Zero-one kinetics

In the zero-one limit, particles can only contain one or no free radicals. The spirit of the original Smith-Ewart formulation is followed by defining No and N as the number of particles containing zero and one radical, respectively. These are normalised so that 

No + Ni = 1, and thus for a zero-one system, n = N. The kinetic equations describing the evolution of the radical population take into account changes due to radical entry (with rate coefficient p, giving the number of entering radicals per particle per unit time) and exit (with rate coefficient k, giving the number of radicals lost per particle per unit time by radicals going from the particle to the aqueous phase). These kinetic equations are  

The next step in developing these equations is to note that entry can be both by radicals that derived directly from initiator, that is, z-mers such as MaSO, and by radicals that have exited from another particle. Recalling that exit is held to arise from transfer to monomer leading to a relatively soluble monomeric radical, it is realised that this exiting radical is chemically quite distinct from a z-mer. Because both z-meric (initiator derived) and exited radicals can enter, it is necessary to take exit into accoimt when considering entry. Now, consideration of the aqueous-phase kinetics of the various radical species (Morrison et al, 1994) shows that the fate of an exited radical is overwhelmingly to enter another particle rather than undergo aqueous-phase termination (except for a few special systems, such as vinyl acetate, De Bruyn et al, 1996). Since the rate of exit is kn, and each exit leads to re-entry, and since = Ni in a zero-one system. Equation 3.5 becomes  

It is apparent why the zero-one limiting case is useful to interpret and predict data it contains only two rate coefficients, p and k. These can be obtained unambiguously by a combination of initiating using y-radiolysis and then following the rate after removal from the radiation source (these data are very sensitive to radical loss processes, especially the exit rate coefficient k), and steady-state rate data in a system with chemical initiator, which is sensitive to both p and k. Moreover, the transfer model for exit makes specific predictions as to the dependence of the value of k on, for example, particle size (Ugelstad Hansen, 1976). Specifically, this model predicts that k should vary inversely with particle area, and that the actual value of k can be predicted a priori from rate parameters measured by quite different means, such as the transfer constant to monomer. This size dependence, and the quantitative accord between model and experiment, have been verified for the exit rate coefficient (Morrison et al., 1994). 

This type of data also yield the rate coefficient for entry, pinit- The dependence of this on initiator concentration can be quantitatively fitted by the Maxwell—Morrison model for exclusive entry by z-mers (Maxwell et al, 1991 van Berkel et al., 2003). Moreover, such experiments have also verified the prediction of this model that pinit should be independent of both particle size (in two systems with the same initiator and particle concentrations but 

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Many emulsion polymerizations can be described by so-called zero-one kinetics. These systems are characterized by particle sizes that are sufficiently small dial entry of a radical into a particle already containing a propagating radical always causes instantaneous termination. Thus, a particle may contain either zero or one propagating radical. The value of n will usually be less than 0.4. In these systems, radical-radical termination is by definition not rate determining. Rates of polymerization are determined by the rates or particle entry and exit rather than by rates of initiation and termination. The main mechanism for exit is thought to be chain transfer to monomer. It follows that radical-radical termination, when it occurs in the particle phase, will usually be between a short species (one that lias just entered) and a long species. 

The values predicted by Eq. 51 agree well with those predicted by Eq. 49 within less than 4%. This type of plot is called a Ugelstad plot and has been applied as a criterion to determine whether a system under consideration obeys either zero-one kinetics (n<0.5) or pseudo-bulk kinetics (n>0.5). 

Fig. 3 Schematic representation of radical transfer between particles of unswollen volume based on zero-one kinetics.

Note that the initiator exponent of 2/5 and the emulsifier exponent of 3/5 are determined simply by the geometrical relation of the surface area to the volume of a sphere and therefore cannot vary so long as zero-one kinetics apply, except that the initiator exponent would differ if the assumption that the panicle nucleation rate was first order in initiator concentration were incorrect. 

The condition for zero-one kinetics to apply (i.e. half the particles contain a growing radical while the other half does not) is that... 

In the first situation (Case 1), where the particles are small or the monomer is substantially water-soluble, and desorption of radicals from the particle is likely, n is very low, and polymerisation is slow. In the second situation. Case 11, radical exit is negligible. When a radical enters a particle, polymerisation occurs until a second radical enters, and both are instantaneously terminated (zero-one kinetics). Under these conditions, n is equal to V2. In the third situation. Case III, the particles are large enough that two or more radicals may coexist within the same particle without... 

