Numerical solution average molecular weight

Equation (32) has been compared with phase boundary concentration data in the following way. For each solution, N of the polymer sample is estimated from Mw or the viscosity-average molecular weight Mv along with the molecular parameters ML and q listed in Table 1, and d is calculated with d from II or 0II/0c data. For systems which lack these data, the values of d from the (partial) specific volume vsp may be substituted. Table 2 lists the resulting values of d from II, 0II/0c, or vsp for various systems. The phase boundary volume fractions vc v ( = vc v v = I and A) are calculated from experimental phase boundary weight fractions (or mass concentrations) with d, Mw (or Mv), and Ml. Finally, with these numerical results, [vc v/dav(d)] — AV(N, d) is computed... 

At present all commercial polystyrene (with average molecular weights between 100,000 and 400,000) is manufactured by radical polymerization, which yields atactic polymers.476 Peroxides and azo compounds are commonly used initiators. The suspension process (usually as a batch process in water at 80-140°C) produces a product with relatively high residual monomer content.223 More important is the continuous solution process (usually in ethylbenzene solvent at 90-180°C), which yields high-purity product. Styrene can be copolymerized with numerous other monomers.477 One of these copolymers, the styrene-divinylbenzene copolymer produced by free-radical polymerization, has a crosslinked stucture and is used in... 

Dynamic light scattering (119) has been used to determine the number average molecular weight (120) and molecular weight distribution of poljuners in solution (121-123). There are several mathematical approaches to determine the molecular weight distribution from PCS data. Comparisons with GPC and numerical simulation have been made (124). The resolution of bimodal distributions has been demonstrated (125). 

Numerous investigations have shown that variations in the synthesis conditions (for example, the value of r, the catalyst type and concentration, the solvent, temperature, and pressure) cause modifications in the structure and properties of the polysilicate products. For example, Sakka and coworkers [28,29,31] observed that hydrolysis of TEOS utilizing r values of 1 to 2 and 0.01 M HCl as a catalyst yields a viscous, spinnable sol (capable of being drawn into a fiber) when aged in open containers exposed to the atmosphere. Subsequent studies showed that spinnable solutions exhibit a strong concentration-dependence of the intrinsic viscosity and a power-law-dependence of the reduced viscosity on the number averaged molecular weight ... 

The resulting equations for heterogeneous polymers assume the same general form, but numerical evaluation of the second coefficient, A 2 or F2, involves formidable summations over the entire distribution. Molecular weights M occurring in the first term of the osmotic expressions must, of course, be replaced by number averages, Mn- Dilute solutions of two chemically different polymer species also have been treated. ... 

This follows from the expansion factor, a is greater than unity in a good solvent where the actual perturbed dimensions exceed the unperturbed ones. The greater the value of the unperturbed dimensions the better is the solvent. The above relationship is an average derived at experimentally from numerous computations. Because branched chains have multiple ends it is simpler to describe them in terms of the radius of gyration. The volume that a branched polymer molecule occupies in solution is smaller than a linear one, which equals it in molecular weight and in number of segments. 

The concentration of the various species in a multi-component mixture may be expressed in numerous ways. In this book the equations are formulated in terms of mass fluxes, thus mass concentrations are used. However, the equations could as well be formulated in terms of molar fluxes, and molar concentrations as usually applied in basic textbooks in Chemical Engineering (e.g [11, 13, 170]). The mass concentration, pc, is the mass of species c per unit of volume of solution. The species c mass density relates to the familiar molar concentration by the simple formula Cc = pc/M c- is the molecular weight of species c. The mass fraction, = pdp, is the mass concentration of species c divided by the mixture mass density of the solution. The corresponding mole fraction, Xc = Cc/C, is the molar concentration of species c divided by the total molar concentration of the solution. The individual chemical species in a mixture moves at different velocities. The Vc denotes the velocity of the species c with respect to stationary coordinate axes. Thus, for a mixture of N species, the local mass-average velocity v is defined as ... 

---