent TMB unit ... 

Table 9-3. Operating conditions for complete separation under Equilibrium Theory. Langmuir adsorption isotherms (see Fig. 9-8).

FIGURE 3.5 Major components of classical receptor theory. Stimulus is the product of intrinsic efficacy (s), receptor number [R], and fractional occupancy as given by the Langmuir adsorption isotherm. A stimulus-response transduction function f translates this stimulus into tissue response. The curves defining receptor occupancy and response are translocated from each other by the stimulus-response function and intrinsic efficacy. 

Langmuir Quantitative theory of adsorption of gases on surfaces... 

For the simulation of SMB-separations efficient software packages,based on the Triangle-Theory, are commercially available. The number of columns, the column dimensions, the theoretical number of plates in the columns, the feed concentration, the bi-Langmuir adsorption isotherm parameters and the number of cycles need to be defined by the user. Then the separation is simulated and values for the flow rate ratios, the flow rates, the switching time and the quality of the separation, purity and yield, are calculated. Based on these values an actual separation can be performed. However, some optimization/further development is usually necessary, since the simulations are based on an ideal model and the derived parameters and results therefore can only be taken as indications for the test runs. 

Assuming that monomer can be bound by the template in a similar manner as in the adsorption process we can apply Langmuir s theory to describe this process. 

ADSORPTION ON SOLID SURFACES 7.9a The Langmuir Equation Theory... 

Theory of Vapor-Particle Partitioning of PAH Compounds in the Atmosphere. Yamasaki et al. (64) treated semivolatile PAHs as inherently vapor-phase materials, existing in the particle phase only because of their tendency to sorb on nonvolatile materials. Under this assumption, the proportion in each phase should vary with the available surface area for sorption and with the ambient temperature. They assumed that sorption followed a Langmuir adsorption isotherm, requiring that there be only a low fractional coverage of the particulate matter with semivolatile material. In this case, competition for sorption sites can be ignored. At equilibrium, the rate of sorption equals the rate of evaporation. They derived this relationship ... 

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). 

Before 1916, adsorption theories postulated either a condensed liquid film or a compressed gaseous layer which decreases in density as the distance from the surface increases. Langmuir (1916) was of the opinion that, because of the rapidity with which intermolecular forces fall off with distance, adsorbed layers are not likely to be more than one molecular layer in thickness. This view is generally accepted for chemisorption and for physical adsorption at low pressures and moderately high temperatures. 

By then, Polanyi had renewed laboratory experiments on adsorption with several co-workers, [23] substituting for his original adsorption theory a picture that covered both unimolecular and multimolecular adsorption. [24] Polanyi began to argue that Langmuir s formula represented too simple an idealization, which was not obeyed in all cases, and that his own revised theory could be used to derive Langmuir s isotherm as a special case. [20, 25]... 

Irving Langmuir General Electric Adsorption theory... 

When gas molecules hit a solid surface, they may be either reflected, or captured by the surface field of force and condense, to evaporate again later. In Langmuir s adsorption theory the factor a expresses the fraction of the impinging molecules which do condense on the surface. In recent years much information as to the manner of reflection or condensation has been obtained experimentally by the use of molecular rays, and theoretically by wave mechanics. Molecular rays are beams of molecules on which a particular direction has been imposed by slits, moving in a vacuum so high that there are practically no collisions between molecules, and the direction of the beams is maintained for long distances. 

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