Solid-liquid equilibrium pure solids

Equifibrium involving a sofid phase, including solid-liquid equilibrium (SLE), solid-solid equilibrium (SSE), and solid-solid-liquid equilibrium (SSLE), can take two forms (1) pure solids, which are immiscible with other species, and (2) solid solutions, which, Hke Hquid solutions, contain more than one species. If the solid phase exists as a pure species, the condition for ph ase equilibrium with a Hquid becomes ... 

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... 

First we write the balanced chemical equation for the reaction. Then we write the equilibrium constant expressions, remembering that gases and solutes in aqueous solution appear in the Kc expression, but pure liquids and pure solids do not. 

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.

Analysis of the growth process by LPE usually stipulates an equilibrium boundary condition at the solid-liquid interface. The solid-liquid phase diagrams of interest to LPE are those for the pure semiconductor and the semiconductor-impurity systems. Most solid alloys exhibit complete mis-... 

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... 

Of interest in crystallization calculations is solid-liquid equilibrium. When the solid phase is a pure component, the following thermodynamic relationship holds ... 

Pure liquids and pure solids are never included in an equilibrium expression because they have an activity of 1. 

The change in the solvent freezing point is a little less obvious. First, consider the triple point—the intersection of the solid-vapor and the liquid-vapor equilibrium curves. It is clear from Figure 6.5-2 that the effect of the vapor pressure lowering is to lower the triple point of the solution relative to pure solvent. If in addition the solid-liquid equilibrium curve for the solution is (like that for the pure solvent) almost vertical, then the freezing point at an arbitrary pressure Po aiso drops—on the diagram, from for the pure solvent to 7 ms for the solution. 

Single-phase tables are essential in uses that involve unstable phases. This includes phases that persist in a metastable region or those that are unstable at all temperatures. The real world abounds in such cases. Single-phase tables are often applied to solid-solid or solid-liquid equilibrium in mixtures, where an unstable phase is stabilized in the presence of other components. Analysis of such equilibria often requires liquid or solid properties that are many hundreds of degrees into the metastable region of each pure phase. Single-phase tables are designed to supply these properties, but their preparation is not trivial. 

If one (or more) of the species in Equation 9-7 is a pure liquid, a pure solid, or the solvent present in excess, no term for this species appears in the equilibrium-constant expression. For example, if Z in Equation 9-6 is the solvent H2O, the equilibrium-constant expression simplifies to... 

Strictly speaking, the letters in brackets represent activities, but we will usually follow the practice of substituting molar concentrations for activities in most calculations. Thus, if some participating species A is a solute, [A] is the concentration of A in moles per liter. If A is a gas, [A] in Equation 18-12 is replaced by p, the partial pressure of A in atmospheres. If A is a pure liquid, a pure solid, or the solvent, its activity is unity, and no term for A is included in the equation. The rationale for these assumptions is the same as that de.scribed in Section 9B-2, which deals with equilibrium-constant expressions. 

Figure 2-20 The solid liquid equilibrium relationship of a solid compound. From right to left, the solid phase at equihbrium can be pure compound A, a mixture of compounds C and A. compound C, a mixture of compounds C and B, and pure compound B. The solution phase at equihbrium can be a mixture of compounds A and B at the corresponding liquid-sohd tie hues.

Figure 2.12 is the classic pressure-temperature (FT) representation of the phase changes of a pure component. There are three primary phases of pure components solid liquid, and vapor solid-solid transitions, liquid crystal phases, and so on, are also possible but will not be considered here. The solid lines represent the sublimation curve (solid —> vapor), the vapor pressure curve (liquid —> vapor) and the melting curve (solid liquid) of the pure component. The triangle represents the triple point, at which a solid, liquid and vapor coexist in equilibrium. The circle represents the pure component critical point, where the supercritical region begins. 

Eq. (1.40) is a general equation for the solubility of any solute in any solvent. We can see from this equation that the solubility depends on the activity coefficient and on the fugacity ratio filft- The standard state fugacity normally used for solid-liquid equilibrium is the fugacity of the pure solute in a subcooled liquid state below its melting point. We can simplify Eq. (1.40) further by assuming that our solid and subcooled liquid have small vapor pressures. We can then substitute vapor pressure for fugacity. If we further assume that the solute and solvent are chemically similar so that 72 = 1, then we can write... 

