Catalytic reactions general rate equation

Equation 8.5-11 applies to a first-order surface reaction for a particle of flat-plate geometry with one face permeable. In the next two sections, the effects of shape and reaction order on p are described. A general form independent of kinetics and of shape is given in Section 8.5.4.5. The units of are such that is dimensionless. For catalytic reactions, the rate constant may be expressed per unit mass of catalyst (k )m. To convert to kA for use in equation 8.5-11 or other equations for d>, kA)m is multiplied by pp, the particle density. 

Rigorous rate equations for multistep catalytic reactions in terms of total amount of catalyst material are enormously cumbersome. Just the reduction to the level complexity of the Christiansen formula calls for the Bodenstein approximation of quasi-stationary behavior of the intermediates, requiring these to remain at trace concentrations, and that formula still entails a lot more algebra than does the general rate equation for noncatalytic simple pathways For a reaction with three intermediates, the Christiansen denominator contains sixteen terms instead of four for a reaction with six intermediates, forty-nine instead of seven Although the mathematics is simple and easy to program for modeling purposes and, usually, some... 

Langmuir-Hinshelwood-Hougen-Watson (LHHW) Rate Equations (1947) Hougen and Watson analyzed several types of catalytic reactions with different ratedetermining steps (adsorption, surface reaction), different types of adsorption (one or more species, dissociative or molecular adsorption), and different types of reactions (mono- or bimolecular, reversible or irreversible). They derived a general rate equation based on three terms ... 

The analytical integration of the continuity equation for the diffusing and reacting component A in a catalyst particle is possible only for a first-order reaction. A catalytic reaction generally has a more complicated rate equation, however, as shown in Chapter 2. It is attractive to extend the application of the simple relation (3.6.1-6 or 3.6.1-14) to any type of rate equation, be it only approximately. This is what led Aris [1965a, b], Bischoff [1965], and Petersen [1965a, b] to the introduction of a generalized modulus. 

Direct Chlorination of Ethylene. Direct chlorination of ethylene is generally conducted in Hquid EDC in a bubble column reactor. Ethylene and chlorine dissolve in the Hquid phase and combine in a homogeneous catalytic reaction to form EDC. Under typical process conditions, the reaction rate is controlled by mass transfer, with absorption of ethylene as the limiting factor (77). Ferric chloride is a highly selective and efficient catalyst for this reaction, and is widely used commercially (78). Ferric chloride and sodium chloride [7647-14-5] mixtures have also been utilized for the catalyst (79), as have tetrachloroferrate compounds, eg, ammonium tetrachloroferrate [24411-12-9] NH FeCl (80). The reaction most likely proceeds through an electrophilic addition mechanism, in which the catalyst first polarizes chlorine, as shown in equation 5. The polarized chlorine molecule then acts as an electrophilic reagent to attack the double bond of ethylene, thereby faciHtating chlorine addition (eq. 6) ... 

However, the mechanisms by which the initiation and propagation reactions occur are far more complex. Dimeric association of polystyryllithium is reported by Morton, al. ( ) and it is generally accepted that the reactions are first order with respect to monomer concentration. Unfortunately, the existence of associated complexes of initiator and polystyryllithium as well as possible cross association between the two species have negated the determination of the exact polymerization mechanisms (, 10, 11, 12, 13). It is this high degree of complexity which necessitates the use of empirical rate equations. One such empirical rate expression for the auto-catalytic initiation reaction for the anionic polymerization of styrene in benzene solvent as reported by Tanlak (14) is given by ... 

The Langmuir Equation for the Case Where Two or More Species May Adsorb. Adsorption isotherms for cases where more than one species may adsorb are of considerable significance when one is dealing with heterogeneous catalytic reactions. Reactants, products, and inert species may all adsorb on the catalyst surface. Consequently, it is useful to develop generalized Langmuir adsorption isotherms for multicomponent adsorption. If 0t represents the fraction of the sites occupied by species i, the fraction of the sites that is vacant is just 1 — 0 where the summation is taken over all species that can be adsorbed. The pseudo rate constants for adsorption and desorption may be expected to differ for each species, so they will be denoted by kt and k h respectively. 

