Equilibrium solid-liquid equilibria

The solid is the more dense phase (Figure 9.7a). The solid-liquid equilibrium line is inclined to the right, shifting away from the y-axis as it rises. At higher pressures, the solid becomes stable at temperatures above the normal melting point In other words, the melting point is raised by an increase in pressure. This behavior is shown by most substances. 

Under certain pressure and temperature conditions, a system can contain two or more phases in equilibrium. An example is the temperature and pressure where solid and liquid are in equilibrium. We refer to this condition as (solid + liquid) equilibrium, and the temperature as the melting temperature. This temperature changes with pressure and with composition. The melting temperature when the... 

Since Raoult s law activities become mole fractions in ideal solutions, a simple substitution of.Y, — a, into equation (6.161) yields an equation that can be applied to (solid + liquid) equilibrium where the liquid mixtures are ideal. The result is... 

Effect of Pressure on Solid + Liquid Equilibrium Equation (6.84) is the starting point for deriving an equation that gives the effect of pressure on (solid + liquid) phase equilibria for an ideal mixture in equilibrium with a pure... 

The melting point of carbon dioxide increases with increasing pressure, since the solid-liquid equilibrium line on its phase diagram slopes up and to the right. If the pressure on a sample of liquid carbon dioxide is increased at constant temperature, causing the molecules to get closer together, the liquid will solidify. This indicates that solid carbon dioxide has a higher density than the liquid phase. This is true for most substances. The notable exception is water. 

Fig. 3.2. A stylized phase diagram for a simple pure substance. The dashed line represents 1 atm pressure and the intersection with the solid-liquid equilibrium line represents the normal boiling point and the intersection with the liquid-vapor equilibrium line represents the normal boiling point.

The initial concentration distribution is therefore simply translated at the velocity of the liquid steady flow and full equilibrium between the liquid and its matrix require that the amount of element transported by the concentration wave is constant. In more realistic cases, either the flow is non-steady due to abrupt changes in fluid advection rate or porosity, or solid-liquid equilibrium is not achieved. These cases may lead to non-linear terms in the chromatographic equation (9.4.35) and unstable behavior. The rather complicated theory of these processes is beyond the scope of the present book. 

Eqn s 13, 14 and 15 represent the structural constraints Eqn s 23 and 24 represent the solid liquid equilibrium and mole fraction constraints respectively. The UNIFAC sub-groups that were used to generate the molecules are given in the appendix. 

Verification through solid liquid equilibrium diagrams... 

The structural constraints used in the first case study namely, Eqn s 27,28 and 29 are used again. The melting point, boiling point and flash point, are used as constraints for both solvent and anti-solvent. Since the solvent needs to have high solubility for solute and the anti-solvent needs to have low solubility for the solute limits of 17 <8 < 19 and 5 > 30 (Eqn s. 33 and 37) are placed on the solubility parameters of solvent and anti-solvents respectively. Eqn.38 gives the necessary condition for phase stability (Bernard et al., 1967), which needs to be satisfied for the solvent-anti solvent pairs to be miscible with each other. Eqn. 39 gives the solid-liquid equilibrium constraint. 

Whether a task can be performed concurrently with other tasks depends on two factors. One is whether the input information for the activity under consideration depends on the output from other activities. The other is the availability of manpower and equipment. Consider a team that has only one chemical engineer to design both the reactor and the crystallizer. Even though reaction kinetics, solid-liquid equilibrium data and crystallization kinetics can be measured in parallel, the total time for these activities is determined by what the single individual can achieve. 

The solid-liquid equilibrium for NaCl is rigorously expressed by ... 

Solid-liquid equilibrium data (1 6) for the HCl-NaCl- 0 system at 25°C were used with Equation 27 to calculate experimental activity coefficients of NaCl. Table 1 shows a comparison between the experimental activity coefficients and those calculated using Equation 12. The agreement between experimental and calculated activity coefficients is very good, and Equation 12 should be useful for predictions of solid-liquid equilibria at other temperatures. 

Equation 27 is similar to the solid-liquid equilibrium relation used for non-electrolytes. As in the case of the vapor-liquid equilibrium relation for HC1, the solid-liquid equilibrium expression for NaCl is simple since the electrolyte is treated thermodynamically the same in both phases. 

CALCULATED ACTIVITY COEFFICIENTS OF NaCl AND EXPERIMENTAL VALUES OBTAINED FROM SOLID-LIQUID EQUILIBRIUM DATA... 

An exceptional case of a very different type is provided by helium [15], for which the third law is valid despite the fact that He remains a liquid at 0 K. A phase diagram for helium is shown in Figure 11.5. In this case, in contrast to other substances, the solid-liquid equilibrium line at high pressures does not continue downward at low pressures until it meets the hquid-vapor pressure curve to intersect at a triple point. Rather, the sohd-hquid equilibrium line takes an unusual turn toward the horizontal as the temperature drops to near 2 K. This change is caused by a surprising... 

A confirmation of this conclusion also is provided by an examination of the solid-liquid equilibrium in the neighborhood of 0 K. As shown in Equation (8.9), a two-phase equilibrium obeys the Clapeyron equation ... 

In Chapter 13 we discussed briefly the solid-liquid equilibrium diagram of a feldspar. Feldspar is an ideal, solid solution of albite (NaAlSiaOg) and anorthite (CaAlSi20g) in the solid state as well as an ideal, liquid solution of the same components in the molten state. The relationships that we have developed in this chapter permit us to interpret the feldspar phase diagram (Figure 13.4) in a quantitative way. 

Using the Gibbs-Helmholtz equation obtained Clapeyron equation for the equilibrium solid liquid. 

It is proposed that in mixed organic base-alkali systems, the presence of the organic base changes the solid-liquid equilibrium and stabilizes larger sol-like aluminosilicate species ( 25 m/ ). The alkali ion affects agglomeration of the sol particles to larger amorphous precipitate particles from 100 to 500 min size which subsequently crystallize to zeolite. 

The data for the phase equilibrium solid-liquid for the binary system cocoa butter-CC>2 and for the equilibrium solubility data of CO2 in the liquid phase of cocoa butter have been presented [70],... 

Empirical equations of the form T = aF + bD + c, expressing the relation between total solids (T), fat (F), and density (D), have been used for years. Such derivations assume constant values for the density of the fat and of the mixture of solids-not-fat which enter into the calculation of the coefficients (a, b, and c). Since milk fat has a high coefficient of expansion and contracts as it solidifies (note that the solid-liquid equilibrium is established slowly), the temperature of measurement and the previous history of the product must be controlled carefully (see Sharp and Hart 1936). Variations in the composition of... 

For solid-liquid equilibrium in a quaternary system, the Gibbs phase rule allows four degrees of freedom. If T, P, xc, and xD (in which x is the mole fraction of component i in liquid solution) are specified, then xA, x, t/, and xAC (in which x is the mole fraction of component ij in solid solution) are determined, and the system is invariant. These variables are defined by the following equations ... 

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. 

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