Polymorphs, entropy differences

The latent heat at a polymorphic transformation equals to TA S where A S is the entropy difference between two structures. The bcc phase has the higher entropy due to the sharper peak in the N E) curve (Figure 13.22). Quite generally, if N E) has a narrow peak in the vicinity of the Fermi level, this peak does not contribute to the heat capacity at high temperatures. 

In some instances, distinct polymorphic forms can be isolated that do not interconvert when suspended in a solvent system, but that also do not exhibit differences in intrinsic dissolution rates. One such example is enalapril maleate, which exists in two bioequivalent polymorphic forms of equal dissolution rate [139], and therefore of equal free energy. When solution calorimetry was used to study the system, it was found that the enthalpy difference between the two forms was very small. The difference in heats of solution of the two polymorphic forms obtained in methanol was found to be 0.51 kcal/mol, while the analogous difference obtained in acetone was 0.69 kcal/mol. These results obtained in two different solvent systems are probably equal to within experimental error. It may be concluded that the small difference in lattice enthalpies (AH) between the two forms is compensated by an almost equal and opposite small difference in the entropy term (-T AS), so that the difference in free energy (AG) is not sufficient to lead to observable differences in either dissolution rate or equilibrium solubility. The bioequivalence of the two polymorphs of enalapril maleate is therefore easily explained thermodynamically. 

Consequently, it can be concluded for the mixtures of LLL-MMM, LLL-PPP, LLL-SSS, MMM-PPP, and PPP-SSS that the TAG binary mixtures are miscible in metastable polymorphs of a and p forms when the difference in the number of carbon atoms of the fatty acid moieties. An, equals 2, whereas immiscible mixtures are found in all polymorphic forms when An is larger than 2. Results obtained for these mixture systems may indicate a relationship between polymorphism and phase behavior of the binary mixtures of the saturated-acid TAGs in such a way that rotational freedom of hydrocarbon chains and entropy of methyl-end stacking are crucial factors determining the polymorph-dependent phase behavior. 

Stacking gap. In contrast, p polymorph has all-trawi-hydrocarbon chains and these rigid chains cannot adjust themselves to their circumstance. Therefore, p polymorph shows a eutectic phase. As for the LLL/PPP and LLL/SSS mixtures, eutectic phases occur for aU polymorphs. Because of large differences in carbon numbers for fatty acid chains between LLL and PPP (An = 4), and between LLL and SSS (An = 6), there are very large methyl-end stacking gaps in these crystals. Therefore, the increased entropy of methyl-end stacking becomes predominant and phase separation must be favored thermodynamically for all polymorphs. 

The low temperature heat capacity has been measured from 54.9 to 296.2 K by Todd (7). The entropy is based on S"(51 K) > 0.62 cal K mol . The high temperature enthalpy has been measured to 1600 K by Pankratz et al. (5). The low and high temperature data were Joined smoothly together by means of a Shomate function plot (8). Since all the aluminum and silicon atoms occupy differently coordinated sites, there is no possibility of any residual entropy of mixing in this polymorph. 

Polymorphism can be detected by the differences in physical properties due to individual characteristics. Based on the fugacity, which relates to the thermodynamic term, entropy of the solid molecule, polymorphism may be defined as monotropic or enantiotropic. Furthermore, combination of these two systems, monotropic and enantiotropic, may yield a third system. The definition of these categories may best be illustrated by the solubility-temperature plots, based on the van t Hoff equation. In a monotropic category as shown in Figure 9, the solubility of form I (the stable form) and that of form II (the metastable form) will not intersect each other at the transition temperature calculated only from the extrapolation of the two curves. In the enantiotropic category (Fig. 10), the solubility of form I (the stable form) and that of form II (the metastable form) will intersect each other at the transition temperature. In the combined category (Fig. 11), for which there are two transition temperatures, the solubility of form III will not intersect any other curves. 

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