Number fraction

Figure 5.5 Number fraction n-mers as a function of n for several values of p.

Dividing both sides of Eq. (6.58) by [M-], the total radical concentration, gives the number fraction of n-mer radicals in the total radical population. This ratio is the same as the number of n-mers in the sample containing a total of N (no subscript) polymer molecules ... 

This expression gives the number fraction or mole fraction of n-mers in the polymer and is thus equivalent to Eq. (5.25) for step-growth polymerization. 

Figure 6.7 Number fraction of n-mers as a function of n for termination by combination. Drawn according to Eq. (6.77) for values of p indicated.

Ethylene is recycled, some of which is purged, to eliminate the accumulation of ethane. The olefins are distilled into even-carbon-number fractions... 

In order to describe these different mechanisms, various breakage functions have been proposed (Hill and Ng, 1995, 1996). For precipitation processes, a breakage function of the form given in equation (6.32) with h(v, Xk) being the discretized number fraction of particles broken from size v into size interval x, seems particularly suitable as both attrition - with a high probability - and particle splitting - with a low probability - are accounted for. 

Run Number Fraction Dead Polymer, (j) Run Number Fraction Dead Polymer, (f)jj... 

FIGURE 13.1 Molecular weight distribution by number fraction for a binary polycondensation going to completion with In — 5 and Iw — 9.8. 

Number fraction of pol5Tner chains having length / 13.8... 

The cumulative number fraction larger than size Vf is ... 

Pi Cumulative number fraction scence rate function... 

Now the function displays the number fraction of molecules with a certain molecular mass. Its integral is 1 by definition. Nevertheless, we still call it the number molecular weight distribution because the factor /N (A/) dM is nothing but a constant. 

Figure 1. Number fraction of ring structures per molecule (Nr) as a function of extent of reaction (p) for bulk, linear, and nonlinear polyurethane-forming reactions with approximately equimolar concentrations of reactive groups (r = [NCO]J [OH]0 ss 1) (2,3). Conditions O-linear polymerization, HDI + poly(ethyleneglycol) at 70°, [NCOfo — 5.111 mol/kg, [OH], = 5.188 mol/kg number-average number of bonds in chain forming smallest ring structure (v) = 25.2, and nonlinear polymerization, HDI and POP triol at 70°C, [NCO] — 0.9073 mol/kg, [OH]0 = 0.9173 mol/kg v = 115. Reproduced with permission from Ref. 5.

On cobalt, the typical shape of the curve of olefin content vs. carbon number fractions in Figure 9.16 shows a low value at Nc = 2, indicating the high reactivity of ethene for hydrogenation and growth. The olefin content declines from Nc = 3... 

FIGURE 9.16 Olefin contents in carbon number fractions primary and secondary olefin selectivity. 

FIGURE 9.17 Contents of a-olefins in carbon number fractions as referring to secondary double-bond shift. 

FIGURE 9.20 Methyl-branched isomers in carbon number fractions at 1.2 bar (left) and 9 bar (right) during FT synthesis on cobalt as indications for spatial constraints on the growth sites. Further legend as in Figure 9.18. 

Increasing pressure affects alcohols selectivity (Figure 9.21). The contents of alcohols in product carbon number fractions generally increase, and the shape of the distribution curve changes, exhibiting a second maximum at higher reaction pressure.29 At Nc = 2 the alcohol content increases drastically from ca. 3 to ca. 55 mole% when the pressure is raised from 1.2 to 33 bar. 

The distributions of products within a certain carbon number fraction are far from equilibrium. In the Cg-fraction, for example, the dimethylhexanes would be thermodynamically favored over the trimethylpentanes, but the latter are predominant. The distribution within the trimethylpentanes is also not equilibrated. 2,2,4-TMP would prevail at equilibrium over the other TMPs, constituting 60-70% of the product, depending on the temperature. Furthermore, 2,2,3-TMP as the primary product is found in less than equilibrium amounts. Qualitatively, the same statement is valid for the other carbon number distributions. Products with a tertiary carbon atom in the 2-position dominate over other isomers in all fractions. 

Mass spectrometry can be used to measure the molar mass distribution (MMD) of a polymer sample by simply measuring the intensity, Nt, of each mass spectral peak with mass m . This is due to the fact that mass spectrometers are equipped with a detector that gives the same response if an ion with mass 1 kDa or 100 Da (actually any mass) strikes against it. In other words, the detector measures the number fraction and this implies that Nt also represents the number of chains with mass m,. Thus, the number-average molar mass, Mn, is given by ... 

Here Nn gives the number fraction of n-mer in the total number of polymer molecules present, N. The weight fraction Wn, similarly, is given by... 

Fig Number-fraction distribution of chain molecules at different extents of reaction in polycondensation... 

A. fraction is a portion of a whole number. Fractions contain two numbers the bottom number (referred to as denominator) and the top number (referred to as numerator). The denominator in the fraction is the total number of parts into which the whole number is divided. The numerator in the fraction is the number of parts we have. 

This chapter reviews fractions and decimals and how to order real numbers. Fractions and decimals are the most common ways that numbers are represented. An understanding of these representations, and how to perform operations on these types of numbers is essential to your success at math. Before you study the lessons in this chapter, take a few minutes to take the following ten-question Benchmark Quiz. These questions are similar to the type of questions that you will find on important tests. When you are finished, check the answer key carefully to assess your results. The quiz will help you assess your prior knowledge of fractions and decimals. You may find that you are successful with one type and need additional help with another. You can then proceed to the lessons with focus. 

The relative number fraction of wet pores as a function of the water content A is given by the law of swelling ... 

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