Phase equilibrium involving solids

The reasoning applies generally to (degenerate) N-phase equilibrium involving N mutually immiscible species. Whence the cited result for solids. 

Phase behavior involving solid-liquid equilibrium is the basis for crystallization in chemical and materials engineering. Binary mixture systems can have up to three degrees of freedom according to the Gibbs phase rule. 

This sugar-syrup example illustrates the principle of phase equilibrium using a liquid-solid system. In many metallurgical and materials systems of interest, phase equilibrium involves just solid phases. In this regard the state of the system is reflected in the characteristics of the microstructure, which necessarily include not only the phases present and their compositions, but, in addition, the relative phase amounts and their spatial arrangement or distribution. 

At high pressures, solid II can be converted (slowly) to solid III. Solid III has a body-centered cubic crystal structure. Line bd is the equilibrium line between solid II and solid III, while line be is the melting line for solid III.P A triple point is present between solid II, solid III, and liquid at point b. Two other triple points are present in this system, but they are at too low a pressure to show on the phase diagram. One involves solid II, liquid, and vapor while the other has solid I, solid II, and vapor in equilibrium. 

Provided the reaction is, in some sense, reversible, so that equilibrium can be attained, and provided the reactants and products arc all gas-phase, solution or solid-state species with well-defined free energies, it is possible to define the free energies for all such reactions under any defined reaction conditions with respect to a standard process this is conventionally chosen to be the hydrogen evolution/oxidation process shown in (1.11). The relationship between the relative free energy of a process and the emf of a hypothetical cell with the reaction (1.11) as the cathode process is given by the expression AC = — nFE, or, for the free energy and potential under standard conditions, AG° = — nFEl where n is the number of electrons involved in the process, F is Faraday s constant and E is the emf. 

Where applications to industrial combustion systems involve a relatively limited set of fuels, fire seeks anything that can bum. With the exception of industrial incineration, the fuels for fire are nearly boundless. Let us first consider fire as combustion in the gas phase, excluding surface oxidation in the following. For liquids, we must first require evaporation to the gas phase and for solids we must have a similar phase transition. In the former, pure evaporation is the change of phase of the substance without changing its composition. Evaporation follows local thermodynamics equilibrium between the gas... 

The fact that the curvature of the surface affects a heterogeneous phase equilibrium can be seen by analyzing the number of degrees of freedom of a system. If two phases a and are separated by a planar interface, the conditions for equilibrium do not involve the interface and the Gibbs phase rule as described in Chapter 4 applies. On the other hand, if the two coexisting phases a and / are separated by a curved interface, the pressures of the two phases are no longer equal and the Laplace equation (6.27) (eq. 6.35 for solids), expressed in terms of the two principal curvatures of the interface, defines the equilibrium conditions for pressure ... 

The worst deviations from the Clapeyron equation occur when one of the phases is a gas. This occurs because the volume of a gas depends strongly on temperature, whereas the volume of a liquid or solid does not. Accordingly, the value of A Vm is not independent of temperature when the equilibrium involves a gas. 

The purpose of this chapter is to outline the simplest methods of arriving at a description of the distribution of species in mixtures of liquids, gases and solids. Homogeneous equilibrium deals with single phase systems, such as electrolyte solutions (e.g., seawater) or gas mixtures (e.g., a volcanic gas). Heterogeneous equilibrium involves coexisting gaseous, liquid and solid phases. 

Esterification is the first step in PET synthesis but also occurs during melt-phase polycondensation, SSP, and extrusion processes due to the significant formation of carboxyl end groups by polymer degradation. As an equilibrium reaction, esterification is always accompanied by the reverse reaction being hydrolysis. In industrial esterification reactors, esterification and transesterification proceed simultaneously, and thus a complex reaction scheme with parallel and serial equilibrium reactions has to be considered. In addition, the esterification process involves three phases, i.e. solid TPA, a homogeneous liquid phase and the gas phase. The respective phase equilibria will be discussed below in Section 3.1. 

Transesterification is the main reaction of PET polycondensation in both the melt phase and the solid state. It is the dominant reaction in the second and subsequent stages of PET production, but also occurs to a significant extent during esterification. As mentioned above, polycondensation is an equilibrium reaction and the reverse reaction is glycolysis. The temperature-dependent equilibrium constant of transesterification has already been discussed in Section 2.1. The polycondensation process in the melt phase involves a gas phase and a homogeneous liquid phase, while the SSP process involves a gas phase and two solid phases. The respective phase equilibria, which have to be considered for process modelling, will be discussed below in Section 3.1. 

