Ideal state diagrams

The translationally ordered state characteristic of crystallites that are large with respect to the wavelengths of X-rays represents perfect phases in the sense of thermodynamics, to which phase diagrams apply. Even this idealized state of matter cannot exist without deviations from perfect ordering, however, because of a requirement of thermodynamics (the entropy of the material at equilibrium must be nonzero for the Gibbs free energy to be minimized). Thus, the material will contain a number of deviations from the ideal arrangement, called defects."... 

Figure 6.13 Idealized example of a part of a state diagram of polymer solutions. tp — net polymer volume fraction, fi — excluded volume parameter, and = correlation length. The critical point for phase separation is denoted by . See text.

The P/Vyr behavior of gases can be described mathematically by means of thermal state diagrams [Gmehling 1992]. The oldest and simplest relationship is the ideal gas law (see Chapter 3). A series of state equations are available for describing real gases, for example ... 

We say a state is well-defined when sufficient property values are specified to locate a system on its state diagram. If, in a well-defined state, the system is at equilibrium, then the condition is said to be an equilibrium state. Consequently, all equilibrium states are well-defined, but well-defined states need not be equilibrium states. In fact, a well-defined state may not be physically realizable—it may be thermodynamically unstable or hypothetical or an idealization. For example, many well-defined states of an ideal gas cannot be realized in a laboratory nevertheless, thermodynamic analyses can be performed on such hypothetical systems. 

Equations (2.1.14) and (2.1.15) are idealizations that are never obeyed exactly by real systems. A reversible change is not a realizable process, it is merely a sequence of equilibrium states on a state diagram (see 1.3 and the Example in 2.1.3). 

Figure 1.5. Concentration dependence of the chemical potential of mixing Ap,f RT (a), the molar Gibbs potential of mixing AGm/RT (6) for regular mixtures with ajRT (the digits at the curves) = 3, 2, 2,1 (endothermic mixtures), 0 (ideal mixture), —I (exothermic mixture). stands for dApifdxj. State diagram (c) bi—binodal, sp—spinodal, C—critical point. ajRTc = 2 at T = Tc...

The best-known examples of phase transition are the liquid-vapour transition (evaporation), the solid-liquid transition (melting) and the solid-vapour transition (sublimation). The relationships between the phases, expressed as a function of P, V and T consitute an equation of state that may be represented graphically in the form of a phase diagram. An idealized example, shown in figure 1, is based on the phase relationships of argon [126]. 

Positive deviations from ideal behaviour for the solid solution give rise to a miscibility gap in the solid state at low temperatures, as evident in Figures 4.10(a)-(c). Combined with an ideal liquid or negative deviation from ideal behaviour in the liquid state, simple eutectic systems result, as exemplified in Figures 4.10(a) and (b). Positive deviation from ideal behaviour in both solutions may result in a phase diagram like that shown in Figure 4.10(c). 

Negative deviation from ideal behaviour in the solid state stabilizes the solid solution. 2so1 = -10 kJ mol-1, combined with an ideal liquid or a liquid which shows positive deviation from ideality, gives rise to a maximum in the liquidus temperature for intermediate compositions see Figures 4.10(h) and (i). Finally, negative and close to equal deviations from ideality in the liquid and solid states produces a phase diagram with a shallow minimum or maximum for the liquidus temperature, as shown in Figure 4.10(g). 

Fig. 1 Schematic drawing to show the concept of system dimensionality (a) bulk semiconductors, 3D (b) thin film, layer structure, quantum well, 2D (c) linear chain structure, quantum wire, ID (d) cluster, colloid, nanocrystal, quantum dot, OD. In the bottom, it is shown the corresponding density of states [A( )] versus energy (E) diagram (for ideal cases).

Fig. 17 Schematic diagram showing the molecular packing and topochemical polymerization of diacetylenes [30] in crystals. There are two idealized packing arrangements either (a), (b) or (c), (d) in crystals and each has two modes of polymerization. The polymerization in the solid state is said to occur smoothly when, v = 240 400 pm and y = 45° (Baughman, 1974 Baughman and Yee, 1978).

In Chapter 13 we discussed briefly the solid-liquid equilibrium diagram of a feldspar. Feldspar is an ideal, solid solution of albite (NaAlSiaOg) and anorthite (CaAlSi20g) in the solid state as well as an ideal, liquid solution of the same components in the molten state. The relationships that we have developed in this chapter permit us to interpret the feldspar phase diagram (Figure 13.4) in a quantitative way. 

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