Complete Phase Diagram

The other phase transitions of a pure substance can also be represented in a p(J) 

This diagram shows the conditions of temperature and pressure under which a 

Below a certain temperature, a condensate forms as a solid and not a liquid. The direct transition from a gaseous to a solid state, or vice versa, is called sublimation and the vapor pressure curve of a solid substance, the sublimation (pressure) curve (Fig. 11.9). Gas and solid are in equilibrium with each other along this phase boundary. As long as pressure remains below this curve at constant temperature during the compression of the gas, only gas will be present in the cylinder. However, if the pressure lies above this, only a solid condensate will appear. 

The almost vertical curve in the illustration (Fig. 11.10), called the melting (pressure) curve, separates the regions at which solid or liquid condensate is stable. The slope of this curve has been greatly exaggerated for the sake of clarity. 

Like the boiling point of a substance, its melting point is not a constant but, to a much lesser extent, also depends upon pressure. The standard melting point Tfi is based upon a pressure of 100 kPa. 

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For Fc = 0 the MR model is mapped onto the DM model (see, e.g.. Fig. 2). The presence of a very small amount of C2 species in the gas phase causes a drastic shift of the first-order IPT with respect to its position in the ZGB phase diagram [96]. Fig. 18 shows the complete phase diagram... 

Of course, LC is not often carried out with neat mobile-phase fluids. As we blend solvents we must pay attention to the phase behavior of the mixtures we produce. This adds complexity to the picture, but the same basic concepts still hold we need to define the region in the phase diagram where we have continuous behavior and only one fluid state. For a two-component mixture, the complete phase diagram requires three dimensions, as shown in Figure 7.2. This figure represents a Type I mixture, meaning the two components are miscible as liquids. There are numerous other mixture types (21), many with miscibility gaps between the components, but for our purposes the Type I mixture is Sufficient. 

By way of example, the Cu-Zn phase diagram shown in Fig. 20.42 exhibits a number of different intermediate phases (j8, 7, 6, etc.) and a number of peritectic reactions and a eutectoid reaction. In many instances it is not necessary to consider a complete phase diagram. Thus Fig. 20.43 illustrates the Al-rich end of the Al-Cu phase diagram and is used below in a discussion... 

Most methods for the determination of phase equilibria by simulation rely on particle insertions to equilibrate or determine the chemical potentials of the components. Methods that rely on insertions experience severe difficulties for dense or highly structured phases. If a point on the coexistence curve is known (e.g., from Gibbs ensemble simulations), the remarkable method of Kofke [32, 33] enables the calculation of a complete phase diagram from a series of constant-pressure, NPT, simulations that do not involve any transfers of particles. For one-component systems, the method is based on integration of the Clausius-Clapeyron equation over temperature,... 

Figure 4.1 The NaF-ZnF2 phase diagram (a) the results of an X-ray investigation into the phases present at 600°C as a function of composition and (b) the complete phase diagram. [Redrawn from B. O. Mysen, Phase Diagrams for Ceramists, Vol VUE, American Ceramic Society, Westerville, Ohio, 1990, p. 337.]...

Also, the phases formed in the course of discharge of an electrode with three or more components may be readily detected by reading the equilibrium cell voltage. As an example, the determination of the quite complex ternary phase diagram of the system Li-In-Sb is shown in Fig. 8.9. In this case, plateaux are observed in the presence of three-phase equilibria. In order to obtain the complete phase diagram it is necessary... 

To determine the complete phase diagram of a ternary system as a function of temperature, at least a three-dimension diagram is necessary. Such diagrams are unfortunately quite difficult to visualize and it is often preferable to reduce the diagrams to two dimensions by keeping the concentration of some of the components constant. Some results for the BE-H2O phase diagram as a function of temperature for fixed quantities of DEC are shown in Fig. 4. In this diagram the mole fraction of BE refers to the binary system... 

Figure 7.14 Complete phase diagram of the water + p-lactoglobulin + gum arabic system at 20 °C and pH = 4.2. Features indicated are , tie-lines , binodal points I, one-phase region II, two-phase region. Reproduced from Schmitt et al. (1999) with permission.

