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Henry’s Law states that, at a constant temperature, the partial pressure of a gaseous species in equilibrium with its liquid form is directly proportional to its mole fraction in dilute solutions. Mathematically, Henry’s Law can be expressed as:

where:

is the partial pressure of the gas,
is the Henry’s Law constant (which varies with temperature),
is the mole fraction of the gas in the liquid.

The law was empirically derived in 1803 by William Henry, an English chemist. While Raoult’s law describes how the vapour pressure of a solvent is lowered when a non-volatile solute is added to an ideal solution, Henry’s law is primarily used to predict the solubility of a gas in real solutions, with important applications in environmental science, chemical engineering and biology. In other words, both the solute and solvent obey Raoult’s law in an ideal solution, while the solute follows Henry’s law in a real dilute solution (see diagram below). This is because, in a real dilute solution, each solute molecule is almost exclusively surrounded by solvent molecules, and its behaviour is determined by interactions with the solvent, not with other solute molecules. The proportionality constant  reflects these solute-solvent interactions.

Despite being an empirical law, Henry’s law can be derived using basic thermodynamics principles. At equilibrium, the chemical potential of the gas in the gas phase must be equal to its chemical potential in the solution. Rearranging eq182, where is defined as 1 atm (the convention for the standard state of gases), gives

Since the exponential factor is a constant for a given temperature, eq188 is equivalent to eq187, with .

A solute-solvent system in which the mole fraction of the solvent is close to 1, and the two species have different intermolecular interactions, is called an ideal-dilute solution (see diagram above). In such a solution, the solvent obeys Raoult’s law (ideal) and the solute follows Henry’s law (dilute). The distinct behaviours exhibited by the solute and solvent in such solutions can be attributed to fundamental differences in their molecular environments.

In a dilute solution, solvent molecules are predominantly surrounded by other solvent molecules. As a result, their immediate molecular environment closely approximates that of the pure liquid solvent. This preservation of their native environment explains why the solvent’s vapour pressure continues to align with Raoult’s law, which describes the vapour pressure of an ideal pure component.

Conversely, solute molecules in a dilute solution are almost exclusively surrounded by solvent molecules. This marks a significant departure from their pure state, where they would typically be surrounded solely by other solute molecules. Due to this altered molecular environment, the solute’s behaviour, particularly its partial vapour pressure, deviates considerably from what might be expected from its pure form. This distinct behaviour is precisely what Henry’s law describes.

Therefore, the solvent behaves effectively as a slightly perturbed pure liquid, while the solute exhibits fundamentally different behaviour. Many naturally occuring systems are ideal-dilute solutions. For example, the small amount of dissolved oxygen in natural waters or blood follows Henry’s Law, while the water (solvent) makes up the majority and its vapour pressure is essentially determined by Raoult’s Law. In carbonated drinks, the dissolution of CO₂ is well-described by Henry’s Law, while the water acts as the Raoultian solvent.

 

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Pressure-composition diagrams are graphical representations that illustrate the relationship between the vapour pressures of components in a liquid mixture and their compositions at a constant temperature. These diagrams are essential tools in physical chemistry and chemical engineering, particularly for understanding phase equilibria in binary mixtures.

 

Ideal solution

Consider two liquids A and B (e.g. benzene and toluene) forming an ideal solution in a closed container (e.g. a container with a movable piston) at a constant temperature above the freezing points of both species. The partial pressures of the components follow Raoult’s law:

where

is the partial vapour pressure of component
is the vapour pressure of pure
is the mole fraction of in the solution

The total vapour pressure of the ideal solution is:

According to Dalton’s law, the mole fractions and of the components in the gas are

Substituting eq189 and eq190 in eq191 results in

Substituting in eq189 into in eq191 gives

Substituting eq193 in of eq192 and rearranging yields

Eq190 and eq194 are plotted in a graph of against to produce the pressure-composition diagram of the ideal solution (see diagram above). Eq190, also known as the liquid composition curve, describes a straight line (blue) along which the liquid begins to vaporise. In other words, the high pressure region above this line corresponds to the liquid phase of the solution. In contrast, eq194 defines the curve (red) along which the last drop of the liquid vaporises. The region below this curve, also called the vapour composition curve, represents the vapour phase. The area between eq190 and eq194 marks the two-phase region where liquid and vapour coexist in equilibrium.

 

 

Consider the process of isothermally lowering the pressure from point a to i (see diagram above), which can be achieved by drawing out the piston. At point a, the pressure is above eq190 at . Under these conditions, the system is entirely in the liquid phase. As the pressure is reduced to point b, the system reaches the liquid composition curve. Here, the first infinitesimal amount of vapour forms, and the system enters a two-phase equilibrium. At this point, the liquid composition is still equal to the overall composition (), but the vapour phase has a different composition corresponding to point f (). A tie-line bf on the diagram connects these two phase compositions, indicating that both phases coexist in equilibrium.

