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Real Mixtures and Solutions

The Thermodynamics of Mixing

Activities

For real mixtures, the average energy of the A-B interactions is not the same as that of the A-A and B-B interactions. The chemical potential of the components in a real mixture therefore depends upon both the mole fractions and the identities of all the other components in the mixture.

In general, we write the expressions for the chemical potential of a A in a real mixture in terms of an activity aA

Since

for any mixture, it follows that

In order to maintain consistency with our earlier expressions for the chemical potentials of the components of an ideal mixture

we can also define the activities as

where is an activity coefficient whose value depends upon the mole fraction xA of A in the mixture and the nature and mole fractions of the other component(s). For an ideal solution, . Thus for a real solution, since as the behaviour of the solution becomes more ideal, .

Excess Thermodynamic Quantities

The change in the interactions means that for a real solution the enthalpy change on mixing,

ΔmixHreal ≠ 0.

If there is ordering of the molecules within the mixture, then the entropy of mixing will also be different from that observed for an ideal solution so that

ΔmixSreal ≠ ΔmixSideal

Deviations from ideality can be identified through changes in the values of other quantities upon mixing. For example, if a small amount of tetrachloroethene is added to cyclopentene, the volume of the resulting mixture is found to be less than the total volume of the separate components.

An excess thermodynamic quantity is defined as the difference between the value measured for a real system and that calculated for an equivalent ideal system. Thus

Excess Enthalpy

Experimental observations show that for many liquid mixtures, the excess enthalpy HE varies with composition as

where β is a dimensionless interaction parameter that varies with temperature. Even better agreement between observed and experimental values can be obtained if β is expanded as a power series in terms of (xA - xB).

Cyclooctane Cyclopentane Excess Enthalpy

Variation of excess enthalpy, HE, with composition for a mixture of cyclopentane and cyclooctane at a temperature of T = 273.15 K. The excess enthalpy is negative for all compositions, implying that the attractive interactions between molecules of different species are, on average, stronger than those between molecules of the same species.

Cyclooctane Cyclopentane Excess Volume

Variation of excess volume, VE, with composition for a mixture of cyclopentane and cyclooctane at a temperature of T = 273.15 K. The volume decreases on mixing for all compositions because the components are stabilised on formation of a solution.

Cyclopentance Tetrachloroethene Excess Enthalpy

Variation of excess enthalpy, HE, with composition for a mixture of cyclopentane and tetrachloroethene at a temperature of T = 273.15 K. The excess enthalpy is positive for all compositions, implying that the attractive interactions between molecules of different species are, on average, weaker than those between molecules of the same species.

Cyclopentance Tetrachloroethene Excess Volume

Variation of excess volume, VE, with composition for a mixture of cyclopentane and tetrachloroethene at a temperature of T = 273.15 K. The volume decreases on mixing for dilute solutions of tetrachloroethene in cyclopentane but increases for the formation of dilute solutions of cyclopentane in tetrachloroethene.

Regular Solutions

A regular solution is defined as being one for which HE ≠ 0 but SE = 0. For such a solution,

the A-B interactions stabilise the mixture over the separate components for all compositions.

If then the excess enthalpy is negative and mixing is an exothermic process. Since, for a regular solution, there is no excess entropy so that the entropy of mixing is always positive, the Gibbs free energy of mixing is negative for all compositions. The A-B interactions always stabilise the solution and mixing is spontaneous.

If, in contrast, β > 0 then the excess enthalpy is positive and mixing is an endothermic process; the A-B interactions destabilise the solution. In this case, the miscibility of the two separate components depends upon the balance between the enthalpy of mixing, which favours separation and the entropy, which usually tends to favour mixing. In turn, this depends upon the composition and the magnitude of the interaction parameter β. Components for which β > 0 are only miscible at certain compositions.

A plot of the Gibbs free energy of mixing ΔmixG against composition can be used to determine the compositions over which the two components are miscible at a particular temperature and pressure.

Free Energy of Mixing for a Regular Solution

Plot of Gibbs free energy of mixing against composition for a regular solution with .

Irregular Systems

Mixtures, such as those between methanol and water, for which the excess entropy of mixing is not negligible are called irregular systems. An excess entropy of mixing implies a change in the order of the system. An negative excess entropy might arise because whilst there is no order in the individual components when pure, the two components may interact in a particular way through, for example, hydrogen bonding. Equally, the introduction of another consitituent may destroy the order of the other component, resulting in a positive excess entropy. As a result, the thermodynamic treatment of irregular systems is more complicated than that of regular systems.

