Langmuir-Hinshelwood kinetic expressions

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. 

The hydrogenation of CO and the formation of each individual C hydrocarbon (e.g., methane) obey Langmuir-Hinshelwood kinetic expressions (32,64,66,67.75)-. 

The proposed Langmuir-Hinshelwood kinetics expression for the gasification reaction (CO2, N2 and CO gas mixture) is given by ... 

Figure 6 Comparison between the light-(0curves assuming either a zero-order or the Langmuir-Hinshelwood kinetic expression. Adiabatic case, no mass and heat transfer limitation. Left plug flow. Right complete mixing.

Although the tested catalyst shows a good intrinsic selectivity for butadiene hydrogenation, the results evidenced the presence of severe diffusion limitations in spite of the thin active shell (230 im). The experimental data were modeled by Langmuir-Hinshelwood kinetic expressions derived from an elementary mechanism. Nine kinetic parameters were reliably estimated by means of a regression analysis and it is concluded that the proposed kinetic model provides a good fitting of the experimental observations. 

Note the similarity to Langmuir-Hinshelwood kinetics.) The rate is expressed on the basis of the instantaneous number of solid carbon atoms, Nc. The rate r (measured at one gas composition) typically goes through a maximum as the carbon is converted. This is the result of a maximum in the intrinsic activity (related to the fraction of reactive carbon atoms, NCJNC) because of both a change in Nq> and a decrease in Nc. 

Based on Langmuir-Hinshelwood kinetics the rate expression for a first order reaction (A —> R) that is surface reaction-controlled becomes equal to the following expression [2] ... 

For the same A R reaction, when kinetics are controlled by adsorption of reactant A rather than by surface reaction, the Langmuir-Hinshelwood rate expression becomes equal to the following [2] ... 

These rate expressions are for Langmuir-Hinshelwood kinetics, which are the simplest forms of surface reaction rates one could possibly find We know of no reactions that are this simple. LH kinetics requires several assumptions ... 

In this rate expression we have lumped C/js into the effective surface rate coefficient by defining k" — CC s- AU sohd reactions have reaction steps similar to those in catalytic reactions, and the rate expressions we need to consider are basically Langmuir-Hinshelwood kinetics, which were considered in Chapter 7. Our use of a first-order irreversible rate expression is obviously a simplification of the more complex rate expressions that can arise from these situations. 

Liquid phase hydrogenation catalyzed by Pd/C is a heterogeneous reaction occurring at the interface between the solid catalyst and the liquid. In our one-pot process, the hydrogenation was initiated after aldehyde A and the Schiff s base reached equilibrium conditions (A B). There are three catalytic reactions A => D, B => C, and C => E, that occur simultaneously on the catalyst surface. Selectivity and catalytic activity are influenced by the ability to transfer reactants to the active sites and the optimum hydrogen-to-reactant surface coverage. The Langmuir-Hinshelwood kinetic approach is coupled with the quasi-equilibrium and the two-step cycle concepts to model the reaction scheme (1,2,3). Both A and B are adsorbed initially on the surface of the catalyst. Expressions for the elementary surface reactions may be written as follows ... 

Langmuir-Hinshelwood rate expressions of all the reactions of the network were used in the kinetic modeling of the HDN of quinoline by Satterfield et al. (80, 81, 88). Their assumption that there is a single catalytic site for all reactions is too simple. Nevertheless, they collected an impressive body of kinetic data and pinpointed the reactions that were close to equilibrium and those which were kinetically significant. Gioia and Lee (100) extended these studies to higher pressures (up to 15 MPa H2). Only one model survived their regression analysis of the kinetic data. In this model, it was necessary to assume that 1,2,3,4-THQ reacted directly not only to o-propylaniline but also to propylbenzene (PB) and propylcyclohexene (PCHE). Their analysis does not appear to be very reliable, however. First,... 

If the system is well represented by Langmuir-Hinshelwood kinetics, then the relative values of the adsorption equilibrium constants of various unsaturated compounds can be obtained from the relative rates determined individually and in competition. The expression for the competitive reaction between two compounds A and B results from the division of the corresponding rate expressions for each compound (equation 3) to give equation (4). The ratio of the individual rates of hydrogenation, k/Jk, permits the abstraction of the ratio K/JK from the competitively determined ratio kkK/Jk K (equation 5). The assumption that the adsorption of the unsaturated compound is rapid and reversible relative to the rates of the surface-catalyzed reactions may not be correct. In that case, KfJK is an apparent relative adsorption equilibrium constant (see Section 3.1.3.2.2). 

In most photocatalysis, Langmuir-Hinshelwood kinetics is observed, suggesting the importance of substrate adsorption in key steps of the reaction [83]. That is, the rate of photocatalysis is proportional to the concentration (or pressure) of the reactant, which in turn can be related to the surface coverage by the Langmuir expression, Eq. 8,... 

