The phenol-water system is a well-studied example of what physical chemists call partially miscible liquids. The extent of miscibility is determined by temperature, as can be seen from the graph below. The inverted U-shaped curve can be regarded as made up of two halves, the one to the left being the solubility curve of phenol in water and the other the solubility curve of water in phenol. The curves meet at the temperature (66°C) where the saturated solutions of water in phenol, and phenol in water, have the same composition.

The thermodynamic forces driving the behavior of the phenol-water system are first visible in the upwardly convex mutual solubility curve, showing that the enthalpy of solution (ΔH_{s}) in the saturated solution is positive i.e. that the system absorbs heat and so solubility increases with temperature in accordance with Le Châtelier’s Principle.

More rigorously, one can ascertain whether solubility of the minor component increases or decreases with temperature by computing

where s is the solubility expressed in mole fraction units, f_{2} is the corresponding activity coefficient and T_{s} is the temperature at which the solution is saturated.

The line of the umbrella curve charts the variation in composition of saturated solutions – phenol in water and water in phenol – with temperature. The area to the left of the curve represents unsaturated solutions of phenol in water and the area to the right represents unsaturated solutions of water in phenol, while the area above the curve represents solutions of phenol and water that are fully miscible i.e. miscible in all proportions.

But what about the area inside the curve, which is beyond the saturation limits of water in phenol and phenol in water? In this region, the system exhibits its most striking characteristic – it divides into two coexistent phases, the upper phase being a saturated solution of phenol in water and the lower phase a saturated solution of water in phenol. The curious feature of these phases is that for a given temperature their composition is fixed even though the total amounts of phenol and water composing them may vary.

To analyze how this comes about, consider the dotted line on the diagram below, which represents the composition of the phenol-water system at 50°C.

Starting with a system which consists of water only we gradually dissolve phenol in it, maintaining the temperature at 50°C, until we reach the point Y on the curve at which the phenol-in-water solution becomes saturated.

Now imagine adding to the saturated solution a small additional amount of phenol. It cannot dissolve in the solution and therefore creates a separate coexistent phase. Since this newly-formed phenol phase contains no water, the chemical potential of water in the solution provides the driving force for water to pass from the aqueous phase into the phenol phase. This cannot happen on its own however since water passing out of a phenol-saturated solution would cause the solution to become supersaturated. This would constitute change from a stable state to an unstable state which cannot occur spontaneously.

What can be postulated to occur is that the movement of water from the solution into the phenol phase simultaneously lowers the chemical potential of phenol in that coexistent phase, allowing phenol to move with the water in such proportion that the phenol-in-water phase remains saturated – as it must do since the temperature remains constant. In other words, saturated solution passes spontaneously from the aqueous phase into the phenol phase, diminishing the amount of the former and increasing the amount of the latter. Because water is the major component of the phenol-in-water phase, this bulk movement will continuously increase the proportion of water in the coexistent water-in-phenol phase until it reaches the saturation point whose composition is given by point Z on the mutual solubility curve.

In terms of chemical potential in the two-phase system, equilibrium at a given temperature will be reached when:

*upper phase = sat. soln. of phenol in water
lower phase = sat. soln. of water in phenol*

**– – – –**

**Applying the Phase Rule**

F = C – P + 2

First derived by the American mathematical physicist J. Willard Gibbs (1839-1903), the phase rule computes the number of system variables F which can be independently varied for a system of C components and P phases in a state of thermodynamic equilibrium.

Applying the rule to the 1 Phase region of the phenol-water system, F = 2 – 1 + 2 = 3 where the system variables are temperature, pressure and composition. So for a chosen temperature and pressure, e.g. atmospheric pressure, the composition of the phase can also be varied.

In the 2 Phase region of the phenol-water system, F = 2 – 2 + 2 = 2. So for a chosen temperature and pressure, e.g. atmospheric pressure, the compositions of the two phases are invariant.

In the diagram below, the compositions of the upper and lower phases remain invariant along the line joining Y and Z, the pressure being atmospheric and the temperature being maintained at 50°C. As we have seen, the upper layer will be a saturated solution of phenol in water where the point Y determines the % weight of phenol (= 11%). Correspondingly, the lower layer will be a saturated solution of water in phenol where the point Z determines the % weight of phenol (= 63%).

**– – – –**

**Relationship between X, Y and Z**

If a mixture of phenol and water is prepared containing X% by weight of phenol where X is between the points Y and Z as indicated on the above diagram, the mixture will form two phases whose phenol content at equilibrium is Y% by weight in the upper phase and Z% by weight in the lower phase.

Let the mass of the upper phase be M_{1} and that of the lower phase be M_{2}. The mass of phenol in these two phases is therefore Y% of M_{1} + Z% of M_{2}. Conservation of mass dictates that this must also equal X% of M_{1} + M_{2}. Therefore

The relative masses of the upper and lower phases change according to the position of X along the line Y-Z. As X approaches Y the upper phase increases as the lower phase diminishes, becoming one phase of saturated phenol-in-water at point Y. Conversely as X approaches Z the lower phase increases as the upper phase diminishes, becoming one phase of saturated water-in-phenol at point Z.

**– – – –**

**Further reading**

*Logan S.R. Journal of Chemical Education 1998*

This paper uses the well-known thermodynamic equation ΔG = ΔH + TΔS as a theoretical basis for determining the circumstances under which spontaneous mixing occurs when two partially miscible liquids are brought together at constant temperature and pressure.

The approach involves the construction of equations for estimating both the enthalpy and entropy of mixing in terms of the mole fraction x of one component, and graphing the change in Gibbs free energy ΔG against x to determine the position of any minimum/minima. The paper goes on to examine the criteria for the existence of two phases on the basis of determining the circumstances under which a system of two phases will have a combined ΔG value that is lower than the corresponding ΔG for a single phase.

The conclusion is reached that the assessment of ΔG on mixing two liquids can provide a qualitative explanation of some of the phenomena observed in relation to the miscibility of two liquids.

The paper is available from the link below (pay to view)

https://pubs.acs.org/doi/abs/10.1021/ed075p339?src=recsys&#

**– – – –**

P Mander November 2018

Thank you ! Help me so much in understanding phase rule. It’s very goodly explained

LikeLike