It is generally observed that the rate of reaction can be altered by the presence of non-reacting or inert ionic species in the solution. This effect is especially great for reactions between ions, where rate of reaction is effected even at low concentrations. The influence of a charged species on the rate of reaction is known as salt effect. The effects are classified as primary and secondary salt effects. The primary salt effect is the influence of electrolyte concentration on the activity coefficient and rate of reaction, whereas the secondary salt effect is the actual change in the concentration of the reacting ions resulting from the addition of electrolytes. Both effects are important in the study of ionic reactions in solutions. The primary salt effect is involved in non-catalytic reactions and has been considered here. The deviation from ideal behaviour can be expressed in terms of Bronsted-Bjerrum equation. 

The oxidation of sulphur dioxide to trioxide is one of the oldest heterogeneous catalytic processes. The classic catalyst based on V2Os has therefore been the subject of numerous investigations which are amply reviewed by Weychert and Urbaneck [346]. These authors conclude that none of the 34 rate equations reported is applicable over a wide range of process conditions. Generally, these equations have the form of a power expression, in which the reverse reaction is taken into account within the limits imposed by chemical equilibrium, viz. 

We shall find that the rate equations of gas-solid heterogeneous catalytic reactions (Chapter 3) also do not, in general, have the same form as equation 1.4. 

For the very restricted conditions where Eq. (5.2) provides a rigorous description of the reaction kinetics, the activation energy, E, is a constant independent of conversion. But in most cases it is found that E is indeed a function of conversion, E (x). This is usually attributed to the presence of two or more mechanisms to obtain the reaction products e.g., a catalytic and a noncatalytic mechanism. However, the problem is in general associated to the fact that the statement in which the isoconversional method is based, the validity of Eq. (5.1), is not true. Therefore, isoconversional methods must be only used to infer the validity of Eq. (5.2) to provide a rigorous description of the polymerization kinetics. If a unique value of the activation energy is found for all the conversion range, Eq. (5.2) may be considered valid. If this is not true, a different set of rate equations must be selected. 

Later, it became clear that the concentrations of surface substances must be treated not as an equilibrium but as a pseudo-steady state with respect to the substance concentrations in the gas phase. According to Bodenstein, the pseudo-steady state of intermediates is the equality of their formation and consumption rates (a strict analysis of the conception of "pseudo-steady states , in particular for catalytic reactions, will be given later). The assumption of the pseudo-steady state which serves as a basis for the derivation of kinetic equations for most commercial catalysts led to kinetic equations that are practically identical to eqn. (4). The difference is that the denominator is no longer an equilibrium constant for adsorption-desorption steps but, in general, they are the sums of the products of rate constants for elementary reactions in the detailed mechanism. The parameters of these equations for some typical mechanisms will be analysed below. 

Parallel reactions are very common in chemistry. They play a key role in catalysis, because the catalyst effectively creates a parallel alternative to the noncatalyzed reaction. In many cases a catalytic reaction will yield several products, because one of the catalytic intermediates can react via two different pathways. The general scheme for two parallel reactions is shown in Eq. (2.46), and the corresponding differential rate equations for B and C are Eq. (2.47). 

If the reaction rate is a function of pressure, then the momentum balance is considered along with the mass and energy balance equations. Both Equations 6-105 and 6-106 are coupled and highly nonlinear because of the effect of temperature on the reaction rate. Numerical methods of solution involving the use of finite difference are generally adopted. A review of the partial differential equation employing the finite difference method is illustrated in Appendix D. Figures 6-16 and 6-17, respectively, show typical profiles of an exothermic catalytic reaction. 

A rate equation in terms of the local composition of reacting fluid in contact with the surface of the cata-lytically active material may be called the "intrinsic rate equation, the coefficients in this equation are "intrinsic rate coefficients. The local concentrations of reactants and products at the catalytic surface in general cannot be observed and have to be inferred from the observable composition at the boundary of a larger system, the observed rate of reaction and the kinetics... 

The characteristic life time of a reaction is a measure of the time required after initiation for it to reach completion. This period is frequently related to the rate constant for the reaction in a veiy clear and specific way. Solutions to some of the common zero-, first- and second-order rate equations are presented in Table 9.5. Examples of zero- and first-order reactions are discussed in this section application of the second-order equations to general catalytic processes will be presented in the section on catalysis. The last column of Table 9.5 lists the relations between r, the characteristic life time of the reactant with respect to the chemical reaction, and the rate constant for the reaction. The meaning of the characteristic life time depends upon the order and reversibility of the reaction. 