Preliminary kinetic studies have been performed. The Langmuir-Hinshelwood rate expression was used to correlate results of experiments as it was indicated by the shape of kinetic curves (see Fig. 6). However, the reaction order with respect to hydrogen appeared to be dependent on temperature, while activation energy depends on pressure (9.6 kJ/mol at 11 bar and 35.5 kJ/mol at 21 bar). Therefore the rate of benzaldehyde consumption was approximated using the following simple power law equation ... 

The applicability of Langmuir-Hinshelwood kinetics has been reported by Cunningham et although the kinetic expression describing... 

An example of a model nonlinear in parameters is Eq. (7-166). Here it is not possible through any number of transformations to obtain a linear form in all the parameters k0, E, K o, Eaa, Km, Ea. Note that for some Langmuir-Hinshelwood rate expressions it is possible to linearize the model in parameters at isothermal conditions and obtain the kinetic constants for each temperature, followed by Arrhenius-type plots to obtain activation energies (see, e.g., Churchill, The Interpretation and Use of Piate Data The Rate Concept, McGraw-Hill, 1974). 

The redox reaction rate equation always implies a monotonic dependence of the rate of reaction upon the concentration of the reacting species. On the other hand kinetic expressions based upon the Langmuir-Hinshelwood kinetics when two of the reactants are competing for the same active sites may show non-monotonic kinetics in certain regions of the parameters. For the reaction between a hydrocarbon and oxygen, the rate equation can be written in the... 

In general, for arbitrary kinetics, a numerical solution of the balance equation taking into account the boundary conditions is necessary. From the obtained concentration profiles one is able to obtain the effectiveness factor. This is completely feasible with the tools of the modern computing technology. Analytical and semi-analytical expressions for the effectiveness factor, rji, are, however, always favoured if they are available, since the numerical solution of the boundary value problem is not a trivial task. The solutions for different types of Langmuir-Hinshelwood kinetics were presented in the literature, for instance by R. Aris and P. Schneider. 

The reactor model presented above has been used in two versions as a conventional model (CM) and as a hybrid model (HM), respectively. In the conventional model the reaction rates r j appearing in the balance equations of Eq. 7 have been calculated following the expressions for Langmuir-Hinshelwood kinetics [13] derived in our previous studies. In the hybrid first principles-neural network model (HM) the conventional kinetic subroutines in the CM algorithm have been replaced by the neural network, so in the HM the reaction rates r- have been supplied by the trained network. 

Rate expressions of the form of Equation 5.153 are known as Hougen Watson or Langmuir-Hinshelwood kinetics [17, This form of kinetic expression is often used to describe the species production rates for heterogeneously catalyzed reactions. We complete the section on the kinetics of elementary surface reactions by returning to the methane synthesis reaction listed in Section 5.2. The development proceeds exactly as outlined in Section 5.2. But now it is necessary to add a site-balance expression (Equation 5,129) in Step 3. 

Unfortunately, a vast portion of the WO works reported in the literature deals with the non-catalyzed oxidation kinetics for single compounds. In a review by Matatov-Meytal and Sheintuch , it was found that pure compounds such as phenol, benzene, dichlorobenzene, and acetic acid obey a first-order rate law with respect to the substrates and mainly half order with respect to the oxygen concentration. A thorough kinetic investigation in an isothermal, differentially operated fixed bed reactor with the oxygen pre-saturated aqueous solutions has revealed that the catalytic oxidation of acetic acid, phenol, chloro-phenol, and nitro-phenol can be well expressed by means of the Langmuir-Hinshelwood kinetic formulation ° , namely... 

Yates and Satterfield [18] determined the kinetics for a Co/MgO/Si02 catalyst operated in a CSTR. They represented their data by a Langmuir-Hinshelwood rate expression ... 

The models mentioned so far are limited in their application as they represent only first order reaction kinetics with Fickian diffusion, therefore do not allow for multicomponent diffusion, surface diffusion or convection. Wood et al. [16] applied the algorithms developed by Rieckmann and Keil [12,44] to simulate diffusion using the dusty gas model, reaction with any general types of reaction rate expression such as Langmuir-Hinshelwood kinetics and simultaneous capillary condensation. The model describes the pore structure as a cubic network of cylindrical pores with a random distribution of pore radii. Transport in the single pores of the network was expressed according to the dusty gas model as... 

The results from this chapter on zeolite catalysis provide a good reference point for the discussion presented later Chapter 8 where we compare heterogeneous catalysis and biocatalysis. The similarity between the Michaelis-Menten kinetic expression for enzyme catalysis and the Langmuir-Hinshelwood kinetic models for heterogeneous catalysis are noted. This ultimately derives from the conservation in the number of active reaction centers for both systems. However, the more refined synergy of the activation of molecular bonds by the enzyme will become apparent as a major difference between the two. 

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