Phase equilibrium crystal-liquid

Measurements of the dependence of partial pressures on temperature along the liquidus (two phase equilibrium between liquid and solid phases) and solidus (solid-solid) lines and partial pressure dependences on system composition at constant temperature give the total pressure and vapour brutto composition as function of temperature and system composition. After these data are obtained, a T X diagram can be transposed to a P T X diagram. Such diagrams are used in applied science e.g. the growth of crystals and films by the vapour deposition method. 

The points R have to be on a straight line terminating in the composition of the methanol hydroquinone clathrate (A) in equilibrium with a-hydroquinone at 25°C. The point B roughly corresponds to the composition of the clathrate obtained by Palin and Powell24 when crystallizing hydroquinone from methanol points between A and B form a continuous range of solid solutions in equilibrium with liquid phases whose compositions lie on the curve CE. It is found that the equilibrium clathrate has a composition corresponding to y — 0.474 at 25°C. 

The non-random two-liquid segment activity coefficient model is a recent development of Chen and Song at Aspen Technology, Inc., [1], It is derived from the polymer NRTL model of Chen [26], which in turn is developed from the original NRTL model of Renon and Prausznitz [27]. The NRTL-SAC model is proposed in support of pharmaceutical and fine chemicals process and product design, for the qualitative tasks of solvent selection and the first approximation of phase equilibrium behavior in vapour liquid and liquid systems, where dissolved or solid phase pharmaceutical solutes are present. The application of NRTL-SAC is demonstrated here with a case study on the active pharmaceutical intermediate Cimetidine, and the design of a suitable crystallization process. 

NUCLEATION. Nucleation creates a new phase that is organizationally more related to the crystal lattice than to the monomeric species that undergoes crystallization. This process permits solutions that are of high relative supersaturation to crystallize and thereby reach equilibrium between liquid and sohd phases . Nucleation occurs when the local concentration of components that will comprise the solid phase exceeds a threshold level as a result of short-range concentration fluctuations in the bulk solution. In this respect, the kinetics of nucle-... 

In the liquid state sulfur and selenium are known to mix in all proportions. The provisional phase diagram shows an eutectic point at 40 mol-% of selenium (m.p. 105 °C). Mixtures with lower selenium content should show freezing points between 105 and 118 °C while those with higher selenium content are expected to have their freezing points at considerably higher temperatures. In practice equilibrium crystallization of the melt is hindered by supercooling and therefore only the melting points can be studied. 

Three-Phase Transformations in Binary Systems. Although this chapter focuses on the equilibrium between phases in binary component systems, we have already seen that in the case of a entectic point, phase transformations that occur over minute temperature fluctuations can be represented on phase diagrams as well. These transformations are known as three-phase transformations, becanse they involve three distinct phases that coexist at the transformation temperature. Then-characteristic shapes as they occnr in binary component phase diagrams are summarized in Table 2.3. Here, the Greek letters a, f), y, and so on, designate solid phases, and L designates the liquid phase. Subscripts differentiate between immiscible phases of different compositions. For example, Lj and Ljj are immiscible liquids, and a and a are allotropic solid phases (different crystal structures). 

Nucleation and Growth (Round 1). Phase transformations, such as the solidification of a solid from a liquid phase, or the transformation of one solid crystal form to another (remember allotropy ), are important for many industrial processes. We have investigated the thermodynamics that lead to phase stability and the establishment of equilibrium between phases in Chapter 2, but we now turn our attention toward determining what factors influence the rate at which transformations occur. In this section, we will simply look at the phase transformation kinetics from an overall rate standpoint. In Section 3.2.1, we will look at the fundamental principles involved in creating ordered, solid particles from a disordered, solid phase, termed crystallization or devitrification. 

Figure 6 shows the phase diagrams plotting temperature T vs c for PHIC-toluene systems with different Mw or N [64], indicating c( and cA to be insensitive to T, as is generally the case with lyotropic polymer liquid crystal systems. This feature reflects that the phase equilibrium behavior in such systems is mainly governed by the hard-core repulsion of the polymers. The weak temperature dependence in Fig. 6 may be associated with the temperature variation of chain stiffness [64]. We assume in the following theoretical treatment that liquid crystalline polymer chains in solution interact only by hardcore repulsion. The isotropic-liquid crystal phase equilibrium in such a solution is then the balance between S and Sor, as explained in the last part of Sect. 2.2. 

More than half a century ago, Bawden and Pirie [77] found that aqueous solutions of tobacco mosaic virus (TMV), a charged rodlike virus, formed a liquid crystal phase at as very low a concentration as 2%. To explain such remarkable liquid crystallinity was one of the central themes in the famous 1949 paper of Onsager [2], However, systematic experimental studies on the phase behavior in stiff polyelectrolyte solutions have begun only recently. At present, phase equilibrium data on aqueous solutions qualified for quantitative discussion are available for four stiff polyelectrolytes, TMV, DNA, xanthan (a double helical polysaccharide), and fd-virus. 

In a binary system more than two fluid phases are possible. For instance a mixture of pentanol and water can split into two liquid phases with a different composition a water-rich liquid phase and a pentanol-rich liquid-phase. If these two liquid phases are in equilibrium with a vapour phase we have a three-phase equilibrium. The existence of two pure solid phases is an often occuring case, but it is also possible that solid solutions or mixed crystals are formed and that solids exists in more than one crystal structure. 