Liquid-Liquid Equilibria

Some liquid-liquid mixtures, such as the acetone-carbon disulphide system, are only partially miscible at low temperatures. The interactions between different molecules destabilise the mixture and the system exists as two separate phases for some compositions. These separate phases are known as solutions. At higher temperatures, however, the thermal energy of the molecules overcomes the tendency of the components to separate and the system forms a single fully miscible homogeneous phase for all compositions. Such systems therefore exhibit an upper critical solution or upper consolute temperature. Above this temperature, the mixture is miscible for all compositions; below this temperature, the mixture may separate out into two conjugate phases.

Carbon Disulfide Acetone Liquid Liquid Phase Diagram

Temperature-composition phase diagram for the acetone-carbon disulphide system showing partial miscibility at low temperatures

Some systems, such as mixtures of triethylamine and water are stabilised at low temperatures by the formation of weakly bound complexes. These systems are miscible for all compositions at low temperatures. At higher temperatures, however, the complexes cease to be stable because of the increased thermal energy of the molecules and the mixture exists as separate phases for some compositions. These systems have a lower critical solution, or lower consolute temperature.

Certain systems, such as the nicotine-water system, exhibit both upper and lower critical solution temperatures. At low temperatures, the nicotine and water are thought to interact to form weak, covalently bonded complexes. The chemical interaction stabilises the mixture and the components form a miscible solution for all compositions. At moderate temperatures the complexes are no longer stable and the solution separates into two distinct phases for some compositions. At high temperatures the system once again forms a solution for all compositions as the increased thermal energy of the molecules dominates any tendency for the nicotine and water molecules to separate.

.

Nicotine Water Phase Diagram

Schematic of the high-pressure temperature-composition phase diagram for the water-nicotine system

Liquid-Vapour Equilibria

Ideal-Dilute Solutions

Many liquid mixtures deviate from the ideal behaviour predicted by Raoult's law. Ideal-dilute solutions are those for which the solvent obeys Raoult's law and the solute Henry's law. The difference in behaviour is because the solvent molecules tend to be surrounded by other solvent molecules so that they behave in an almost ideal way. However, the solute molecules tend also to be surrounded by solvent molecules so that their environment and therefore their thermodynamic behaviour is very unlike that of the pure substance.

Acetone Chloroform Partial Pressure Composition

Partial pressure-composition diagram for the ideal-dilute solution formed between acetone and chloroform

Deviations from ideality may have a significant effect on the temperature-composition phase diagram for a mixture. If the interactions between A and B stabilise the liquid phase then the vapour pressure will be depressed below its ideal value and the boiling temperature raised.

Such negative deviations from Raoult's law may even result in a maximum, called an azeotrope, in the temperature-composition phase diagram. When a mixture with the azeotropic composition boils, it produces vapour of exactly the same composition. For all other liquid compositions, however, the composition of the vapour will be different to that of the liquid that remains.

Acetone Chloroform Pressure Composition Phase Diagram

Pressure-composition diagram for the ideal-dilute solution formed between acetone and chloroform showing large negative deviations from Raoult's law

Acetone Chloroform Temperature Composition Phase Diagram

Temperature-composition diagram for the ideal-dilute solution formed between acetone and chloroform showing large negative deviations from Raoult's law

Alternatively, if the A-B interactions destabilise the liquid mixture then the vapour pressure will be increased above its ideal value and the liquid will boil more easily. Positive deviations from Raoult's law may result in an azeotropic minimum in the temperature-composition phase diagram.

Methanol Benzene Pressure Composition Phase Diagram

Pressure-composition diagram for the ideal-dilute solution formed between methanol and benzene, showing large positive deviations from Raoult's law

Methanol Benzene Temperature Composition Phase Diagram

Temperature-composition diagram for the ideal-dilute solution formed between methanol and benzene, showing large positive deviations from Raoult's law

Distillation

If the deviations from ideal behaviour are so great that an azeotrope is formed, then fractional distillation can no longer be used to separate the mixture into the pure components. Instead, fractional distillation of such a mixture yields a pure sample of one of the components and another sample with the azeotropic composition.

Partially Miscible Solutions

The form of the liquid-vapour phase diagram for systems that are only partially miscible in the liquid phase depends on the relative positions of the critical and azeotropic points.

For certain systems, with relatively low upper critical solution temperatures, the liquids become fully miscible before the azeotropic temperature is reached and any vapour is formed.

However, for other systems especially those with relatively high upper critical solution temperatures, two separate liquid phases always exist for certain compositions.

Solid, Solid-Liquid and Solid-Liquid-Vapour Equilibria

The same thermodynamic principles may be used to explain the features of the phase diagrams of solid, solid-liquid and solid-liquid-vapour mixtures. The lowest temperature at which liquid is formed is called the eutectic point.

Reactive Mixtures

The phase diagrams of mixtures where the components may react to form a chemical compound are only slightly more complicated. Some compounds only exist as solids. In these cases the solid compound dissociates into its liquid components on melting in a process known as congruent melting.