The logarithmic term has the same form as the equilibrium constant for the reaction. The term is called Q, the reaction quotient, when the concentrations (rigorously, the activities) are not the equilibrium values for the reaction. As in any equilibrium constant expression, pure liquids and pure solids have activities equal to 1, so they are omitted from the expression. If the values of R, T (25°C = 298 K), and F are inserted into the equation and the natural logarithm is converted to log to the base 10, the Nemst equation reduces to... 

AC or BC, which melts at a higher temperature than either of the pure elements (except for the InSb-Sb case). The binary phase diagram consists of two simple eutectic systems on either side of the compound (e.g., the A-AC and the AC-C systems). The third binary phase diagram represents solid-liquid equilibrium between elements from the same group. In Figure 1 the A-B portion of the ternary phase diagram is depicted as being isomorphous... 

With many million pure substances now known, an essentially infinite number of mixtures can be formed, resulting in a diversity of phase behavior that is overwhelming. Consider just two components not only can binary mixtures exhibit solid-gas, liquid-solid, and liquid-gas equilibria, but they might also exist in liquid-liquid, solid-solid, gas-gas, gas-liquid-liquid, solid-liquid-gas, solid-solid-gas, solid-liquid-liquid, solid-solid-liquid, and solid-solid-solid equilibria. That s a dozen different kinds of phase equilibrium situations— just for binary mixtures. For multicomponent mixtures the possibilities seem endless. 

It appears that the lower the temperature the higher the nu-eleation rate and eonsequently it is very difficult to get the solidification of a pure liquid near its bulk solid-liquid equilibrium temperature. Another less obvious consequence is that the temperature at which a given sample will solidify is not unique. Therefore, only a most probable temperature ean be given and this temperature T appears to be volume dependent and of eourse it also depends on the sample composition as it is shown thereafter. 

At temperature Ty, a solid-liquid equilibrium is expected, pure ice being the solid (composition x "=0,point A/j ) and a more concentrated solution being the liquid (composition x point /nj ). The proportion of ice in the system is given by e ratio of the lengths of the segmentsA/ M andA/ A/. ... 

The picture is quite different when a sample is reg ularly cooled in so far as solid-liquid equilibrium is not reached at any stage of the cooling. Kinetic aspects have to be taken into account as has been done for the pure material in Sec. 

T," solid-liquid equilibrium melting temperature of the pure solvent... 

The freezing point of a solution is the temperature at which the first crystals of pure solvent form in equilibrium with the solution. Recall from Section 11.6 that the line representing the solid-liquid equilibrium rises nearly vertically from the triple point. It is easy to see in Figure 13.21 that the triple-point temperature of the solution is lower than that of the pure liquid, but it is also true for all points along the solid-Uquid equilibrium curve the freezing point of the solution is lower than that of the pure liquid. 

FIGURE 9.1 The chemical potential of paracetamol in CCl at varying concentrations (in mole fractions x) at 298 K. The intersection with the solid-state chanical potential defines the solid-liquid equilibrium (SLE) and corresponds to the solnbiUty of the dmg in this particnlar solvent. The difference between the pure liquid and the solid-state chemical potential is the free energy of fusion Also shown is the liquid-liquid equilibrium (LLE) where the virtually supercooled liquid is at equilibrium with the dissolved drug. 

Besides the pure component parameters, in particular the mixture parameters, for example of a g -model or an equation of state, should be checked carefully prior to process simulation. The procedure is shown in Figure 11.4 for the binary system acetone-cyclohexane, which may be one of the binary key systems of a multicomponent mixture. From the results shown in Figure 11.4, it can be concluded that the VLE behavior of the binary system can be reliably described in the temperature range 0-50 C with the Wilson parameters used. But from the poor -results, it seems that an extrapolation to higher or lower temperature may be dangerous, as already can be seen from the solid-liquid equilibrium (SLE) results of the eutectic system in the temperature range 0 to —lOO C and also from the incorrect temperature dependence of the calculated azeotropic data. 

Remember that concentrations or pressures for pure liquids and pure solids do not appear in the equilibrium constant expressions. 

Similarly, the triple point S has been moved to S, and the solid-liquid equilibrium curve now extends up from S instead of S. The solid-vapor equilibrimn curve is unafFected because the solid that freezes from the solution is, in most cases, pure solvent. The freezing point ( ( ) is lower than it was (ff) for the pure solvent. The depression of the freezing point tf — t/) is, like the boiling-point elevation, a colligative property. 

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