Liquid-Solid Phase Equilibrium

Solid-liquid equilibrium phase diagrams play an important role in the design of industrial crystallization processes. The calculation of phase diagrams can be used to validate the activity coefficient model used for process simulation. 

Figure 5. Isothermal solid-liquid equilibrium phase diagram for the solvent + lauric acid + myristic acid system, (ideal solution)...

The concentration of solute that remains in the liquid solution is dependent on solid-liquid equilibrium. Phase diagrams can be of many types, depending on the system. The maximum amount of solute that can be dissolved in a given volume of solvent at equilibrium and at a given temperature is called solubility. While it is typical for solubility to increase with increasing temperature, that is not always the case for example, see Ce CSO jj, as seen in Figure 7.10. 

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... 

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... 

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 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 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. 

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. 

An alternative explanation concerns the existence of two equilibria. As the vapour/liquid equilibrium is disturbed by the passage of air, the concentration of dissolved compounds in the liquid phase falls, disturbing the solid /liquid equilibrium. The kinetics of transfer across this latter phase boundary are much slower than for the liquid/vapour transfer, so that the extraction of odour becomes limited by the rate of diffusion into the liquid phase. 

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. 

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. 

A Quadruple Point Figure 14.20 shows phase diagrams for (water + acetonitrile) at five different pressures.16 The diagram in (a) at / = 0.1 MPa for this system is very similar to the (cyclohexane + methanol) diagram shown in Figure 14.19a that we described earlier, with a (liquid-I-liquid) equilibrium region present above the (solid + liquid) equilibrium curve for water. 

Figure 14.23 gives the (solid + liquid) phase diagram for (tetrachloromethane + 1,4-dimethylbenzene).19 The maximum in the (solid + liquid) equilibrium curve at X2 = 0.5 results from the formation of a solid addition compound with the formula CCl4-l,4-C6H4(CH3)2 that melts at the temperature corresponding... 

PI4.2 Given the following (solid + liquid) equilibrium temperatures for phase changes in the JC1CFCI3 + X2HCON(CH3)2 system 7... 

An interesting case of solid-liquid equilibrium is one in which a solvent dissociates at least to some extent in the liquid phase and a solute is one of the species formed by the dissociation. We show in Section 10.20 that the experimental temperature-composition curve has a maximum at the composition of the pure solvent. We consider here that the solid phase is the pure, undissociated component, designated by the subscript 1 that this component dissociates in the liquid phase according to the reaction... 

The Earth s core is thought to be mainly iron, and seismic data indicate that the inner core is solid and the outer core is liquid. The pressure at the center of the Earth is 3.6 x 1011 Pa, and at this pressure, iron melts at 6350 K. From this information, what can you infer about the solid-liquid equilibrium boundary in the iron high-pressure phase diagram (Pressure and temperature both increase toward the Earth s center.)... 

The line BC maps the points (Tb, P) at which the equilibrium between liquid and gas exists whilst the line AD maps the points (Pm, P) where the solid/liquid equilibrium exists and therefore correspond to temperatures and pressures at which the two phases will co-exist together with one another. 

Equation (25.7) now gives an expression for the rate of change of pressure with temperature (= dP/dT ) which corresponds to the gradient of the line representing the solid-liquid equilibrium in the phase diagram (gradient of AD Figure 25.1). This quantitative equation (25.7) now enables us to rationalise that since ... 

This is one form of the Clapeymn Equation which gives the slope of the P, T line (AD) for the solid-liquid equilibrium in the phase diagram (Figure 25.1, Frame 25). 

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