Phase equilibria ideal solutions

At equilibrium, a component of a gas in contact with a liquid has identical fugacities in both the gas and liquid phase. For ideal solutions Raoult s law applies ... 

Equilibrium between different phases in ideal solutions... 

Following Newman, Boynton3 also took the liquid rates Lj to be the independent variables, and for each choice of these variables the temperatures required to satisfy the component-material balances, and equilibrium relationships were found by successive application of the Newton-Raphson method. The results so obtained were then used in the enthalpy balances to compute a new set of liquid rates by use of one application of the Newton-Raphson method. For the case where both the vapor and liquid phases form ideal solutions, Boynton s method constitutes an exact application of the Newton-Raphson method. All of the matrix equations solved by Boynton were of order N. 

In the following development of the equations needed for the 0 method of convergence and the Newton-Raphson method which follows, it is supposed that both the vapor and liquid phases form ideal solutions throughout the column. For any component /, the N equilibrium relationships for a column having a partial condenser (yu = A, )... 

This form of the equilibrium relationship is used because the behavior of a large number of mixtures can be approximated by use of ideal solutions. For the case where both the vapor and liquid phase form ideal solutions, it is shown in the next section that yf = 1 and y = 1. Consequently, for ideal solutions, Eq. (14-39) reduces to... 

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. 

Raoult s Law. The molar composition of a liquid phase (ideal solution) in equilibrium with its vapor at any temperature T is given by... 

In our discussion of (vapor + liquid) phase equilibria to date, we have limited our description to near-ideal mixtures. As we saw in Chapter 6, positive and negative deviations from ideal solution behavior are common. Extreme deviations result in azeotropy, and sometimes to (liquid -I- liquid) phase equilibrium. A variety of critical loci can occur involving a combination of (vapor + liquid) and (liquid -I- liquid) phase equilibria, but we will limit further discussion in this chapter to an introduction to (liquid + liquid) phase equilibria and reserve more detailed discussion of what we designate as (fluid + fluid) equilibria to advanced texts. 

Figure 8.23 (Solid + liquid) phase diagram for (. 1CCI4 +. yiCHjCN), an example of a system with large positive deviations from ideal solution behavior. The solid line represents the experimental results and the dashed line is the ideal solution prediction. Solid-phase transitions (represented by horizontal lines) are present in both CCI4 and CH3CN. The CH3CN transition occurs at a temperature lower than the eutectic temperature. It is shown as a dashed line that intersects the ideal CH3CN (solid + liquid) equilibrium line.

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). 

Historically, an ideal solution was defined in terms of a liquid-vapor or solid-vapor equilibrium in which each component in the vapor phase obeys Raoult s law. 

Consider now two practically immiscible solvents that form two phases, designated by and ". Let the solute B form a dilute ideal solution in each, so that Eq. (2.19) applies in each phase. When these two hquid phases are brought into contact, the concentrations (mole fractions) of the solute adjust by mass transfer between the phases until equilibrium is established and the chemical potential of the solute is the same in the two phases ... 

The distribution ratio of a solute between two liquid phases at equilibrium is a constant, provided that the solute forms a dilute ideal solution in each phase. 

The mixture CMC is plotted as a function of monomer composition in Figure 1 for an ideal system. Equation 1 can be seen to provide an excellent description of the mixture CMC (equal to Cm for this case). Ideal solution theory as described here has been widely used for ideal surfactant systems (4.6—18). Equation 2 can be used to predict the micellar surfactant composition at any monomer surfactant composition, as illustrated in Figure 2. This relation has been experimentally confirmed (ISIS) As seen in Figure 2, for an ideal system, if the ratio XA/yA < 1 at any composition, it will be so over the entire composition range. In classical phase equilibrium thermodynamic terms, the distribution coefficient between the micellar and monomer phases is independent of composition. 

