Schulz—Flory distribution

The simple one-moment degree of polymerization averages are related to each other by 

Consequently, the distributions become increasingly narrower for increasing degrees of coupling. The Schulz-Flory distribution can be distinguished from the logarithmic normal distributions via Equations (8-37) and (8-31). For the Schulz-Flory distribution, the following, of course, holds  

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The FTS mechanism could be considered a simple polymerization reaction, the monomer being a Ci species derived from carbon monoxide. This polymerization follows an Anderson-Schulz-Flory distribution of molecular weights. This distribution gives a linear plot of the logarithm of yield of product (in moles) versus carbon number. Under the assumptions of this model, the entire product distribution is determined by one parameter, a, the probability of the addition of a carbon atom to a chain (Figure 4-7). ... 

Reaction mechanisms and molar mass distributions The molar mass distribution of a synthetic polymer strongly depends on the polymerization mechanism, and sole knowledge of some average molar mass may be of little help if the distribution function, or at least its second moment, is not known. To illustrate this, we will discuss two prominent distribution functions, as examples the Poisson distribution and the Schulz-Flory distribution, and refer the reader to the literature [7] for a more detailed discussion. 

Figure 1 Poisson and Schulz-Flory distribution with identical (N)n = 50. The arrows indicate (,N)w = 51 (Poisson) and (N)w = 100 (Schulz-Flory).

Figure 9.4. Typical Schulz-Flory distribution fory=0.85...

Figure 9.5. Comparison of Poisson and Schulz-Flory distribution, see text...

The sensitivity of the product distribution for small changes of these parameters can also be exploited to our advantage the product distribution can be easily adjusted to changes of the demand for certain oligomers, but only within the limits of Schulz-Flory distributions ... 

Already in the study of linear chain molecules it has become evident that the shape of the molar mass distribution and its width provide a valuable guide to the mechanism of chain formation. Best known are the most probable (or Schulz-Flory) distribution and the narrow Poisson distribution. The former is often... 

Zirconia-modified silica impregnated with Co2(CO)s and activated under H2 at 300 °C renders a catalyst more active and selective to diesel fraction, in the CO hydrogenation reaction, than that conventionally prepared from a nitrate salt solution. The selectivity patterns followed a Schulz-Flory distribution and catalysts prepared from carbonyl precursor exhibited low water-gas shift activity [146]. 

Note 2 In the literature, this distribution is sometimes referred to as the Flory distribution or the Schulz-Flory distribution. 

The termination step for 1-alkene formation is now the reaction of the surface alkenyl with surface H instead of the p-elimination step. Chain branching can proceed by the involvement of allylic intermediates. Since this new mechanism involves different types of reactions to form C2 and C2< hydrocarbons, it is not expected that the amounts of C2 products will lie on the normal curve of the Ander-son-Schulz-Flory distribution. 

There exists some confusion on the naming of this distribution. The most probable distribution is called by statisticians the geometric distribution see Ref. 99, pp. 268. We shall use the names either most probable or Schulz-Flory distribution. 

Thus, Equation 27 is in this case a possible distribution function. It is of the type of the Schulz-Flory (25) distribution function. The expressions p and alternating polymerization (chain termination). The validity of the Schulz-Flory distribution function in this example of a polymerization with reversible propagation steps is evident. This type of distribution is always present if the distribution of the chain lengths... 

Because in some cases the production of methane does not obey the Schulz-Flory distribution, kj is allowed to differ from kj. The Schulz-Flory constant is defined as a = rn/rn i, for n 3,... 

The minimum value for the rate constant k on ruthenium is calculated from the data published by Dautzenberg et al. for the on iron the data of Vannice are used. For the ruthenium case it is assumed that the methane production meets the Schulz-Flory distribution (kj = k3). The value of the rate constant of propagation, where the reaction is completely determined by this constant is shown for the different models in table II. In the first column of this table minimum k values are shown for ruthenium. The data used fpom the work of Dautzenberg and coworkers were ... 

The oligomerization product thus formed is a mixture of olefins with a Schulz-Flory distribution of molecular weights (see also Section 4.7.2), whose composition can be modified, for example by adding an excess of tertiary phosphine or by changing the ancillary ligand. 

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