Homogeneous rate equations

It is possible to apply the homogeneous rate equation to some Ziegler-Natta catalysts. This can be expressed as ... 

Basic Unstructured Kinetic Models for Growth and Substrate Utilization (Homogeneous Rate Equations)... 

After the rates have been determined at a series of reactant concentrations, the differential method of testing rate equations is applied. Smith [3] and Carberry [4] have adequately reviewed the designs of heterogeneous catalytic reactors. The following examples review design problems in a plug flow reactor with a homogeneous phase. 

Chemical reactions obey the rules of chemical kinetics (see Chapter 2) and chemical thermodynamics, if they occur slowly and do not exhibit a significant heat of reaction in the homogeneous system (microkinetics). Thermodynamics, as reviewed in Chapter 3, has an essential role in the scale-up of reactors. It shows the form that rate equations must take in the limiting case where a reaction has attained equilibrium. Consistency is required thermodynamically before a rate equation achieves success over tlie entire range of conversion. Generally, chemical reactions do not depend on the theory of similarity rules. However, most industrial reactions occur under heterogeneous systems (e.g., liquid/solid, gas/solid, liquid/gas, and liquid/liquid), thereby generating enormous heat of reaction. Therefore, mass and heat transfer processes (macrokinetics) that are scale-dependent often accompany the chemical reaction. The path of such chemical reactions will be... 

Equilibrium Compositions for Single Reactions. We turn now to the problem of calculating the equilibrium composition for a single, homogeneous reaction. The most direct way of estimating equilibrium compositions is by simulating the reaction. Set the desired initial conditions and simulate an isothermal, constant-pressure, batch reaction. If the simulation is accurate, a real reaction could follow the same trajectory of composition versus time to approach equilibrium, but an accurate simulation is unnecessary. The solution can use the method of false transients. The rate equation must have a functional form consistent with the functional form of K,i,ermo> e.g., Equation (7.38). The time scale is unimportant and even the functional forms for the forward and reverse reactions have some latitude, as will be illustrated in the following example. 

These component balances are conceptually identical to a component balance written for a homogeneous system. Equation (1.6), but there is now a source term due to mass transfer across the interface. There are two equations (ODEs) and two primary unknowns, Og and a . The concentrations at the interface, a and a, are also unknown but can be found using the equilibrium relationship, Equation (11.4), and the equality of transfer rates. Equation (11.5). For membrane reactors. Equation (11.9) replaces Equation (11.4). Solution is possible whether or not Kjj is constant, but the case where it is constant allows a and a to be eliminated directly... 

The rates of propagation and termination in the aqueous phase were also calculated. The radical entry rate, radical generation rate, and aqueous propagation rate were then used to develop an algebraic equation for the rate of formation of primary precursors. This equation is an extension to copolymers of the homogeneous nucleation equation derived by Hansen and Ugelstad (7.) for a homopolymer. 

Enzymes also are homogeneous catalysts, although they are sometimes attached to solid surfaces without degradation. They possess a different form of rate equation, for which the development may be found in problem P2.03.02. Their behavior is especially sensitive to temperature and to substrate concentration. 

Experiment shows that the primary reaction in the homogeneous decomposition of nitrous oxide proceeds with stoichiometry N20 => N2 + 0.5 02 and rate equation... 

Here the pseudo-homogeneous rate r is related to the surface reaction rate r" through the area of active catalyst per unit volume of reactor. Assuming further a plug-flow regime, the integration of the mass balance equation for this simple rate expression gives an expression for CO conversion ... 

Catalyst decomposition ( die-out ) during the catalytic reaction is a common phenomenon also distorting the kinetic measurements. If the decomposition reaction obeys a rate equation in a well-behaved manner, one can include the decomposition reaction in the kinetics, but usually one will prefer the use of a stable catalyst. Catalyst decomposition is an import issue in applied catalysis although it has received relatively little attention in literature as far as homogeneous catalysis is concerned [5],... 

Rate equation analyses for classical size exclusion chromatography have been based on treating the porous matrix as a homogeneous, spherical medium within which radial diffusion of the macromolecular solute takes place (e.g. (28,30,31)) or If mobile phase lateral dispersion Is considered Important, a two dimensional channel has been used as a model for the bed (32). In either case, however, no treatment of the effects to be expected with charged Brownian solute particles has been presented. As a... 

Theoretical studies are primarily concentrated on the treatment of flame blow-off phenomenon and the prediction of flame spreading rates. Dunskii [12] is apparently the first to put forward the phenomenological theory of flame stabilization. The theory is based on the characteristic residence and combustion times in adjoining elementary volumes of fresh mixture and combustion products in the recirculation zone. It was shown in [13] that the criteria of [1, 2, 5] reduce to Dunskii s criterion. Longwell et al. [14] suggested the theory of bluff-body stabilized flames assuming that the recirculation zone in the wake of the baffle is so intensely mixed that it becomes homogeneous. The combustion is described by a second-order rate equation for the reaction of fuel and air. 

In a homogeneous isothermal liquid polymerization, 20% of the monomer disappears in 34 minutes for initial monomer concentration of 0.04 and also for 0.8 mol/liter. What rate equation represents the disappearance of the monomer ... 

So far we have considered the mathematical expression called the rate equation which describes the progress of a homogeneous reaction. The rate equation for a reacting component i is an intensive measure, and it tells how rapidly component i forms or disappears in a given environment as a function of the conditions there, or... 

The method of searching for a rate equation is analogous to that of homogeneous reactions start with the simplest kinetic form and see if it fits the data. If it doesn t, try another kinetic form and so on. The main complication here is that we have an extra factor, the activity, to contend with. Nevertheless, the strategy is the same always start by trying to fit the simplest rate expression. 

The use of tracers in this manner has also been considered by Le Cardinal et al (56), with special reference to homogeneous systems, and discussed by Happel (57) and Le Cardinal (55). Such an approach parallels the viewpoint of Aris and Mah (42) in which they distinguished between the kinematics and kinetics of overall reactions. Rates of change of species are considered without reference to their correlation in terms of rate equations related to particular physical conditions. 

Two important ways in which heterogeneously catalyzed reactions differ from homogeneous counterparts are the definition of the rate constant k and the form of its dependence on temperature T. The heterogeneous rate equation relates the rate of decline of the concentration (or partial pressure) c of a reactant to the fraction / of the catalytic surface area that it covers when adsorbed. Thus, for a first-order reaction,... 

As with homogeneous aldol reactions, simple power-type rate equations have been frequently used to describe the kinetics of solid-catalysed condensations. For several liquid phase reactions, second-order kinetics was established, viz. 

---