The activity coefficient with dilution (along with pressure drop in case of gases) tends to 1, i.e. highly dilute solutions are close to ideal ones.

Limited Mutual Solubility of Liquids

If two liquids are close to each other by their properties, for example, hexane and heptane, then the interaction forces between the similar and different molecules do not differ significantly. The formation of solutions from such two liquids is not accompanied by the volume change, and also by the absorption or production of heat. Such solutions are considered to be ideal as well along with highly dilute solutions. The vapor pressure of components over a solution in this case can be expressed as follows (the generalized Raoult’s law):

for a binary solution, where p0i is the partial pressure of saturated vapors of a component over the pure component, pi is the vapor pressure of a component over a solution, and Ni is the component’s molar fraction. It means that the partial vapor pressure of a component over a solution equals its molar fraction in a solution multiplied by the vapor pressure over pure component. Consequently, the total vapor pressure over a binary solution can be expressed by the formula:

Both positive and negative deviations from the Raoult’s law are often observed in practice. This occurs when liquids differing strongly by their nature and properties are mixed together. Intermolecular forces between similar and diverse molecules in the resultant solutions can differ essentially. When diverse molecules attract each other weaker than similar molecules, the mutual dissolution of two liquids weakens intermolecular interactions. Therefore the energy of system increases, and, as a result, the

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transition of molecules from liquid to vapors is facilitated. The vapor pressure over a solution in this case will be higher than would be expected from the Raoult’s law, and positive deviation from the last is observed. On the contrary, negative deviation from the Raoult’s law is observed in the case of strongly interacting diverse molecules.

The Dutch scientist Johannes van Laar suggested the formula uniting both the Raoult’s law and deviations from it:

where E° is the so-called interchange energy taking into account a difference in the interaction of similar and different molecules. Positive deviations from the Raoult’s law take place at E° > 0 and negative ones at

where E1–1 and E2–2 are the interaction energies between the solvent’s molecules and between the solute’s molecules, respectively (the interaction energy between similar molecules), and E1–2 is the interaction energy between the molecules of a solute and solvent (the interaction energy between diverse molecules).

As found upon the analysis of the dependence of the saturated vapor pressure of a component on the E°/RT value (Fig. 3.1.), several values of the concentration, N, of a component correspond to the same p2 value at

E°/RT ≥ 2. It is possible only under the condition that there are an equilibrium between two fluid phases in the system, one of which is a solution of the first component in the second one, and the other phase is a solution of the second component in the first one. The temperature, at which E°/RT becomes equal to 2, is the upper critical temperature, above which liquids unlimitedly dissolve in each other.

Actually, there are pairs of liquids which are almost insoluble in each other, e.g., water and mercury, water and oil, etc. There are also pairs of liquids that are soluble in a limited range of concentrations, e.g., water and butanol, water and aniline, etc. Absolutely insoluble liquids do not exist at all.

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When two liquids with limited solubility are mixed together, in a certain range of temperatures and concentrations they can form a homogeneous phase (solution) without limitations.

In the other temperature and concentration ranges, the system breaks up into two homogeneous phases which are involved in equilibrium. In general, the last systems are heterogeneous, and the composition of each of the equilibrium layers remains constant at a constant temperature. The temperature, above or below which the components show unlimited mutual solubility, is called the critical temperature of dissolution, i.e. the temperature, at which the

compositions of two equilibrium fluid phases are identical. In order to visualize the temperature dependence of the mutual solubility of liquids at a constant pressure, the phase diagrams in the temperature–structure coordinates (solubility diagrams) are plotted.

The following kinds of systems with the limited mutual solubility of liquids exist:

1) The systems with the upper critical temperature of dissolution, e.g., phenol–water, aniline–water, etc. The schematic solubility diagram for such systems is shown in Fig. 3.2. The akb curve on this diagram, called the layering curve (aliquation curve), divides the areas of the existence of heterogeneous (under the aliquation curve) and homogeneous (above the aliquation curve) phases. The compositions of the equilibrium liquid phases are determined by the rule of a connecting straight line: the figurative points corresponding to the composition of the entire system and to the compositions of separate equilibrium phases lie on a straight line named the node. So, for example, the system with the composition, x2(m) (x is the concentration of a component in molar, weight, or volume fractions), indicated on the diagram by the point, m, at the temperature, tm, breaks up

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into the two equilibrium phases, am and bm, with the compositions, x2 (liq1) and x2 (liq2).

The characteristic feature of the layered systems is that upon changing the composition of the whole system the compositions of separate phases in heterogeneous area remain the same at a constant temperature, and only their quantitative ratio, which is defined by the lever rule, varies. An increase in temperature leads to an increase in the

mutual solubility of liquids, i.e. the compositions of two separate liquid phases become close to each other, so, the infinite mutual solubility of liquids is reached at a certain temperature, for example, at tcr = 168°С for the water–aniline system and at tcr = 65,8°С for the phenol–water system (tcr is the critical temperature of dissolution).

2) Systems with the lower critical temperature of dissolution

(triethylamine – water, 2,4,6-trimethylpyridine – water). An increase in temperature in such systems gives rise to a decrease in the mutual solubility of liquids. The full mutual solubility of liquids is observed at temperatures below the critical temperature of dissolution, and the system is biphasic above the critical temperature.

3) Systems with the upper and lower critical temperatures of dissolution

(water – nicotine). There is an area between the two critical temperatures where the system is heterogeneous. For example, the corresponding critical temperatures of dissolution for the water–nicotine system are 208 and

61°С, respectively.

The last two types of systems are much rarer than the systems with the lone upper critical point of dissolution.

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