Thermodynamics between phases

The new phases were discovered by the combination of exploratory synthesis and a phase compatibility study. As commonly practised, the new studies were initially made through the chemical modification of a known phase. Inclusion of salt in some cases is incidental, and the formation of mixed-framework structures can be considered a result of phase segregation (for the lack of a better term) between chemically dissimilar covalent oxide lattices and space-filling, charge-compensating salts. Limited-phase compatibility studies were performed around the region where thermodynamically stable phases were discovered. Thus far, we have enjoyed much success in isolating new salt-inclusion solids via exploratory synthesis. 

The problem of transport of molecules through swollen gels is of general interest. It not only pertains to catalysis, but also to the field of chromatographic separations over polymeric stationary phases, where the partition of a solute between the mobile phase (liquid phase) and a swollen polymeric stationary phase (gel phase) is a process of the utmost importance. As with all the chemical and physicochemical processes, the thermodynamic and the kinetic aspect must be distinguished also in partition between phases. 

The discussion of moisture uptake by hygroscopic materials must include a description of the thermodynamics of vapor-liquid equilibria. For gas (g) and liquid (1) phases to be in equilibrium, the infinitesimal transfer of molecules between phases (dng and dn ) must lead to a free energy change of zero. 

On heating from a crystalline phase, DOBAMBC melts to form a SmC phase, which exists as the thermodynamic minimum structure between 76 and 95°C. At 95°C a thermotropic transition to the SmA phase occurs. Finally, the system clears to the isotropic liquid phase at 117°C. On cooling, the SmC phase supercools into the temperature range where the crystalline solid is more stable (a common occurrence). In fact, at 63°C a new smectic phase (the SmF) appears. This phase is metastable with respect to the crystalline solid such phases are termed monotropic, while thermodynamically stable phases are termed enantiotropic. The kinetic stability of monotropic LC phases is dependent upon purity of the sample and other conditions such as the cooling rate. However, the appearance of monotropic phases is typically reproducible and is often reported in the phase sequence on cooling. It is assumed that phases appearing on heating a sample are enantiotropic. 

While until now we have considered relatively simple phase diagrams and the fundamentals of the connection between phase diagrams and thermodynamics, we are here going to consider a somewhat more complex example, but only briefly. 

The only potential that varies significantly is the phase boundary potential at the membrane/sample interface EPB-. This potential arises from an unequal equilibrium distribution of ions between the aqueous sample and organic membrane phases. The phase transfer equilibrium reaction at the interface is very rapid relative to the diffusion of ions across the aqueous sample and organic membrane phases. A separation of charge occurs at the interface where the ions partition between the two phases, which results in a buildup of potential at the sample/mem-brane interface that can be described thermodynamically in terms of the electrochemical potential. At interfacial equilibrium, the electrochemical potentials in the two phases are equal. The phase boundary potential is a result of an equilibrium distribution of ions between phases. The phase boundary potentials can be described by the following equation ... 

When isotopes are fractionated kinetically during chemical reactions, the isotope ratio shift of the reaction products relative to the reactants often depends on reaction mechanisms and rates. This contrasts with isotopic fractionations between phases in isotopic equilibrium, where the isotopic differences are thermodynamic quantities and thus do not depend on reaction mechanisms or rates. In this section, we briefly review the well-developed theory for kinetic isotope effects that appears in the S isotope literature. This background serves as a guide for interpreting and predicting Se and Cr isotope systematics. 

The standard Gibbs transfer energy can only be found thermodynamically, for example, by using distribution equilibria, both for nonelectrolytes and for electrolytes as a whole. The distribution coefficient for substance X between phases a and /3 is given by the equation... 

Design of extraction processes and equipment is based on mass transfer and thermodynamic data. Among such thermodynamic data, phase equilibrium data for mixtures, that is, the distribution of components between different phases, are among the most important. Equations for the calculations of phase equilibria can be used in process simulation programs like PROCESS and ASPEN. 

Ganguly and Ruitz (1986) investigated the significance of closure temperature in terms of simple equilibrium thermodynamics. Assuming Rb-Sr exchanges between phases a and jS to be representable in terms of the equilibrium... 

Firstly, the optimisation shows how well the various thermodynamic quantities are matched and the excellent agreement with the experimentally observed phase diagram. It also shows a clear discrepancy between one set of experimental results and the optimised values for the mixing enthalpy in the liquid, emphasising the point that the combined thermodynamic and phase-diagram optimisation has been able to differentiate between conflicting experiments. 