In Chapter 3 we described the structure of interfaces and in the previous section we described their thermodynamic properties. In the following, we will discuss the kinetics of interfaces. However, kinetic effects due to interface energies (eg., Ostwald ripening) are treated in Chapter 12 on phase transformations, whereas Chapter 14 is devoted to the influence of elasticity on the kinetics. As such, we will concentrate here on the basic kinetics of interface reactions. Stationary, immobile phase boundaries in solids (e.g., A/B, A/AX, AX/AY, etc.) may be compared to two-phase heterogeneous systems of which one phase is a liquid. Their kinetics have been extensively studied in electrochemistry and we shall make use of the concepts developed in that subject. For electrodes in dynamic equilibrium, we know that charged atomic particles are continuously crossing the boundary in both directions. This transfer is thermally activated. At the stationary equilibrium boundary, the opposite fluxes of both electrons and ions are necessarily equal. Figure 10-7 shows this situation schematically for two different crystals bounded by the (b) interface. This was already presented in Section 4.5 and we continue that preliminary discussion now in more detail. 

Stability Limit 1, With the exception of helium and certain apparent exceptions discussed below. Fig. I gives a universal phase diagram liir all pure compounds The triple point of one P and one T is the single point at which all three phases, crystal, liquid, and gas. are in equilibrium. The triple point pressure is normally below atmospheric. Those substances, c.g.. CO . / - 3H85 mm. 7, = -5ft.fi C. for which it lies above, sublime without melting ai atmospheric pressure. 

However, the formal differences between microemulsions and macroemulsions are well defined. A microemulsion is a single, thermodynamically stable, equilibrium phase a macroemulsion is a dispersion of droplets or particles that contains two or more phases, which are liquids or liquid crystals (48). 

Partial Miscibility in the Solid State So far, we have described (solid + liquid) phase equilibrium systems in which the solid phase that crystallizes is a pure compound, either as one of the original components or as a molecular addition compound. Sometimes solid solutions crystallize from solution instead of pure substances, and, depending on the system, the solubility can vary from small to complete miscibility over the entire range of concentration. Figure 14.26 shows the phase diagram for the (silver + copper) system.22 It is one in which limited solubility occurs in the solid state. Line AE is the (solid -I- liquid) equilibrium line for Ag, but the solid that crystallizes from solution is not pure Ag. Instead it is a solid solution with composition given by line AC. If a liquid with composition and temperature given by point a is... 

A more complete procedure to compute the average thicknesses of the water and oil layers will be presented below. It is based on eq 3c, mass balances of components, and phase equilibrium equations. The calculations indicated that the interfacial tension of lamellar liquid crystals is very low, of the order of 10 5 N/m. It will be shown that y = 0 is always an excellent approximation of eq 3 c. 

In principle, the clinker liquid can behave in any of three ways during cooling, with intermediate possibilities. In the first, equilibrium between the liquid and the pre-existing solid phases (alite and belite) is continuously maintained. This implies the possibility of material transfer between these solids and the liquid in either direction for as long as any liquid remains. We shall call this situation equilibrium crystallization . In the second, the liquid does not crystallize but forms a glass. In the third, the liquid crystallizes independently, i.e. without interacting with the solid phases already present. Intermediate modes include, for example, equilibrium crystallization down to a certain temperature and independent crystallization below it. 

A mixed triacylglyceride system with N components crystallizing into P phases at equilibrium (a liquid phase and P - 1 solid phases) must satisfy the equality (Prausnitz 1986) ... 

A brief discussion of solid-liquid phase equilibrium is presented prior to discussing specific crystallization methods. Figures 22-1 and 22-2 illustrate the phase diagrams for binary solid-solution and eutec-... 

Two phases in contact with each other are said to be in equilibrium when the temperature, pressure, composition, and all other variables that characterize each phase do not change with time. Many chemical process operations—particularly separation processes such as distillation, absorption, crystallization, liquid extraction, and adsorption—work by distributing mixture components between two phases and then separating the phases. An essential step in analyzing such processes is determining how the feed mixture components distribute themselves between the two phases at equilibrium. This chapter summarizes common procedures for making this determination,... 

The first equation represents the equilibrium between hydrated Ag+ ions and Ag atoms in a single-crystal configuration. Alternatively, we may say that there is a heterogeneous thermodynamic equilibrium between Ag+ ions in the solid phase (where they are stabilized by the gas of free electrons) and Ag+ ions in the liquid phase (stabilized by interaction with water molecules). The forward reaction step corresponds to the anodic dissolution of a silver crystal. On an atomic level, one may say that a Ag" " core ion is transferred from the metallic phase to the liquid water phase. In an electrochemical cell, an electron flows from the Ag electrode (the working electrode) to the counter electrode each time that one Ag+ ion is transferred from the solid to the liquid phase across the electrochemical double layer. Although the electron flow is measured in the external circuit between the working... 

Phase Equilibrium in a Rigid-Chain Polymer-Solvent System 2.1 Lyotropic and Thermotropic Liquid Crystals... 

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