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 a pressure of 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 liquid-liquid phase separation begins.
Just to the right of point f, the system consists of two liquid phases: a hexane-rich phase and a nitrobenzene-rich phase. Since Hexane has a lower density than nitrobenzene, the hexane-rich phase forms the upper layer, while the nitrobenzene-rich phase forms the lower layer.
As more nitrobenzene is added between f and g, the amount of the nitrobenzene-rich layer increases, while that of the hexane-rich layer decreases. The compositions of the two layers remain constant at equilibrium because the temperature and pressure are fixed. Specifically, the composition of the upper, hexane-rich phase corresponds to point f, while that of the lower, nitrobenzene-rich phase corresponds to point g.
The lever rule can be applied to determine the relative amounts of the two phases between points f and g. For example, if the mole fraction of nitrobenzene is at point f and
at point g, then at an overall mole fraction of
, the ratio of the amounts of the hexane-rich phase to the nitrobenzene-rich phase is
Thus, the amount of the nitrobenzene-rich phase is about 16.3% more than the hexane-rich phase at an overall mole fraction of at
.
Question
Explain, using thermodynamic principles, why the equilibrium compositions of the two layers remain constant at a given temperature and pressure, and why the amount of the nitrobenzene-rich layer increases while that of the hexane-rich layer decreases as more nitrobenzene is added between two nodal points.
Answer
When a binary liquid system attains equilibrium at a specific temperature and pressure, the chemical potential of each component must be equal in both phases:
and
, where subscripts 1 and 2 denote hexane and nitrobenzene, and superscripts
and
denote the hexane-rich and nitrobenzene-rich layers. Specifically, the chemical potential of a component
in a real liquid mixture is defined mathematically as:
where
is the chemical potential of pure component
at the system temperature and pressure.
is the activity of component
, which is defined as
, where
is the mole fraction of component
, and
, a non-linear function of
,
and
, is the activity coefficient accounting for non-ideal molecular interactions.
Substituting the activity expression into the two equilibrium equations gives and
. Since the mole fractions of components in each phase sum to 1, we have
and
, yielding
As depends on
,
and
, it becomes a function of
alone when
and
are held constant. It follows that the two equilibrium equations form a pair of simultaneous equations in two independent variables:
and
(note that
and
depend on
and
respectively, and hence also depend on
and
because of the constraints
and
). Importantly, the total moles of either component does not appear explicitly in the equilibrium equations. This indicates that the addition of either component to the system at constant temperature and pressure does not affect the equilibrium mole fractions in either layer. Therefore, solving the simultaneous equations gives a distinct set of values for
and
(and hence
and
) at a given
and
, maintaining constant compositions for the
-layer (characterised by
and
) and the
-layer (characterised by
and
).
To explain why the amount of the nitrobenzene-rich layer increases, while that of the hexane-rich layer decreases as more nitrobenzene is added between two nodal points at a given temperature, we refer to the lever rule
where and
are the total moles in the hexane-rich and nitrobenzene-rich layers respectively, and
is the mole fraction of nitrobenzene in the overall system.
As nitrobenzene is added, increases. Consequently, the numerator
decreases, while the denominator
increases. The ratio
therefore decreases, indicating that the hexane-rich layer decreases in amount relative to the nitrobenzene-rich layer, whose relative amount increases. This trend continues until the overall composition reaches the nitrobenzene-rich nodal point, where the hexane-rich layer disappears and the system becomes a single liquid phase.
Beyond point g, the hexane-rich phase disappears, leaving a single homogeneous, nitrobenzene-rich liquid phase. Further addition of B causes the overall composition to shift towards pure nitrobenzene at point h, where .
Notably, the above liquid-liquid phase diagram, like most binodal curves, can be constructed from empirical data. 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.
Question
Describe a typical experiment to determine the coordinate points on a temperature-dependent binodal curve.
Answer
A common method for locating binodal points is the cloud-point method, in which the mixture composition is changed until the first sign of phase separation appears, or until turbidity disappears. Begin by placing a known amount of pure A in a sealed vessel at the desired temperature and pressure. Add B in small, measured increments while stirring. When the mixture first becomes persistently cloudy or separates into two liquid phases, record the amount of B added. This gives the first binodal point on the A-rich side.
Continue adding B until the two-phase mixture becomes homogeneous again, forming a single liquid phase. Record the total amount of B added at this second cloud point. This gives the other binodal point on the B-rich side. Repeat the procedure at several temperatures while maintaining the required pressure. Each temperature generally produces a different pair of cloud-point compositions. Finally, calculate the mole fraction of B at each cloud point using the total amounts of A and B present in the mixture. Plot the mole fraction of B against temperature to obtain the temperature-dependent binodal curve.
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.