Schulz-Flory chain length distribution

T he higher hydrocarbons were predominantly straight-chain ones, with a close to linear Schulz-Flory chain length distribution. Therefore, it was concluded that the carbidic intermediates CH, leading to methane [cf. Eq. (43)] were also participating in chain growth. Moreover, the extent of incorporation into the longer chains was, just as with methane, indicative of... 

Figure 7 depicts the chain length distribution in liquid product, collected with repeated CO/Hj pulses (interrupted by hydrogenation pulses) with durations of 8 and 20 min. Steady-state distributions obtained with ruthenium exhibit highly linear Schulz-Flory plots, and the distribution in Fig. 7 is therefore substantially off steady state. 

Figure 4. Flory-Schulz chain-length distributions. Equation 3, for v= 100 (left) and v = 1 000 (right), Fy, = 1 (smaller unbroken) andFy, = 0 (taller dashed).

The initiator free radicals are not all produced at the same time in free radical polymerization. Consequently, the polymer chains do not all grow at the same time. In addition, a given polymer free radical can terminate another polymer free radical of any desired size, which leads to a random chain length distribution. The derivation given in Appendix A. 20 of this chapter gives what is called a Schulz-Flory distribution of chain lengths. The distribution obtained for termination by disproportionation is identical to the normal distribution occurring in equilibrium polycondensations chains are formed randomly in both cases. 

The catalysts belonging to the second class are especially reactive towards ethylene, and afford mixtures of nearly pure linear a-olefins ranging from C4 to C30 (chain length distribution of the Schulz—Flory type). These do not catalyze a double-bond shift. 

The chain lengths of the hydrocarbons obey a statistical distribution named after Anderson, Schulz, and Flory, given by... 

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... 

The shape of the Schulz-Flory distribution and the chain length of the a-olefins are controlled by the geometric chain-growth factor K, defined as K = n(C +2)/ n(C ) (see Figure 2). For the economy of the whole process it is very important that the /f-factor can easily be adjusted by varying the catalyst composition. Usually the value is between 0.75 and 0.80. 

The shape of the Schulz-Flory distribution and the chain length of the a-al-kenes are controlled by the geometric chain-growth factor K, defined as K = n(C +2)ln C ) (Figure 1). 

The hypothesis of multiple build-in where a chain can interact with a chain Cj leads one to reflect on the possibility of a chain termination by combination. If reactions were occurring in which termination could occur by simple desorption and also by combination, two peaks would be observed. The second maxima would have a center at approximately twice the value of the flrst, as doubling of the most prevalent adsorbed chain lengths is likely (25). Furthermore, secondary events such as those discussed above or chain transfer could cause the distributions of the two peaks to be different from one another. Thus the fact that secondary reactions during Fischer-Tropsch synthesis occur and that multiple build-in and termination by combination are viable propositions help rationalize distributions that do not follow the Schulz-Flory law and appear with more than a single maximum. 

Zein el Deen et al. (30) studied the kinetics of the FTS on sintered oxides of iron and manganese. They observed, too, that the rate is independent of the CO partial pressure. Bub et al. (20) developed empirical expressions for the production rate of CO2 and Ci to C4 hydrocarbons on a Mn/Fe catalyst which could be used to successfully describe the conversion and selectivity in a pilot plant fixed bed reactor (2 cm ID by 80 cm length). If a catalyst like Mn/Fe gives a Schulz-Flory product distribution the hydrocarbon fraction can be calculated from the overall conversion rate and the chain growing probability a... 

The aforementioned expression is the geometric distribution or the Flory-Schulz distribution. The results can be illustrated by plotting the mole fraction of chain length for different values of conversion, p. 

Equations that describe several types of molecular weight distribution were presented in Chapter 2. One of these is the most-probable or Schulz-Flory distribution that is derived assuming that each catalyst site is equally active and that the probability of termination does not depend on chain length. This distribution has a polydispersity index of two, and... 

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