Ideal solubility of a solid

The ideal solubility of a solid at a given temperature and pressure is the solubility calculated on the assumptions that (1) the liquid is an ideal liquid mixture, and (2) the molar differential enthalpy of solution equals the molar enthalpy of fusion of the solid (Asoi,B7/=Afus,B )-These were the assumptions used to derive Eq. 12.5.4 for the freezing-point curve of an ideal liquid mixture. In Eq. 12.5.4, we exchange the constituent labels A and B so that the solid phase is now component B  

From the freezing behavior of benzene-toluene mixtures shown by the open circles in Fig. 12.4 on page 384, we can see that solid benzene has close to ideal solubility in liquid toluene at temperatures not lower than about 20 K below the melting point of benzene. 

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Subscripts A and B are also used to indicate different states of the system. Very occasionally other subscripts are used and these are defined within the section id which they appear, for example xjd, the ideal solubility of a solid in a liquid, referred to... 

If the liquid.mixture is ideal, so that y = 1, we have the case of ideal solubility of a solid in a liquid, and the solubility can be computed from only thermodynamic data and ACp) for the solid species near the melting point. For nonideal solutions, yi must be estimated from either experimental data or a liquid solution model, for example, UNIFAC. Alternatively, the regular solution theory estimate for this activity coefficient is... 

According to Equation (4.5) the ideal solubility of a compound is only dependent upon the heat of fusion, the difference in heat capacity of the solid and supercooled liquid and the melting point of the compound. Since there are no properties of the solvent included in the ideal solubility equation, the solubility of a compound should be the same in all solvents. This equation overlooks all solute-solvent and solvent-solvent interactions. 

When we consider the solubility of a solid component in a solvent, the emphasis is placed on obtaining the mole fraction or other composition variable as a function of the temperature. Thus, Equation (10.96) gives the solubility as a function of the temperature in this interpretation. The solubility in an ideal solution is given by... 

Equation (3.6) illustrates that the solubility of a solid in a liquid depends on the enthalpy change at Tm and the melting temperature of the solid. Equation (3.6) is a valid one when T > Tm because the liquid solute in an ideal solution is completely miscible in all proportions. Table 3.1 shows the ideal solubilities of compounds and their heat of fusion. Equation (3.6) is the equation for ideal solubility. The relationship of In x2 (ideal or nonideal solubility) vs. 1/T is shown in Figure 3.1. 

Equation 1 shows that the solubility of a solid in SCF depends among others on the fugacity coefficient large values of the solubility. These solubilities are much larger than those in ideal gases, and enhancement factors of 10 —10 are not uncommon. They are, however, still relatively small and usually do not exceed several mole percent. 

As in a previous paper [Int. J. Pharm. 258 (2003) 193-201], the Kirkwood-Buff theory of solutions was employed to calculate the solubility of a solid in mixed solvents. Whereas in the former paper the binary solvent was assumed ideal, in the present one it was considered nonideal. A rigorous expression for the activity coefficient of a solute at infinite dilution in a mixed solvent [Int. J. Pharm. 258 (2003) 193-201] was used to obtain an equation for the solubility of a poorly soluble solid in a nonideal mixed solvent in terms of the solubilities of the solute in the individual solvents, the molar volumes of those solvents, and the activity coefficients of the components of the mixed solvent. 

In a previous paper (Ruckenstein and Shulgin, 2003), the Kirkwood-Buff theory of solutions (Kirkwood and Buff, 1951) was employed to obtain an expression for the solubility of a solid (particularly a drug) in binary mixed (mainly aqueous) solvents. A rigorous expression for the composition derivative of the activity coefficient of a solute in a ternary solution (Ruckenstein and Shulgin, 2001) was used to derive an equation for the activity coefficient of the solute at infinite dilution in an ideal binary mixed solvent and further for the solubility of a poorly soluble solid. By considering that the excess volume of the mixed solvent depends on composition, the above equation was modified empirically by including one adjustable parameter. The modified equation was compared with the other three-parameter equations available in the literature to conclude that it provided a better agreement. 

The prediction of the solubility of poorly soluble substances of environmental significance in multicomponent (ternary and higher) aqueous mixed solvents is a difficult task because it requires the knowledge of the activity coefficient of a solute in a multicomponent mixed solvent. The method most often used for the solubility of a solid in ternary and multicomponent mixed solvents is the combined nearly ideal binary solvent/Redlich - Kister equation (33). That equation was applied to the solubility of a solid in ternary... 

Integration of equation (16) provides the relationship known as the van t Hoff equation, which expresses the temperature dependence of the solubility of a solid solute (identified as species B) in an ideal solution ... 

Recalling, first, ideal solution theory, the solubility of a solid in a liquid is related to the heat of fusion of the solid and the temperature of the solution, ignoring ACp and AK terms. 

The simplest type of phase behavior to understand is the solubility of a solid solute, such as naphthalene, in a supercritical fluid. When the solute is a crystalline solid, the solid phase may be assumed to be pure and only the supercritical phase is a mixture. Imagine solid naphthalene in a closed vessel under one atmosphere of carbon dioxide at 40°C. The reduced temperature and reduced density of CO2 are 1.03 and 3.7x10 respectively. At this pressure, the gas phase is ideal and the naphthalene solubility is determined by its vapor pressure. As the container volume is decreased isothermally, the solubility initially decreases when the gas phase is still nearly ideal. As the pressure is increased further, however, the gas phase density becomes increasingly nonideal and approaches the mixture critical density (near the critical density of CO2 because the gas phase is still mostly CO2). The reduced density of CO2 increases rapidly near the critical region as shown in Figure 2. The solvent power of CO2 is related to the density which leads to a rapid solubility increase. A brief description of intermolecular interactions is helpful in understanding this behavior. 