In general, a polymerization process model consists of material balances (component rate equations), energy balances, and additional set of equations to calculate polymer properties (e.g., molecular weight moment equations). The kinetic equations for a typical linear addition polymerization process include initiation or catalytic site activation, chain propagation, chain termination, and chain transfer reactions. The typical reactions that occur in a homogeneous free radical polymerization of vinyl monomers and coordination polymerization of olefins are illustrated in Table 2. 

The assumptions inherent in this procedure should not be forgotten. They have been mentioned, but it is worthwhile to emphasize one, the con- stancy of C . This corresponds to an assumption that the maximum value of 9 is unity, or that the total number of active sites is constant. Also, note that we are considering a general approach to the formulation of rate equations for fluid-solid catalytic reactions. In several specific cases sufficient data have been obtained to permit a more detailed analysis of the mechanism and rates of the adsorption and surface-reaction steps. An example is the... 

It is important to note that catalysts are generally high surface area, three dimensional surfaces, quite different in that respect from the macroscopic two dimensional single crystals most commonly used in surface science studies. The reason such high area materials are used can be seen from the consideration of the rate equation for the simplest form of catalytic reaction, an isomerisation under conditions of pre-equilibrium ... 

This chapter is devoted to numerical integration, and more specifically to the integration of rate expressions encountered in chemical kinetics. For simple cases, integration yields closed-form rate equations, while more complex reaction mechanisms can often be solved only by numerical means. Here we first use some simple reactions to develop and calibrate general numerical integration schemes that are readily applicable to a spreadsheet. We then illustrate several non-trivial applications, including catalytic reactions and the Lotka oscillator. 

The isotherm we have developed in equation 3.29 is the Langmuir Isotherm which, in principle, applies only to sets of sites of uniform strength. However, the surfaces of catalysts usually contain sites with a distribution of strengths of adsorption, a fact that would be reflected in the activation energy of the constant k., and in the heat of adsorption in KA. It has been found, however, that real catalytic rate equations based on the equilibrium isotherm are generally satisfactory in fitting the rates of catalytic reactions without taking this complication into account. Nevertheless, as in all such cases, we should bear this approximation in mind so we can be alert to the appearance of a counter-example. 

If there are more than two steps and the mechanism of a heterogeneous catalytic reaction follows the linear Christiansen sequences, then the general kinetic expressions are given by eq. (5.94-5.95). Some particular cases of Christiansen sequences with 3, 4, 6 and 8 steps were presented for homogeneous catalysis by metal complexes, e.g., equations (5.72, 5.76, 5.84) and (5.88) respectively. It should be stressed that in the case of heterogeneous catalysis equations for the reaction rates are exactly the same, which is not surprising as similar kinetic steps describe the reaction mechanisms. 

In case of reactions involving adsorbed intermediates the electrode material affects the rates in a similar way as in the case of catalytic reactions at the gas-solid interface. However, the potential dependence of the rates of reactions producing adsorbed intermediates results in a potential-dependent coverage of the latter. In general, for a reaction sequence with a rate-controlling step involving an adsorbed intermediate, the kinetic equation may be expressed as... 

Hydroformylation (Equation (14)) is one of the very largest homogeneous catalytic reactions carried out by industry making over 15 billion pounds of aldehyde products each year. These are subsequently hydrogenated to alcohols or oxidized to carboxylic acids. There are several recent excellent reviews on hydroformylation catalysis (cf. Refs 7,7a-7c). Industry is generally more interested in the linear aldehyde product, and much of hydroformylation catalyst development work has been directed at increasing the linear to branched regioselectivity (L B, also referred to as normal to iso), reaction rates, and catalyst stability (lifetime). There are three main hydroformylation catalysis... 

In gas-liquid or liquid-liquid reactions. Equation 4.3a based on total reactor volume, which is identical to Equation 4.2 for homogeneous reactions, may be used to give rj. Alternatively, the rate can be expressed as rsi given by Equation 4.3d for catalytic reactions but based on gas-liquid or liquid-liquid (instead of the catalyst) interfacial area. In this case (see Part IV) we generally use the symbol r(. 

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