Equation weight average molecular

For a polymer with a molecular weight distribution the proper molecular weight average to use in equations (T) and (lO) can be determined using the following considerations. In the case of a heterogeneous polymer it has been shown that (7 19.20). 

Ordinate values of the molecular weight distribution rather than molecular weight averages obtained from size exclusion chromatography were used in the modelling (3.9). [Pn] in Equation (1) can be transformed into (log M), the ordinate of the molecular weight distribution using (2) ... 

In the past three decades, industrial polymerization research and development aimed at controlling average polymer properties such as molecular weight averages, melt flow index and copolymer composition. These properties were modeled using either first principle models or empirical models represented by differential equations or statistical model equations. However, recent advances in polymerization chemistry, polymerization catalysis, polymer characterization techniques, and computational tools are making the molecular level design and control of polymer microstructure a reality. 

A few very important points have been neglected by some authors in their evaluation of chain transfer constants by means of kinetic measurements. Frequently, a retardation of the overall rate is to be observed in the presence of chain transfer agents. A correct value of the chain transfer constant can result only if the reactions which lead to this retardation are properly considered in the kinetic scheme. In addition, the equation which one must use to calculate the chain transfer constant depends on the type of molecular weight average which is measured. Failure to... 

From a theoretical point of view, the equations which relate the three different molecular weight averages to the chain transfer constant are equally precise. Practically, however, it is recommended that one uses the viscosity average, because it can be measured most accurately. 

These two types of molecular weight averages are representative of the type called absolute methods, in that well-established thermodynamic equations can be used to convert the experimental data directly into a value of the molecular weight. However, some other methods require calibration. The most important of these indirect methods involves a measurement of the intrinsic viscosity. This quantity is a measure of the extent to which a polymer molecule increases the viscosity of the solvent in which it is dissolved. The viscosity method can be calibrated to yield a viscosity-average molecular weight, defined by... 

The molecular weight of the ith species is then obtained from the calibration curve at point V, and the molecular weight averages can be calculated from Equations 12-58 ... 

J. = Mfy], from the universal calibration curve. We have assumed that the intrinsic viscosity of a branched polymer is equal to a branching function g expressed in terms of a parameter A equal to the (number) average number of branch points per unit molecular weight (see Equation 12-68). 

In addition to termination, transfer reactions to all other components of the polymerization occur. These transfer reactions do not affect the polymerization rate. However, they diminish the average molecular weight. The equation for transfer reactions is... 

Equation (2-4) defines the arithmetic mean of the distribution of molecular weights. Almost all molecular weight averages can be defined from this equation. 

A mathematical model for styrene polymerization, based on free-radical kinetics, accounts for changes in termination coefficient with increasing conversion by an empirical function of viscosity at the polymerization temperature. Solution of the differential equations results in an expression that calculates the weight fraction of polymer of selected chain lengths. Conversions, and number, weight, and Z molecular-weight averages are also predicted as a function of time. The model was tested on peroxide-initiated suspension polymerizations and also on batch and continuous thermally initiated bulk polymerizations. 

A digital-computer program was written to solve the equations and calculate the conversion at any time in a batch polymerization. Polymerization rates, instantaneous and cumulative molecular-weight distributions, and molecular-weight averages are also calculated at this time. 

Equations (17), (21), (32), (33) and those of Table 1 completely determine the instantaneous polymerization kinetics and molecular weight averages. Simple relationships are available... 

To illustrate the calculation of molecular weight averages using moment equations, let us consider a free radical polymerization of vinyl monomers. Table 3 shows the kinetic equations based on the kinetic... 

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