Pure coexisting phases

Let us consider a rock at temperature T whose chemical composition q (recipe) is expressed as the vector of all the molar fractions x0 of s elements or oxides. It is assumed that it can be made by an arbitrarily large number p s of mineral phases exclusive of solid solution. B is the component matrix of these minerals for the selected set of elements or oxides. Let nj be the number of moles of mineral j and gj its Gibbs free energy of formation AGf T estimated when formed from either the elements or the oxides. The function to be minimized is the Gibbs free energy G given by 

In an s-dimensional space, s vectors at most can be independent. At equilibrium, a rock made of s elements cannot consist of more than s minerals, which implies that at least p—s of the p mole numbers are zero. In order to find the set of independent vectors that minimize the energy, we first rearrange the order of variables and split the vector n into two parts. The first part is the vector nB made of s base variables, and the second part is the vector F of (p —s) free variables. Provided the base variables are non-negative, the non-negativity constraints can be satisfied by setting the free variables to zero. For the vector n to be a feasible solution, it should also satisfy the recipe equation, i.e., 

For nF = 0, we immediately get the relationship between nB and q. We now want to change both uB and nF in a direction that decreases G. More precisely, we will exchange one free for one base variable at a time as long as the Gibbs free energy can be decreased. The last equation can be differentiated as 

The first element k of B reaching zero is found by selecting the smallest positive components of nB/ui (the ratio being understood as the vector obtained by element to element ratio). At this point, the fcth mineral of nB is exchanged with the ith mineral of nF, which simply amounts to exchanging their corresponding row in both B and g, and the calculation restarted from the beginning. The calculation stops when 8G cannot be decreased any further, i.e., when each component of gF — BfBb lgB is positive. 

The mole numbers of oxide minerals are obtained from equation (6.3.20) as 

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In equilibrium the chemical potential must be equal in coexisting phases. The assumption is that the solid phase must consist of one component, water, whereas the liquid phase will be a mixture of water and salt. So the chemical potential for water in the solid phase fis is the chemical potential of the pure substance. However, in the liquid phase the water is diluted with the salt. Therefore the chemical potential of the water in liquid state must be corrected. X refers to the mole fraction of the solute, that is, salt or an organic substance. The equation is valid for small amounts of salt or additives in general ... 

Here scalar order parameter, has the interpretation of a normalized difference between the oil and water concentrations go is the strength of surfactant and /o is the parameter describing the stability of the microemulsion and is proportional to the chemical potential of the surfactant. The constant go is solely responsible for the creation of internal surfaces in the model. The microemulsion or the lamellar phase forms only when go is negative. The function/(<))) is the bulk free energy and describes the coexistence of the pure water phase (4> = —1), pure oil phase (4> = 1), and microemulsion (< ) = 0), provided that/o = 0 (in the mean-held approximation). One can easily calculate the correlation function (4>(r)(0)) — (4>(r) (4>(0)) in various bulk homogeneous phases. In the microemulsion this function oscillates, indicating local correlations between water-rich and oil-rich domains. In the pure water or oil phases it should decay monotonically to zero. This does occur, provided that g2 > 4 /TT/o — go- Because of the < ), —<(> (oil-water) symmetry of the model, the interface between the oil-rich and water-rich domains is given by... 

Phase solubility analysis is a technique to determine the purity of a substance based on a careful study of its solubility behavior [38,39]. The method has its theoretical basis in the phase mle, developed by Gibbs, in which the equilibrium existing in a system is defined by the relation between the number of coexisting phases and components. The equilibrium solubility of a material in a particular solvent, although a function of temperature and pressure, is nevertheless an intrinsic property of that material. Any deviation from the solubility exhibited by a pure sample arises from the presence of impurities and/or crystal defects, and so accurate solubility measurements can be used to deduce the purity of the sample. 

The thermodynamic aspects of hydride formation from gaseous hydrogen are described by means of pressure-composition isotherms in equilibrium (AG = 0). While the solid solution and hydride phase coexist, the isotherms show a flat plateau, the length of which determines the amount of H2 stored. In the pure P-phase, the H2 pressure rises steeply vfith increase in concentration. The two-phase region ends in a critical point T, above which the transition from the a- to the P-phase is continuous. The equilibrium pressure peq as a function of temperature is related to the changes AH° and AS° of enthalpy and entropy ... 

The temperature dependence of the FE and glassy volume fractions was determined for the four different compositions mentioned above (Fig. 10). Whereas D-RADP-0.20 exhibits a quasi-continuous sequence of local PE-FE phase transitions with a coexistence range of about 20 K and a pure FE phase state throughout the whole crystal below 135 K [17], in D-RADP-0.25 part of the crystal remains in the PE or glass state, respectively, down to very low temperatures. This is also observed in D-RADP-0.30, while in D-RADP-0.35 (not shown) no FE polarization could be observed at all. 

The first of these can be recognized as the ordinary Clapeyron equation for a pure two-phase system (usually written for equimolar phases Af(1) = Af(2) = 1 cf. Sections 7.2.2 and 11.11), and the second is an analogous equation determining the slope of the coexistence curve in the pi-T plane. These equations in turn determine the slope of the coexistence line in the pi-P plane ... 

