Langmuir kinetic models

The BET treatment is based on a kinetic model of the adsorption process put forward more than sixty years ago by Langmuir, in which the surface of the solid was regarded as an array of adsorption sites. A state of dynamic equilibrium was postulated in which the rate at which molecules arriving from the gas phrase and condensing on to bare sites is equal to the rate at which molecules evaporate from occupied sites. 

J. W. Evans. Kinetic phase transitions in catalytic reaction models. Langmuir 7 2514-2519, 1991. 

Fig. 8.7. Kinetic model for batch fermentation, Langmuir-Hanes plot. Reprinted from Najafpour et al. (2004).18 Copyright with permission from Elsevier.

A Langmuir-Hanes plot based on the Monod rate equation is presented in Figure 8.7. The Monod kinetic model can be used for microbial cell biocatalyst and is described as follows ... 

The preferred kinetic model for the metathesis of acyclic alkenes is a Langmuir type model, with a rate-determining reaction between two adsorbed (complexed) molecules. For the metathesis of cycloalkenes, the kinetic model of Calderon as depicted in Fig. 4 agrees well with the experimental results. A scheme involving carbene complexes (Fig. 5) is less likely, which is consistent with the conclusion drawn from mechanistic considerations (Section III). However, Calderon s model might also fit the experimental data in the case of acyclic alkenes. If, for instance, the concentration of the dialkene complex is independent of the concentration of free alkene, the reaction will be first order with respect to the alkene. This has in fact been observed (Section IV.C.2) but, within certain limits, a first-order relationship can also be obtained from many hyperbolic models. Moreover, it seems unreasonable to assume that one single kinetic model could represent the experimental results of all systems under consideration. Clearly, further experimental work is needed to arrive at more definite conclusions. Especially, it is necessary to investigate whether conclusions derived for a particular system are valid for all catalyst systems. 

Examples of Hougen-Watson kinetic models, which are also called Langmuir-Hinshelwood models, can be derived for a great variety of assumed surface mechanisms. See Butt and Perry s Handbook (see Suggestions for Further reading in Chapter 5) for collections of the many possible models. The models usually have numerators that are the same as would be expected for a homogeneous reaction. The denominators reveal the heterogeneous nature of the reactions. They come in almost endless varieties, but all reflect competition for the catalytic sites by the adsorbable species. 

Langmuir-Hinshelwood (LH) and Eley-Rideal (ER) kinetic models... 

Loukidou et al. (2005) fitted the data for the equilibrium sorption of Cd from aqueous solutions by Aeromonas caviae to the Langmuir and Freundlich isotherms. They also conducted, a detailed analysis of sorption rates to validate several kinetic models. A suitable kinetic equation was derived, assuming that biosorption is chemically controlled. The so-called pseudo second-order rate expression could satisfactorily describe the experimental data. The adsorption data of Zn on soil bacterium Pseudomonas putida were fit with the van Bemmelen-Freundlich model (Toner et al. 2005). 

Mann, Thurgood, and coworkers—Langmuir-Hinshelwood kinetic model for methanol steam reforming and WGS over Cu/Zn. Mann et al.335 published a complex Langmuir-Hinshelwood model for CuO/ZnO catalysts based on what one would encounter for a methanol steam reformer (MSR) for fuel cell applications. The water-gas shift rate, containing all MSR terms, was determined to be ... 

Wheeler, Schmidt, and coworkers—kinetic model for Pt/Ce at short contact times over medium to high T range. In 2004, Wheeler and coworkers422 reported on the water-gas shift reaction over Pt/ceria at short contact times (0.008-0.05 sec) for temperatures between 300 and 1000 °C. The reactant composition for CO, H2, and H20 was 1/2/4. A Langmuir-Hinshelwood kinetic model was used to adequately fit the medium and high temperature shift data ... 

Kinetic models proposed for sorption/desorption mechanisms including first-order, multiple first-order, Langmuir-type second-order, and various diffusion rate laws are shown in Sects. 3.2 and 3.4. All except the diffusion models conceptualize specific sites to or from which molecules may sorb or desorb in a first-order fashion. The following points should be taken into consideration [ 181,198] ... 

