Concentration of active centers

Reactions that are catalyzed by solids occur on the surfaces of the solids at points of high chemical activity. Therefore, the activity of a catalytic surface is proportional to the number of active centers per unit area. In many cases, the concentration of active centers is relatively low. This is evident by the small quantities of poisons present (material that retards the rate of a catalytic reaction) that are sufficient to destroy the activity of a catalyst. Active centers depend on the interatomic spacing of the solid structure, chemical constitution, and lattice structure. 

This assumption derives rate equations from which terms involving the concentrations of active centers can be eliminated. 

In this regime expressions (Eq. 83) for finding the concentrations of active centers are... 

The equilibrium concentration of each kind of imperfection depends on the biography of the sample and on the temperature. Poltorak lOS, 104) has shown that in real crystals, characterized by non-equilibrium forms of boundary faces, this concentration may be quite considerable corresponding in order of magnitude to the concentration of active centers, as determined from the catalytic data. 

The straight lines In the plot of log [M0]/[M] vs time, (Figure 5) Indicate that the kinetic order Is equal to 1 with respect to monomer concentration and that the concentration of active centers remains constant. 

From the order 1 with respect to the concentration of active centers and taking Into account the agreement between measured and theoretical molecular weights, we can deduce that true active species are In a predominant concentration. 

Compared to ordinary kinetic experiments the study of flame velocity has disadvantages we cannot simultaneously and independently vary the temperature and concentration of all the reacting components peculiar conditions interrelate, for example, the temperature and concentration of a deficient component. However these disadvantages are expiated by the tremendous expansion of the field of kinetic studies in a flame we study reactions which occur at 1500-3000°K and which are so rapid that the reaction time lies within the limits 10 3-10 6 sec. This expansion of the kinetic experimental region, undoubtedly, will yield a number of new laws of particular interest in connection with the fact that at high temperatures the equilibrium concentration of active centers of the reaction becomes significant. 

Figure 2.2. Correlation between molecular weights in synthesis of polydodecanamide (PA-12), calculated by concentration of active centers of polymerization (I) and measured by intrinsic viscosity (II).

An example illustrating the application of Eq. (2.4) for calculating Mn is shown in Fig. 2.2. The position of the line at 45° reflects the exact equality of the Mn values calculated from the concentration of active centers and those determined by measurement of intrinsic viscosity, [r ]. Bearing in mind possible experimental errors and the accuracy of the equation relating [tj] and Mn, we can conclude that the correspondence between the two series of Mn values is satisfactory. 

This produces a rise in the concentration of active centers and a corresponding increase in the propagation rate. Chains produced at this stage are longer, and this leads to a broadening of the molar mass distribution. The term gel effect is widely used to describe this effect, although no gel is actually formed in the system. The effect is also called the Trommsdorff effect (see Chapter 5). 

A case classically associated with radical chain polymerization for which a (pseudo)steady state is assumed for the concentration of active centers this condition is attained when the termination rate equals the initiation rate (the free-radical concentration is kept at a very low value due to the high value of the specific rate constant of the termination step). The propagation rate, is very much faster than the termination rate, so that long chains are produced from the beginning of the polymerization. For linear chains, the polydispersity of the polymer fraction varies between 1.5 and 2. 

To obtain a behavior similar to case 2 for free-radical polymerizations, it is necessary to decrease the influence of the bimolecular termination reaction by decreasing the concentration of active centers. This is possible by producing a reversible combination between a growing chain (a polymer radical P ) with a stable radical N to form an adduct P-N, which behaves as a dormant species ... 

To obtain a solution of this equation, Danby et al. (78) assumed that the rate of removal of the gas from the air stream is proportional to both the concentration of active centers and the concentration of gas in the air stream. Hence — dx/dt = kcN, where A is a constant and N is the number of active centers per cubic centimeter of material. If No is the number of active centers present initially, then (No — n) = N. By substituting kcN for — dx/dt, one can simplify the equation to... 

These values are not absolute velocity constants since transfer to alkyl-aluminum occurs during the reaction. However, the transfer reactions will be proportional to the concentration of active centers. Hence, at similar conversions the fractions of active alkylmetal bonds are likely to be approximately the same, and the constants will be proportional to the true velocity constants. Determining relative propagation velocity constants is complicated by the participation of a termination reaction. At low temperatures the polymer is insoluble and the catalyst is embedded in a semi-solid mass, resulting in very slow rates of polymerization. At temperatures of 41°-60° C. reasonably good first-order reactions with respect to monomer are found, but at higher temperatures there is a rapid fall-off in reaction rate with time (Figure 4). The velocity constants in Table III were calculated from the linear portion of the rate-time curve, and no account was taken of termination reactions. 

Note that v, when divided by the total concentration of active centers [K]o, gives the very important catalyst parameter of the catalyst activity that is the turnover frequency of the active center, TOP ... 

In the polymerization of 1,3-dioxolane and tetrahydrofuran it has been shown additionally that concentration of active centers is constant throughout the polymerization (both by direct determination and from analysis of polymerization kinetics). In some other polymerizations, believed to proceed as living processes, only the moderate molecular weights regions (M < 105) were studied thus, for example, no very high molecular weight polymers were obtained in the polymerization of oxazolines. 

Contrarily to what has been reported by Guastalla and Giannini, Soga124) observed a decrease in catalyst activity with an increase of the hydrogen partial pressure. This has been attributed to a decrease of the concentration of active centers, C, due to the slow realkylation of the Ti—H bond by the monomer. However, it must be pointed out that Soga s data refer to the predominantly stereospecific... 

Rates of ionic polymerizations are by and large much faster than in free-radical processes. This is mainly because termination by mutual destruction of active centers occurs only in free-radical systems (Section 6.3.3). Macroions with the same charge will repel each other and concentrations of active centers can be much higher in ionic than in free-radical systems. Rate constants for ionic propagation reactions vary but some are higher than those in free-radical systems. This is particularly true in media where the ionic active center is free of its counterion. 

Siloxanolate Catalysts. The initial step for the study of the kinetics of base-catalyzed siloxane equilibration reactions was the preparation of a number of well-defined siloxanolate catalysts. The catalysts were prepared separately, prior to the equilibration reactions, so that a homogeneous moisture-free system with a known concentration of active centers might be obtained. The catalysts studied included potassium, tetramethylammonium, and tetrabutylphosphonium siloxanolate. 

Where kp is the propagation rate constant, C the initial concentration of active centers, D the monomer diffusivity, and S the initial radius of the catalyst particle. 

Since atomic oxygen is chosen as an active species, its density on the surface is important and is assumed to be determined by the overall reversible thermodynamics of oxygen chemisorption. In the above model, it was assumed that the dissociative chemisexption of oxygen at adjacent vacant sites (Dj) is governed by the concentration of active centers ... 

C , = total concentration of active centers, in moles per gram of catalyst = specific reaction-rate constant for surface reaction Kqo = adsorption equilibrium constant for CO... 

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