Solvent adsorption Langmuir equation

The more dispersive solvent from an aqueous solvent mixture is adsorbed onto the surface of a reverse phase according to Langmuir equation and an example of the adsorption isotherms of the lower series of aliphatic alcohols onto the surface of a reverse phase (9) is shown in figure 9. It is seen that the alcohol with the longest chain, and thus the most dispersive in character, is avidly adsorbed onto the highly dispersive stationary phase, much like the polar ethyl acetate is adsorbed onto the highly polar surface of silica gel. It is also seen that... 

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 ... 

This result is not surprising, namely, the original Langmuir equation was derived for the case of single-gas adsorption, and adsorption from solution occurs in the presence of solvent molecules and of other solutes that are potential competitors for the adsorption sites. A reaction analogous to Eq. (5.2) occurs for each competitor, and the apparent adsorption capacity for the adsorbate of interest is a result of occupation of surface sites by the competitors. The apparent adsorption capacity, i.e. the plateau in the adsorption isotherm (measured in the presence of competitors) does not have specific physical sense. The concentration of surface sites is seldom considered as a fully adjustable parameter, because there are actual atoms or groups of atoms on the surface of the adsorbent responsible for adsorption. The number of these atoms or groups is proportional to the concentration of the adsorbent (solid to liquid ratio). 

The Langmuir equation may be applicable for adsorption on fluid surfaces under ideal conditions. The parameter 0 should then be interpreted as being equal to F/F >, where r, is the maximum surface excess attainable. Comparison with an actual adsorption isotherm (taken from Figure 10.6) in Figure 10.7 shows marked differences. These must be due to deviations from ideality. The most important causes of nonideality are (a) a difference in molecular size between surfactant and solvent, and (b)... 

The second and fourth assumptions are seldom valid, as solvent molecules do adsorb to the monolayer of solvent attached to the aaivated adsorbent — forming layers. But, in the same way that the ideal gas law is useful, so also is the Langmuir Equation (isotherm). It is often the first choice for a model of the adsorption process because it has been shown to be valid for, and produces information about, the number of activated sites on (not within) a monolayer of an adsorbent. 

This is not the total capadty of the activated carbon to hold solvent because the BET Equation allows for multilayer adsorption As used in the Langmuir Equation, this parameter can take on values no greater than s ax (as solvent Is present only in a monolayer)... 

The Langmuir equation is theoretically valid when the adsorbent is homogeneous, when there are no solute-solute or solute-solvent interactions either in solution or in surface layers and under conditions of monolayer adsorption [42]. 

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. 

This equation is the mono-layer adsorption isotherm of solvent (B) and is exactly the same as the previously derived Langmuir adsorption equation but somewhat differently expressed. 

First let us note that experiment revealed long ago that not all ions prefer the bulk to the interface [8]. Gibbs adsorption equation predicts that the surface tension increases with the electrolyte concentration when the total surface excess of ions is negative. The conventional picture, that the ions prefer the bulk, is probably due to Langmuir, who noted that the increase in the surface tension of aqueous solutions of simple salts with increasing concentration can be explained by assuming a surface layer of pure solvent with a thickness of about 4 A [9]. However, because the aqueous solutions of some simple acids (such as HC1) possess surface tensions smaller than that of pure water [8], Gibbs adsorption equation indicates a positive total... 

When one of the solvents has a limited and low solubility, C, we find the classic Langmuir absorption isotherm is obtained with a sli t modification to the activity axis as shown in Figure 9.13. Such a solubility limit can be obtained by precipitation cn- micellization in the case of surfactants. Micellization is the association of ionized surfactant molecules into structures where the hydrophobic parts of the surfactant molecules expel water. Micelles have different structures (i.e., spheres, cylinders, and lamellar structures) depending on the surfactant molecule and its concentration of surfactant in solution. Each structure has a different maximum concentration, C , which limits adsorption. In Figure 9.13, the activity is replaced by concentration in the dilute solution case and the ooncentraticm axis C2 becomes dimensionless by division by the solubility limit when the constant b is replaced by 6C in equation 9.40. 

Two papers deal with the kinetics of the oxidation on TS-1 andTi,Al- 3 [110, 111]. Equation 18.11 illustrates the general rate law valid for both catalysts, according to a Langmuir-Hinshelwood model. It is consistent with the competition of alcohol/solvent for Ti sites and the adsorption of the substrate and the oxidant on the same site. The inhibitory effect of water, implicit in the competitive adsorption of the solvent, was proved by oxidation tests performed in acetonitrile containing variable quantities of water [77, 111]. 

The Langmuir isotherm has been readily extended to Hquid-soHd equilibria, first on an empirical basis, then on a more fundamental one. This problem is discussed in the next section (Section 3.2.1.1). The Freimdlich isotherm (Section 3.2.2.4), first used for gas-solid isotherms, has also been extended to liquid-solid equilibria. These isotherms have permitted a correct description of experimental results in a variety of experimental studies involving dilute solutions of a strongly adsorbed component in a pure solvent. The pressure is replaced by the concentration in the equation of the isotherm. As expected from the derivation already discussed, the Langmuir isotherm appears to accoxmt fairly well for adsorption data acquired at low or moderate concentrations. At high concentrations, on the other hand, the activity coefficients of the species in solution are concentration dependent and systematic deviations from Langmuir adsorption behavior are observed. 

