Nonideal solutions solid-liquid

The case of binary solid-liquid equilibrium permits one to focus on liquid-phase nonidealities because the activity coefficient of solid component ij, Yjj, equals unity. Aselage et al. (148) investigated the liquid-solution behavior in the well-characterized Ga-Sb and In-Sb systems. The availability of a thermodynamically consistent data base (measurements of liquidus, component activity, and enthalpy of mixing) provided the opportunity to examine a variety of solution models. Little difference was found among seven models in their ability to fit the combined data base, although asymmetric models are expected to perform better in some systems. 

Density and Specific Gravity For binary or pseudobinary mixtures of liquids or gases or a solution of a solid or gas in a solvent, the density is a function of the composition at a given temperature and pressure. Specific gravity is the ratio of the density of a noncompress-ible substance to the density of water at the same physical conditions. For nonideal solutions, empirical calibration will give the relationship between density and composition. Several types of measuring devices are described below. 

Two theoretical techniques worthy of serious review here, perturbation and Green function methods, can be considered complementary. Perturbation methods can be employed in systems which deviate only slightly from regular shape (mostly from planar geometry, but also from other geometries). However, they can be used to treat both linear and nonlinear PB problems. Green function methods on the other hand are applicable to systems of arbitrary irregularity but are limited to low surface potential surfaces for which the use of the linear PB equation is permitted. Both methods are discussed here with reference to surfactant solutions which are a potentially rich source of nonideal surfaces whether these be solid-liquid interfaces with adsorbed surfactants or whether surfactant self-assembly itself creates the interface. 

Nonideal solid-liquid TX diagram at 1 atm for Cu and Al (only about the left half of the diagram is shown). The two-phase regions are indicated. There is a very limited solubility of Cu in Al this is phase a. There is similarly a limited solubility of Cu in the stoichiometric phase or intermetallic compound CuAI2 (called the 6 phase). The liquid solution of Al in Cu freezes at the lowest possible temperature ( 540°C) for 32 mass % Cu this is the eutectic point (which is technologically useful in solders). 

The activity coefficient has to be estimated for nonideal solutions. There is no general method for predicting activity coefficients of solid solutes in liquid solvents. For nonpolar solutes and solvents, however, a reasonable estimate can frequently be made with the regular solution theory, or the Scatchard-Hildebrand relation. 

If the liquid.mixture is ideal, so that y = 1, we have the case of ideal solubility of a solid in a liquid, and the solubility can be computed from only thermodynamic data and ACp) for the solid species near the melting point. For nonideal solutions, yi must be estimated from either experimental data or a liquid solution model, for example, UNIFAC. Alternatively, the regular solution theory estimate for this activity coefficient is... 

For substances that form nonideal solutions in both the liquid and solid phases, the analogous results (Problem 12.4-2) are, for the liquidus line. 

In Fig. 9.26, the thermodynamic equilibrium, solid-liquid phase diagram of a binary (species A and B) system is shown for a nonideal solid solution (i.e., miscible liquid but immiscible solid phase). The melting temperatures of pure substances are shown with Tm A and Tm B. At the eutectic-point mole fraction, designated by the subscript e, both solid and liquid can coexist at equilibrium. In this diagram the liquidus and solidus lines are approximated as straight lines. A dendritic temperature T and the dendritic mass fractions of species (p)7(p)s and (p)equilibrium partition ratio kp is used to relate the solid- and liquid-phase mass fractions of species (p)7(p)J and (p)f/(p)f on the liquidus and solidus lines at a given temperature and pressure, that is,... 

T which represents the ratio of the nonidealities in the liquid phase to those in the solid solution. 

In 10.1 we present the basic thermodynamic relations that are used to start phase-equilibrium calculations we discuss vapor-liquid, liquid-liquid, and liquid-solid calculations. We have seen that the most interesting phase behavior occurs in nonideal solutions, but when we describe nonidealities using an ideal solution as a basis, we must select an appropriate standard state. Common options for standard states are discussed in 10.2 they include pure-component standard states and dilute-solution standard states. 

An important first step in any model-based calculation procedure is the analysis and type of data used. Here, the accuracy and reliability of the measured data sets to be used in regression of model parameters is a very important issue. It is clear that reliable parameters for any model cannot be obtained from low-quality or inconsistent data. However, for many published experimentally measured solid solubility data, information on measurement uncertainties or quality estimates are unavailable. Also, pure component temperature limits and the excess GE models typically used for nonideality in vapor-liquid equilibrium (VLE) may not be rehable for SEE (or solid solubility). To address this situation, an alternative set of consistency tests [3] have been developed, including a new approach for modehng dilute solution SEE, which combines solute infinite dilution activity coefficients in the hquid phase with a theoretically based term to account for the nonideality for dilute solutions relative to infinite dilution. This model has been found to give noticeably better descriptions of experimental data than traditional thermodynamic models (nonrandom two liquid (NRTE) [4], UNIQUAC [5], and original UNIversal Eunctional group Activity Coefficient (UNIEAC) [6]) for the studied systems. 

Doing this gets complicated, because we have gaseous, liquid and solid solutions, a variety of concentration scales, nonideal solutions, and several different standard states that q,° refers to. That is, the quantity q.,- -/r° need not always refer to the difference between i in solution and i in its pure state. At the same time, the form of Equation (7.34) is very convenient, and we want to retain it for all these conditions. We do this by defining the activity, already mentioned in 7.4.3, as... 