Figure 9 illustrates the behavior and solubility of a solid in a compressed gas. At low pressure (e.g., < 1 MPa for CO2), the solubility of a solid in a gas follows ideal behavior and is inversely proportional to the pressure. As the pressure approaches the sublimation pressure of the solid, the mole fraction solubility tends toward 1 below the sublimation pressure the solid completely vaporizes. Beyond the ideal region, the solubility experiences a minimum before becoming proportional to the... 

As deductions from equation (8 63) we have (a) the solubility of a solid may be expected to increase with rise of temperature (6) the solubility of a solid may be expected to be the greater the lower is its melting-point and the smaller is its enthalpy of melting. These results, although they are based on the supposition of ideal solutions, are in fairly general agreement with experience. 

From Eq. (13), solubility expressed in mole fraction of the solid x, is dependent on the heat of fusion A which can be related to the sublimation pressure P " of the solid, and the melting temperature Pm (exactly triple-point temperature Prr) of the solid. Assuming an ideal solution, with an activity coefficient /i of unity, the solubility of a solid in a liquid can be calculated. In the present case, the magnitude of separation of two species will depend principally on the difference in their melting temperatures (APm = Pmi — 7m2). Modifying the solvent will produce a nonideal solution with activity coefficients different from unity. In such a case, separation is also dependent on the difference in activity coefficient of both species (A/ = /i - /2). 

Solubility equilibria are described quantitatively by the equilibrium constant for solid dissolution, Ksp (the solubility product). Formally, this equilibrium constant should be written as the activity of the products divided by that of the reactants, including the solid. However, since the activity of any pure solid is defined as 1.0, the solid is commonly left out of the equilibrium constant expression. The activity of the solid is important in natural systems where the solids are frequently not pure, but are mixtures. In such a case, the activity of a solid component that forms part of an "ideal" solid solution is defined as its mole fraction in the solid phase. Empirically, it appears that most solid solutions are far from ideal, with the dilute component having an activity considerably greater than its mole fraction. Nevertheless, the point remains that not all solid components found in an aquatic system have unit activity, and thus their solubility will be less than that defined by the solubility constant in its conventional form. 

Solid-Fluid Equilibria The solubility of the solid is very sensitive to pressure and temperature in compressible regions, where the solvent s density and solubility parameter are highly variable. In contrast, plots of the log of the solubility versus density at constant temperature often exhibit fairly simple linear behavior (Fig. 20-19). To understand the role of solute-solvent interactions on sofubilities and selectivities, it is instructive to define an enhancement factor E as the actual solubihty divided by the solubility in an ideal gas, so that E = ysP/Pf, where P is the vapor pressure. The solubilities in CO2 are governed primarily by vapor pressures, a property of the solid... 

The data of Table III show that the surface layer of the solid particles is indistinguishable from pure fluorapatite in all equilibrations at x = 0.110, 0.190 and 0.435 and 0.595. However, some equilibrations at x = 0.763 and all at x = 0.868 do deviate significantly from the behavior of pure fluorapatite. A peculiar aspect is that the activity of fluorapatite becomes significantly larger than 1. Simutaneously, the activity of hydroxyapatite approaches unity. This would mean that at all values of x both activities would become smaller than 1, and thus an ideal behavior of the solid solutions would not explain the observed solubility behavior. 

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. 

Even under ideally controlled laboratory conditions using pure chemicals, the dissolution of a solid compound in water may involve several complex reactions and the formation of numerous dissolved species. As an example, the dissolution of slightly soluble AS2S3 (orpiment) in water can be investigated in a laboratory closed system at 25°C and 1 bar pressure. Nordstrom and Archer (2003), 9 proposed the following reaction to describe the dissolution of orpiment, which initially forms H3ASO30 and H2S° ... 

This phenomenon can also be looked at in terms of the dissolution of solid A in the solution (even though A, because it is present at a greater concentration, would usually be considered the solvent of this solution). As the temperature is lowered, the solubility of A in the solution decreases. If the solution is ideal, we can use Eq. (55) with aA = xA to calculate the solubility of A in the solution at temperatures below the freezing point of A ... 

For the ideal solubilities of gases in liquids, a similar approach to that taken in Section 3.1 for the ideal solubilities of solids in liquids can be used and thus, Equation (3.69), analogous to Equation (3.8), is obtained ... 

The present paper deals with the application of the fluctuation theory of solutions to the solubility of poorly soluble drugs in aqueous mixed solvents. The fluctuation theory of ternary solutions is first used to derive an expression for the activity coefficient of a solute at infinite dilution in an ideal mixed solvent and, further, to obtain an equation for the solubility of a poorly soluble solid in an ideal mixed solvent. Finally, this equation is adapted to the solubility of poorly soluble drugs in aqueous mixed solvents by treating the molar volume of the mixed solvent as nonideal and including one adjustable parameter in its expression. The obtained expression was applied to 32 experimental data sets and the results were compared with the three parameter equations available in the literature. 

Inserting expressions (15-17) into Eq. (22) yields the following equation for the solubility of a poorly soluble solid in an ideal mixed solvent ... 

The combination of Eq. (8) with Eqs. (5)-(7) yields an expression for the solubility of a poorly soluble solid in an ideal (n — l)-component mixed solvent in terms of its solubilities in the ideal (n — 2)-component mixed solvents I and II and their molar volumes. 

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