Along with dissolved hydrogen and some admixtures, a liquid reaction mixture contains one or several reactants, reaction products, and sometimes also a solvent consisting of one or several compounds. All these compounds may affect the kinetics of the hydrogenation reaction. For the purposes of this study, the aim of which consists in a quantitative description of changes in composition of the reaction mixture, it is sufficient to employ the Langmuir-Hinshelwood kinetics (7) in its simplest form, where the process of hydrogenation is characterized by the rate constant and adsorption coefficient obtained from the simplest kinetic equations. 

A higher form of interpretation of the effect of solvents on the rate of heterogeneously catalyzed reactions was represented by the Langmuir-Hinshelwood kinetics (7), in the form published by Hougen and Watson (2), where the effect of the solvent on the reaction course was characterized by the adsorption term in the kinetic equation. In catalytic hydrogenations in the liquid state kinetic equations of the Hougen-Watson type very frequently degrade to equations of pseudo-zero order with respect to the concentration of the substrate (the catalyst surface is saturated with the substrate), so that such an interpretation is not possible. At the same time, of course, also in these cases the solvent may considerably affect the reaction. As is shown below, this influence is very adequately described by relations of the LFER type. 

Of the cases described so far, the presence of a third compound had the strongest effect on the selectivity of hydrogenation of two olefinic substrates in the pair olefin-unsaturated alcohol. This influence appeared both in cases where the third compound was unsaturated, was adsorbed competitively and reacted on the catalyst surface, and in cases where the third compound was represented by an inert solvent not undergoing competitive adsorption (was not entering equations of the Langmuir-Hinshelwood type, which were degraded to pseudo-zero order) and obviously operated through interactions of molecules from the bulk phase with adsorbed molecules of the substrate. 

The study of a particular adsorption process requires the knowledge of equilibrium data and adsorption kinetics [4]. Equilibrium data are obtained firom adsorption isotherms and are used to evaluate the capacity of activated carbons to adsorb a particular molecule. They constitute the first experimental information that is generally used as a tool to discriminate among different activated carbons and thereby choose the most appropriate one for a particular application. Statistically, adsorption from dilute solutions is simple because the solvent can be interpreted as primitive, that is to say as a structureless continuum [3]. Therefore, all equations derived firom monolayer gas adsorption remain vafid. Some of these equations, such as the Langmuir and Dubinin—Astakhov, are widely used to determine the adsorption capacity of activated carbons. Batch equilibrium tests are often complemented by kinetics studies, to determine the external mass transfer resistance and the effective diffusion coefficient, and by dynamic column studies. These column studies are used to determine system size requirements, contact time, and carbon usage rates. These parameters can be obtained from the breakthrough curves. In this chapter, I shall deal mainly with equilibrium data in the adsorption of organic solutes. 

However, the treatment used by Cameron and his co-workers is physically incorrect. Equations 10-2 and 10-6 apply to a Langmuir-type adsorption equilibrium in which the solvent is merely the inert carrier for the solute and does not compete with it for the surface of the adsorbent. We have examined evidence in this chapter that demonstrates this is not always the case, particularly if the solvent has the linear long-chain structure of the higher n-alkanes used by Cameron zt al. The equilibria for the two-component adsorption are given by... 

Aside from the relative position of the profile, the shape of the effluent profile contains information concerning the kinetics of the adsorption process. All concentrations of protein from zero to cQ are brought into contact with the column surface as the protein solution flows through the column, as a function of the position of the profile, and therefore as a function of time. Working with small molecules, previous researchers have shown that compounds exhibiting Langmuir isotherms produce sharp fronts, and diffuse tails, if pure solvent is used to desorb the column (21,22). However, Equation 7 shows that both diffusional and adsorption effects can alter the shape of the effluent profile. The former effect includes both normal molecular diffusion, and also diffusion due to flow properties in the column (eddy diffusion), which broadens (decreases the slope) the affluent profiles. To examine the adsorption processes, apart from the diffusional effects, the following technique can be applied. 

The thermodynamics and dynamics of interfacial layers have gained large interest in interfacial research. An accurate description of the thermodynamics of adsorption layers at liquid interfaces is the vital prerequisite for a quantitative understandings of the equilibrium or any non-equilibrium processes going on at the surface of liquids or at the interface between two liquids. The thermodynamic analysis of adsorption layers at liquid/fluid interfaces can provide the equation of state which expresses the surface pressure as the function of surface layer composition, and the adsorption isotherm, which determines the dependence of the adsorption of each dissolved component on their bulk concentrations. From these equations, the surface tension (pressure) isotherm can also be calculated and compared with experimental data. The description of experimental data by the Langmuir adsorption isotherm or the corresponding von Szyszkowski surface tension equation often shows significant deviations. These equations can be derived for a surface layer model where the molecules of the surfactant and the solvent from which the molecules adsorb obey two conditions ... 

The mathematical models that have been applied to the physical adsorption from liquid solutions are generally extensions of the theories that have been developed to describe the sorption of gases on solid surfaces with modifications to account for the competition between the solute and solvent for the adsorption sites. Two of these models have been applied to the adsorption isotherms of nonelectrolytes from solution they are the Langmuir model and the Brunauer, Emmett, and Teller (BET) model in addition the Freundlich empirical equation has also been used. In the Langmuir model it is assumed that the adsorbed species forms a monolayer on the surface of the adsorbent, that the adsorbed molecules... 

No data on solutions are included either. Although there is considerable information in the literature on certain polymers, it is dependent on the particular choice of solvent and not amenable to systematic tabulation. The same is true of the wealth of adsorption from solution onto solids and spread film Langmuir trough data. Equations are available for calculating the surface tension of simple liquid mixtures that could be applied to polymers [14] and for calculating polymer solvent solution [15] surface tensions. 

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