Our initial design divides the process into the three unit operations cooling the seawater to form an ice/brine" slush, separating the ice from the brine, and melting the ice to yield water. In practice the process is not as simple. What have we trivialized Have you ever tried to separate ice from a water solution The ice retains water. Solids will retain liquids in most solid/liquid separations, unless special efforts are made. Therefore, in contrast to the vapor/liquid separator used in the process that evaporated the water, we must allow for the nonideality of the ice/brine separator. Assume that the slush formed in our process retains 1.0 wt% liquid. Our process is now as depicted in Figure 3.8. 

From Eq. (13), solubility expressed in mole fraction of the solid x, is dependent on the heat of fusion A which can be related to the sublimation pressure P " of the solid, and the melting temperature Pm (exactly triple-point temperature Prr) of the solid. Assuming an ideal solution, with an activity coefficient /i of unity, the solubility of a solid in a liquid can be calculated. In the present case, the magnitude of separation of two species will depend principally on the difference in their melting temperatures (APm = Pmi — 7m2). Modifying the solvent will produce a nonideal solution with activity coefficients different from unity. In such a case, separation is also dependent on the difference in activity coefficient of both species (A/ = /i - /2). 

For ideal gases, (and ideal solutions of liquids, solids or nonideal gases, see Chapter 7) AA ixing = 0, so that... 

If, as shown above, for ideal gas mixtures the fugacity of one species in the mixture is equal to its partial pressure, then we would like to extend that simple idea to nonideal gas mixtures, and to solutions of liquids and solids. We can, using the definition of an ideal solution. An ideal solution is like an ideal gas in the following respects ... 

In summary, the reference state for species i in the liquid (or solid) phase is no more than a particular state, real or hypothetical, at a given P and Xi (usually that of the system) and at the temperature of the system. We choose the reference state to be that of an ideal solution in which the fugacity is linearly proportional to mole fraction. While the concept of an ideal solution was conjured up in analogy to an ideal gas, there are some interesting differences. A pure gas can be a nonideal gas, while a pure liquid cannot be a nonideal solution because all intermolecular forces in a pure liquid are the same Additionally, an increase in pressure leads to deviations from ideal gas behavior, whereas deviations from ideal solution are caused by changes in composition because nonideal behavior results primarily from the chemical differences of species in a mixture, even at low pressures. 

This case of diffusion in polymers is described by ideas drawn from both diffusion in liquids and diffusion in solids. The theoretical development takes place in two steps. First, the binary diffusion coefficient D is corrected for the nonideal solution... 

Although modeling of supercritical phase behavior can sometimes be done using relatively simple thermodynamics, this is not the norm. Especially in the region of the critical point, extreme nonidealities occur and high compressibilities must be addressed. Several review papers and books discuss modeling of systems comprised of supercritical fluids and solid or liquid solutes (rl,r4—r7,r9,r49,r50). 

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. 

Equation (3.6) illustrates that the solubility of a solid in a liquid depends on the enthalpy change at Tm and the melting temperature of the solid. Equation (3.6) is a valid one when T > Tm because the liquid solute in an ideal solution is completely miscible in all proportions. Table 3.1 shows the ideal solubilities of compounds and their heat of fusion. Equation (3.6) is the equation for ideal solubility. The relationship of In x2 (ideal or nonideal solubility) vs. 1/T is shown in Figure 3.1. 

For nonideal liquid and solid solutions, two identical equations of chemical potential can be used ... 

In addition to the physical state of reactants, it should be remembered that the ideal behavior is encountered only in the gaseous state. As the polymerization processes involve liquid (solution or bulk) and/or solid (condensed or crystalline) states, the interactions between monomer and monomer, monomer and solvent, or monomer and polymer may introduce sometimes significant deviations from the equations derived for ideal systems. The quantitative treatment of thermodynamics of nonideal reversible polymerizations is given in Ref. 54. 

Nonideal Behavior. The discussion of phase behavior up to this point represents the ideal case. A number of factors cause deviation from ideality. The phases present may include liquid crystals, gels, or solid precipitates in addition to the oil, brine, and microemulsion phases (39, 40). The high viscosities of these phases are detrimental to oil recovery. To control the formation of these phases, the practice has been to add low-molecular-weight alcohols to the micellar solution these alcohols act as cosolvents or in some cases as cosurfactants. 

Note that the relative distribution of the solutes depends on nonidealities in both the liquid and solid phases, in addition to pure-component properties. Also, if one of the solutes is an impurity, then crystallization of the other solute from a solution may result in crystals containing significant levels of the impurity. 

The main differences between adsorption from the gas phase and that from liquid phase are as follows [3]. First, adsorption from solution is essentially an exchange process, and hence, molecules adsorb not only because they are attracted by solids but also because the solution may reject them. A typical illustration is that the attachment of hydrophobic molecules on hydrophobic adsorbents from aqueous solutions is mainly driven by their aversion to the water and not by their attraction to the surface. Second, isotherms from solution may exhibit nonideality, not only because of lateral interactions among adsorbed molecules but also because of nonideality in the solution. Third, multilayer adsorption from solution is less common than from the gas phase, because of the stronger screening interaction forces in condensed fluids. 

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