An example of this phase-coexistence is shown in Fig. 13. We will call this macroscopic phase boundary a bottleneck due to its shape. In fact, this coexistence includes three phases, i.e. swollen gel, shrunken gel and pure solvent phases surrounding the gel, and has been called triphasic equilibrium in the... 

Avery common occurrence is that, in the liquid phase, the components are completely miscible, whereas in the sohd phase, the components are only partially miscible, usually in small ranges around the pure components. This is illustrated in Fig. 10. Except for the single-phase sohd solution regions in the vicinity of the pure solid components, this diagram is similar to Fig. 10 of Chapter 8. It shows a eutectic, which freezes to a mixture of fine crystals of the two solid solutions. These three coexisting phases are represented by a horizontal line on the phase diagram. 

It is shown as a solid bar. This solid solution series coexists with metallic molybdenum, M02.06S3, and MoS2 (molybdenite). Copper coexists with a Cu-rich member i.e., pure X-phase) up to a Y-content of approximately 18%. Iron coexists with the solid solution series from 18 to 75% Y-content, whereas the remainder of the solid solution series forms divariant regions with both Fe and Fe3Mo2. Fe3Mo2 is also stable with Cu and with an iron-rich member of the bornite solid solution series. A large portion of the X—Y solid solution is in equilibrium with the metal-rich portion of the chalcocite-bornite solid solution series. The more Y-rich members are stable with pyrrhotite or FeS (ie., troilite). 

Since coexisting phases of saturated liquid and saturated vapor are in < librium, the equality of fugacities as expressed by Eqs. (11.22) and (11.24) is criterion of vapor/liquid equilibrium for pure species. 

As shown in Fig. 7.7, a pure perovskite phase was obtained after calcining the milled mixture at 850°C for 4 h in air. On the other hand, a cubic pyrochlore (P3N4) phase was predominant with the coexistence of perovsWte and PbO (litharge) on the calcined products from non-milled mixture. When MgO was used instead of Mg(OH)2, no significant change of... 

We briefly extend the preceding discussion to systems in which one or more pure condensed phases coexist with an ideal homogeneous mixture in gaseous, liquid, or solid form. It is now expedient to distinguish between pure condensed phases, subscript 5, and species involved in the solution, subscript i. For the reaction, written as -f- v,-A,- = 0 we write out the equilibrium condition as... 

Coexistence of binary systems. Coexisting phases are characterized by different figures of the order parameter M. In pure fluids, one identifies M with the density difference of the coexisting phases. In solutions, M is related to some concentration variable, where theory now advocates the number density or the closely related volume fraction [101]. At a quantitative level, these divergences are described by crossover theory [86,87] or by asymptotic scaling laws and corrections to scaling, which are expressed in the form of a so-called Wegner series [104], The two branches of the coexistence curve are described by... 

Solid lines represent the liquid-vapor phase equilibria of the two pure components that end in critical points marked by arrows. When a small amount of solvent is added to the pure pol3uner, the liquid-vapor coexistence shifts and so does the critical point. The loci of critical points for the binary system form a critical line that is shown by the dashed line with squares for = 1 and triangles for = 0.886. In the former case - phase behavior of t3q>e I -the critical line connects the critical points of the two pure components and the two coexisting phases gradually change from vapor and solvent-rich liquid... 

A different behavior is observed [76] for bilayers composed of partially miscible polymers below their critical temperature Tc. In this case two pure blend components interdiffuse until the equilibrium of two coexisting phases is established. The above equilibrium state is characterized by the coexistence compositions ( q and (]>2 and the interfacial width w. The relaxation of the initial interface between pure constituents involves two processes (see Fig. 3) ... 

Coexistence conditions of high polymer mixtures may be determined directly with the advent of the novel approach [74,75] focused on two coexisting phases confined in a thin film geometry and forming a bilayer morphology. Such equilibrium situation is obtained in the course of relaxation of an interface between pure blend components or in late stages of surface induced spinodal decomposition. It is shown that both methods lead to equivalent results [107] (Sect. 2.2.1). 

The phase diagram of the two-component eutectic system with the formation of a solid solution of one component has five planes (Figure 3.21). The plane L represents the region of homogeneous solution of components A and B. The plane Ass + L is the region of coexistence of crystals of the saturated solid solution of the component B in component A and the melt saturated with the component A. The plane B+L is the region of coexistence of the crystals of the pure solid phase B and the melt saturated with the component B. The plane Ass is the region of non-saturated solid solutions of the component B in component... 

In this section we consider how one uses an equation of state to identify the states of vapor-liquid equilibrium in a pure fluid. The starting point is the equality of molar Gibbs energies in the coexisting phases. 

Since the pure component liquid fugacities on both sides of the equation are the same, they cancel. (Why is this different from vapor-liquid equilibrium ) The compositions of the coexisting phases are the sets of mole fractions x, x, , xl, xf, x ,..., x that simultaneously satisfy Eqs. 11.2-2 and... 

Thus the ratio of the Henry s constant to the solvent fugacity must be evaluated in each coexisting phase to start the integration. This quantity can be measured in a simulation of pure component 1 by performing test identity changes of individual molecules from species 1 to species 2. Details of this calculation, as well as the limiting formulas appropriate to osmotic-ensemble integrations, have been presented elsewhere [17]. 

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