Gas-liquid relationships, in the geochemical sense, should be considered liquid-solid-gas interactions in the subsurface. The subsurface gas phase is composed of a mixture of gases with various properties, usually found in the free pore spaces of the solid phase. Processes involved in the gas-liquid and gas-solid interface interactions are controlled by factors such as vapor pressure-volatilization, adsorption, solubility, pressure, and temperature. The solubility of a pure gas in a closed system containing water reaches an equilibrium concentration at a constant pressure and temperature. A gas-liquid equilibrium may be described by a partition coefficient, relative volatilization and Henry s law. 

In contrast to the detonation of gaseous materials, the detonation process of explosives composed of energetic solid materials involves phase changes from solid to liquid and to gas, which encompass thermal decomposition and diffusional processes of the oxidizer and fuel components in the gas phase. Thus, the precise details of a detonation process depend on the physicochemical properties of the explosive, such as its chemical structure and the particle sizes of the oxidizer and fuel components. The detonation phenomena are not thermal equilibrium processes and the thickness of the reachon zone of the detonation wave of an explosive is too thin to identify its detailed structure.[i- i Therefore, the detonation processes of explosives are characterized through the details of gas-phase detonation phenomena. 

In applying equation 33, Cpsl (the constant-pressure molar heat capacity of the stoichiometric liquid) is usually extrapolated from high-temperature measurements or assumed to be equal to Cpij of the compound, whereas the activity product, afXTjafXT), is estimated by interjection of a solution model with the parameters estimated from phase-equilibrium data involving the liquid phase (e.g., solid-liquid or vapor-liquid equilibrium systems). To relate equation 33 to an available data base, the activity product is expressed... 

A general formulation of the problem of solid-liquid phase equilibrium in quaternary systems was presented and required the evaluation of two thermodynamic quantities, By and Ty. Four methods for calculating Gy from experimental data were suggested. With these methods, reliable values of Gy for most compound semiconductors could be determined. The term Ty involves the deviation of the liquid solution from ideal behavior relative to that in the solid. This term is less important than the individual activity coefficients because of a partial cancellation of the composition and temperature dependence of the individual activity coefficients. The thermodynamic data base available for liquid mixtures is far more extensive than that for solid solutions. Future work aimed at measurement of solid-mixture properties would be helpful in identifying miscibility limits and their relation to LPE as a problem of constrained equilibrium. 

The potential of supercritical extraction, a separation process in which a gas above its critical temperature is used as a solvent, has been widely recognized in the recent years. The first proposed applications have involved mainly compounds of low volatility, and processes that utilize supercritical fluids for the separation of solids from natural matrices (such as caffeine from coffee beans) are already in industrial operation. The use of supercritical fluids for separation of liquid mixtures, although of wider applicability, has been less well studied as the minimum number of components for any such separation is three (the solvent, and a binary mixture of components to be separated). The experimental study of phase equilibrium in ternary mixtures at high pressures is complicated and theoretical methods to correlate the observed phase behavior are lacking. 

Because the activity of condensed phases is not appreciably affected by the amount of the condensed phase or the pressure, equilibria involving condensed phases are achieved by variation of gas-phase concentrations. For example, in the equilibrium NH4Cl(solid) NH .(g,ls ) + HCl, if the amounts of NH3 and HC1... 

The mass action law assumes that the reaction medium is homogeneous. In heterogeneous reactions (involving different substances in multiple phases), the densities and effective concentrations of pure condensed phases (liquids or solids) are constant. The concentrations of such species are set to unity in the equilibrium constant expression for such reactions. For example, given the following decomposition,... 

If a saturated solution is cooled, the solubility of the solute generally decreases in order for the cooled solution to return to equilibrium, some solute must come out of solution as solid crystals. The crystallization rate may be slow, however, so that a metastable condition can exist in which the concentration of the solute is higher than the equilibrium value at the solution temperature. Under such conditions, the solution is said to be supersaturated and the difference between actual and equilibrium concentrations is referred to as supersaturation. Ail problems involving solid-liquid separations in this text assume that equilibrium exists between the solid and liquid phases, so that supersaturation need not be considered. 

---