Continuing to lower the pressure brings the system to point e, which lies between the two curves. This is the two-phase region, where both liquid and vapour coexist in significant amounts. The overall composition remains fixed at , but the compositions of the individual phases are (point c) and (point g) for the liquid phase and vapour phase respectively.

​When the system reaches point h on the vapour composition curve, the last drop of liquid evaporates. At this point, the liquid phase composition is (), and the vapour phase composition matches the overall composition (). Finally, as the pressure is reduced even further to point i, the system moves below the vapour composition curve, where it exists as a single-phase vapour. No liquid remains and the composition remains constant at . The line abehi is called an isopleth.

 

 

Interestingly, even though the compositions of the phases are fixed at a given pressure (e.g. and at ), the overall mole fraction of A can vary continuously between them, depending on the relative amounts of each phase. To understand this, let

be the number of moles of A in the liquid phase
be the number of moles of A in the vapour phase
be the total number of moles of A and B in the liquid phase
be the total number of moles of A and B in the vapour phase
be the total number of moles of A and B in the closed container

Clearly,

Substituting eq195 and eq196 into eq197 and rearranging gives

Since , the range of is , which corresponds to the domain of a tie-line at a particular pressure and temperature. Multiplying both sides of eq198 by  and rearranging yields

Substituting into eq199 and rearranging results in the lever rule:

Here, represents the length from the left end of a tie-line to the intersection of the tie-line and with the isopleth, while represents the length from that intersection to the right end of the tie-line. Eq200 allows us to determine the relative amounts of the two phases in equilibrium by measuring these two lengths. This principle has important applications in material science and chemical engineering — for instance, in predicting phase amounts, which is critical for the design of distillation columns.

 

Non-ideal solution

Pressure-composition diagrams for non-ideal binary solutions typically exhibit two characteristic shapes, depending on the degree and nature of deviation from Raoult’s law. When the deviation is small, the liquid composition curve is monotonic, with no stationary points. In such cases, the diagram features two smooth curves that intersect only at and (see diagram below). A typical example is the carbon tetrachloride-toluene system, where carbon tetrachloride is the more volatile component.

In contrast, when the deviation from Raoult’s law is significant, the liquid composition curve may develop a stationary point. The nature of this point is governed by the intermolecular interactions between components A and B. A maximum point occurs when A-B interactions are weaker than A-A and B-B interactions, while a minimum point arises when A-B interactions are stronger than A-A and B-B interactions.

Now consider a liquid composition curve with a maximum point. Suppose the corresponding vapour composition curve intersects the liquid composition curve only at and . In this case, a tie-line drawn just below the maximum point would connect two points on the liquid composition curve (see diagram above), implying two coexisting liquid phases at different compositions — a physical impossibility for a binary system in vapour–liquid equilibrium. This contradiction indicates that the vapour composition curve must also touch the liquid composition curve at the maximum point, as shown in the diagram below. A solution that behaves this way is called an azeotrope. The ethanol–water system is an example of an azeotrope with a maximum point, while the HCl–water system forms an azeotrope with a minimum point. These diagrams are typically constructed using empirical data.

 

Distillation

A rotary evaporator (see diagram below) separates the components of a binary mixture by reducing the pressure above the mixture with a vacuum pump. Pressure-composition diagrams can help predict the operating pressure required for distilling the more volatile component when using a rotary evaporator.

Pressure-composition diagrams are also used in flash distillation, which involves the partial vaporisation of a liquid mixture to separate its components based on differences in volatility. In a flash distillation process, a pre-heated liquid mixture is introduced into a flash drum or separator operating at a reduced pressure (see diagram below). Upon entry, a portion of the liquid “flashes” into vapour due to the sudden pressure drop. This generates two phases in equilibrium: a vapour phase enriched in the more volatile component, and a liquid phase enriched in the less volatile one. These two phases are then physically separated, with the vapour removed at the top and the liquid removed at the bottom. The phases can undergo further distillation, depending on the desired separation efficiency and purity of the components.

Pressure-composition diagrams are crucial in this context because they show how the equilibrium compositions of the vapour and liquid phases vary with pressure at a fixed temperature. These diagrams help determine the operating conditions (such as temperature and pressure) at which the desired separation can occur, as well as predict the relative amounts of vapour and liquid formed.

Flash distillation is widely used in the chemical and petroleum industries — for example, in crude oil refining — where it serves as a rapid, energy-efficient method for achieving partial separation of multi-component mixtures without the complexity of a full distillation column.

