Lattice, coordination energy

Since A Aw is the change in interaction energy per 1,2 pair, it can be expressed as some multiple of kT per pair or of RT per mole of pairs. It is also conventional to consolidate the lattice coordination number and x into a single parameter x, since z and x are not measured separately. With this change of notation Az Aw is replaced by its equivalent xRT, and Eq. (8.41) becomes... 

There have been many attempts to describe the process of mixing and solubility of polymer molecules in thermodynamic terms. By assuming that the sizes of polymer segments are similar to those of solvent molecules, Flory and Huggins derived an expression for the partial molar Gibbs free energy of dilution that included the dimensionless Flory Higgins interaction parameter X = ZAH/RT, where Z is the lattice coordination number. It is now... 

For Knudsen diffusion, d is the pore diameter and the activation energy for diffusion is zero. For activated gas translational diffusion d is the diffusional length between adjacent low-energy sites [37] or the pore diameter [38], and the activation energy is between 10 and 60 kJ/mol [37]. The parameter z, the lattice coordination number, depends on the zeolite type, z is equal to 4 for ZSM-5 and to 6 for 5 A, compared to 3 for Knudsen diffusion. 

Fig. 21. Schematic dependence of lattice free energies on a structural coordinate or on the ionicity q. The drawing indicates the high activation energies for structural modifications and the smaller changes with q...

The NRTL equation was developed by Renon and Prausnitz.54 The expression for the excess energy function gl (see Table 14-12) was developed on the basis of a two-liquid theory. In the theoretical development, the A,/s are the Gibbs interaction energies and the a,/s are the reciprocals of the lattice coordination numbers. In practice, the quantities (Ajt - A ), (A0 — A ), and atj are taken to be three adjustable parameters per binary pair in the mixture and are obtained by a regression of the equilibrium data. The NRTL equation has been used extensively. It is superior to the Wilson equation in that it can be used to represent liquid-liquid systems (al7 < 0.426). However, the NRTL equation requires three parameters per binary pair, whereas the Wilson equation requires only two parameters per binary pair. 

The first term of Eq 2.48 is the H-F combinatorial entropy. The second term (where the subscript v indicates the free volume fraction) represents the free volume contribution to the entropy of mixing. The third term represents the non-combinatorial contribution [g ((t) ) is the non-combinatorial energy on a molten state of polymer i having the free volume fraction (()y]. The fourth term represents the energetic contribution originating from interaction between unlikely species, i j. Here gj.( (()j ) is the interaction term expressed as polynomial with coefficients that depend on the structure of the polymer chains. These coefficients are computed as double expansions in 1/z ( z is the lattice coordination number), and e jj/kgT (e j is the van der Waals interaction energies between groups a and b). 

Lattice stabilization energy for 4-, 6-, and 8-coordinate geometries as a function of the radius ratio r/R. [Biatt Communications.]... 

Here c and v are measures of the maximum attraction energy and the segmental repulsion volume, respectively, qz = s z — 2) + 2, v a specific volume for a molar segmental mass Ms, R = 8.314462 [JK moP ] is the universal gas constant, and z = 12 is the lattice coordination number. The data analyzed by Walsh et al. (1992) were based on the mixtures of commercial HDPE, PP, and EPR copolymers containing... 

Here cj) and Ni are the volume fraction and number of united atom groups, z is the lattice coordination number, and = n + 22 12 is the blend exchange energy. The parameter NJj expresses the number of dissimilar sets of i sequential bonds in... 

The crystal lattice coordinates, a, b, c, a, p, and y were taken as the variables for energy minimization, and in the case of acetic acid, the packing was studied by changing the methyl hydrogen coordinates as a function of methyl rotation. 

In the lattice model, the driving force for partitioning depends on the lattice coordination number z, which serves as a measure of the surface area of contact between the solute molecule and surrounding medium. It follows that Pj (A) -p j (B) should be a linear function of the sizes of solutes that partition between different media. Figure 16.10 shows examples in which the partitioning free energy increases linearly with the solute size. For the same reason, the Henry s law constant should increase with solute size. Figure 16.12 shows this result. 

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