Langmuir model parameters

Figure 10 Values of kinetic and Langmuir model parameters [see Equation (19)] vs. the absorbed photon flow per unit mass of catalyst. , K[ A, Nl , k. The empty symbols refer to dark conditions (Yurdakal et at, 2008b).

Another limitation of tire Langmuir model is that it does not account for multilayer adsorption. The Braunauer, Ennnett and Teller (BET) model is a refinement of Langmuir adsorption in which multiple layers of adsorbates are allowed [29, 31]. In the BET model, the particles in each layer act as the adsorption sites for the subsequent layers. There are many refinements to this approach, in which parameters such as sticking coefficient, activation energy, etc, are considered to be different for each layer. 

For a Langmuir model where a diatomic gas dlssoclatlvely adsorbs, the relationships between a j, 3j and the kinetic parameters are given by ... 

Another problem which can appear in the search for the minimum is intercorrelation of some model parameters. For example, such a correlation usually exists between the frequency factor (pre-exponential factor) and the activation energy (argument in the exponent) in the Arrhenius equation or between rate constant (appears in the numerator) and adsorption equilibrium constants (appear in the denominator) in Langmuir-Hinshelwood kinetic expressions. 

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. 

Although the assumptions of the Langmuir model are generally not fulfilled for molecular sieve adsorbents, this equation has been found to provide a useful empirical representation of the isotherms (2). The parameters b and qB must, however, be regarded simply as empirical constants. 

On a more basic level, since the matrix model implicitly requires a somewhat inconsistent interpretation for the various model parameters in Eq (A-l) and Eq (A-2), it becomes primarily an empirical means of reproducing the observed pure component data with no fundamental basis for generalization to mixtures. One could, of course envision several extensions based on additional a terms in the denominator of Eq (A-l) and additional 8 terms in Eq (A-2). Such an approach to mixture permeation analyses would be completely empirical and mimic the generalization of Eq (2) and Eq (7) however, without any physical justification. The generalizations of Eq (2) and Eq (7) were natural outgrowths of the fundamental physical basis of the Langmuir isotherm. The fact that the mixture data are so consistent with Eq (7) and Eq (9) provides strong support for the physical basis of the dual mode model. 

Figure 1. Dependence of surface pressure on total Langmuir trough area for silanized glass particles 75 15 pm in diameter with different hydrophobisation degree (thin lines) according to Horvolgyi et al. 27 theoretical calculations used Eq. (14) (bold solid line) and Eq. (15) (bold dashed line) with model parameters given in Table 1.

The Langmuir model describes, for a uniform surface and a non-self-interacting adsorbate, the relationship between amount adsorbed and exposure concentration. The parameters of the model are the maximum amount adsorbed as a full monolayer and the equilibrium constant for the adsorption-desorption process which indirectly reflects the strength of the adsorbate-substrate interaction. For the present situation the analysis is modified in the following ways ... 

The fitting of the Langmuir, Freundlich, and R-P models to the data has been firstly applied to the photoreactivity results obtained from runs carried out at equal mass of cafalysf and lamp power. For the Langmuir model, the following procedure has been followed. In order to have an estimate of parameters values, the data at high initial concentration of benzyl alcohol have been fitted to Equation (A6) (see Appendix Al) and those at low initial concentration to Equation (A13). The parameters obtained by these fitting procedures have been used to determine Ng by means of Equation... 

Figure 23 Chondrite-normalized abundances of REEs in representative harzburgites from the Oman ophiolite (symbols—whole-rock analyses), compared with numerical experiments of partial melting performed with the Plate Model of Vemieres et al. (1997), after Godard et al. (2000) (reproduced by permission of Elsevier from Earth Planet. Set Lett. 2000, 180, 133-148). Top melting without (a) and with (b) melt infiltration. Model (a) simulates continuous melting (Langmuir et al., 1977 Johnson and Dick, 1992), whereas in model (b) the molten peridotites are percolated by a melt of fixed, N-MORB composition. Model (b) is, therefore, comparable to the open-system melting model of Ozawa and Shimizu (1995). The numbers indicate olivine proportions (in percent) in residual peridotites. Bolder lines indicate the REE patterns of the less refractory peridotites. In model (a), the most refractory peridotite (76% olivine) is produced after 21.1% melt extraction. In model (b), the ratio of infiltrated melt to peridotite increases with melting degree, from 0.02 to 0.19. Bottom modification of the calculated REE patterns residual peridotites due to the presence of equilibrium, trapped melt. Models (c) and (d) show the effect of trapped melt on the most refractory peridotites of models (a) and (b), respectively. Bolder lines indicate the composition of residual peridotites without trapped melt. Numbers indicate the proportion of trapped melt (in percent). Model parameters...