 

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Colligative properties are physical properties of solutions that depend solely on the number of solute particles present, not on their chemical identity. The term “colligative” comes from the Latin word colligatus, meaning “bound together”, reflecting the idea that these properties arise from the collective presence of solute particles, regardless of their type, size or chemical nature.

The classical treatment of colligative properties rests on two key assumptions:

• • The solute is non-volatile — only the solvent contributes to the vapour phase, so any change in vapour pressure (and related properties) comes solely from the dilution of the solvent.
  • The solution behaves ideally — solute and solvent molecules interact only through simple physical mixing, with no strong intermolecular forces altering the behaviour of the components. Dilute non-ideal solutions behave nearly ideally at low solute concentrations and also exhibit colligative properties.

The four principal colligative properties are vapour pressure lowering, boiling point elevation, freezing point depression and osmotic pressure. As vapour pressure lowering has been discussed in the article on Raoult’s law, we will elaborate on the remaining properties.

 

Boiling point elevation

The normal boiling point of a pure liquid or solution is the temperature at which its vapour pressure equals 1 atm. According to Raoult’s law, the addition of a non-volatile solute to a pure liquid lowers the vapour pressure of the solvent. As a result, a higher temperature is required for the solution’s vapour pressure to reach 1 atm. Therefore, the normal boiling point of the solution is elevated relative to that of the pure solvent.

To explain this mathematically, consider a volatile solvent and a nonvolatile solute in a solution at equilibrium with the gaseous solvent at a constant pressure. Raoult’s Law states that at any given temperature, the vapour pressure of the solvent in the solution is related to the vapour pressure of the pure solvent at that same temperature. Using eq178 at the solution’s boiling point , we have

where is the external pressure and denotes the vapour pressure of the solution at .

Since the pure solvent boils at , we have , which when combined with eq189 gives

As the Clausius-Clapeyron equation describes how the vapour pressure of a substance changes with temperature, we can express eq169 as:

Substituting eq190 into eq191 yields

For dilute solutions, . So, the Taylor expansion of becomes , and

where .

Assuming that for a dilute solution and is independent of temperature, eq192 becomes

where .

 

 

 

Freezing point depression

Freezing is the process by which a liquid transforms into a solid with an ordered, crystalline structure. When a non-volatile solute is added to the solvent, it effectively dilutes the solvent and increases the entropy of the solution. The reduction in the concentration of solvent molecules, along with the enhanced molecular randomness, opposes the tendency of the solution to freeze (freezing lowers the entropy of the system). As a result, a lower temperature is required for freezing to occur.

Consider a pure solid (A) at equilibrium with a solution containing the dilute solute (B). Since the chemical potentials of A in the two phases are equal at equilibrium, eq181 becomes

Dividing eq194 by and differentiating it with respect to at constant gives

 

 

Therefore, eq195 becomes , or equivalently,

where .

Assuming is independent of temperature and integrating both sides of eq196 with respect to yields,

The limits of integration represent a transition from the normal freezing temperature of pure solvent A (where ) to the depressed freezing point of the solution (where ). Therefore,

For dilute solutions, . So, the Taylor expansion of and

Assuming that for a dilute solution,

where is the freezing point depression and .

 

Osmotic pressure

Osmosis is the tendency of pure solvent molecules A to diffuse across a semipermeable membrane into a solution of A and B. Consider two equal chambers separated by a rigid, thermally conducting, semipermeable membrane that allows only molecules of solvent A to pass through, but not solute B (see diagram below). In the left chamber is pure solvent A. In the right chamber, we have a solution of B in A. Initially, the heights of the liquids in the two capillary tubes connected to the chambers are equal, so the pressures in both chambers are the same (). Furthermore, the membrane conducts heat and thermal equilibrium is maintained.

The chemical potential of pure solvent A in the left chamber is denoted by . In the right chamber, the presence of solute B lowers the chemical potential of A, so . The resulting chemical potential difference indicates that the system, being out of equilibrium, will spontaneously evolve to minimise its Gibbs energy. This is achieved by solvent A flowing from the pure solvent side through the semipermeable membrane into the solution, thereby diluting the solute and raising the chemical potential of A in the right chamber.

As solvent flows into the right chamber, the liquid level in its capillary tube rises, increasing the hydrostatic pressure in that chamber. This pressure buildup continues until the chemical potentials of the solvent in both chambers are equal. At that point, equilibrium is restored and the additional pressure  generated in the right chamber is called the osmotic pressure. The magnitude of this pressure reflects the extent of osmosis that has occurred in the system and enables quantification of that extent.