The change of lyotropic l.c. behavior of a monomer in the hexagonal phase by polymerization has been described for the first time by Friberg et al. I07,108). A change from the hexagonal monomer phase to the lamellar phase of the polymer was observed and carefully identified. Complete phase diagrams of monomer and polymer, however, were not compared. 

Table 2.1 lists equilibrium ratios for the reduction of selected metal oxides [4], while Figure 2.2 provides a complete phase diagram for the reduction of iron oxide at different temperatures [3, 5], In order to reduce bulk iron oxide to metallic iron at 600 K, the water content of the hydrogen gas above the sample must be below a few percent, which is easily achieved. However, in order to reduce Cr2C>3, the water content should be as low as a few parts per billion, which is much more difficult to realize. The data in Table 2.1 also illustrate that, in many cases, only partial reduction to a lower oxide may be expected. Reduction of Mn2C>3 to MnO is thermodynamically allowed at relatively high water contents, but further reduction to manganese is unlikely. 

Urusova MA, Valyashko VM. Tricritical phenomena in the NaCl-Na2B407-H20 system and the transformation laws of complete phase diagrams. Russ J Inorg Chem 1998 43 948-955. 

It is essential to realize that any thermodynamic evaluation of this solubility "maximum" with standard reference conditions in the form of the three pure components in liquid form is a futile exercise. The complete phase diagram. Fig. 2, shows the "maximum" of the solubility area to mark only a change in the structure of the phase in equilibrium with the solubility region. The maximum of the solubility is a reflection of the fact that the water as equilibrium body is replaced by a lamellar liquid crystalline phase. Since this phase.transition obviously is more. related to packing constraints — than enthalpy of formation — a view of the different phases as one continuous region such as in the short chain compounds water/ethanol/ethyl acetate. Fig. 3, is realistic. The three phases in the complete diagram. Fig. 2, may be perceived as a continuous solubility area with different packing conditions in different parts (Fig. 4). 

At present, inclusion of the hquid into such calculations requires use of fitted free energy curves, such as those used in the more empirical approach described below, but as liquid alloy theory advances, this situation will improve. Thus, complete phase diagram calculations based on pseudopotential calcnlations have been presented by Hafher. ... 

In addition to the importance of the M41S materials for size- and shape-selective applications, these materials have been also regarded as a suitable mesoporous model adsorbent for testing theoretical predictions of pore condensation. Pore condensation represents a first order phase transition from a gas-like state to a liquid-like state of a pore fluid in presence of a bulk fluid reservoir, which occurs at a pressure p less than the saturation pressure po at gas-liquid coexistence of the bulk fluid [6,7]. In this sense pore condensation can be regarded as a shifted gas-liquid bulk phase transition due to confinement of a fluid to a pore. Recent work has shown that in fact the complete phase diagram of the confined fluid is shifted to lower temperature and higher mean density as compared with the bulk coexistence curve [e.g., 8,9]. 

In addition to ij, and there are a number of crystalline polymorphs stable only under pressure, though the complete phase diagram is not yet established (Fig. 15.1). There are five distinct structures (differing in arrangement of O atoms) and there are low-temperature forms of two of them the numbering is now unfortunately unsystematic ... 

Thus I can construct a complete phase diagram for the mixture using only vapor-pressure data for the pure components. 

Extrapolation of Equation 1 to Vni=100 corresponds to T=1360°C for the liquidus. Extrapolation of Equation 3 to Vm=0 yields T=1119°C for the disappearance of liquid, the solidus temperature. This is a complex system for which complete phase diagrams are not available pseudoternary diagrams such as those presented by Grove et a l. (2) for similar compositions are generally applicable to this composition. 

A more detailed representation of phase equilibrium in a pure fluid, including the presence of a single solid phase, js given in the three-dimensional PVT phase diagram of Fig. 7.3-5. Such complete phase diagrams are rarely available, although data may be available in the form of Fig. 7.3-4, which is a projection of the more complete diagram onto the P-V plane, and Fig. 7.3-6, which is the projection onto the P-T plane. 

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