Solution phase models ideal solutions

In the pseudo-phase model, ideal solutions are assumed2 so that 72 = 1 and 02 — am. (18.71)... 

Liquid-State Activity-Coefficient Models. If the conditions of the unit operation are far from the critical region of the mixture or that of the major conponent and if experimental data are available for the phase equilibrium of interest (VLE or LEE), then a liquid-state activity-coefficient model is a reasonable choice. Activity coefficients (y,) correct for deviations of the liquid phase from ideal solution behavior, as shown in Equation fl3.ll. 

In this section, we explore how we can calculate fugacity in the liquid phase. In doing so, we will develop an appropriate reference state for the liquid phase, the ideal solution, and then correct for real behavior through the activity coefficient. We will then learn a type of empirical model that correlates experimental data efficiently. The methodology developed in this section can also be applied to solids. Finally, an alternative approach to calculating fugacity in the liquid using PvT equations of state is discussed. This approach is similar to that used for the vapor phase in Section 7.3. 

The combination of non-ideal phase behaviour of solutions, the non-linearity of particle formation kinetics, the multi-dimensionality of crystals, their interactions and difficulties of modelling, instrumentation and measurement have conspired to make crystallizer control a formidable engineering challenge. Various aspects of achieving control of crystallizers have been reviewed by Rawlings etal. (1993) and Rohani (2001), respectively. 

Any convenient model for liquid phase activity coefficients can be used. In the absence of any data, the ideal solution model can permit adequate design. 

Phase Equilibria Models Two approaches are available for modeling the fugacity of a solute in a SCF solution. The compressed gas approach includes a fugacity coefficient which goes to unity for an ideal gas. The expanded liquid approach is given as... 

While experiment and theory have made tremendous advances over the past few decades in elucidating the molecular processes and transformations that occur over ideal single-crystal surfaces, the application to aqueous phase catalytic systems has been quite limited owing to the challenges associated with following the stmcture and dynamics of the solution phase over metal substrates. Even in the case of a submersed ideal single-crystal surface, there are a number of important issues that have obscured our ability to elucidate the important surface intermediates and follow the elementary physicochemical surface processes. The ability to spectroscopically isolate and resolve reaction intermediates at the aqueous/metal interface has made it difficult to experimentally estabhsh the surface chemistry. In addition, theoretical advances and CPU limitations have restricted ab initio efforts to very small and idealized model systems. 

The most important aspect of the simulation is that the thermodynamic data of the chemicals be modeled correctly. It is necessary to decide what equation of state to use for the vapor phase (ideal gas, Redlich-Kwong-Soave, Peng-Robinson, etc.) and what model to use for liquid activity coefficients [ideal solutions, solubility parameters, Wilson equation, nonrandom two liquid (NRTL), UNIFAC, etc.]. See Sec. 4, Thermodynamics. It is necessary to consider mixtures of chemicals, and the interaction parameters must be predictable. The best case is to determine them from data, and the next-best case is to use correlations based on the molecular weight, structure, and normal boiling point. To validate the model, the computer results of vapor-liquid equilibria could be checked against experimental data to ensure their validity before the data are used in more complicated computer calculations. 

By using a thermodynamic model based on the formation of self-associates, as proposed by Singh and Sommer (1992), Akinlade and Awe (2006) studied the composition dependence of the bulk and surface properties of some liquid alloys (Tl-Ga at 700°C, Cd-Zn at 627°C). Positive deviations of the mixing properties from ideal solution behaviour were explained and the degree of phase separation was computed both for bulk alloys and for the surface. 

The most common model for describing adsorption equilibrium in multi-component systems is the Ideal Adsorbed Solution (IAS) model, which was originally developed by Radke and Prausnitz [94]. This model relies on the assumption that the adsorbed phase forms an ideal solution and hence the name IAS model has been adopted. The following is a summary of the main equations and assumptions of this model (Eqs. 22-29). 

The use of an ideal-solution model meant that there were a number of instances where calculated and experimental results were quantitatively at variance. However, the approach very successfully predicted the general form of most of the phase diagrams, for example whether they were peritectic or eutectic, and accounted for the appearance of intermediate phases in systems such as Cr-Rh. That the approach could do this using such simple and internally self-consistent models is a demonstration of the inherent power of CALPHAD methods. The importance of this first step therefore cannot be overestimated, although its significance was not... 

In many cases the CALPHAD method is applied to systems where there is solubility between the various components which make up the system, whether it is in the solid, liquid or gaseous state. Such a system is called a solution, and the separate elements (i.e., Al, Fe...) and/or molecules (i.e., NaCl, CuS...) which make up the solution are defined as the components. The model description of solutions (or solution phases) is absolutely fundamental to the CALPHAD process and is dealt with in more detail in chapter S. The present chapter will discuss concepts such as ideal mixing energies, excess Gibbs energies, activities, etc. 

Calculations including the vapour phase were ftien made to determine the extent of release of various components during the reaction. Two types of calculation were made, one where ideal mixing in the solution phases was considered and the other where non-ideal interactions were taken into account. For elements such as Ba, U and, to a certain extent. Si, the calculations were relatively insensitive to the model adopted. However, the amoimt of Sr in the gas was 24 times higher in the full mo r in comparison to the ideal model. This led to the conclusion that sensitivity analysis was necessary to determine the extent to which accuracy of the thermodynamic parameters used in die model affected die final outcome of the predictions. 