When equilibrium is restored, . Using eq181, we have

Integrating eq173 at constant , we have  or

Substituting eq198 into eq199 yields

For dilute solutions, . So, the Taylor expansion of and

Substituting and for dilute solutions, and in eq200 gives

Eq201 is called the van’t Hoff law, which describes the osmotic pressure of an ideal-dilute solution. For non-ideal solutions, the osmotic pressure can be approximated using a power series expansion analogous to the virial expansion for real gases:

where , are osmotic virial coefficients that capture non-ideal interactions.

 

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Raoult’s Law states that the partial vapour pressure of each volatile component in an ideal solution at equilibrium is equal to the product of its mole fraction in the liquid phase and the vapour pressure of the pure component at the same temperature. Mathematically:

where:

is the partial vapour pressure of component
is the mole fraction of component  in the liquid
is the vapour pressure of pure component

It is named after François-Marie Raoult, a French chemist who formulated it in the 1880s. Through his experiments on the vapour pressures of solutions, Raoult discovered that the presence of a non-volatile solute lowers the vapour pressure of the solvent in proportion to its mole fraction. His work laid the groundwork for colligative properties and was pivotal in the development of solution thermodynamics.

Let’s explain Raoult’s law in an intuitive way before we derive it from thermodynamic principles. Consider a sealed container with a pure volatile liquid A (see diagram below). At a given temperature, some molecules of A evaporate from the liquid surface into the gas phase, while other molecules of A from the gas phase condense back into the liquid. Eventually, a dynamic equilibrium is established, where the rate of evaporation equals the rate of condensation. The pressure exerted by the gaseous A at this equilibrium is the vapour pressure of pure A, .

If we add another volatile liquid component B to the container, forming an ideal solution (where A and B have similar intermolecular interactions), the surface of the liquid is now shared by molecules of both A and B. This means that, for any given area of the liquid surface, fewer molecules of A can escape into the vapour phase than when A was pure. When a new equilibrium is attained, the reduced partial pressure of A is expected to be equal to the product of its mole fraction in the liquid and . The same is true for B. This is the essence of Raoult’s law. The total vapour pressure in the container is then the sum of the partial pressures of A and B:

Similarly, if we add a non-volatile solute to a container of pure A to form an ideal solution, . However, the total vapour pressure is now equal to just , because the solute is non-volatile. Therefore, Raoult’s law can also be defined as follows: the vapour pressure of a solvent in a solution is directly proportional to its mole fraction in the liquid phase. In other words, when a non-volatile solute is added to a volatile solvent, the solvent’s vapour pressure decreases in proportion to the amount of solute present: .

Raoult’s law can be derived from thermodynamic principles. Consider a multi-component solution with a volatile liquid A at equilibrium with its vapour. At constant temperature eq148 becomes or its molar form . Substituting the ideal gas law in the molar form and integrating gives:

which leads to:

Substituting eq149a in eq179 yields

The equivalent expression for the chemical potential of the liquid-phase of A is

where is the mole fraction of A in the solution (see below for a complete derivation of eq181).

When a liquid and its vapour are in equilibrium, their chemical potentials are equal. Hence, , or

which rearranges to give the Raoult’s law because .

 

 

Raoult’s Law is most accurate when applied to ideal solutions, where the intermolecular forces between unlike molecules closely resemble those between like molecules. A classic example is the binary mixture of benzene and toluene, which closely follows Raoult’s Law. When the vapour pressure of toluene is plotted against its mole fraction in the solution, the resulting graph is a straight line, consistent with eq178. A similar linear plot can be made for benzene, using the inverse relationship between its mole fraction and that of toluene. The total vapor pressure of the ideal solution is obtained by summing the partial vapour pressures of both components, as illustrated in the diagram below.

However, the law fails or becomes less accurate for non-ideal solutions, in which the components interact more strongly or more weakly with each other than with themselves. An example is the acetone-chloroform system, where hydrogen bonding causes deviations from ideal behaviour (see diagram below).

Vapour pressure diagrams allow for a straightforward determination of the partial vapour pressure of each component and the total vapour pressure of the solution at a specific temperature. They also provide a basis for plotting boiling point diagrams and ultimately phase diagrams of mixtures.

 

 

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Temperature-composition diagrams are graphical representations that show the relationship between the boiling point (temperature) of a liquid mixture and the composition of the mixture at a constant pressure.