Abstract Removal of the pesticide metobromuron from aqueous solutions by adsorption at the high area activated carbon cloth was investigated. Kinetics of adsorption was followed and adsorption isotherms of the pesticide was also be determined. In kinetic studies a special V-shaped cell with an UV cuvette attached to it was used for adsorption processes. With this cell it was possible to follow the concentration of pesticide molecule by in situ UV spectroscopy as it is adsorbed at the activated carbon cloth. The obtained absorbance vs time data were converted into concentration vs time data and these data were treated according to pseudo-first-order and psendo-second-order kinetic models. Adsorption of that pesticide was fonnd to follow second-order kinetic model with k 87.35 g mol min. Adsorption isotherms were derived at 25°C on the basis of batch analysis. Isotherm data were treated according to Langmuir and Freundlich models. The fits of experimental data to these equations were examined and founded that the adsorption isotherm was well represented by Frenndlich model. 

The solution of the simplest kinetic model for nonlinear chromatography the Thomas model [9] can be calculated analytically. The Thomas model entirely ignores the axial dispersion, i.e., 0 =0 in the mass balance equation (Equation 10.8). For the finite rate of adsorption/desorption, the following second-order Langmuir kinetics is assumed... 

Using kinetic models of typical catalytic mechanisms (Eley-Rideal and Langmuir-Hinshelwood (LH) mechanisms) as examples, we found parametric domains, in which the hypergeometric representation is an excellent approximation... 

In remarkable progress of catalytic industry in 1930-1950s, a kinetic model was considered as a basis of reactor design. Langmuir and Hinshelwood... 

For SR of higher hydrocarbons, Rostrup-Nielsen " and Tottrup " postulated a Langmuir-Hinshelwood-Houghen-Watson (LHHW) kinetic model. It was assumed that the hydrocarbon chemisorbs on a dual catalytic site, followed by successive a-scission of the C-C bond. The resulting Ci species react with adsorbed steam to form H2 and CO. The expressions were lit to data for SR of n-Cv on a Ni/MgO catalyst at 500°C the overall rate expression is " " ... 

Recently, Praharso et al also developed a Langmuir-Hinshelwood type of kinetic model for the SR kinetics of i-Cg over a Ni-based catalyst. In their model, it was assumed that both the hydrocarbon and steam dissociatively chemisorb on two different dual sites on the catalyst surface. The bimolecular surface reaction between dissociated adsorbed species was proposed as the ratedetermining step. The following generalized rate expression was proposed ... 

The model followed mass action kinetics with Langmuir-Hinshelwood adsorption. 

The theoretical calculations described have recently been supported by an extraordinary kinetic analysis conducted by Vanrysellberghe and Froment of the HDS of dibenzothiophene (104). That work provides the enthalpies and entropies of adsorption and the equilibrium adsorption constants of H2, H2S, dibenzothiophene, biphenyl, and cyclohexylbenzene under typical HDS conditions for CoMo/A1203 catalysts. This work supports the assumption that there are two different types of catalytic sites, one for direct desulfurization (termed a ) and one for hydrogenation (termed t). Table XIV summarizes the values obtained experimentally for adsorption constants of the various reactants and products, using the Langmuir-Hinshelwood approach. As described in more detail in Section VI, this kinetic model assumes that the reactants compete for adsorption on the active site. This competitive adsorption influences the overall reaction rate in a negative way (inhibition). 

A large number of authors describe the oxidation kinetics by Langmuir—Hinshelwood type models. Depending on the particular L—H model selected, the mathematical difference between L—H models and redox models can be very small, although the former always contains more... 

Several authors have studied the kinetics in detail and have proposed kinetic models. According to Trimm and Gabbay [329], the kinetics of the reaction over a Sn/Sb = 1/4 catalyst is best described by a Langmuir— Hinshelwood type of model, which reflects that hydrogen abstraction from butene, involving dissociatively adsorbed oxygen, is rate-controlling viz. 

However, simple kinetic models, especially of the Langmuir—Hinshel-wood type, can serve with advantage for correlation of experimental data in spite of simplifying assumptions which are necessary for their derivation. Experience shows that heterogeneous acid—base catalysis is the very field where they fit best. Their most frequent general form... 

The kinetic model developed in this study can be used to design and analyze various chemical reactors for the hydrogenation of benzaldehyde. Although it is based on a Langmuir-Hinshelwood mechanism, it does not prove that this is the correct mechanism. 

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