It is therefore essential that thermodynamic modeling be applied to obtain a simultaneous quantitative fit to the phase diagram and thermodynamic data in order to evaluate the internal consistency of the various published data. Then a reliable framework can be established for smoothing, interpolating, and extrapolating experimental data that are costly and laborious to obtain. In Chapter 3 an associated solution model is presented. This model provides a good fit to the data for the Hg-Cd-Te system as well as for the Ga-In-Sb system, which is closer to the simpler picture of an ideal solution. 

This description is elaborated below with an idealized model shown in Figure 17. Imagine a molecule tightly enclosed within a cube (model 10). Under such conditions, its translational mobility is restricted in all three dimensions. The extent of restrictions experienced by the molecule will decrease as the walls of the enclosure are removed one at a time, eventually reaching a situation where there is no restriction to motion in any direction (i.e., the gas phase model 1). However, other cases can be conceived for a reaction cavity which do not enforce spatial restrictions upon the shape changes suffered by a guest molecule as it proceeds to products. These correspond to various situations in isotropic solutions with low viscosities. We term all models in Figure 17 except the first as reaction cavities even... 

Analytical solutions for x and y as functions of the bed-length, z, and time, t, are available [45,52], The expressions are a useful extension of two-phase model applied to plug-flow. These two models are appropriate in describing the extraction of crushed or broken seeds to recover the seed oil, either in shallow beds or in plug flow. As shown by Sovova [52], applying the plug-flow model requires corrections for non-ideal residence-time distribution (non-plug flow) of the fluid in contact with the solid. 

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.).

The need to abstract from the considerable complexity of real natural water systems and substitute an idealized situation is met perhaps most simply by the concept of chemical equilibrium in a closed model system. Figure 2 outlines the main features of a generalized model for the thermodynamic description of a natural water system. The model is a closed system at constant temperature and pressure, the system consisting of a gas phase, aqueous solution phase, and some specified number of solid phases of defined compositions. For a thermodynamic description, information about activities is required therefore, the model indicates, along with concentrations and pressures, activity coefficients, fiy for the various composition variables of the system. There are a number of approaches to the problem of relating activity and concentrations, but these need not be examined here (see, e.g., Ref. 11). 

Integration of a time-dependent thermal-capillary model for CZ growth (150, 152) also has illuminated the idea of dynamic stability. Derby and Brown (150) first constructed a time-dependent TCM that included the transients associated with conduction in each phase, the evolution of the crystal shape in time, and the decrease in the melt level caused by the conservation of volume. However, the model idealized radiation to be to a uniform ambient. The technique for implicit numerical integration of the transient model was built around the finite-element-Newton method used for the QSSM. Linear and nonlinear stability calculations for the solutions of the QSSM (if the batchwise transient is neglected) showed that the CZ method is dynamically stable small perturbations in the system at fixed operating parameters decayed with time, and changes in the parameters caused the process to evolve to the expected new solutions of the QSSM. The stability of the CZ process has been verified experimentally, at least... 

Determination of T y. In the formulation of the phase equilibrium problem presented earlier, component chemical potentials were separated into three terms (1) 0, which expresses the primary temperature dependence, (2) solution mole fractions, which represent the primary composition dependence (ideal entropic contribution), and (3) 1, which accounts for relative mixture nonidealities. Because little data about the experimental properties of solutions exist, Tg is usually evaluated by imposing a model to describe the behavior of the liquid and solid mixtures and estimating model parameters by semiempirical methods or fitting limited segments of the phase diagram. Various solution models used to describe the liquid and solid mixtures are discussed in the following sections, and the behavior of T % is presented. 

As shown in Figure 23, the value of TGaSb in the Al-Ga-Sb system is significantly different from unity as a result of nonidealities in the liquid solution. The value of rGaSb in this system is also nearly independent of both composition and temperature. Thus this system can be modeled as an ideal solution in both phases if the value of 0GaSb was suitably adjusted. 

On the basis of the DLP model, the solid is expected to be an ideal solution for systems in which both components have similar lattice constants. This case is true for the AlAs-GaAs system (Figure 5), in which the solid-phase composition equals the gas-phase composition. Compounds in which the two components have very different behavior will show highly nonlinear composition variations, and miscibility gaps may occur (90, 91). As an example, Figure 5 also shows that the solid-phase composition of In As Sb is a nonlinear function of the gas-phase composition. 

It should be noted that the use of the KVSj charts implies that both the gas phase and the hydrate phase can be represented as ideal solutions. This means that the Kvsi of a given component is independent of the other components present, with no interaction between molecules. While the ideal solution model is approximately acceptable for hydrocarbons in the hydrate phase (perhaps because of a shielding effect by the host water cages), the ideal solution assumption is not accurate for a dense gas phase. Mann et al. (1989) indicated that gas gravity may be a viable way of including gas nonidealities as a composition variable. 

Thus in the ideal solution model of a two phase field, a knowledge of the enthalpies of fusion of the pure components at their respective melting points allows simultaneous solution of these two equations for the two unknowns, NA(l> and NA(aj at the temperature of interest. 

When combined with the ideal-gas and ideal-solution models of phase behavi the criterion of vapor/liquid equilibrium produces a simple and useful equati known as Raoult s law. Consider a liquid phase and a vapor phase, both compris of N chemical species, coexisting in equilibrium at temperature T and pressil P, a condition of vapor/liquid equilibrium for which Eq. (10.3) becomes... 

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