Many emulsion polymerizations can be described by so-called zero-one kinetics. These systems are characterized by particle sizes that ate sufficiently small that entry of a radical... 

In Mampel s treatment [447] of nucleation and growth reactions, eqn. (7, n = 3) was found to be applicable to intermediate ranges of a, sometimes preceded by power law obedience and followed by a period of first-order behaviour. Transitions from obedience of one kinetic relation to another have been reported in the literature [409,458,459]. Equation (7, n = 3) is close to zero order in the early stages but becomes more strongly deceleratory when a > 0.5. 

On the other hand, very few ncdels for nulticonponent systans have been reported in the literature. Apart from models for binary systems, usually restricted to "zero-one" systans (5) (6), the most detailed model of this type has been proposed by Hamielec et al. (7), with reference to batch, semibatch and continuous emilsion polymerization reactors. Notably, besides the usual kinetic informations (nonomer, conversion, PSD), the model allows for the evaluation of IWD, long and short chain brandling frequencies and gel content. Comparisons between model predictions and experimental data are limited to tulK and solution binary pwlymerization systems. 

The heat profile shows that the reaction has zero order kinetics at first, and then switches to positive order kinetics. The benzophenone hydrazone reacts first only when it is completely consumed, the reaction involving hexylamine begins. Samples were taken and analyzed by and NMR. One sample was taken when the aryl halide conversion was low, at about 5%, and the profile was overall zero order the second when the profile had switched to positive order and the conversion of the halide was greater than 50%. 

We studied the competitive amination of two amines (benzophenone hydrazone and -hexylamine) and one aryl halide (3-bromobenzotrifluoride), catalyzed by Pd(BlNAP). We showed that, when reacting alone at the same conditions, n-hexylamine is considerably more reactive and shows positive order kinetics benzophenone hydrazone shows zero order kinetics and forms a very stable intermediate, the BlNAP(Pd)Ar(amine) we also observed by NMR. During the competitive reaction of the two amines, the benzophenone hydrazone reacts first and only when it is completely consumed, the hexylamine starts to react. In this case it is the stability of the major intermediate, and not the relative reactivity, which dictates the selectivity. 

The other state variables are the fugacity of dissolved methane in the bulk of the liquid water phase (fb) and the zero, first and second moment of the particle size distribution (p0, Pi, l )- The initial value for the fugacity, fb° is equal to the three phase equilibrium fugacity feq. The initial number of particles, p , or nuclei initially formed was calculated from a mass balance of the amount of gas consumed at the turbidity point. The explanation of the other variables and parameters as well as the initial conditions are described in detail in the reference. The equations are given to illustrate the nature of this parameter estimation problem with five ODEs, one kinetic parameter (K ) and only one measured state variable. 

One may use the same general approach when the reaction kinetics are other than first-order. However, except in the case of zero-order kinetics, it is not possible to obtain simple closed form expressions for CAN, particularly if unequal reactor volumes are used. However, the numerical calculations for other reaction orders are not difficult to make for the relatively small number of stages likely to be encountered in industrial practice. The results for zero-order kinetics may be determined from equation... 

The state of the metal ion is of great importance to the behavior of an electroless solution. On one hand, adequate complexation ensures a relatively stable solution with respect to precipitation of hydrolyzed metal ion species and homogeneous decomposition (plating out) on the other hand, too high a concentration of certain complexants may effectively decrease to zero the kinetics of deposition. A brief examination of metal ion complexation in the context of electroless solutions will be helpful in understanding the role of the complexant. 

Even relatively complex reactions can behave very simply, and 99 percent of the time, understanding simple first-, second-, and zero-order kinetics is more than good enough. With very complicated mechanistic schemes with multiple intermediates and multiple pathways to the products, the kinetic behavior can get very complicated. But more often than not, even complex mechanisms show simple kinetic behavior. In complex mechanisms, one step (called the rate-determining step) is often much slower than all the rest. The kinetics of the slow step then dictates the kinetics of the overall reaction. If the slow step is simple, the overall reaction appears simple. 

The energy of the electron gas is composed of two terms, one Hartree-Fock term (T)hp) and one correlation term (Hartree-Fock term comprises the zero-point kinetic energy density and the exchange contribution (first and second terms on the right in equation 1.148, respectively) ... 

The polymerization kinetics will be first-order with respect to [A,=o], and the polymerization reaction will exhibit zero-order kinetics for any set of rate measurements conducted at one fixed concentration of monomer. 

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