In a real solution, the activity coefficient, γ_i, depends on both temperature and composition, but, in an ideal solution, γ_i equals 1 for all components in the mixture. For an ideal binary mixture then, the above equation becomes, for components 1 and 2, y₁P = x₁P₁° and y₂P = x₂P₂°, respectively. Upon adding these equations—recalling that x₁ + x₂ = 1 and y₁ + y₂ = 1—the total pressure, P, is shown to be expressed by the equation P = x₁P₁° + x₂P₂° = x₁[P₁° - P₂°] + P₂°, which is a linear function of x₁.

Assuming γ₁ = γ₂ = 1, equations for y₁P and y₂P express what is commonly known as Raoult’s law, which states that at constant temperature the partial pressure of a component in a liquid mixture is proportional to its mole fraction in that mixture (i.e., each component exerts a pressure that depends directly on the number of its molecules present). It is unfortunate that the word law is associated with this relation, because only very few mixtures behave according to the equations for ideal binary mixtures. In most cases the activity coefficient, γ_i, is not equal to unity. When γ_i is greater than 1, there are positive deviations from Raoult’s law; when γ_i is less than 1, there are negative deviations from Raoult’s law.

An example of a binary system that exhibits positive deviations from Raoult’s law is represented in Figure 3, the partial pressures and the total pressure being related to the liquid-phase composition; if Raoult’s law were valid, all the lines would be straight, as indicated by the dashed lines. As a practical result of these relationships, it is often possible by a series of repeated vaporizations and condensations to separate a liquid mixture into its components, a sequence of steps called fractional distillation.

When the vapour in equilibrium with a liquid mixture has a composition identical to that of the liquid, the mixture is called an azeotrope. It is not possible to separate an azeotropic mixture by fractional distillation because no change in composition is achieved by a series of vaporizations and condensations. Azeotropic mixtures are common. At the azeotropic composition, the total pressure (at constant temperature) is always either a maximum or a minimum with respect to composition, and the boiling temperature (at constant pressure) is always either a minimum or a maximum temperature.

Partial miscibility

Only pairs of liquids that are completely miscible have been considered so far. Many pairs of liquids, however, are only partially miscible in one another, the degree of miscibility often depending strongly on temperature. In most cases, rising temperature produces enhanced solubility, but this is not always so. For example, at 50° C the solubility (weight percent) of n-butyl alcohol in water is 6.5 percent, whereas that of water in n-butyl alcohol is 22.4 percent. At 127° C, the upper consolute temperature, complete miscibility is attained: above 127° C the two liquids mix in all proportions, but below 127° C they show a miscibility gap. Thus, if n-butyl alcohol is added to water at 50° C, there is only one liquid phase until 6.5 weight percent of the mixture is alcohol; when more alcohol is added, a second liquid phase appears the composition of which is 22.4 weight percent water. When sufficient alcohol is present to make the overall composition 77.6 weight percent alcohol, the first phase disappears, and only one liquid phase remains. A qualitatively different example is the system water-triethylamine, which has a lower consolute temperature at 17° C. Below 17° C the two liquids are completely miscible, but at higher temperatures they are only partially miscible. Finally, it is possible, although rare, for a binary system to exhibit both upper and lower consolute temperatures. Above 128° C and below 49° C butyl glycol and water are completely miscible, but between these temperatures they do not mix in all proportions.

Rise in boiling point

It was observed that, whenever one component in a binary solution is present in large excess, the partial pressure of that component is correctly predicted by Raoult’s law, even though the solution may exhibit departures from ideal behaviour in other respects. When Raoult’s law is applied to the solvent of a very dilute solution containing a nonvolatile solute, it is possible to calculate the mole fraction of the solute from an experimental determination of the rise in boiling point that results when the solute is dissolved in the solvent. Since the separate weights of solute and solvent are readily measured, the procedure provides a simple experimental method for the determination of molecular weight. If a weighed amount of a nonvolatile substance, w₂, is dissolved in a weighed amount of a solvent, w₁, at constant pressure, the increase in the boiling temperature, ΔT_b1, the gas constant, R (derived from the gas laws), the heat of vaporization of the pure solvent per unit weight, l₁^vap, and the boiling temperature of pure solvent, T_b1, are related in a simple product of ratios equal to the molecular weight of the solute, M₂. The equation is:

The essence of this technique follows from the observation that, in a dilute solution of a nonvolatile solute, the rise in boiling point is proportional to the number of solute molecules, regardless of their size and mass.

Decrease in freezing point

Another colligative property of solutions is the decrease in the freezing temperature of a solvent that is observed when a small amount of solute is dissolved in that solvent. By reasoning similar to that leading to equation (5), the freezing-point depression, ΔT_f , the freezing temperature of pure solvent, T_{f 1}, the heat of fusion (also called the heat of melting) of pure solvent per unit weight, l₁^fusion, and the weights of solute and solvent in the solution, w₂ and w₁, respectively, are so related as to equal the molecular weight of solute, M₂, in the equation

A well-known practical application of freezing-point depression is provided by adding antifreeze to the cooling water in an automobile’s radiator. Water alone freezes at 0° C, but the freezing temperature decreases appreciably when ethylene glycol is mixed with water.