Binary solid solutions

A brief discussion of sohd-liquid phase equihbrium is presented prior to discussing specific crystalhzation methods. Figures 20-1 and 20-2 illustrate the phase diagrams for binary sohd-solution and eutectic systems, respectively. In the case of binary solid-solution systems, illustrated in Fig. 20-1, the liquid and solid phases contain equilibrium quantities of both components in a manner similar to vapor-hquid phase behavior. This type of behavior causes separation difficulties since multiple stages are required. In principle, however, high purity... 

The dominant mechanism of purification for column crystallization of solid-solution systems is recrystallization. The rate of mass transfer resulting from recrystallization is related to the concentrations of the solid phase and free liquid which are in intimate contact. A model based on height-of-transfer-unit (HTU) concepts representing the composition profile in the purification section for the high-melting component of a binary solid-solution system has been reported by Powers et al. (in Zief and Wilcox, op. cit., p. 363) for total-reflux operation. Typical data for the purification of a solid-solution system, azobenzene-stilbene, are shown in Fig. 20-10. The column crystallizer was operated at total reflux. The solid line through the data was com-putecfby Powers et al. (op. cit., p. 364) by using an experimental HTU value of 3.3 cm. 

Models can be constructed in this manner (e.g., Bourcier, 1985), but most modelers choose for practical reasons to consider only minerals of fixed composition. The data needed to calculate activities in even binary solid solutions are, for the... 

The distribution of componentsof binary solid solutions over the solid phase and the aqueous phase has been studied for a number of systems. Table I contains a summary of some of these systems with references. This literature review is not complete more data are available especially for rare earth and actinide compounds, which primarily obey type I Equations to a good approximation. In the following sections, the theory above will be applied to some special systems which are relevant to the fields of analytical chemistry, inorganic chemistry, mineralogy, oceanography and biominerals. 

Figure 8. Free enthalpy of mixing G of binary solid solutions (regular) Ca,. (PO ) OH F as a function of x at W/2.303 RT = 1.4.

By examining the compositional dependence of the equilibrium constant, the provisional thermodynamic properties of the solid solutions can be determined. Activity coefficients for solid phase components may be derived from an application of the Gibbs-Duhem equation to the measured compositional dependence of the equilibrium constant in binary solid solutions (10). 

Example 4.1. Suppose olivine and garnet are in contact and olivine is on the left-hand side (x<0). Ignore the anisotropic diffusion effect in olivine. Suppose Fe-Mg interdiffusion between the two minerals may be treated as one dimensional. Assume olivine is a binary solid solution between fayalite and forsterite, and garnet is a binary solid solution between almandine and pyrope. Hence, Cpe + CMg= 1 for both phases, where C is mole fraction. Let initial Fe/(Fe- -Mg) = 0.12 in olivine and 0.2 in garnet. Let Xq = (Fe/Mg)gt/ (Fe/Mg)oi = 3, >Fe-Mg,oi = 10 ° mm+s, and Dpe-Mg,gt =... 

Spinodal decompositions, often observed in binary solid solutions of metals and in glasses, on the other hand, arise from thermodynamic instabilities caused by composition (Cahn, 1968). A special feature of this type of solid state transformation is the absence of any nucleation barrier. There is a class of transformation called eutectoid decomposition in which a single phase decomposes into two coupled phases of different compositions, the morphology generally consisting of parallel lamellae or of rods of one phase in the matrix of the other. 

A hydride formed in the reaction of a binary solid-solution alloy with hydrogen can be considered as a solid solution of two binary hydrides and will have properties related to the properties of the constituent binary hydrides. An intermetallic-compound hydride, however, formed in accordance with Reaction... 

Defect thermodynamics is more complicated when applied to binary (or multi-component) compound crystals. For binary systems, there is one more independent thermodynamic variable to control. In the case of extended binary solid solutions, one would normally choose a composition variable for this purpose. For compounds with very narrow ranges of homogeneity (i.e., point defect concentrations), however, the composition is obviously not a convenient variable. The more natural choice is the chemical potential of one of the two components of the compound crystal. In practice one will often use the vapor pressure ( activity) of this component. 

In many cases, p is rather insensitive to the composition (NAO) because both A21 and B2+ are rendered mobile by the same vacancies in the same sublattice. In deriving Eqn. (8.11), we have assumed that (A, B)0 is an ideal quasi-binary solid solution. Analogous to Eqn. (8.6), Eqn. (8.11) has to be integrated under the restricting condition of the conservation of cation species A and B. There is no analytical solution to this problem, but a numerical solution has been presented in [H. Schmalzried, et al. (1979)]. 

