The Three-Dimensional Diagram

Above hexaphosphate compositions all other polyphosphates are amorphous, until the very long chain polyphosphate such as Kurrol s salts, Maddrell s salts, insoluble metaphosphate, IMP, and the like are reached. There is therefore an amorphous region of the phase diagram within the interior of which no known crystalline polyphosphates are known to exist. Perhaps these will be prepared as crystalline salts some time in the future. 

a line between poly- and ultraphosphates represents metaphosphate compositions. These are long-chain polyphosphates or ring compounds. Both types have been discovered and this line is very important for preparations of phosphate fibers. Above metaphosphate compositions are ultraphosphates. These are rich in P2O5 and all compositions in this area contain cross-linked phosphates, or triply 

With the current knowledge of melts it is impossible to predict whether or not a system will crystallize or remain amorphous when cooled. Before seeds are found 

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Figure 24 shows the ternary phase diagram (solubility isotherm) of an unsolvated conglomerate that consists of physical mixtures of the two enantiomers that are capable of forming a racemic eutectic mixture. It corresponds to an isothermal (horizontal) cross section of the three-dimensional diagram shown in Fig. 21. Examples include A-acetyl-leucine in acetone, adrenaline in water, and methadone in water (each at 25°C) [141].

In the three-dimensional diagram, the curve formed by the intersection of the two surfaces represents all of the equilibrium points of the two-phase system. Such a curve is obtained for each type of a two-phase equilibrium existing in a single-component system. At any triple point of the system three such curves meet at a point, giving the temperature and pressure of the triple point. 

The Woodward-Hoffmann rules give a more satisfying description and we shall follow the routine outlined for cycloadditions. Note that for stage 3, we can use the three-dimensional diagram we have already made. 

The adsorbed amount of cyanide is proportional to the clay, humic acid, and iron content. It is difficult to tell which soil component is mainly responsible for cyanide adsorption because, in the case of examined samples, clay, humus, and Fe content increase simultaneously. More informative is the three-dimensional diagram, in which the amount of adsorbed cyanide versus clay number and humus content is plotted (Figure 3.9). 

Because of the complexity of Fig. 12.1, the detailed characteristics of binary VLB are usually depicted by two-dimensional graphs that display what is seen on various planes that cut the three-dimensional diagram. The three principal planes, each perpendicular to one of the coordinate axes, are illustrated in Fig. 12.1. Thus a vertical plane perpendicular to the temperature axis is outlined as ALBDEA. The lines on this plane represent a Pxy phase diagram at constant T, of which we have already seen examples in Figs. 10.1, 11.7, 11.9, and 11.11. If the lines from several such planes are projected on a single parallel plane, a diagram like Fig. 12.2 is obtained. It shows Pxy plots for three different temperatures. The one for represents the section of Fig. 12.1 indicated by ALBDEA. 

The phase diagrams of two quaternary mixtures made of sodium dodecylsulfate (SDS)-water-dodecane and hexanol (system A) or pentanol (system B) have been investigated in detail [22,23]. In both cases, sections of the three-dimensional diagram with constant water/surfactant ratio have been examined. These cuts were chosen because they allow a good description of the oil region and also because the water/SDS ratio, termed X in the following, fixes the size of the droplets in the inverse microemulsion phase and the thickness of the bilayers in the oil-rich lamellar phase. In the description of the quaternary mixtures, we emphasize the details of the evolution of the phase equilibria as X is varied. We have focused our attention not only on the characterization and the location of the boundaries of the various phases but also on the equilibria between the phases. 

In certain intervals ACp is a continuous function of the composition x (Fig. 3) [6], This implies that ACp can be obtained by interpolation at any concentration within this interval, which helps in plotting the concentration dependence of the heat capacity of the system at various temperatures and the three-dimensional diagram of Cp(x, T). 

Since the composition of real flue gases is subject to great fluctuations, it was decided to vary the SO2 input concentration and H2O vapour portion in the tests. The three-dimensional diagram in Fig. 9 shows the effects of these two parameters in the reaction temperature range of 100 to 140°C. 

Fig. 6 and the remarks made about it in 2.2.1. bring out the essential features of liquid-liquid equilibria in polymer solutions. Given the appropriate A G function for the system considered, the location of the miscibility gap is determined by the temperature. Introducing T on a third axis, we obtain the three-dimensional diagram in Fig. 50. Coexisting... 

Although these curves are in the same plane, we must remember that on the three-dimensional diagram they actually lie in different planes and consequently do not intersect. The point of intersection (R ) of curves 1 and 2 in Figure 5.2 is the point of intersection not of the curves themselves but of their projections on the same plane. In actual practice, we may have a set of different values of q for R =... 

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