As has become clear adsorption phenomena play an important, if not, decisive role in this behaviour, and good data and modelling of adsorption are mandatory, too, to serve as the input parameters for the permeation description. This should not be l ted to the T.angmnir model, but other theories like the IAS (ideal adsorbed solution) and NIAS (non-ideal) should be considered, since they sometines work well for binary systems where the Langmuir model fails. 

In this review we concentrate on the studies that attempt to elucidate the importance of carbon surface properties in controlling the equilibrium uptakes of aromatic and aliphatic adsorbates. Rather than comparing model parameters, such as Langmuir or Freundlich constants, we examine the uptakes at comparable equilibrium concentrations and attempt to rationalize the differences observed under different conditions and on different adsorbents. 

Interestingly, a fully appropriate model was developed at the same time as the Langmuir model using a similar basic approach. This is the Michaelis-Menten equation which has proved to be so useful in the interpretation of enzyme kinetics and, thereby, understanding the mechanisms of enzyme reactions. Another advantage in using this model is the fact that a graphical presentation of the data is commonly used to obtain the reaction kinetic parameters. Some basic concepts and applications will be presented here but a more complete discussion can be found in a number of texts. ... 

All processes are modeled as series of countercurrent equilibrium cells. Parameters were determined experimentally (section 3). A liquid-phase reaction is accounted for by Da = (rate constant)x(cell volume)/(solid flow rate). Adsorption is described by the bi-Langmuir model. All equations were implemented in the simulation environment DIVA [3] details on the implementation of a largely analogous model can be found in [1,4]. The following set of performance parameters were used to evaluate each process ... 

The isotherm data acquired from frontal analysis over a broad concentration range fitted well to the bi-Langmuir model, see Figure 18, demonstrating that the adsorption on Kromasil CHI-TBB is heterogeneous with two types of sites. The saturation capacity of site II obtained from the bi-Langmuir isotherm parameters were qs>n = 130 mM for (R)-(-)-2-phenylbutyric acid and qs,n= 123 mM for (S)-(+)-2-phenylbutyric. 

The model shown in Eq. 2.42 has three independent fitting parameters, qS3t, e and b, which allow independent control of slope and curvature. For e =1 it approaches the Langmuir model. 

A Langmuir isotherm describes the adsorption. All model parameters are summarized in Appendix B.4. 

Similar adsorption data have also been reported for the adsorption isotherms of many compounds in various systems. For example, the adsorption data of several jS-blockers, particularly those of propranolol acquired in a 1 to 7000 relative concentration range, on an immobilized cellulase. Cel 7A, fit very well to the bi-Langmuir model, as illustrated in Figure 3.13 [47,55]). The enantioselective site was identified as a pair of amino acid residues in the tunnel formed by the main chain of the protein. The parameters of the isotherm depend on the pH as illustrated in Figure 3.14. A bi-Langmuir model was also found to accoimt well for the separation of pairs of enantiomers on polymers molecularly imprinted with one of the enantiomers [56]. Note, however, that there are also many systems in which the adsorption isotherms of enantiomers are not accounted for by a bi-Langmuir model showing that enantioselectivity is often achieved by a complex... 

Finally, it should be emphasized that the successful use of a bi-Langmuir isotherm model (as of any other combination of models that multiplies the number of model parameters) to account for a set of experimental adsorption data requires that these data are acquired in a wide concentration range [55]. This is even more... 

This isotherm is similar to the Langmuir model, to which it becomes identical for t = 1. The parameters b and t permit independent adjustment of the initial slope and curvature of the isotherm. This model has been used successfully to account for experimental isotherm data regarding gas-solid adsorption [33]. It was used to account for the adsorption behavior of theophylline on a Kromasil Cig column eluted with an aqueous solution of methanol [11]. It is frequently used to account... 

The agreement that was observed between the experimental results and the prediction of a competitive Langmuir model based on the use of single-component Langmuir isotherms in the case of the adsorption of enantiomeric derivatives of amino acids on immobilized serum albumin [26] is imusual. It demonstrates the validity of the competitive Langmuir model based on the use of the parameters of the single-component Langmuir model. However, as explained before, the experimental conditions are exceptionally favorable since the column saturation capacities for the two enantiomers are equal. Nevertheless, Zhou et ah have shown that it is possible, in certain favorable cases, to derive the equilibrium isotherms of the pure enantiomers and to calculate isotherm equilibrium data for any mixture of... 

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