Properties Relative to Ideal Solutions

In Chapter 4 we used differences and ratios to relate the conceptuals of real substances to those of ideal gases. To compute values for those differences and ratios, we use the equations given in 4.4 together with a volumetric equation of state. Such equations of state are available for many mixtures, particularly gases however, few of those equations reliably correlate properties of condensed-phase mixtures. Although some equations of state reproduce the behavior of condensed phases of complex substances, those equations are complicated and applying them can require considerable computational skill and resources. This is particularly true when we attempt to apply equations of state to mixtures of liquids. 

Therefore we seek ways for computing conceptuals of condensed phases while avoiding the need for volumetric equations of state. One way to proceed is to choose as a basis, not the ideal gas, but some other ideality that is, in some sense, doser to condensed phases. By closer we mean that changes in composition more strongly affect properties than changes in pressure or density. The basis exploited in this chapter is the ideal solution. We still use difference measures and ratio measures, but they will now refer to deviations from an ideal solution, rather than deviations from an ideal gas. 

We start the development in 5.1 by defining ideal solutions and giving expressions for computing their conceptual properties. In 5.2 we introduce the excess properties, which are the differences that measure deviations from ideal-solution behavior, and in 5.3 we show that excess properties can be computed from residual properties. In 5.4 we introduce the activity coefficient, which is the ratio that measures deviations from ideal-solution behavior, and in 5.5 we show that activity coefficients can be computed from fugacity coefficients. This means that deviations from ideal-solution behavior are formally related to deviations from ideal-gas behavior, but in practice, one kind of deviation may be easier to compute than the other. Traditionally, activity coefficients have been correlated by fitting excess-property models to available experimental data simple forms for such models are introduced in 5.6. Those few simple models are enough to allow us to exercise many of the relations presented in this chapter however, more thorough discussions of models for excess properties and activity coefficients must be found elsewhere [1, 2]. 

---

Figure 5.2 Composition dependence of excess properties (relative to Lewis-Randall ideal solutions) in representative binary liquid mixtures, (a) (kyi) n-hexane(l)-cyclohexane(2) at 20°C, (b) (right) chloroform(l)-acetone(2) at 25°C. Note different scales on ordinates. Redrawn from plots in [5].

Figure 5.4 Effect of temperature on the excess properties for liquid mixtures of water(l) and ethanol(2). Note that g IRT is only weakly affected by these changes in T, while h /RT changes sign. Excess properties relative to Lewis-Randall ideal solution. From data tabulated in [7].

Figure 5.5 Excess properties for gaseous mixtures of methane and sulfur hexafluoride at 60°C and 20 bar computed from the virial equation (5.3.3) using (5.3.9)-(5.3.11). Excess properties relative to Lewis-Randall ideal solution (5.1.6).

The column should permit the modulation of retention behavior over a very wide range of conditions. This requirement in fact means that the stationary phase is inert, that is it does not facilitate specific interactions with certain molecular functions of solute molecules with the concomitant advantage of a relatively clean and rapid adsorption-desorption kinetics. Preferably then the stationary phase has no functional groups such as fixed charges that would have strong affinity to counterionic solutes and exclude solutes of co-ionic nature. In this regard the properties of well-prepared hydrocarbonaceous bonded phases indeed approach those that we would expect from an ideal phase. 

A general formulation of the problem of solid-liquid phase equilibrium in quaternary systems was presented and required the evaluation of two thermodynamic quantities, By and Ty. Four methods for calculating Gy from experimental data were suggested. With these methods, reliable values of Gy for most compound semiconductors could be determined. The term Ty involves the deviation of the liquid solution from ideal behavior relative to that in the solid. This term is less important than the individual activity coefficients because of a partial cancellation of the composition and temperature dependence of the individual activity coefficients. The thermodynamic data base available for liquid mixtures is far more extensive than that for solid solutions. Future work aimed at measurement of solid-mixture properties would be helpful in identifying miscibility limits and their relation to LPE as a problem of constrained equilibrium. 

The inclusion of activity coefficients into the simple equations was briefly considered by Purlee (1959) but his discussion fails to draw attention to the distinction between the transfer effect and the activity coefficient (y) which expresses the non-ideal concentration-dependence of the activity of solute species (defined relative to a standard state having the properties of the infinitely dilute solution in a given solvent). This solvent isotope effect on activity coefficients y is a much less important problem than the transfer effect, at least for fairly dilute solutions. For example, we have already mentioned (Section IA) that the nearequality of the dielectric constants of H20 and D20 ensures that mean activity coefficients y of electrolytes are almost the same in the two solvents over the concentration range in which the Debye-Hiickel limiting law applies. For 0-05 m solutions of HC1 the difference is within 0-1% and thus entirely negligible in the present context. Of course, more sizeable differences appear if concentrations are based on the molality scale (Gary et ah, 1964a) (see Section IA). 

The aqueous activity coefficient of a compound describes the effect of molecular structure on aqueous solubility. If a compound mixes with water and forms an ideal solution, then the activity coefficient would be taken as unity and the last term in Eq. (1) would become zero. In such a case, then the solid phase, if any, would be the sole physical property inhibiting solubility. Most drugs are relatively nonpolar and do not form ideal solutions with water. Therefore in order to understand the extent by which the inherent molecular structure is limiting solubility, it is helpful to obtain an estimate for the aqueous activity coefficient. 

The standard state of a substance is a reference state that allows us to obtain relative values of such thermodynamic quantities as free energy, activity, enthalpy, and entropy. All substances are assigned unit activity in their standard state. For gases, the standard state has the properties of an ideal gas, but at one atmosphere pressure. It is thus said to be a hypothetical state. For pure liquids and solvents, the standard states are real states and are the pure substances at a specified temperature and pressure. For solutes In dilute solution, the standard state is a hypothetical state that has the properties of an infinitely dilute solute, but at unit concentration (molarity, molality, or mole fraction). The standard state of a solid is a real state and is the pure solid in its most stable crystalline form. 

---