Phase diagrams three-dimensional

Figure 1.14 Complete phase diagram (three-dimensional p-T-x scheme (a), T-x (b), p-T (c), p-X (d) projections) for binary system A-B of type 1 without liquid-liquid immiscibility (type la) (Reproduced by permission of M AIK / Nauka Interperiodica).

The integrals are over the full two-dimensional volume F. For the classical contribution to the free energy /3/d([p]) the Ramakrishnan-Yussouff functional has been used in the form recently introduced by Ebner et al. [314] which is known to reproduce accurately the phase diagram of the Lennard-Jones system in three dimensions. In the classical part of the free energy functional, as an input the Ornstein-Zernike direct correlation function for the hard disc fluid is required. For the DFT calculations reported, the accurate and convenient analytic form due to Rosenfeld [315] has been used for this quantity. 

A. Ciach, J. S. Hoye, G. Stell. Microscopic model for microemulsion. II. Behavior at low temperatures and critical point. J Chem Phys 90 1222-1228, 1989. A. Ciach. Phase diagram and structure of the bicontinuous phase in a three dimensional lattice model for oil-water-surfactant mixtures. J Chem Phys 95 1399-1408, 1992. 

Figure 7.2 A three-dimensional phase diagram for a Type I binary mixture (here, CO2 and methanol). The shaded volume is the two-phase liquid-vapor region. This is shown ti uncated at 25 °C for illustration purposes. The volume surrounding the two-phase region is the continuum of fluid behavior.

Moore and Jorgenson eombined the rapid two-dimensional separation aehieved by LC-CZE with SEC to make the first eomprehensive three-dimensional separation involving an eleetrodriven eomponent in 1995. Size exelusion ehromatography separated the analytes over a period of several hours while the reverse phase HPLC-CZE eombination separated eomponents in only 7 min. A sehematie diagram of the three-dimensional SEC-reverse phase HPLC-CZE instrument is shown in Eigure 9.9 (18). A dilution tee was plaeed between the SEC eolumn and the reverse phase HPLC injeetion loop in order to dilute the eluent from the SEC eolumn, sinee it eon-tained more methanol than was optimal for the reverse phase HPLC eolumn. 

In each of the composition diagrams in Fig. 14.2, the numbers represent a series of reactions run at a defined composition and temperature. These are isometric sulfur slices through three-dimensional K/P/RE/S quaternary phase diagrams. As just one example of what we have studied. Table 14.1 identifies the compositions at each point and the resulting phase(s). We have rigorously studied how phase formation is dependent upon the compositions of reactions for the rare-earth elements Y, Eu, and La and we have also discovered key structural relationships between the rare-earth elements, indicating a significant dependence on rare-earth and alkali-metal size for sulfides and selenides. 

The melting temperature of A is higher them that of B. Therefore, the melting temperature of a drops as the composition becomes richer in B. At specific temperatures on the diagram (see 1. 2.), a two-phase system appears, that of a Uquid plus that of a. Finally, as the temperature rises, the melt is homogenous and the solid, a, has melted. In the three-phase system, only the relationship between A, B and C can be illustrated on a two-dimensioncil drawing. A three-dimensional diagram would be required to show the effect of temperature as well. 

Other diagrams could be devised indicating the influence of pH, electrolyte concentration, etc., on the appearance of one or two phases. Three-dimensional diagrams might also be of great aid. 

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].

Figure 2.8 Three-dimensional p,T,Vrepresentation of a single component phase diagram visualizing the critical point.

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