Second-order Langmuir kinetics

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... 

The solid and liquid film linear driving force models can be written under the same general form of a second order Langmuir kinetic model [1]. We can insert the Langmuir isotherm equation q = qsbC)/ l bC)) in the partial differential equation of tire solid fihn linear driving force model (Eq. 14.3)... 

Cavazzini et at. showed that the above Monte Carlo model of nonlinear chromatography is equivalent to the Thomas kinetic model of second order Langmuir kinetics [70]. The solution of the Thomas model for a Dirac impulse injection is given by Eq. 14.65. When the chromatographic process is modeled at the molecular level with the stochastic model, the Thomas model becomes [70] ... 

The adsorption of methylene blue by coir pith carbon was carried out by varying the parameters such as agitation time, dye concentration, adsorbent dose, pH and temperature. Equilibrium adsorption data obeyed Langmuir isotherm. Adsorption kinetics followed a second order rate kinetic model. The adsorption capacity was found to be 5.87 mg dye per g of the adsorbent. There was no significant change in the per cent removal with pH. The pH effect and desorption studies suggest that chemisorption might be the major mode of the adsorption process. 

A general analytical solution for a nonlinear (Langmuir) system with a pseudo second-order reaction kinetic rate law has been found by Thomas. His model is summarized in Table 8.3. The numerical values of t( j8) have... 

Thomas (1944) has provided a general analytical solution for a non-linear Langmuir system with a pseudo second-order reaction kinetic law. The results, which are given in graphical form by Hiester and Vermeulen (1952), provide a means of assessing the importance of a mass transfer resistance in any system for which the rate constant and equilibrium parameters are... 

The data of Loukidou et al. (2004) for the equilibrium biosorption of chromium (VI) by Aeromonas caviae particles were well described by the Langmuir and Freundlich isotherms. Sorption rates estimated from pseudo second-order kinetics were in satisfactory agreement with experimental data. The results of XAFS study on the sorption of Cd by B. subtilis were generally in accord with existing surface complexation models (Boyanov et al. 2003). Intrinsic metal sorption constants were obtained by correcting the apparent sorption constants by the Boltzmann factor. A 1 2 metal-ligand stoichiometry provides the best fit to the experimental data with log K values of 6.0 0.2 for Sr(II) and 6.2 0.2 for Ba(II). 

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). 

The limitations of analytical solutions may also interfere with the illustration of important features of reactions and of reactors. The consequences of linear behavior, such as first-order kinetics, may be readily demonstrated in most cases by analytical techniques, but those of nonlinear behavior, such as second-order or Langmuir-Hinshelwood kinetics, generally require numerical techniques. 

In principle, the FIAM does not imply that the measured flux. / s should be linear with the metal ion concentration. The linear relationship holds under submodels assuming a linear (Henry) isotherm and first-order internalisation kinetics [2,5,66], but other nonlinear functional dependencies with for adsorption (e.g. Langmuir isotherm [11,52,79]) and internalisation (e.g. second-order kinetics) are compatible with the fact that the resulting uptake is a function (not necessarily linear) of the bulk free ion concentration cjjjj, as long as these functional dependencies do not include parameters corresponding with the speciation of the medium (such as or K [11]). 

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 catechol and resorcinol from aqueous solutions by adsorption onto high area activated carbon cloth (ACC) was investigated. Kinetics of adsorption was followed by in-situ uv-spectroscopy and the data were treated according to pseudo-first-order, pseudo-second-order and intraparticle drfiusion models. It was fotmd that the adsorption process of these compotmds onto ACC follows pseudo-second-order model. Furthermore, intraparticle drfiusion is efiective in rate of adsorption processes of these compoimds. Adsorption isotherms were derived at 25°C on the basis of batch analysis. Isotherm data were treated according to Langmuir and Freundhch models. The fits of experimental data to these equations were examined. 

For a formal kinetic description of vapour phase esterifications on inorganic catalysts (Table 21), Langmuir—Hinshelwood-type rate equations were applied in the majority of cases [405—408,410—412,414,415]. In some work, purely empirical equations [413] or second-order power law-type equations [401,409] were used. In the latter cases, the authors found that transport phenomena were important either pore diffusion [401] or diffusion of reactants through the gaseous film, as well as through the condensed liquid on the surface [409], were rate-controlling. 

If we assume a Langmuir isotherm, with sorption kinetics described by a second-order rate constant (kf) that depends on both dissolved contaminant concentration and number of available sorption sites, the following expression results [3,21] ... 

In this paper, we present the unusual adsorption properties of ETS-10 towards heavy metal ions Pb and Cu. Adsorption equilibrium and kinetic data are reported. Fitting of the experimental equilibrium results to both Langmuir and Freundlich isotherms and the kinetic data to both pseudo-first- and pseudo-second-order kinetic models is described. 

The present stu(fy shows that the coir pith carbon is an effective adsorbent for the removal of methylene blue from aqueous solution. Adsorption follows Langmuir isotherm. Kinetic data follow second order kinetic model. The adsorption capacity was found to be 5.87mg/g. The results would be useful for the fabrication and designing of wastewater treatment plants for the removal of dye. As the raw material, coir pith is discarded as waste in coir industries, the treatment method using coir pith carbon is expected to be economical. 

Jaulmes and Vidal-Madjar [51] studied the influence of the mass transfer kinetics on band profiles, using a Langmuir second-order kinetics, and a constant axial dispersion coefficient, D. They derived numerical solutions using a finite difference algorithm. The influence of the rate constant on the band profile at various sample sizes is illustrated in Figure 14.18. As the mass transfer kinetics slows down, the band broadens and the shock layer becomes thicker. When the sample size increases, however, the influence of thermodynamics on the profile becomes more dominant, as shown by the change in shock layer thickness which decreases with increasing sample size. 

According to the lUPAC, the Langmuir-Hinshelwood mechanism is defined as a mechanism for surface catalysis in which the reaction occurs between species that are adsorbed on the surface. This mechanism is expected to exhibit a second order kinetics with respect to the surface coverage of the two reactants. 

First-order kinetics was chosen in writing Eq. (11-46), so that an analytical solution could be obtained. Numerical solutions for rj vs have been developed for many other forms of rate equations. Solutions include those for Langmuir-Hinshelwood equations with denominator terms, as derived in Chap. 9 [e.g., Eq. (9-32)]. To illustrate the extreme effects of reaction, Wheeler obtained solutions for zero- and second-order kinetics for a fiat plate of catalyst, and these results are also shown in Fig. 11-7. For many catalytic reactions the rate equation is approximately represented... 

Example 9.8. Parallel and sequential deactivation in a hypothetical reaction. The principle of mechanistic modeling can be illustrated by the oversimplified example of a single-step isomerization reaction A — P with Langmuir-Hinshelwood kinetics, rate control by the surface reaction, and slow second-order deactivation. 

Equations (1.30-1.32) can be applied for batch reactors independent of the which type of catalysis is operative, if reactions could be described by zero, first or second order. More complicated cases for Langmuir kinetics or Michaelis-Menten kinetics will be considered further. 

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