Extensions of the Langmuir equation

The mass adsorbed is proportional to the fractional coverage of the surface. The proportionality factor is embodied in the constant of Eq 6.7. 

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An extension of the Langmuir equation (equation (6.3)) has been proposed for phosphate adsorption to soils and metal oxides in order to relate adsorption to the electrostatic potential of mineral surfaces and to the bulk solution characteristics, such as pH, electrolyte concentration, temperature and competing ions (Barrow et al., 1980 Barrow, 1983, 1985, 1993 Bolan et al., 1986) ... 

The multisite Langmuir (MSL) model [80] is an extension of the Langmuir equation to adsorption on multiple sites. Component i is assumed to occupy a, sites (a, is not necessarily an integer) in the adsorbed phase. The specific saturation capacity of component i is and for different components on the same adsorbent it satisfies... 

With the development of improved numerical methods for solution of differential equations and faster computers it has recently become possible to extend the numerical simulation to more complex systems involving more than one adsorbable species. Such a solution for two adsorbable species in an inert carrier was presented by Harwell et al. The mathematical model, which is based on the assumptions of plug flow, constant fluid velocity, a linear solid film rate expression, and Langmuir equilibrium is identical with the model of Cooney (Table 9.6) except that the mass transfer rate and fluid phase mass balance equations are written for both adsorbable components, and the multicomponent extension of the Langmuir equation is used to represent the equilibrium. The solution was obtained by the method of orthogonal collocation. 

Engineering theories for multicomponent adsorption can be roughly divided into three categories extensions of the Langmuir equation, the thermodynamic approach (ideal and real adsorbed solution theories, lAST and RAST) by Myers and Prausnitz (1965) and finally the potential adsorption theory, especially as extended to multi-component systems by Shapiro and co-workers (Shapiro and Stenby, 1998 Monsalvo and Shapiro, 2007a, 2009a,b). 

The theory of Brunauer, Emmett and Teller167 is an extension of the Langmuir treatment to allow for multilayer adsorption on non-porous solid surfaces. The BET equation is derived by balancing the rates of evaporation and condensation for the various adsorbed molecular layers, and is based on the simplifying assumption that a characteristic heat of adsorption A Hi applies to the first monolayer, while the heat of liquefaction, AHL, of the vapour in question applies to adsorption in the second and subsequent molecular layers. The equation is usually written in the form... 

By introducing a number of simplifying assumptions, Brunauer, Emmett and Teller (1938) were able to extend the Langmuir mechanism to multilayer adsorption and obtain an isotherm equation (the BET equation), which has Type II character. The original BET treatment involved an extension of the Langmuir kinetic theory of monomolecular adsorption to the formation of an infinite number of adsorbed layers. 

An extension of the Langmuir approach to multilayers adsorption was made by Brunauer, Emmett, and Teller, BET [17]. They assumed that the Langmuir equation applies to each layer. The heat of adsorption of the first layer was assumed to have a special value, but for the subsequent layers, the heat of adsorption was assumed to be equal to the heat of condensation of the gas. The volume adsorbed is then a summation of the adsorbed volumes of each layer. Upon evaluation of the summation, the BET equation results ... 

No equation of state is known that leads to an adsorption isothenu wliich in general fits experimental data over the entire range of n from zero to full monolayer coverage. Isotlieniis tliat find practical use are often 3-parameter empirical extensions of the Langmuir isothenu. An example is the Toth equation ... 

Of the three theories we will discuss, the Langmuir multicomponent expression is a more or less direct extension of a theory we have previously seen, i.e. of the Langmuir equation for binary systems that was presented in Chapter 7. 

Bi-layer adsorption is not uncommon and the development of the bi-layer adsorption isotherm equation is a simple extension of that used for the mono-layer equation. The Langmuir equation for bi-layer adsorption is as follows ... 

The double-reciprocal Langmuir model has been extensively used in site assessment projects for elemental adsorption data. The double-reciprocal Langmuir is an adaptation of the traditional equation for elemental sorption of solid phases exhibiting two primary adsorbing surface sites. The double-reciprocal Langmuir model is as follows ... 

Equations 21 and 22 present the useful extension of the Szyszkowski-Langmuir model to the adsorption with two orientational states at the interface. If the molecular interactions are considered, a similar simphfied model with P = 2 = P and b = b2 = b can be obtained from Eqs. 10 and 11, giving... 

Equations 27 and 28 present the extension of the Szyszkowski-Langmuir model to the adsorption of one-surfactant systems with aggregation at the interface. For the formation of dimmers on the surface, n = 2 and Eqs. 27 and 28 can be expanded to obtain the Frumkin equation of adsorption state. In general, the surface aggregation model described by Eqs. 27 and 28 contains four free parameters, including coi, n, b and Fc, which can be obtained by regression analysis of the data for surface tension versus surfactant concentration in the solution. 

Problem P8.04.01 takes the case where ES is not in equilibrium. Extensions of the M-M equation take into account the presence of additional substrates and inhibitors, and even other mechanisms. Some of these alternates are treated in the Problems. Clearly the M-M equation is a special case of the Langmuir-Hinshelwood equations of Chapter 6. 

The original derivation of the BET equation was an extension and generalization of Langmuir s treatment of monolayer adsorption. This derivation is based on kinetic considerations—in particular on the fact that at equilibrium the rate of condensation of... 

Equation 2.41 represents an additive extension of the modified Langmuir isotherm. The second Langmuir term covers the adsorption of the solute molecules on a second, completely independent group of adsorption sites of the adsorbent. Those... 

Finally, it can be shown that the multicomponent competitive Langmuir isotherm (Eq. 4.5) does not satisfy the Gibbs-Duhem equation if the column saturation capacities are different for the components involved [13]. This profound inconsistency may explain in part why this model does not accoimt well for experimental results. There are two very different alternative approaches to the problem of competitive Langmuir isotherms when the saturation capacities for the two pure compounds are different. Before discussing this important problem and an interesting extension of the competitive Langmuir isotherm, we must first present the competitive bi-Langmuir isotherm model. 

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... 

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]. 

The work discussed in this report is concerned with these two groups. The surface of type (a) will be referred to as patchwise heterogeneous after Ross and Olivier and type (b) as random heterogeneous after Hill. The development of the patchwise heterogeneous model followed an extension of the concepts discussed by Langmuir. It was realized that equation (5) could be written in a more general form ... 

Can the Langmuir equation represent the adsorption of butanol on water over an extensive concentration range ... 

There have been smdies of the impact of heterogeneity and improvements have been proposed but they lead to cumbersome equations. Consequently, the Langmuir equation is typically used in the simple form shown here. For a more detailed discussion of the various extensions of Langmuir equation, see the review by Shapiro and Stenby (2002). 

Ross and Olivier [55], in their extensive development of the van der Waals equation of state model have, however, provided a needed balance to the Langmuir picture. 

One important direetion of study has been to use empirieal adsorption data, together with the preassumed model for loeal adsorption, and attempt to extraet information about the form of x(e) [13,14]. The ehoiee of the model for loeal adsorption, whieh is an important input here, has been eustomarily treated quite easually, assuming that it has rather limited influenee on the form and properties of the evaluated EADFs. Usually, one of so many existing equations developed for adsorption on uniform surfaees is used as the loeal adsorption isotherm. The most often used forms of 0 p, T,e) are the Langmuir [6] and the Fowler-Guggenheim [15] equations for loealized adsorption. Ross and Olivier [4] extensively used the equation for mobile adsorption, whieh results from the two-dimensional version of the van der Waals theory of fluids. The most radieal solution has been... 

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