Average Functionality Approach

Average Functionality Approach As we derived earlier [see Eq. (5.43)], the number-average degree of polymerization of the reaction mixture is given by 

Equation (5.61), which reduces to Eq. (5.30) for /av = 2 and is also referred to as the Carothers equation, relates the extent of reaction and the degree of polymerization to the average functionality of the reaction mixture. 

Since the number-average degree of polymerization of gel can be considered to be infinite (see Problem 1.5), i.e., Xn 00, Eq. (5.61) gives the extent of reaction at the gel point, called the critical extent of reaction or critical conversion 

for a glycerol-phthalic acid (2 3 molar ratio) stoichiometric system, /av = (2x3 + 3x2)/(2-h 3) = 2.4 and, hence, pc = IjlA = 0.833. For applying Eq. (5.62) to nonstoichiometric mixtures, however, /av must be calculated as the average useful functionality of the reaction mixture. 

Problem 5.23 Can the following alkyd recipe be reacted to complete conversion of the limiting reactant without gelation  

---

Two approaches can be used for the experimental determination of average functionalities. 

These defects of the Hartree SCF method were corrected by Fock (Section 4.3.4) and by Slater2 in 1930 [8], and Slater devised a simple way to construct a total wavefunction from one-electron functions (i.e. orbitals) such that will be antisymmetric to electron switching. Hartree s iterative, average-field approach supplemented with electron spin and antisymmetry leads to the Hartree-Fock equations. 

It is of course realized that several different Ions and Ion pairs may be present under these conditions (8,10). Indeed, In precise work, one needs to address the kinetic parameters in terms of their exact Ionic nature (e.g., k , k-, etc.) and not simply in terms of a "global rate constant. In any event, it is possible to prepare low molecular weight (2,000-4,000) polymers that are predominately hydroxyl terminated with one primary and one secondary group. This general approach has been widely utilized for some years as a route for the Important intermediates used In polyurethane foams. Since it Is often desired that these foams be chemically crossllnked, one must consider methods that can produce average functionalities higher than 2 in order to generate the desired network behavior. 

Approach II uses the average functionality as described above under Type... 

Approach II uses average functionality /av defined (Rudin, 1982) by... 

Another simulation-based approach to stochastic optimization is based on the following idea. Let be the sample average function defined in (22), based on a sample of size N. Consider the optimization problem... 

The MPIB and VIB [35] models attempt to improve the aeeuraey of the earlier models and to overcome some of the difficulties associated with the use of Hartree-Fock wave functions. We have already stated some of the advantages of using the density functional approach to obtain ionic wave-functions they were amply demonstrated by the PIB model which replaced the Hartree-Fock equation with a density functional implementation of the Dirac equation [21]. The MPIB is so called because it also adopts the density functional approach to obtain ionic charge densities (specifically anon-relativistic version derived from the Herman-Skillman [48],but replaces the potential inside the Watson shell with the spherical average of the potential due to the rest of the material, IF (r)[36] ... 

This is not to say that systematic errors cannot happen. It is clearly recognized that these failures do occur and that they do impact safety integrity. One field failure study done by one of the authors traced instrument failure reports to specific end user sites. The results showed that failure rates for the same instrument varied by over an order of magnitude from site to site. There is no doubt that this is significant. But the site specific and even person specific variables preclude an "average" probabilistic approach. That is why it is so important to understand and follow all the procedures, techniques and measures presented in the functional safety standards to avoid and control systematic failures. It is so important to have a "well designed system" for any safety instrumented function. 

---