Number chain length distribution weight average

Henry constant for absorption of gas in liquid Free energy change Heat of reaction Initiator for polymerization, modified Bessel functions, electric current Electric current density Adsorption constant Chemical equilibrium constant Specific rate constant of reaction, mass-transfer coefficient Length of path in reactor Lack of fit sum of squares Average molecular weight in polymers, dead polymer species, monomer Number of moles in electrochemical reaction Molar flow rate, molar flux Number chain length distribution Number molecular weight distribution... 

Figure 1 Effect of multiple site types and mass and heat transfer resistances on the microstructure of polypropylene made with heterogeneous Ziegler-Natta and metallocene catalysts. The overall MWD and CCD are assumed to result from the superposition of individual MWDs and CCDs for three site t)rpes (T = temperature, M = number average molecular weight, = hydrogen, CjH = propylene, C2H4 = ethylene, Fj = molar fraction of propylene in copolymer, /(F,) == copolymer composition distribution, r = chain length, wix) = weight chain length distribution).

By combination of such moments one can easily calculate mean values for the degree of polymerization, P, which characterize the chain length distribution. The distribution is only fully described if all moments are known. However, in practice there are two mean values calculated by the first three statistical moments, which are extensively used the number-average degree of polymerization, P, and the weight-average degree of polymerization,... 

The ratio of the weight-average and the number-average degree of polymerization, Pvi/Pn, describes the polydispersity of a chain length distribution. It becomes unity if all chains have the same length—called a monodisperse distribution— and values greater than one, if the distribution exhibits a broader shape. 

Exchange reactions without formation of by-products, such as the acidolysis and aminolysis reactions present in polyamidations, do not increase number-average molecular weight, but can be an important cause of relaxation of chain length distributions toward the equilibrium. Kotliar has presented a general review of these reactions [19]. 

In Eq. (15), w r) is the weight distribution of chains of length r and r is the number-average chain length of the polymer population. Therefore, from a polymer microstructure point of view, a coordination catalyst is considered to have only one site type if it produces polymer with a chain length distribution that follows Eq. (15) instantaneously. 

It is usually straightforward to detect the presence of multiple-site types on a coordination catalyst because these catalysts will produce polymer with polydispersity higher than 2 even under invariant polymerization conditions. The simplest way to visualize this phenomenon is to assume that every different site type on a multiple-site catalyst produces polymers that follow a distinct Flory s distribution that is, those with a distinct number-average chain length, [38]. In this way, the chain length distribution for the whole polymer is a combination of distinct Flory s distributions weighted by the mass fraction of polymer made on each site type, mj [Eq. (24)]. 

The number- and weight-fraction chain length distributions are interchangeable. The former can also be calculated from the latter by n(r) = (i >(r)/r)/y, (w(r)/r). The number-average chain length ftg is thus... 

We wish to compute as functions of 0 < /> < 1 the chain length distribution [P ], the number (DP ) and weight (DP ) averaged chain lengths, and their ratio (thepolydispersity T disp), which is a measme of the breadth of the distribution. If Edisp = 1, all chains are of the same length, and if Pdisp 1 there is considerable variation in chain length. 

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