 

Ideal solution

Consider two liquids A and B (e.g. benzene and toluene) forming an ideal solution in a closed container at a constant pressure . The partial pressures of the components follow Raoult’s law:

where

is the partial vapour pressure of component
is the vapour pressure of pure as a function of
is the mole fraction of in the solution

The total vapour pressure of the ideal solution is equal to the fixed pressure: , which rearranges to

Using known values of at different , eq201 is used to plot the liquid composition curve for a graph of against (see diagram below). Substituting eq201 into yields

Eq202 describes the vapour composition curve. Unlike the pressure-composition diagram of an ideal solution, the vapour composition curve lies above the liquid composition curve (also known as the boiling point curve). The vapour phase corresponds to the region above the vapour composition curve, while the liquid phase lies below the liquid composition curve. Furthermore, neither curve is linear.

 

 

Similar to pressure-composition diagrams, the horizontal axis, after plotting the two curves, is relabelled as , representing the overall mole fraction of A in the system, including both the liquid and vapour phases. Consider the process of isobarically increasing the temperature from point a to i (see diagram above). At point a, the system is entirely in the liquid phase. As the temperature is increased to point b, the system reaches the liquid composition curve. Here, the first infinitesimal amount of vapour forms, and the system enters a two-phase equilibrium. At this point, the liquid composition is still equal to the overall composition (), but the vapour phase has a different composition corresponding to point f (). A tie-line bf on the diagram connects these two phase compositions, indicating that both phases coexist in equilibrium. The lever rule, as discussed in the article on pressure-composition diagrams, can likewise be used to determine the relative equilibrium amounts of the two phases present at any stage of heating, provided the temperature and overall composition fall between the liquid and vapour lines.

Continuing to increase the temperature brings the system to point e, which lies between the two curves. This is the two-phase region, where both liquid and vapour coexist in significant amounts. The overall composition remains fixed at , but the compositions of the individual phases are (point c) and (point g) for the liquid phase and vapour phase respectively.

 

 

When the system reaches point h on the vapour composition curve, the last drop of liquid evaporates. At this point, the liquid phase composition is (), and the vapour phase composition matches the overall composition (). Finally, as the temperature is increased even further to point i, the system exists as a single-phase vapour. No liquid remains and the composition remains constant at . The line abehi is called an isopleth.

 

 

 

Non-ideal solution

Temperature-composition diagrams for non-ideal binary solutions typically exhibit two characteristic shapes, depending on the degree and nature of deviation from Raoult’s law. When the deviation is small, the diagram features two smooth monotonic curves intersecting only at and (see diagram below). A typical example is the diethyl ether–ethanol system. Such a diagram has the same structure as that of an ideal solution, but differs in the curvature of the curves.

In contrast, when the deviation from Raoult’s law is significant, the curves may develop a stationary point. The nature of this point is governed by the intermolecular interactions between components A and B. When A-B interactions are weaker than A-A and B-B interactions, a minimum point may occur, as the mixture exhibits higher vapour pressures than predicted, resulting in lower boiling points. Conversely, if A-B interactions are stronger, a maximum point may form due to lower vapour pressures, which lead to higher boiling points than expected. Mixtures with such characteristics are called azeotropes. The ethanol–water system is an example of an azeotrope with a minimum point, while the HCl–water system has a maximum point (see diagram below). These diagrams are typically constructed using empirical data.

 

 

 

Distillation

Temperature-composition diagrams are essential tools in understanding and designing simple and fractional distillation processes.

Consider the ethanol–water system (see above diagram). In simple distillation, the mixture is heated until it reaches its boiling point and begins to vaporise. At this stage, the vapour composition of ethanol is (point b), which is richer in ethanol. This vapour is then condensed and collected as the distillate, effectively separating it from the original mixture. Since this process involves just a single equilibrium between the liquid and vapour phases, it is typically considered a one-step separation. By using the diagram, one can estimate how effective the process will be for a single distillation step. However, simple distillation has its limitations. It provides only a partial enrichment of the more volatile component. For mixtures requiring higher purity, more advanced techniques like fractional distillation are necessary.

The fractional distillation apparatus used in the laboratory includes a fractionating column filled with glass beads, which provide a large surface area for repeated condensation and vaporisation cycles (see diagram above). Suppose the initial overall mole fraction of ethanol in the mixture is (with reference to the minimum azeotrope diagram above). When the temperature reaches , the vapour in equilibrium with the liquid has an ethanol composition of , which is richer in ethanol due to its greater volatility. As this vapour rises through the fractionating column, it encounters a cooler region at temperature , where part of it condenses into a liquid with the same composition . The remaining uncondensed vapour becomes even more enriched in ethanol, now with composition . This ethanol-riched vapour continues to ascend, repeating the cycle: it partially condenses into liquid of composition , and leaves behind a vapour of even higher ethanol content, .

