Flory—Schulz plot

Fig. 30. Panel (a) shows the pressure-temperature projection of mixtures of n-alkanes and CO2 for several chain lengths. At high pressnres, all such lines become almost vertical, a featnre characteristic of hquid-hquid immiscibihty. In panel (b) the critical temperatures at 1600 bar are used to plot a Flory—Schulz plot in order to show the occurrence of a high pressure 0 point...

The description of the product distribution for an FT reaction can be simplified and described by the use of a single parameter (a value) determined from the Anderson-Schulz-Flory (ASF) plots. The a value (also called the chain growth probability factor) is then used to describe the total product spectrum in terms of carbon number weight fractions during the FT synthesis. In the case... 

The total product spectrum for a typical precipitated iron catalyst in an LTFT process is shown in Figure 13.3. Constructing an Anderson-Schulz-Flory (ASF) plot from the total product spectrum does not give a straight line and can conveniently be separated in two distinct regions, one from C, to C8 and another from C20 onward (as shown in Figure 13.4). The light olefins and... 

Figure 3. A Schulz-Flory-type plot for catalyst derived from CpFe(CO)2 2-NaY adduct initial pressure, 20 bar H2CO = 4/1 reaction temperature, 250°C...

Equations 2-86 and 2-89 give the number- and weight-distribution functions, respectively, for step polymerizations at the extent of polymerization p. These distributions are usually referred to as the most probable or Flory or Flory-Schulz distributions. Plots of the two distribution functions for several values of p are shown in Figs. 2-9 and 2-10. It is seen that on a... 

The steady state Anderson-Schulz-Flory (ASF) plot shows a single alpha value for the C1-C20 products however for the sample collected for synthesis for the longest period time there is a distinct two-alpha ASF plot (figure 16). A plot of the lower alpha (for C1-C9 products) versus the time on stream clearly shows that this alpha value changes with periodic operation (figure 17). Furthermore, as the period length increases the alpha value for the C1-C9 products decreases. With a return to steady state operations for the last four sample collections, the alpha value returned to the value that it had prior to the periodic operation. 

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. 

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). ... 

Figure 5. Examples of moment free energy (70) for Flory-Huggins theory of length-polydisperse polymers, with one moment density, p, retained. The parent is of the Schulz form (65), with pf = 0.03, Lu = 100 (hence p = p, /Lv = 3 x 10-4), and a = 2 (hence Lw = 150) the point pt = pj° is marked by the filled circles. In plot (a), the value of x = 0.55 is sufficiently small for the parent to be stable The moment free energy is convex. Plot (b) shows the cloud point, % 0.585, where the parent lies on one endpoint of a double tangent the other endpoint gives the polymer volume fraction p, in the shadow phase. Increasing x further, the parent eventually becomes spinodally unstable [x 0.62, plot (c)]. Note that for better visualization, linear terms have been added to all free energies to make the tangent at the parent coincide with the horizontal axis.

Indeed, the formation of esters of the type RO—(COCE CE ),—H was observed when the solvent for the copolymerization reaction was ROH (R=CH3, C2H5). The polyketoesters corresponding to n = 1-5 could be seperated and quantified, and gave well-behaved Schulz-Flory plots indicating the validity of the mechanism involving a single mode of stepwise chain growth as shown in Scheme 1. 

Meyerhoff (182)] but also in the short-chain region [Marzolph and Schulz (176)]. The decrease in the slope of the double-logarithmic plot at low molecular weights is naturally and faithfully reproduced. This would be equally true of the Flory-Fox theory, as represented by Eqs. (5) to (8). There seems to be absolutely no evidence for partial draining of the solvent through the polymer coils, such as would be predicted by the theories of Brinkman (45 ), Debye and Bueche (79), Kirkwood and... 

Figure 15 Anderson-Schulz-Flory plot for CO hydrogenation (dashed line theoretical plot for polymerization of Cj species solid line, experimentally observed values. See M. L. Turner, H. C. Long, A. Shenton, P. K. Byers, P. M. Maitlis, Chem. Eur. J., 1995, l,8j.

The experimentally obtained Anderson-Schulz-Flory (ASF) distribution (solid line) follows the theoretical values closely and was an early indication that the reaction to form the hydrocarbons was a type of polymerization, and indeed of Ci species. An interesting feature of the ASF plot is that it is not quite smooth but has a kink at A = 2 which comes below the curve (see Figure 15). The reason why substantially less ethane and ethylene than expected is formed has been widely debated it can occur if fewer free C2 species are produced or if the C2 fraction preferentially undergoes further reaction. The former explanation seems to be the more accepted one, in other words the rate at which surface-attached C2 undergoes further polymerization is faster than the rate of liberation of the free C2 hydrocarbons from the surface. 

FIGURE 5 Anderson-Schulz-Flory distribution of the linear hydrocarbons, linear oxygenates (n-alcohols, n-aldehydes, and linear carboxylic acids), and methyl alkyl ketones formed in the Fischer-Tropsch synthesis on an iron-containing Fischer-Tropsch catalyst operating at a temperature of 498 K (plotted using log(lO)). 

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 105 Three-dimensional plots representing the minimum number of Schulz-Flory site distributions needed to reproduce the MW distributions of polymers made with the two low-temperature activated catalysts in Table 35. On the left (Cr/silica), about half of the branches are in an island at low MW and high branch concentration, which has been eliminated on the right (Cr/silica-titania). Adapted with permission from Ref. [435]. 

Figure 220 is a plot of the logarithm of the mol% a-olefin produced in the presence of BEt3 against the carbon number of the olefin. In this format, the Schulz-Flory distribution stands out clearly as a straight line on which the data for all the olefins but 1-hexene fall. The 1-hexene product clearly stands out as a spike off the Schulz-Flory background. 

FIGURE 220 Plot of the distribution of a-olefins obtained in situ with BEt3, showing the prominent 1-hexene spike on a Schulz-Flory background line. 

Fig. 21. Fischer-Tropsch activity of LaRhO) at different temperatures, as Schulz-Flory plots. (Reprinted by permission from Ref. 41.)...

A plot of In M P)/P) versus P yields a value of a from either the slope or the ordinate intercept. Agreement between the slope and intercept is used as a criterion of the soundness of Schulz-Flory fit. A typical product distribution of CO hydrogenation that reflects the polymerization kinetics is shown in Figure 7.34. 

An exponentially decreasing curve is obtained for A = 1 when the mole fraction is plotted against the degree of polymerization (see Figure 8-3). For this reason, and not because an exponential function appears in Equation (8-35), the Schulz-Flory distribution is called an exponential distribution. 

Fig. 2 Schulz-Flory plot of FT products from slurry reactor (Satterfield and Huff, 1981)...

Recently, Satterfield and Huff (13) determined carefully product distributions of the FTS in a slurry well-mixed reactor and found striking agreement with Schulz-Flory distribution. As shown in Fig. 2 the product slates obtained under different conditions for a conventional Fe based ammonia catalyst give a chain growing probability CC of about 0.67. The same value of 01. was reported for a Mn/Fe catalyst which was studied in fixed bed and slurry phase (14,15). Deviations from Schulz-Fbory plot are often reported in literature. But in the case of conventional catalysts such deviations are probably caused by insufficient analytical techniques and nonconsideration of oxygenates (13). 

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