Phase equilibria supersaturation

Variables . L-S phase equilibrium, supersaturation, metastability, and crystal growth rate... 

If we could prevent the mixture from separating into two phases at temperatures below Tc, we would expect the point of inflection to develop into curves similar to those shown in Figure 8.17 as the dotted line for /2, with a maximum and minimum in the fugacity curve. This behavior would require that the fugacity of a component decreases with increasing mole fraction. In reality, this does not happen, except for the possibility of a small amount of supersaturation that may persist briefly. Instead, the mixture separates into two phases. These phases are in equilibrium so that the chemical potential and vapor fugacity of each component is the same in both phases, That is, if we represent the phase equilibrium as... 

The technique employed to generate supersaturation in a solution is referred to as the mode of operation. The mode chosen by the designer is strongly influenced by the phase-equilibrium characteristics of the system, and it dictates the material and energy balance requirements of the system. The common techniques for producing solids (or generating supematuration) from a solution include ... 

Systems of type 2/ (with metastable three-phase (L,-LrG) immiscibility and critical phenomena L = G in solid saturated solutions) (Figure 1.30). In systems of this type, the entire three-phase equilibrium (L1-L2-G) is in the metastable region of supersaturated solutions. The metastable immiscibility equilibria (Li = L2, L1-L2) become stable only for temperatures at and above the second critical endpoint Q (Li = L2-Sb). The metastable equilibria in systems of type 2d and 2d", shown in Figures 1.28 and 1.30, cannot be observed experimentally. In the case of type 2d" there are stable equilibria L1-L2-G and L1-L2-G-S indicating an existence of immiscibility phenomena hidden in part by the occurrence of a solid phase. Such indicators are absent in the systems of type 2d, moreover the stable equilibria in the types 2d and 2a are very similar and therefore difficult to tell these phase behavior apart. However, the presence of... 

If co-crystals are to solve solubility problems one must assess their true or thermodynamic solubility so that development strategies are guided by the fundamental properties of co-crystals. Measuring the solubility of co-crystals that generate supersaturation of the parent drug is often experimentally impossible due to conversion. Eutectic points, described in Section 11.4, provide a measure of co-crystal solubility under thermodynamic equilibrium conditions. The solution at the eutectic point is saturated with co-crystal and solution concentrations represent experimentally accessible thermodynamic solubility values. Once co-crystal solubility is determined at the eutectic, the solubility under different solution conditions (pH, co-former, micelle concentration) can be obtained from solubility models that consider the appropriate solution phase equilibrium expressions. 

Fig. 6.67 Stages in the reaction of two solid phases a and / to yield phase 7. The initial interdiffusion of the components at contact a/P leads to a supersaturation with respect to phase 7 (see right hand diagram). Here concentrations are assumed that correspond at contact to a virtual phase equilibrium between a and / (i.e. x P) and x (a) are the concentrations that would be arrived at if no product formation were to occur (G- — oo)). The situation then corresponds to a strong supersaturation with respect to the formation of 7. This is shown in the right hand diagram. The supersaturation (e.g. x (7) — x (/i)) can be regarded as the driving force. From Ref. [8j.

The two quantities, t] and Ap, can be considered as measures for the deviation from the state of stable thermodynamic equilibrium, but the mere fact that the parent phase is supersaturated does not mean that a phase... 

Crystal Formation There are obviously two steps involved in the preparation of ciystal matter from a solution. The ciystals must first Form and then grow. The formation of a new sohd phase either on an inert particle in the solution or in the solution itself is called nucle-ation. The increase in size of this nucleus with a layer-by-layer addition of solute is called growth. Both nucleation and ciystal growth have supersaturation as a common driving force. Unless a solution is supersaturated, ciystals can neither form nor grow. Supersaturation refers to the quantity of solute present in solution compared with the quantity which would be present if the solution were kept for a veiy long period of time with solid phase in contac t with the solution. The latter value is the equilibrium solubility at the temperature and pressure under consideration. The supersaturation coefficient can be expressed... 

Of the generic aluminium alloys (see Chapter 1, Table 1.4), the 5000 series derives most of its strength from solution hardening. The Al-Mg phase diagram (Fig. 10.1) shows why at room temperature aluminium can dissolve up to 1.8 wt% magnesium at equilibrium. In practice, Al-Mg alloys can contain as much as 5.5 wt% Mg in solid solution at room temperature - a supersaturation of 5.5 - 1.8 = 3.7 wt%. In order to get this supersaturation the alloy is given the following schedule of heat treatments. 

The great importance of the solubility product concept lies in its bearing upon precipitation from solution, which is, of course, one of the important operations of quantitative analysis. The solubility product is the ultimate value which is attained by the ionic concentration product when equilibrium has been established between the solid phase of a difficultly soluble salt and the solution. If the experimental conditions are such that the ionic concentration product is different from the solubility product, then the system will attempt to adjust itself in such a manner that the ionic and solubility products are equal in value. Thus if, for a given electrolyte, the product of the concentrations of the ions in solution is arbitrarily made to exceed the solubility product, as for example by the addition of a salt with a common ion, the adjustment of the system to equilibrium results in precipitation of the solid salt, provided supersaturation conditions are excluded. If the ionic concentration product is less than the solubility product or can arbitrarily be made so, as (for example) by complex salt formation or by the formation of weak electrolytes, then a further quantity of solute can pass into solution until the solubility product is attained, or, if this is not possible, until all the solute has dissolved. 

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