Phase rule mixing

The isolation of crystalline products having mixed polymorphic compositions (often referred to as concomitant polymorphism) remains a topic of interest, even though the phase rule predicts that a system at equilibrium consisting two components (solvent + solute) and three phases (solution + Form I + Form II) is uni variant. Hence, for crystallizations performed at a fixed pressure (typically atmospheric) the system becomes nonvariant and genuine equilibrium can exist at only one temperature. Therefore, concomitant products must be obtained under nonequilibrium conditions. Flexibility in molecular conformation was attributed to the concomitant polymorphs of a spirobicyclic dione [34] and of 3-acetylcoumarin [35],... 

Fig. 1.8 Lever rule in phase diagrams. In Fig. 1.7, the composition X5 shows two phase mixtures, the compositions of which are Xi (M(O) phase) and Xj (MO phase). The mixing ratio of the two phases equals (x2 — X5)/(x5 — Xj), (b), which is...

Since we have briefly introduced the consideration of mixed systems, we will mention, in passing, Gibbs1 famed Phase Rule. Its basic equation, which is generally applicable, is ... 

In a mathematical sense, / represents the difference between the number of independent variables (including T and P) and the number of constraints (equations). If the number of equations equals the number of unknown variables, we can solve for all the concentrations using equilibrium equations. For more discussion of the phase rule, see Stumm and Morgan (1981). Sillen s approach was to mix components, pick a reasonable set of phases that might be present, and then see how many degrees of freedom there are to be fixed. 

The positive heats of mixing for lecithin-cholesterol mixtures indicate that interactions between unlike molecules are smaller than the interactions between like molecules, i.e., the hydrocarbon chain interactions with cholesterol are smaller than in each of the pure phases. If the excess heats of mixing become large enough, phase separation will occur. It may occur when the surface pressure is increased (i.e., as the films are compressed). The point at which phase separation occurs is difficult to predict, measure, or detect however, evidence of phase separation can be deduced from the following experiment. If excess amounts of two lipids are placed in water, the equilibrium surface pressure should reflect whether the surface film is a mixture. According to the phase rule (11,12, 13,14), if two bulk lipid phases are present, only one surface phase can be present at the air—water surface. Thus the composition of the equi-... 

The electroreduction of zirconium halides in alkali halide melts has led to the measurement of reversible potentials (Table XXV) in the temperature range 670°-750°C (550). Phase rule studies of the mixed systems preceded the cell studies and revealed that the phase diagrams of the KCl-ZrCl2 and NaCl-ZrCl2 systems were of the simple eutectic type. The liquidus curves of these binary systems were established by freezing point measurements. The melting point of pure zirconium dichloride was found to be 722° 1°C. In the potassium chloride-zirconium dichloride system, the eutectic is found at 698° 1°C at... 

A very useful technique is a study of miscibility of different substances [2]. As a rule, only identical phases are mixed with each other (nematic with nematic, smectic A (SmA) with SmA, SmC with SmC etc.). Therefore, using a well investigated substance as a reference, one can make a preliminary conclusion about a structure of a new compound not doing X-ray and other cumbersome structural studies. For instance, by mixing with a reference liquid crystal, it was concluded... 

When two film-forming components are immiscible, then according to the phase rule, their mixed monolayers collapse at the same surface pressure regardless of their composition e.g. the component of the mixed film that has a lower equilibrium spreading pressure relative to the other film-forming component is squeezed out from the monolayer as the surface pressure reaches a value corresponding to its own collapse pressure [16-18]. 

The definition of solubility permits the occurrence of a single solid phase which may be a pure anhydrous compound, a salt hydrate, a non-stoichiometric compound, or a solid mixture (or solid solution, or "mixed crystals"), and may be stable or metastable. As well, any number of solid phases consistent with the requirements of the phase rule may be present. Metastable solid phases are of widespread occurrence, and may appear as polymorphic (or allotropic) forms or crystal solvates whose rate of transition to more stable forms is very slow. Surface heterogeneity may also give rise to metastability, either when one solid precipitates on the surface of auiother, or if the size of the solid particles is sufficiently small that surface effects become important. In either case, the solid is not in stable equilibrium with the solution. See (21) for the modern formulation of the effect of particle size on solubility. The stability of a solid may also be affected by the atmosphere in which the system is equilibrated. 

Transitions in two-component systems have one more degree of freedom, — i.e., one more variable of state must be specified. The additional variable is the concentration. The phase rule of Fig. 3.7 shows the relationship between number of phases, degrees of freedom and number of components. The equation can be verified using reasoning analogous to that for the one-component system. In this section the thermodynamics of the first-order transition in systems involving one pure and one mixed phase for small molecules will be treated. Other systems, especially those involving macromolecules, are treated in Chapters 4 and 5. Under certain conditions of temperature, pressure, and concentration, the transitions can be sharp and thermometry can yield useful information. 

The thermodynamic equilibria of amphiphilic molecules in solution involve four fundamental processes (1) dissolution of amphiphiles into solution (2) aggregation of dissolved amphiphiles (3) adsorption of dissolved amphiphiles at an interface and (4) spreading of amphiphiles from their bulk phase directly to the interface (Fig. 1.1). All but the last of these processes are presented and discussed throughout this book from the thermodynamic standpoint (especially from that of Gibbs s phase rule), and the type of thermodynamic treatment that should be adopted for each is clarified. These discussions are conducted from a theoretical point of view centered on dilute aqueous solutions the solutions dealt with are mostly those of the ionic surfactants with which the author s studies have been concerned. The theoretical treatment of ionic surfactants can easily be adapted to nonionic surfactants. The author has also concentrated on recent applications of micelles, such as solubilization into micelles, mixed micelle formation, micellar catalysis, the protochemical mechanisms of the micellar systems, and the interaction between amphiphiles and polymers. Fortunately, almost all of these subjects have been his primary research interests, and therefore this book covers, in many respects, the fundamental treatment of colloidal systems. 

In two-component systems, the Gibbs phase rule allows one degree of freedom, usually in the form of a temperature vs. composition line. If there is little or no excess heat of mixing and the components are highly compatible, i.e., nearly same atomic radius and electronegativity, same crystal structure, and same valence (Hume-Rothery rules), the system can have complete solid and liquid solubility over the entire range of composition. Such systems are said to be isomorphous. However, even isomorphic systems do not solidify... 

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