Early Kinetic Models

A number of simple kinetic models [12,13] have been developed for catalysts that have relatively low activities and are characterized by Unetic rate-time profiles of the type shown in Fig. 9.6 (a to f). These models are based on the assumption that the total concentration of active centers, C, remains constant throughout the polymerization consisting of three steps chain initiation, chain propagation, and chain transfer. 

This step is assumed to be the insertion of the first monomer molecule, M, into a transition metal-carbon bond in an active center, Cat-R, resulting in the formation of a polymerization center Cat-P  

The chain propagation step is regarded as the insertion of a monomer molecule into a transition metal-carbon bond in a polymerization center, Cat-P  

In this scheme, all polymerization centers are regarded as equally active and having the same propagation rate constant, kp, independent of their geometric location and of their degree of polymerization. However, since it is established that all centers are not equally active, the A p in this scheme must be regarded as an average. 

Where no chain transfer agent has been added to the polymerization system three transfer reactions [Eq. (9.9)-(9.12)] are usually considered, viz., chain 

---

The early kinetic models for copolymerization, Mayo s terminal mechanism (41) and Alfrey s penultimate model (42), did not adequately predict the behavior of SAN systems. Copolymerizations in DMF and toluene indicated that both penultimate and antepenultimate effects had to be considered (43,44). The resulting reactivity model is somewhat compHcated, since there are eight reactivity ratios to consider. 

The early kinetic model by Smith and Ewart was based on Harkin s mechanistic understanding of the batch process. The particle population balances were written for a stationary state assuming that the rate of formation of particles with n radicals equals the rate of their disappearance (see equation at the bottom of this page). Where / , is the rate of radical entry into a particle (m /sec) is the rate constant for radical exit (m/sec) S is the particle surface area (m ) ktp is the rate constant for bimolecular termination in the particles (m /sec) and o is the particle volume. According to Smith and Ewart three limiting cases can be identified ... 

We classify kinetic models according to the chemical entities that makeup the model. Typically, the entities or lumps are boiling point lumps or yield lumps, grouped chemical lumps and full chemical lumps. Early kinetic models consist entirely of yield lumps, which represent the products that refiner collects from the main fractionator following the FCC unit Figure 4.4 shows a typical kinetic model based on yield lumps by Takatsuka et al. [9]. Many similar models have appeared in the literature. The models differentiate themselves based on their number of lumps. Models may contain as few as two [10] or three lumps [11] and as many as fifty lumps [12]. We note that models with more lumps do not necessarily have more predictive capabilities than models with fewer lumps [6]. 

Micro-kinetic modeling represents the state of the art in describing the kinetics of catalytic reactions. It was pioneered by Stoltze and Norskov in the mid-1980s and was further explored by Dumesic and coworkers in the early 1990s [J.A. Dumesic,... 

The kinetic models just described are only a few of those that have been found to represent reactions in solids. Moreover, it is sometimes observed that a reaction may follow one rate law in the early stages of the reaction, but a different rate law may apply in the later stages. Because many of the rate laws that apply to reactions in solids are quite different from those encountered in the study of reactions... 

Some early kinetic studies on the enzymic reaction indicated that LADH exhibits pre-steady state half-of-the-sites reactivity. Bernard et al. reported that two distinct kinetic processes, well separated in rate, were observed for the conversion of reactants into products under conditions of excess enzyme.1367 They also reported that each of the two phases corresponded to conversion of exactly one half of the limiting concentration of substrate being converted to products. On the basis of this they proposed two possible models, the favoured one based on catalytically non-equivalent but interconvertible states of the two binding sites, with the possibility that the asymmetry of the sites may be induced by coenzyme binding. Further evidence for this non-equivalence of the subunits was obtained in similar subsequent studies using a chromophoric nitroso substrate, p-nitroso-A,JV-dimethylaniline with limiting NADH concentrations.1368... 

The enzyme from B. stearothermophilus is an a4 tetramer of subunit Mr 33 900. Early kinetic studies indicated that the enzyme acts in a manner that is qualitatively consistent with an MWC two-state model. The enzyme acts as a A system i.e., both states have the same value of kcal but different affinities for the principle substrate. In the absence of ligands, the enzyme exists in the T state that binds fructose 6-phosphate more poorly than does the R state. In the absence of ADP, the binding of fructose 6-phosphate is highly cooperative, and h = 3.8. The positive homotropic interactions are lowered on the addition of the allosteric effector ADP, with h dropping to 1.4 at 0.8-mM ADP.52 ADP thus binds preferentially to the R state. The allosteric inhibitor phosphoenolpyruvate binds preferentially to the T... 

The kinetic models all allow for evaporation-condensation to be a significant mechanism in surface reconstruction. In particular, as noted earlier, it was frequently suggested that a metal or metal oxide would evaporate preferentially from certain planes, leading to a surface (presumably equilibrium) consisting of planes with the lowest evaporation rates. Net weight loss was anticipated. Yet, no evidence of weight loss is available from the early literature. 

The determination of the evolution of concentrations of different species and functional groups enables one to discern different paths present in the reaction mechanism. For example, Fig. 5.13 shows that as the molar ratio of styrene to polyester C=C double bonds (MR) increases from 1/1 to 4/1, the curves tend to shift downward. For MR = 4/1 there is a very low styrene consumption until the polyester double bonds are converted to 40%. On the other hand, SEM (scanning electron microscopy) shows phase separation of a UP-rich phase in the early stages of the polymerization. Most radicals are probably trapped in this phase, which explains the higher initial conversion of the UP double bonds than styrene double bonds. A kinetic model would have to take this observation into account. 

It has been shown that changes in the UV and IR absorbance of unplasticized Cellophane films subjected to accelerated aging in a dry oven at 140 °C follow the behavior predicted by a first-order kinetic model, except for deviations in the early aging period, and that these deviations are most likely caused by oxidation products in the films. It has also been shown that, for Cellophane films, the changes in UV and IR absorbance follow the same kinetics as color change, and that these kinetics are nearly identical with those for rayon and cotton cloths aged under similar conditions. 

Thermodynamics deals with relations among bulk (macroscopic) properties of matter. Bulk matter, however, is comprised of atoms and molecules and, therefore, its properties must result from the nature and behavior of these microscopic particles. An explanation of a bulk property based on molecular behavior is a theory for the behavior. Today, we know that the behavior of atoms and molecules is described by quantum mechanics. However, theories for gas properties predate the development of quantum mechanics. An early model of gases found to be very successftd in explaining their equation of state at low pressures was the kinetic model of noninteracting particles, attributed to Bernoulli. In this model, the pressure exerted by n moles of gas confined to a container of volume V at temperature T is explained as due to the incessant collisions of the gas molecules with the walls of the container. Only the translational motion of gas particles contributes to the pressure, and for translational motion Newtonian mechanics is an excellent approximation to quantum mechanics. We will see that ideal gas behavior results when interactions between gas molecules are completely neglected. 

Berner, R.A. (1974) Kinetic models for early diagenesis of nitrogen, sulfur, phosphorus, and silicon in anoxic marine sediments. In The Sea (Goldberg, E.D., ed.), pp. 427 -50, John Wiley, New York. 

---