Activity from solid solution composition

Finally, it is not appropriate to derive thermodynamic properties of solid solutions from experimental distribution coefficients unless it can be shown independently that equilibrium has been established. One possible exception applies to trace substitution where the assumptions of stoichiometric saturation and unit activity for the predominant component allow close approximation of equilibrium behavior for the trace components (9). The method of Thorstenson and Plummer (10) based on the compositional dependence of the equilibrium constant, as used in this study, is well suited to testing equilibrium for all solid solution compositions. However, because equilibrium has not been found, the thermodynamic properties of the KCl-KBr solid solutions remain provisional until the observed compositional dependence of the equilibrium constant can be verified. One means of verification is the demonstration that recrystallization in the KCl-KBr-H20 system occurs at stoichiometric saturation. 

C, 1 bar is 10 bar. The pyrrhotite in this equilibrium is Feo.gaS, which may be considered as a solid solution composition in the system FeS — Sa. The activity of FeS in this pyrrhotite is 0.46 based on a standard state of pure stoichiometric FeS at the same P and T. The pyrite is pure stoichiometric FeSa. Calculate ArG° for the reaction forming pyrite from pyrrhotite and Sa gas at this P, T. 

The partial molar entropy of a component may be measured from the temperature dependence of the activity at constant composition the partial molar enthalpy is then determined as a difference between the partial molar Gibbs free energy and the product of temperature and partial molar entropy. As a consequence, entropy and enthalpy data derived from equilibrium measurements generally have much larger errors than do the data for the free energy. Calorimetric techniques should be used whenever possible to measure the enthalpy of solution. Such techniques are relatively easy for liquid metallic solutions, but decidedly difficult for solid solutions. The most accurate data on solid metallic solutions have been obtained by the indirect method of measuring the heats of dissolution of both the alloy and the mechanical mixture of the components into a liquid metal solvent.05... 

Substitutional Disorder In Regular Solid Solutions. Most simple ionic solutions in which substitution occurs in one sublattice only are not ideal, but regular 2, J3) Most complex ionic solid solutions in which substitution occurs in more than one sublattice are not only regular in the sense of Hildebrand s definition (15) but also exhibit substitutional disorder. The Equations describing the activities of the components as a function of the composition of their solid solutions are rather complex ( 7, V7, 1 ), and these can be evaluated best for each individual case. Both type II and type III distributions can result from these conditions. 

Calculation of the extreme values of the activities at the spinodal compositions xgp for variable values of W/2.303 RT results in the data presented in Figure 10. It appears that values as high as log ap/ = 2 are reached in the range xgp >0.63. Thus, the assumption of a subregular behavior of the solid solutions of OHA and FA explains the observed solubility behavior qualitatively. It follows further from the calculations that W/2.303 RT. 8 so that W, > 4.6 104 J mol"1. 

Thermodynamic calculations based on the compositional dependence of the equilibrium constant are applied to solubility data in the KCl-KBr-H20 system at 25°C. The experimental distribution coefficient and activity ratio of Br /Cl in solution is within a factor of two of the calculated equilibrium values for compositions containing 19 to 73 mole percent KBr, but based on an assessment of uncertainties in the data, the solid solution system is clearly not at equilibrium after 3-4 weeks of recrystallization. Solid solutions containing less than 19 and more than 73 mole percent KBr are significantly farther from equilibrium. As the highly soluble salts are expected to reach equilibrium most easily, considerable caution should be exercised before reaching the conclusion that equilibrium is established in other low-temperature solid solution-aqueous solution systems. 

Although nearly identical solid-aqueous solution compositions are observed in recrystallization from two directions under conditions of total constant composition, this alone is insufficient proof of the establishment of equilibrium. In order to test for equilibrium, the solid solution activity coefficients must be determined and used to compare observed solid and aqueous solution compositions with the appropriate values expected at equilibrium. 

This permits provisional calculation of the compositional dependence of the equilibrium constant and determination of provisional values of the solid phase activity coefficients (discussed below). The equilibrium constant and activity coefficients are termed provisional because it is not possible to determine if stoichiometric saturation has been established without independent knowledge of the compositional dependence of the equilibrium constant, such as would be provided from independent thermodynamic measurements. Using the provisional activity coefficient data we may compare the observed solid solution-aqueous solution compositions with those calculated at equilibrium. Agreement between the calculated and observed values confirms, within the experimental data uncertainties, the establishment of equilibrium. The true solid solution thermodynamic properties are then defined to be equal to the provisional values. 