The condensation and vaporisation cycle continues as long as the column is tall enough to provide sufficient equilibrium stages. Eventually, the vapour composition reaches , which corresponds to the azeotropic point — the minimum boiling point of the ethanol–water system. At this composition, the mixture behaves like a pure substance and boils at a constant temperature, . No further separation by distillation is possible beyond this point. Each individual equilibrium step in the fractional distillation process is referred to as a theoretical plate. For an ethanol-water mixture to reach azeotropic purity, typically 8 to 10 theoretical plates are needed. This can be achieved in the lab using a well-packed fractionating column of 50 to 70 cm in height.

At the top of the column, the azeotropic vapour is directed into the condenser, while the condensed liquid within the column flows back down towards the boiling flask, diluting the boiling mixture. The final composition of the distilled ethanol depends on this azeotropic limit, which is approximately 95.6% ethanol by mass at atmospheric pressure ().

 

 

Temperature-composition diagrams are essential tools in understanding and optimising fractional distillation, particularly in complex mixtures like petroleum. In an industrial fractional distillation column (see above diagram), crude oil is first heated in a furnace until it partially vaporises. The resulting vapour enters the base of the column (reboiler section) and rises through a series of trays or packing. Components with lower boiling points condense on trays higher up in the column (where temperatures are cooler), while components with higher boiling points condense on trays lower down (where it is hotter).

Although temperature–composition diagrams are typically illustrated for binary mixtures, the same principles can be conceptually applied to the multi-component mixture of crude oil. At different heights in the column, the mixture can be approximated as a “pseudo-binary” system, where one component (A) represents a heavier fraction and the other component (B) represents a lighter fraction. At a given temperature (i.e. at a particular tray), the vapour phase will be richer in the more volatile component B (lower boiling point), while the liquid phase will be richer in component A (higher boiling point). This vapour–liquid equilibrium process repeats over many stages, progressively enriching the vapour in more volatile components and enabling effective separation based on differences in volatility.

 

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Liquid-liquid phase diagrams are graphical representations that show the conditions under which two (or more) partially or fully immiscible liquids coexist in equilibrium. The most common type of liquid-liquid phase diagram is the temperature-composition diagram.

A typical temperature-composition diagram of two partially miscible liquids features a binodal curve (with two nodes at the ends of a tie-line) that divides the diagram into two regions (see diagram below). The region outside the curve corresponds to a single liquid phase, while the area enclosed by the curve represents a two-phase region, where the liquids separate into two immiscible layers

To interpret this diagram, consider starting at point e, which represents pure liquid A (hexane) at a fixed temperature and 1 atm. As small amounts of liquid B (nitrobenzene) are gradually added, it dissolves completely in A to form a single-phase solution between e and f. Upon further addition of B, the system reaches the limit of mutual solubility, and phase separation occurs (between points f and g, the system exists as two liquid phases). Along the tie line connecting f and g, each end corresponds to the composition of one of the two coexisting phases. A mixture composition closer to f indicates that the phase richer in A is more abundant, while the phase richer in B (composition at g) is less abundant. The opposite holds for mixtures closer to g. The lever rule can be applied to determine the relative amounts of each phase. At point g, enough B has been added to dissolve all of A, resulting in a saturated single-phase solution of A in B. Further addition of B beyond this point leads to continuous dilution of A, and the system remains in the single-phase region until pure B is reached at point h, where .

A liquid-liquid phase diagram like the one above is constructed with empirical data. It is interesting to note that the compositions of the two phases at equilibrium for the hexane-nitrobenzene system vary with temperature. The narrowing of the curve at higher temperatures indicates an increasing miscibility of the two components as temperature rises. The upper critical solution temperature is the temperature above which the system exists as a single phase, with the two liquids miscible in all proportions.

Some binary liquid systems exhibit greater miscibility as temperature decreases (see diagram below). An example is the water–triethylamine system, in which the two components form a weak complex at lower temperatures. In such cases, a lower critical solution temperature marks the temperature below which the system becomes fully miscible and exists as a single phase.

Occasionally, a system displays a combination of both behaviours, exhibiting both an upper and a lower critical solution temperature (see diagram above). In such systems, the components are completely miscible at both high and low temperatures, but become partially miscible over an intermediate temperature range. This results in a closed-loop miscibility gap on the temperature–composition diagram, bounded above by the upper critical solution temperature and below by the lower critical solution temperature. The unusual phase behaviour is typically due to competing molecular interactions, such as complex formation at low temperatures and thermal disruption at higher temperatures. Examples include nicotine–water and m-toluidine–glycerol systems.

Liquid–liquid diagrams are commonly used in extractive distillation processes, where the goal is to exploit both vapour–liquid equilibrium and liquid–liquid immiscibility to achieve better separation. The diagram above shows a mixture of two partially miscible liquids that form a low-boiling azeotrope. An example is the water–isobutanol system. If a water–isobutanol mixture of composition at point a is heated from to , the resulting vapour, upon cooling to will condense into a distillate consisting of two phases: one with composition and the other .