If the electrostatic forces between charged surfactant head groups are sufficiently high, vesicles can also be a thermodynamically stable phase and be... 

We do not need to regard YBa2Cus07 as a solid solution, but it is not a thermodynamically stable phase at any temperature/pressure condition (11). Thermodynamic stability exists in the YBa2Cus06+x system in a certain region of temperature and oxygen pressure, but only for values of x between zero and about 0.6. Even these compositions appear not to be thermodynamically stable at room temperature and below. 

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. 

A change in size on scale-up is not the sole determinant of the seal-ability of a unit operation or process. Scalability depends on the unit operation mechanism(s) or system properties involved. Some mechanisms or system properties relevant to dispersions are listed in Table 2 (59). In a number of instances, size has little or no influence on processing or on system behavior. Thus, scale-up will not affect chemical kinetics or thermodynamics although the thermal effects of a reaction could perturb a system, e.g., by affecting convection (59). Heat or mass transfer within or between phases is indirectly affected by changes in size while convection is directly... 

The condition for thermodynamic equilibrium between phases is that the species chemical potentials are equal in each of the phases. Thus, at equilibrium,... 

Gas chromatography involves chemical equilibria between phases to bring about a particular separation. Thus, a brief discussion of phase equilibria is pertinent at this point. Phase equilibria separations can be understood with the use of the second law of thermodynamics. The phase rule states that if we have a system of C components which are distributed between. P phases, the composition of each of these phases will be completely defined by C-l concentration terms. Thus, to have the compositions of P phases defined it is necessary to have P(C-l) concentration terms. The temperature and pressure also are variables and are the same for all the phases. Assuming no other forces influence the equilibria it follows that. 

The thermodynamics of phase equilibria is reviewed in Chapter 17 and the fundamental thermodynamic differences between conserved and nonconserved order parameters are reinforced with a geometrical construction. These order parameters are used in the kinetic analyses of continuous and discontinuous phase transformations. 

O. Penrose and P.C. Fife. On the relation between the standard phase-field model and a thermodynamically consistent phase-field model. Physica D, 69(1-2) 107-113, 1993. 

Figure 1 shows changes in the system phase behavior as its HLB value is systematically adjusted. The left side of the diagram represents a two-phase system with micellar-solubilized oil in equilibrium with an excess oil phase (Winsor Type I) (Winsor 1954). The right side of the diagram represents a different two-phase system with reversed micellar-solubilized water. In-between these two systems a third phase coemerges which contains enriched surfactant with solubilized water and oil. This new thermodynamically stable phase is known as a Winsor Type HI middle phase microemulsion. 

Most speciation modelling is based on the assumption of thermodynamic equilibrium between phases, so it is necessary to describe the various equations that are used to quantify these chemical reactions. 

Thorstenson and Plummer (1977), in an elegant theoretical discussion (see section on The Fundamental Problems), discussed the equilibrium criteria applicable to a system composed of a two-component solid that is a member of a binary solid solution and an aqueous phase, depending on whether the solid reacts with fixed or variable composition. Because of kinetic restrictions, a solid may react with a fixed composition, even though it is a member of a continuous solid solution. Thorstenson and Plummer refer to equilibrium between such a solid and an aqueous phase as stoichiometric saturation. Because the solid reacts with fixed composition (reacts congruently), the chemical potentials of individual components cannot be equated between phases the solid reacts thermodynamically as a one-component phase. The variance of the system is reduced from two to one and, according to Thorstenson and Plummer, the only equilibrium constraint is IAP g. calcite = Keq(x>- where Keq(x) is the equilibrium constant for the solid, a function of... 

The FACTSage thermochemical database was used to identify the thermodynamically stable phases that could exist in a system comprised of a pure metal, oxygen, HC1 and Cl2 at 500°C (Suppiah, 2008). The predominant Fe, Ni, Cu and Cr phases in an 02/HCl/Cl2 environment were determined. The equilibrium reaction boundary was plotted as a function of the partial pressures of 02 and HC1, for a constant Cl2 partial pressure. The resulting predominance diagrams were plotted over an 02 and HC1 partial pressure range of 10-20 to 1 atm for Cl2 partial pressures between 10-6 and 1 atm. The predominant Ni and Cr species are solids, suggesting that a corrosion resistant protective layer could be formed on the metal. 

Experimentally it has been observed that the ratio of concentrations in 2 phases is constant if the concentrations of the chemical in both phases are sufficiently low (thermodynamic equilibrium). In this case, at equilibrium conditions the reversible distribution between phases can be described by a constant, which is known as the distribution coefficient ... 

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