Equations molecular weight moment

The mathematical modeling of polymerization reactions can be classified into three levels microscale, mesoscale, and macroscale. In microscale modeling, polymerization kinetics and mechanisms are modeled on a molecular scale. The microscale model is represented by component population balances or rate equations and molecular weight moment equations. In mesoscale modeling, interfacial mass and heat transfer... 

In general, a polymerization process model consists of material balances (component rate equations), energy balances, and additional set of equations to calculate polymer properties (e.g., molecular weight moment equations). The kinetic equations for a typical linear addition polymerization process include initiation or catalytic site activation, chain propagation, chain termination, and chain transfer reactions. The typical reactions that occur in a homogeneous free radical polymerization of vinyl monomers and coordination polymerization of olefins are illustrated in Table 2. 

The first three leading molecular weight moment equations are derived as follows. 

The necessary (Gj — 29g) and (H — //298) functions can be calculated very accurately by statistical methods, using molecular parameters such as the molecular weight, moments of inertia, and vibrational frequencies. The equations needed to make the calculation, along with necessary molecular data, are summarized in Appendix 6, while values for the thermodynamic functions for a number of substances are given in Appendix... 

Using this, the following differential equations have been obtained for the overall molecular weight moments and the average number of long chain branches per polymer molecule. 

The value of the equilibrium constant may thus be derived from AHS for the reaction, and the molecular weights, moments of inertia, and the symmetry numbers of the substances taking part. Equations of the type of (33.56) have been employed particularly in the study of isotopic exchange reactions, where the error due to the cancellation of the vibrational partition functions is very small, especially if the temperatures are not too high. ... 

To calculate the molecular weight averages, the polymer molecular moment equations can be derived with the k-th molecular weight moments of live and... 

This result shows that the square root of the amount by which the ratio M /M exceeds unity equals the standard deviation of the distribution relative to the number average molecular weight. Thus if a distribution is characterized by M = 10,000 and a = 3000, then M /M = 1.09. Alternatively, if M / n then the standard deviation is 71% of the value of M. This shows that reporting the mean and standard deviation of a distribution or the values of and Mw/Mn gives equivalent information about the distribution. We shall see in a moment that the second alternative is more easily accomplished for samples of polymers. First, however, consider the following example in which we apply some of the equations of this section to some numerical data. 

Mathematical models of the reaction system were developed which enabled prediction of the molecular weight distribution (MWD). Direct and indirect methods were used, but only distributions obtained from moments are described here. Due to the stiffness of the model equations an improved numerical integrator was developed, in order to solve the equations in a reasonable time scale. 

Also, the zeroth moment of the differential molecular weight distribution, DMWD, may be obtained by integration of the simplified equation ... 

The present section analyzes the above concepts in detail. There are many different mathematical methods for analyzing molecular weight distributions. The method of moments is particularly easy when applied to a living pol5mer polymerization. Equation (13.30) shows the propagation reaction, each step of which consumes one monomer molecule. Assume equal reactivity. Then for a batch polymerization. 

The first moment of the distribution is Pt0T the total, cumulative molar concentration of polymeric material. As the molecular weight of polymeric species increases, branching and crosslinking reactions yield a thermoset resin. Chromatography analysis of epoxy resin extracts confirms the expected population density distribution described by Equation 4, as is shown in Figure 2. Formulations and cure cycles appear in Table II. 

This area is a development in the usage of NDDO models that emphasizes their utility for large-scale problems. Structure-activity relationships (SARs) are widely used in the pharmaceutical industry to understand how the various features of biologically active molecules contribute to their activity. SARs typically take the form of equations, often linear equations, that quantify activity as a function of variables associated with the molecules. The molecular variables could include, for instance, molecular weight, dipole moment, hydrophobic surface area, octanol-water partition coefficient, vapor pressure, various descriptors associated with molecular geometry, etc. For example, Cramer, Famini, and Lowrey (1993) found a strong correlation (r = 0.958) between various computed properties for 44 alkylammonium ions and their ability to act as acetylcholinesterase inhibitors according to the equation... 

Dipole moment in benzene solution at 20 °C. b AH is the enthalpy of the reaction in equation (42). CIR = infrared spectrum NMR = H NMR spectrum M = molecular weight A = electrical conductance. 

The prediction of the MWD of emulsion polymers proved to be a relatively intractable problem even after the advent of the Harkins-Smith-Ewart theory. Perhaps the most successful early attack on the problem was that of Katz, Shinnar and Saidel (2). They considered only two microscopic events entry and bimolecular termination by combination. Their theory resulted in a set of partial integrodifferential equations, whose numerical solution provided the lower moments of the molecular weight distribution function. Other attempts to predict the MWD of emulsion polymers include those of Parts and Wat ter son (3 ), Sundberg and Eliassen (4), Min and Ray (5) and Gardon (6). 

The differential equations used to calculate the evolution in time of the moments of the molecular-weight distribution will be presented in Section 13.6.2. 

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