Beyond separation, liquid–liquid diagrams also play a crucial role in other scientific fields. In polymer blends and materials engineering, they help predict phase compatibility and guide the design of homogeneous or phase-separated materials with tailored properties. In food science, liquid–liquid behaviour underpins the formation and stabilisation of oil–water emulsions, which are essential in products like dressings, creams and sauces. Similarly, in environmental science, such diagrams aid in understanding the partitioning of pollutants between aqueous and organic phases, which is key to modelling contaminant transport and designing remediation strategies.

 

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Solid-liquid phase diagrams illustrate the relationship between temperature and composition in mixtures as they transition between solid and liquid states. These diagrams are essential for understanding melting behavior, phase equilibrium and solubility in binary or multi-component systems. They typically feature regions representing pure solid, pure liquid, and a mixture of both, separated by boundary lines such as the liquidus (liquid composition curve) and solidus (solid composition curve).

 

Ideal liquid solution and ideal solid solution

For instance, the silicon-germanium solid-liquid temperature-composition diagram resembles the liquid–vapour diagram of an ideal liquid solution (see above diagram). This similarity arises because the two elements form both an ideal solid solution and an ideal liquid solution. The region below the solidus corresponds to a single-phase solid solution, while the area above the liquidus represents a single-phase liquid solution. Between these two curves, the solid and liquid phases coexist in equilibrium, and the lever rule can be applied to determine the relative amounts of each phase at a given temperature.

 

 

 

Eutectics

The temperature-composition diagram of the gold-copper system is shown in the diagram below. In this system, gold and copper are completely miscible in both the liquid and solid phases, but they do not form an ideal solid solution due to differences in atomic interactions. The diagram features a eutectic point and resembles a minimum azeotrope in shape. The eutectic point corresponds to the lowest melting temperature of the alloy, occurring at a specific composition known as the eutectic composition. At this point, the homogeneous liquid solidifies into a single solid phase with a well-defined composition. Because the eutectic temperature is lower than the melting points of either pure component, this point is particularly useful in applications such as casting and welding. The term “eutectic” was coined in 1884 by British physicist and chemist Frederick Guthrie, derived from the Greek words eu (“well” or “good”) and têxis (“melting”), meaning “easily melted.”

Unlike the gold-copper system, the silver-copper system features complete miscibility of its components in the liquid phase but only partial miscibility in the solid phase. Based on the gold-copper phase diagram, we would expect the solid solution region to divide into multiple regions due to the limited solid-state solubility of silver and copper at lower temperatures (see diagram below).

Indeed, the temperature-composition diagram of the silver-copper system includes a single-phase liquid solution region (), two single-phase solid solution regions ( and ), a two-phase solid region (), and two two-phase regions where a liquid solution is in equilibrium with a solid solution ( and ). The eutectic point k, at 779^oC, represents the equilibrium state where the solid solutions and coexist with the liquid phase .

When a liquid solution with composition at point d is cooled, the solid solution phase begins to precipitate at point e, with its composition corresponding to point h on the tie-line. As the two-phase mixtrure of cools to point f, it reaches the eutectic composition. At this stage, the solid solution also begins to form.

The Ag–Cu phase diagram is a powerful tool for understanding, designing and processing silver–copper alloys. It helps engineers to select the right alloy composition for specific properties (e.g. ductility and conductivity), and guide thermal treatments for practical applications in electronics, jewelry and joining technologies.

 

 

Compound formation (congruent melting)

Consider two substances, A and B, that react to produce a compound C in a 1:1 ratio. If C separately forms eutectic mixtures with both A and B, the solid-liquid phase diagram of A and B (e.g. aniline and phenol) will appear as follows:

Aniline and phenol combine to form an adduct , which is a stable complex with its own characteristic melting point and crystal structure distinct from the individual components. Aniline and phenol are miscible in the liquid phase at higher temperatures, while the adduct does not exist in the liquid state. Furthermore, the solid forms of all three species are completely insoluble in one another. As a result, solid phenol coexists with the solid adduct in a two-phase region between at lower temperatures. When , all phenol molecules will react with aniline, leaving a two-phase region consisting of solid adduct and solid aniline. In other words, the maximum amount of adduct is formed when . Since the adduct forms eutectic mixtures with both aniline and phenol, the phase diagram can be viewed as two eutectic phase diagrams positioned side by side. Points a and b correspond to the eutectic points of the phenol-adduct and the adduct-aniline mixtures respectively.