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). 

Fig. 2. Logarithmic activity diagram depicting equilibrium phase relations among aluminosilicates and sea water in an idealized nine-component model of tire ocean system at the noted temperatures, one atmosphere total pressure, and unit activity of H20. The shaded area represents (lie composition range of sea water at the specified temperature, and the dot-dash lines indicate the composition of sea water saturated with quartz, amotphous silica, and sepiolite, respectively. The scale to the left of the diagram refers to calcite saturation foi different fugacities of CO2. The dashed contours designate the composition (in % illite) of a mixed-layer illitcmontmorillonitc solid solution phase in equilibrium with sea water (from Helgesun, H, C. and Mackenzie, F T.. 1970. Silicate-sea water equilibria in the ocean system Deep Sea Res.).

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. 

As more salt is added, excess salt is present in the solid phase and the solution composition is invariant. Therefore, the pH is constant and the product of the cation and anion activities equals the solubility product, as deLned in Equation 15.5, in the absence of cation or anion from other sources including molecular complex forms (Amis, 1983). At this point, more salt will not dissolve, and the salt concentration represents the solubility of the drug in the speciLc salt form. To conLrm that the salt solubility has been reached, it should be veriLed (Anderson and Conradi, 1985) that the solid salt phase in equilibrium with the solution has not been contaminated with the uncharged form precipitate. 

The mechanism of action of some Co porphyrins has been investigated [339]. It has been suggested that Co (I) is responsible for the reduction of water so that a sort of surface redox catalysis is operative. The principle of the activity is not very different from that of composite solid materials, e.g., oxides. Various solvents have been tested, and it has been found that the catalyst is especially active in neutral solution since the redox potential of the Co(II)/Co(I) couple approaches the potential of the H+/H2 couple. 

Nonstoichiometric solid solutions are substances whose composition approximates that of stoichiometric compounds, but which have a range of compositions. The problem of applying thermodynamics to such substances is primarily how to express the composition of the solution. The simplest choice would be to use the mole fractions or atom fractions in terms of the components. In such a case the effects of the formation of the compound from the components would be contained in the values of the activity coefficients or excess chemical potentials. 

A basic premise of solubility considerations is that a solution in contact with a solid can be in an equilibrium state with that solid so that no change occurs in the composition of solid or solution with time. It is possible from thermodynamics to predict what an equilibrium ion activity product should be for a given mineral for a set of specified conditions. As will be shown later in this chapter, however, it is not always possible to obtain a solution of the proper composition to produce the equilibrium conditions if other minerals of greater stability can form from the solution. It shall also be shown that while it is possible to calculate what mineral should form from a solution based on equilibrium thermodynamics, carbonate minerals usually behave in a manner inconsistent with such predictions. 

One of the exceptions was the discovery of high ionic conductivity in appropriately doped FaGa03.128 129 As in the other oxide ion conductors, its ionic conductivity depends on both the dopant level as well as on the nature of the dopant. A major difference to ceria and zirconia is the presence of two cations that can be substituted the detailed defect chemistry of such solid solutions is far from being fully understood. Co-doping of Sr on A sites and Mg on B-sites leads to an ionic conductivity of ca. 0.12—0.17 S cm 1 at 800°C,130-133 which is similar to doped ceria but considerably exceeds the value of YSZ (ca. 0.03 S cm 1 at 800°C80 81). The activation energy also varies with composition and can be as low as ca. 0.6 eV.130 131 At about 600-700°C, the... 

From the R and C values of the time constants a-c in the model, it was possible to estimate the thickness and resistivity of layers comprising the compact part of the surface films. The temperature dependence of these three time constants (e.g., linear Arrhenius plots for the different resistivities calculated that reflect different activation energy for Li+ ion migration in each layer), as well as their dependence on the solution composition and the experimental conditions, revealed that the model has a solid physicochemical ground [48,49,186],... 

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