Long-range order is possible in some binary solid solutions having compositions corresponding to simple ratios of the number of A and B atoms. One species of atom may tend to occupy certain lattice positions. Figure 8.1 shows several ordered structures and Table 8.1 lists compositions that can form these ordered structures. 

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... 

Chalcocite forms a limited binary solid solution series (Cu2 S) extending into the ternary diagram beyond a composition of bornite. The central portion of the system is dominated by the biconvex chalcopyrite stability field. Pyrrhotite displays binary as well as ternary solid solubility, in both cases >4 wt.%, whereas pyrite has... 

Equation 11.32 is used to model a single-phase liquid in a ternary system, as well as a ternary substitutional-solid solution formed by the addition of a soluble third component to a binary solid solution. The solubility of a third component might be predicted, for example, if there is mutual solid solubility in all three binary subsets (AB, BC, AC). Note that Eq. 11.32 does not contain ternary interaction terms, which ate usually small in comparison to binary terms. When this assumption cannot, or should not, be made, ternary interaction terms of the form xaXbXcLabc where Labc is an excess ternary interaction parameter, can be included. There has been httle evidence for the need of terms of any higher-order. Phase equilibria calculations are normally based on the assessment of only binary and ternary terms. 

A brief discussion of solid-liquid phase equilibrium is presented prior to discussing specific crystallization methods. Figures 22-1 and 22-2 illustrate the phase diagrams for binary solid-solution and eutec-... 

The central problem in carrying out quantitative analysis of a mixture of phases by X-ray powder diffractometry is that the intensity of the peaks is not linear with concentration. For quantitative analysis of binary solid solutions, see the section on solid solutions. Here we concern ourselves with mixtures of two or more phases. In such cases the intensity depends upon the absorption of the sample. We assume that the two phases have already been identified from prior knowledge or from the powder pattern. The expression for the intensity of a reflection from a set of parallel planes hkl, hu is a very complex equation. For our purpose we will use a simplified form. i... 

Kirkwood, J. G., Order and disorder in binary solid solutions. J. Chem. Phys. 6, 70 (1938). 

The thermodynamic relationships for equilibrium between a binary solid solution and a binary liquid solution are given by... 

Similarly, in the case of equilibrium between a binary solid solution (mixed crystals) and liquid, if the liquidus and solidus curves meet, then this point must be at an extreme value of the temperature of coexistence. Conversely, if the two curves pass through an extreme value, this point must be common to the two curves and the two phases have the same composition. 

The lattice parameter of a binary solid solution of B in A depends only on the percentage of B in the alloy, as long as the solution is unsaturated. This fact can be made the basis for chemical analysis by parameter measurement. All that is needed is a parameter vs. composition curve, such as curve be of Fig. 12-8(b), which can be established by measuring the lattice parameter of a series of previously analyzed alloys. This method has been used in diffusion studies to measure the change in concentration of a solution with distance from the original interface. Its accuracy depends entirely on the slope of the parameter-composition curve. In alpha brasses, which can contain from 0 to about 40 percent zinc in copper, an accuracy of +1 percent zinc can be achieved without difficulty. 

Assume that Henry s law applies to minor component B and Raoult s law to major component A of binary solid solution, Aand that A and B have the same charge. Given also that = 10, and that the solubility products of pure AX and BX are = to and K p BX) = 10"and that the composi-... 

Kittaka, S., Isoelectric point of ALO, CisO, and FejO,. IL Binary solid solution, J. Colloid Intetf. Sci., 48, 334, 1974. 

Fig. 2.25. Polymorphous Clapeyron diagram in T-P-c space for a binary solid solution displaying three isothermal sections to the surface T(P, c)...

The number of solid state energy transfer studies is smaller than in previous years. Energy diffusion in binary solid solutions of 1,4-dibromonaphthalene and 1-brorao-4-chloronaphthalene is one... 

The effect of substitutional impurities on the stability and aqueous solubility of a variety of solids is investigated. Stoichiometric saturation, primary saturation and thermodynamic equilibrium solubilities are compared to pure phase solubilities. Contour plots of pure phase saturation indices (SI) are drawn at minimum stoichiometric saturation, as a function of the amount of substitution and of the excess-free-energy of the substitution. SI plots drawn for the major component of a binary solid-solution generally show little deviation from pure phase solubility except at trace component fractions greater than 1%. In contrast, trace component SI plots reveal that aqueous solutions at minimum stoichiometric saturation can achieve considerable supersaturation with respect to the pure trace-component end-member solid, in cases where the major component is more soluble than the trace. 

Thermodynamic equilibrium in a system with a binary solid-solution Bx.xCxA can be defined by the law-of-mass-action equations ... 

Table I. Estimated Mixing Parameters for Binary Solid-Solutions at 25 C...

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