Total concentration of active

Note that under steady-state conditions the rate of each reaction step equals the overall net rate, 0, 9a, and 6b represent the fractions of the total number of sites that are vacant, or occupied by A and B, respectively. Afr represents the total concentration of active sites. Conservation of the total number of active sites leads to the site balance expression ... 

Langley employs the Flory criterion to evaluate the number of active strands bounded by chemical crosslinks. The total concentration of active strands, including strands bounded by trapped entanglement junctions, can be expressed (255)... 

P ], = total concentration of active centres = [C]0, the initial catalyst concentration. 

Such an expression is based on the simple kinetic model of olefin polymerisation and is valid for systems with non-supported Ziegler-Natta catalysts (of moderate activity). Some important limiting expressions can be deduced when [M] is high and when k kp km, and Cp = C, where C denotes the total concentration of active sites, and Pn denotes the number-average degree of polymerisation ... 

We can simplify this unwieldy equation by making two variable changes. If we represent the total concentration of active species (as assayed after substantial dilution) by Pa, then we may write that... 

Equations 10 are equivalent to those of eq 9, in which only two are independent. Using this steady state assumption, generally one or more of these unknowns can be eliminated, but one remains. The second assumption is the site balance. The total concentration of active sites is constant and equal to Nt. ... 

Spillover may have a dramatic effect on this analysis. If spillover of A exists and new active sites are created, Cs, the total concentration of active sites, may be divisible into Cs0, the initial concentration of sites on the surface, and Csi, the concentration of sites created by spillover. Even if the mechanism and specific activity are the same, a separate term in the equation may be included since Cs - Cs0 -I- Csi. As assumed, spillover can modify the effective surface concentration of A. This may involve including another term in addition to the product KAPA, or maybe just a modification of KA is needed. The actual... 

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 ... 

Trapping agents, such as malonate anions, naphthoxides, and phosphines have been used to determine the concentration of chain carriers in controlled/living and other carbocationic systems [85,249,250]. These strong nucleophiles react with all sufficiently electrophilic species, including not only carbocations but also onium ions and covalent esters. Thus, the discussed measurements can provide only the total concentration of active and dormant end groups. In principle, the kinetics of formation of the product in the trapping experiments could resolve more and less active species but only if they are present at comparable concentrations. As discussed before, carbocations are present in ppm quantities in comparison with dormant species. 

Here kl is a surface intermediate formed by adsorption of A onto a vacant surface site 1. The total concentration of active sites on the catalytic surface, in moles per unit amount of catalyst, is... 

The total molar concentration of active sites per imit mass of catalyst is equal to the number of active sites per imit mass divided by Avogadro s number and will be labeled C, (mol/g-cat.) The molar coneentration of vacant sites, C , is the number of vacant sites per imit mass of eatalyst divided by Avogadro s number. In the absence of catalyst deactivation we assume that the total concentration of active sites remains constant. Some further definitions include ... 

In designing fixed and ideal fluidized-bed catalytic reactors, we have assumed up to now that the activity of the catalyst remains constant throughout the catalyst s life. That is, the total concentration of active sites, C accessible to the reaction does not change with time. Unfortunately, Mother Nature is not so kind as to allow this behavior to be the case in most industrially significant catalytic reactions. One of the most insidious problems in catalysis is the loss of catalytic activity that occurs as the reaction takes place on the catalyst. A wide variety of mechanisms have been proposed by Butt and Petersen, to explain and model catalyst deactivation. 

In the polymerization of styrene initiated by anhydrous perchloric acid the rate of initiation is assumed to be very fast by analogy with initiation by sulphuric acid, but, in contrast to the latter system, true termination does not seem to occur, and the main molecular weight limiting processes are transfer reactions. Thus to a good approximation the total concentration of active centres is always equal to the initial concentration of perchloric acid. In ethylene dichloride and methylene dichloride solutions, the reaction proceeds to high conversions according to the experimental rate law... 

The total concentration of active centres, eqn. (2), will be the sum of uncoordinated and coordinated sites, viz. 

Aggregation of ion-pairs has been demonstrated in the polymerization of ethylne oxide (... -CH206 K in THF solvent) apparently cyclic trimers of ion pairs dominate, formed with the equilibrium constants equal approx, to 10 107 l2.mol"2. This value was determined from the analysis of the kinetics of polymerization (30). Polymerization of e-caprolactone with Na as counterion in THF solvent also shows the 1/3 dependence of the rate of polymerization on the total concentration of active species... 

Figure 4 shows the time-conversion curves for monomer and the total concentrations of active species [P ]. The following... 

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. 

At any instant there will be polymerization centers generally represented by Cat—P (n = 1, 2, ) and different initiation centers Cat—R, Cat—R" and Cat—R " [cf. Eqs. (9.18)-(9.20)]. Thus the total concentration of active centers, C, is given by the sum of the concentration of polymerization centers, C, and the concentrations of different initiation centers, Q, at which initiation takes place at that particular moment ... 

The concentration of active chains is found from analyzing the total concentration of active centers ... 

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