Langmuir model, sorption

The Langmuir isotherm (or Langmuir model) provides an improvement over the K( and Freundlich approaches by maintaining a mass balance on the sorbing sites (Stumm and Morgan, 1996). The model, for this reason, does not predict that species can sorb indefinitely, since the number of sites available is limited. When the calculation carries reactions for the sorption of more than one aqueous species, furthermore, it accounts for competition such a calculation is known as a competitive Langmuir model. 

To parameterize a Langmuir model, you determine from experimental measurements not only the equilibrium constant K, but the surface s sorption capacity (or exchange capacity). The latter value is a measure of the number of sorbing sites and is commonly reported in moles or electrical equivalents, per gram of dry sediment. Multiplying the sorption capacity in moles by the mass of sediment in a system gives the mole number of sorbing sites, which is... 

Several investigators [7,123] suggested the use of a Langmuir-type saturation model in addition to the electrostatic model to account for saturation effects. The Langmuir model implies that there are a finite number of localised sorption sites [15] ... 

Another widely used sorption model is the Langmuir equation. It was developed by Irving Langmuir [140] to describe the adsorption of gas molecules on a planar surface. It was first applied to soils by Fried and Shapiro [ 141 ] and Olsen and Watanabe [142] to describe phosphate sorption on soils. Since that time, it has been heavily employed in many environmental fields to describe sorption on various solid surfaces [19,65]. The general Langmuir model is... 

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 ... 

The Langmuir and Brumnauer, Emmett, and Teller (BET) models also have been used to describe nonlinear sorption behavior for environmental solids, particularly for mineral dominated sorption (Ruthven, 1984 Weber et al., 1992). The Langmuir model assumes that maximum adsorption corresponds to a saturated monolayer of solute molecule on the absorbent surface, that there is no migration of the solute on the surface phase, and that the energy of adsorption is constant. The BET model is an extension of the Langmuir model that postulates multilayer sorption. It assumes that the first layer is attracted most strongly to the surface, while the second and subsequent layers are more weakly held. 

Many other equilibrium relationships have been applied to model sorption. For example, the Langmuir (Eq.44) and the Polanyi-Dubinin (Eq.45) isotherms have been widely applied to adsorption in zeolites [9] ... 

Adsorption and desorption processes of F in soils and soil components has been studied extensively. Bar-Yosef et al. (1988) studied F sorption by montmorillonite and kaolinite and reported that F affinity was greatest in K-kaolinite and least with Ca-montmorillonite. These authors also concluded that the Langmuir model satisfactorily fit F sorption by these adsorbents. Fluoride sorption in soils has been described by both Langmuir and Freundlich isotherms (Omuetti and Jones, 1977 Chhabra et al., 1980 Morshina, 1980 Gupta et al., 1982). Peek and Volk (1985) reported that the Langmuir isotherm is applicable only for limited concentrations of F in soils. 

If we pursue the model of the catalytic surface as a collection of a fixed number of sites S capable of forming chemical bonds with adsorbed species, then we may expect to find that substances which can compete for the catalytic sites may inhibit the reaction. Furthermore, the effectiveness of the inhibition will depend on the relative pressures of the two adsorbates as well as their sorption constants on the surface. Let us consider as an example the simple Langmuir-type sorption of two sorbates A and B on a surface of S° sites ... 

Dent s Surface Sorption Theory. The Dent sorption theory or model (52), in the simple form, is a modification of the BET model, which is itself an extension of the earlier Langmuir model (56). The Langmuir model assumes that a gas (water vapor in the case of wood) is sorbed onto sorption sites on the substrate or sorbent in a mono-layer only. The fraction of sorption sites occupied by the vapor or sorbate is a function of the vapor pressure of the sorbate and approaches unity as the vapor pressure increases. 

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... 

A nonlinear local isotherm model is clearly required for description of sorption reactions between the TCB and the shale isolate. A variety of conceptual and empirical models for representing nonlinear sorption equilibria, exists (2). The Langmuir model is one of the ideal limiting-condition-type models cited earlier. It is predicated on a uniform surface affinity for the solute and prescribes a nonlinear asymptotic approach to some maximum sorption capacity. 

Natural surfaces generally are too complex to be characterized as having uniform solute interaction energies, however. It is therefore not unexpected that sorption equilibrium data for natural soils and sediments are rarely described adequately by the Langmuir model. Such data are commonly described more satisfactorily by the empirical Freundlich isotherm model, which has the form... 

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... 

Fundamentals of sorption and sorption kinetics by zeohtes are described and analyzed in the first Chapter which was written by D. M. Ruthven. It includes the treatment of the sorption equilibrium in microporous sohds as described by basic laws as well as the discussion of appropriate models such as the Ideal Langmuir Model for mono- and multi-component systems, the Dual-Site Langmuir Model, the Unilan and Toth Model, and the Simphfied Statistical Model. Similarly, the Gibbs Adsorption Isotherm, the Dubinin-Polanyi Theory, and the Ideal Adsorbed Solution Theory are discussed. With respect to sorption kinetics, the cases of self-diffusion and transport diffusion are discriminated, their relationship is analyzed and, in this context, the Maxwell-Stefan Model discussed. Finally, basic aspects of measurements of micropore diffusion both under equilibrium and non-equilibrium conditions are elucidated. The important role of micropore diffusion in separation and catalytic processes is illustrated. 

The total solubility is the sum of the contributions of both the dissolution mode based on Henry s law and the binding mode using the Langmuir model the dual mode sorption [11], At equilibrium, the total solubility of olefin is shown by Eq. (9-4), where C is the total concentration of the olefin gas absorbed in the sample, p is the applied olefin pressure, ku is the solubility coefficient of the olefin gas for Henry s law mode, K is the olefin binding equilibrium constant of Eq. (9-2), as defined in Eq. (9-3), and Q is the saturated amount of the olefin gas bound to the silver complex. 

Recent developments in spectroscopic techniques offer the opportunity to increase our understanding of oxyanion surface speciation and binding. This understanding is essential to properly use mechanistic sorption models, such as the constant capacitance model and the triple layer model. A recent criticisms of these models is that selection of the surface species and reaction from the sorption data alone results in an empirical model which could be replaced with the traditional Langmuir model (7). However determination of the surface species and reactions will constrain the parameterization and allow for mechanistic evaluation of the sorption models. Knowledge of the actual species and reaction should also enable more generalized prediction of sorption behavior outside the range of the actual experiment, which is not possible at present. 

The Langmuir model is based on the assumption of ideal localized adsorption without interaction on a set of identical sites as outlined in Section 2.4. The special case of sorption of CH4 or Ar in sodalite was noted in Section 3.5 as an example of a system for which the basic assumptions of the Langmuir model are in fact fulfilled and for which the isotherms conform, as expected, to the Langmuir equation (Eq. (2.28)]. Sorption of normal tri- or tetradecane in 5A zeolite is another example of a zeolitic system in which each cage can accommodate only one sorbate molecule. Approximate conformity... 

The data of Barrer and Wasilewski " for the sorption of I2 on 5A and 13X zeolites (Figure 4.2) are remarkable since they show conformity with the simple Langmuir model over very wide concentration ranges. More detailed analysis reveals that the heat of sorption varies strongly with sorbate concentration, as may be seen from Figure 4.2c. A constant heat of sorption is one of the requirements of the Langmuir model so the variation of heat of adsorption... 

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