The orientation dependent parameter p defined by Eq. (11) becomes unity in the isotropic state, and decreases as the polymers are uniaxially oriented. Therefore, it follows from Eqs. (9) and (10) that the wormlike hard spherocylinder system has a smaller translational entropy loss from the ideal solution in the liquid crystal state than in the isotropic state. This difference drives the system to form a liquid crystal phase. However, in order to determine the equilibrium orientation of the system, the orientation dependence of Sor has to be formulated, and this is done in Sect. 2.3. 

Equation (1) may be applied to the equilibrium between vapor and liquid of a pure substance (X = vapor pressure) or to the equilibrium between an ideal dilute solution and the pure phase of a solute X = solubility) or to the equilibrium of a chemical reaction (X = equilibrium constant). 

Compositions and Quantities of the Equilibrium Gas and Liquid Phases of an Ideal Solution... 

To understand why a solute lowers the vapor pressure, we need to look at the thermodynamic properties of the solution. We saw in Section 8.2, specifically Eq. 1, that at equilibrium, and in the absence of any solute, the molar free energy of the vapor is equal to that of the pure solvent. We now need to consider the molar free energies of the solvent and the vapor when a solute is present. We shall consider only nonvolatile solutes, which do not appear in the vapor phase, and limit our considerations to ideal solutions. 

The thermodynamic quantity 0y is a reduced standard-state chemical potential difference and is a function only of T, P, and the choice of standard state. The principal temperature dependence of the liquidus and solidus surfaces is contained in 0 j. The term is the ratio of the deviation from ideal-solution behavior in the liquid phase to that in the solid phase. This term is consistent with the notion that only the difference between the values of the Gibbs energy for the solid and liquid phases determines which equilibrium phases are present. Expressions for the limits of the quaternary phase diagram are easily obtained (e.g., for a ternary AJB C system, y = 1 and xD = 0 for a pseudobinary section, y = 1, xD = 0, and xc = 1/2 and for a binary AC system, x = y = xAC = 1 and xB = xD = 0). 

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. 

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. 

Chapters 17 and 18 use thermodynamics to describe solutions, with nonelectrolyte solutions described in Chapter 17 and electrolyte solutions described in Chapter 18. Chapter 17 focuses on the excess thermodynamic properties, with the properties of the ideal and regular solution compared with the real solution. Deviations from ideal solution behavior are correlated with the type of interactions in the liquid mixture, and extensions are made to systems with (liquid + liquid) phase equilibrium, and (fluid -I- fluid) phase equilibrium when the mixture involves supercritical fluids. 

In an ideal solution, the maximum solubility of a drug substance is a function of the solid phase in equilibrium with a speciLed solvent system at a given temperature and pressure. Solubility is an equilibrium constant for the dissolution of the solid into the solvent, and thus depends on the strengths of solute solvent interactions and solute solute interactions. Alteration of the solid phase of the drug substance can inLuence its solubility and dissolution properties by affecting the solute solutc molecular interactions. 

Martin has considered the chemical and physical factors affecting partition coefficients.20 Restricting his discussion to ideal solutions, he considered a solute, A, distributed between two phases in equilibrium with each other. The partition coefficient, a, of the solute A is related to the free energy required to transport one mole of A from one phase to... 

Fundamental studies on the adsorption of supercritical fluids at the gas-solid interface are rarely cited in the supercritical fluid extraction literature. This is most unfortunate since equilibrium shifts induced by gas phase non-ideality in multiphase systems can rarely be totally attributed to solute solubility in the supercritical fluid phase. The partitioning of an adsorbed specie between the interface and gaseous phase can be governed by a complex array of molecular interactions which depend on the relative intensity of the adsorbate-adsorbent interactions, adsorbate-adsorbate association, the sorption of the supercritical fluid at the solid interface, and the solubility of the sorbate in the critical fluid. As we shall demonstrate, competitive adsorption between the sorbate and the supercritical fluid at the gas-solid interface is a significant mechanism which should be considered in the proper design of adsorption/desorption methods which incorporate dense gases as one of the active phases. 

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