Consider the isopleth defg. As the mixture cools from point e to point g, negligible amounts of phenol will be present, meaning that the tie-lines extend only from onwards. On the other hand, if a solution with is cooled, only pure solid adduct will separate out, without any change in composition. If the process is reversed, the liquid formed will have the same composition as the solid. This phenomenon is known as congruent melting.

In pharmaceutical and chemical industries, the diagram can help in purification processes or the synthesis of the adduct of aniline and phenol, which is a valuable intermediate in the production of various compounds, such as dyes, drugs, and plastics

 

Compound formation (incongruent melting)

Some stable solids of the form A₂B, each melts to give a liquid with a different composition. We call such a process incongruent melting. The phase diagram of an A₂B alloy resembles that of the aniline-phenol system, except that the melting point of one component is much higher the other (see diagram below).

The vertical line at in the phase diagram marks the composition of the pure solid compound A₂B. When the alloy at is heated, it begins to melt at temperature , producing pure solid A and a single-phase liquid containing both A and B. The resulting liquid has a mole fraction of A () that differs from that of A₂B.

An example of such a system is the Na₂K alloy, which is used in liquid metal coolants and metallic lubricants. Understanding the melting behavior of Na₂K is important for selecting suitable compositions in these applications. Although the phase diagram shows the presence of Na₂K, it is still a binary system diagram for the Na-K system. The compound Na₂K is simply an intermetallic phase that appears as part of the equilibrium between Na and K. In contrast, ternary phase diagrams involve three independent components.

 

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A ternary system refers to a mixture composed of three components. Its phase diagram requires at least three dimensions, typically representing either temperature or pressure , along with two independent mole fractions.

 

 

Since , the perpendicular distances DE, DF and DG becomes , and respectively if we set . This normalisation allows us to represent the ternary system in an equilateral triangle, assuming both and are held constant (see diagram above). For instance, the red dot in the diagram corresponds to mole fractions , and . In fact, once two independent mole fractions are specified, the third is automatically determined due to the constraint . This effectively reduces the degrees of freedom to two, allowing the system to be represented on a two-dimensional diagram.

The ternary phase diagram below represents the acetone-water-diethyl ether system at 30^oC and 1 atm. Under these conditions, water and diethyl ether are only partially miscible, while the other two binary pairs are fully miscible. The region under the curve consists of two liquid phases in equilibrium, whereas the region above the curve represents a single-phase liquid.

Since temperature and pressure are constant, tie lines must remain in the plane of the diagram, but they are not required to be parallel. Their orientations are determined experimentally. The compositions of the two coexisting phases at a given point under the curve are found at the ends of the tie line that intersects the point. For example, point f lies within the two-phase region and corresponds to a water-rich, ether-poor phase () of composition e, and a water-poor, ether-rich phase () of composition g. Moreover, point e is richer in acetone than point g. Finally, the lever rule also applies here: the relative amounts of the two phases are inversely proportional to the lengths of the tie line segments, i.e. .

 

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The phase rule, formulated by Josiah Willard Gibbs, describes the number of degrees of freedom (independent variables) needed to define a multiphase, multicomponent system at equilibrium. It is given by

where

is the number of degrees of freedom (i.e. independent intensive variables such as temperature, pressure or mole fraction). This value is a non-negative integer.
is the number of independent components that can describe the composition of a point in a phase diagram.
is the number of phases present in equilibrium.

For example, we need two independent mole fractions to describe a liquid mixture of three fully miscible components at constant temperature and pressure (single-phase ternary system) because and .

To derive eq203, assume that every chemical species is present in every phase of the system. Each phase has a composition defined by the mole fractions of components. Since mole fractions must sum to 1, each phase has independent composition variables. For phases, the total number of composition variables is . Including temperature and pressure, the total number of variables is .

However, not all of these variables are independent because the system is at equilibrium. At equilibrium, the chemical potential of each component must be equal in all phases. For example, for component 1,

This gives equations per component, resulting in a total of independent equations. These equations represent constaints that reduce the number of independent variables because each chemical potential is a function of temperature, pressure and the mole fractions of all components in each phase, e.g. . If say, and , then one variable becomes dependent on the others when we equate . Even if the functions are more complex, the value of any one variable remains dependent on the others.

Therefore, the number of independent variables needed to define a multiphase, multicomponent system at equilibrium is , which rearranges to eq203. In this derivation, we assumed that every chemical species is present in every phase of the system. What if one or more species is absent from one or more phases? Suppose species is absent from phase . In this case, the number of variables is reduced by one because is no longer a variable. At the same time, the number of independent chemical potential equations is also reduced by 1, e.g. becomes . Therefore, the phase rule still holds.

 

 

 

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