Posts Tagged ‘Phase Rule’

Dmitiri Konovalov (1856-1929) was a Russian chemist who made important contributions to the theory of solutions. He studied the vapor pressure of solutions of liquids in liquids and in 1884 published a book on the subject which gave a scientific foundation to the distillation of solutions and led to the development of industrial distillation processes.

On the subject of partially miscible liquids forming conjugate solutions, Konovalov in 1881 established the following fact: “If two liquid solutions are in equilibrium with each other, their vapor pressures, and the partial pressures of the components in the vapor, are equal.”

J. Willard Gibbs in America had already developed the concept of chemical potential to explain the behavior of coexistent phases in his monumental treatise On the Equilibrium of Heterogeneous Substances (1875-1878). Konovalov was unaware of this work, and independently found a proof on the basis of this astutely reasoned thought experiment:

«Consider Figure 77 shown above. Two liquid layers α and β in coexistent equilibrium are contained in a ring-shaped tube, and above them is vapor. If the pressure of either component in the vapor were greater over α than over β, diffusion of vapor would cause that part lying over β to have a higher partial pressure of the given component than is compatible with equilibrium. Condensation occurs and β is enriched in the specified component. By reason of the changed composition of β however, the equilibrium across the interface of the liquid layers is disturbed and the component deposited by the vapor will pass into the liquid α. The whole process now commences anew and the result is a never-ending circulation of matter round the tube i.e. a perpetual motion, which is impossible. Hence the partial pressures of both components are equal over α and β and therefore also their sum i.e. the total vapor pressure.»

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Equivalence of vapor pressure and chemical potential

Konovalov showed that the condition of equilibrium in coexistent phases was equality of vapor pressure p for each component. This is consistent with the concept of ‘generalized forces’, a set of intensive variables which drive a thermodynamic system spontaneously from one state to another in the direction of equilibrium. Vapor pressure is one such variable, and chemical potential is another. Hence Gibbs showed that chemical potential μ is a driver of compositional change between coexistent phases and that equilibrium is reached when the chemical potential of each component in each phase is equal. In shorthand the equilibrium position for partially miscible liquids containing components 1 and 2 in coexistent phases α, β and vapor can be stated as:

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P Mander, July 2020

Exploring the mutual solubility of phenol and water at the Faculty of Pharmacy, National University of Malaysia.

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 (ΔHs) 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, f2 is the corresponding activity coefficient and Ts 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

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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%).

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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 M1 and that of the lower phase be M2. The mass of phenol in these two phases is therefore Y% of M1 + Z% of M2. Conservation of mass dictates that this must also equal X% of M1 + M2. 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.

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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)

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P Mander November 2018


The Phase Rule formula was first stated by the American mathematical physicist Josiah Willard Gibbs in his monumental masterwork On the Equilibrium of Heterogeneous Substances (1875-1878), in which he almost single-handedly laid the theoretical foundations of chemical thermodynamics.

In a paragraph under the heading “On Coexistent Phases of Matter”, Gibbs gives the derivation of his famous formula in just 77 words. Of all the many Phase Rule proofs in the thermodynamic literature, it is one of the simplest and shortest. And yet textbooks of physical science have consistently overlooked it in favor of more complicated, less lucid derivations.

To redress this long-standing discourtesy to Gibbs, CarnotCycle here presents Gibbs’ original derivation of the Phase Rule in an up-to-date form. His purely prose description has been supplemented with clarifying mathematical content, and the outmoded symbols used in the single equation to which he refers have been replaced with their modern equivalents.

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Gibbs’ derivation

Gibbs begins by introducing the term phase to refer solely to the thermodynamic state and composition of a body (solid, liquid or vapor) without regard to its quantity. So defined, a phase cannot be described in terms of extensive variables like volume and mass, since these vary with quantity. A phase can only be described in terms of intensive variables like temperature and pressure, which do not vary with quantity.

To derive the Phase Rule, Gibbs chooses as his starting point equation 97 from his treatise, now known as the Gibbs-Duhem equation

This general thermodynamic equation, which relates to a single phase, connects the intensive variables temperature T, pressure P, and chemical potential μ where μn is the potential of the nth component substance. Any possible variations of these quantities sum to zero, indicating a phase in internal equilibrium.

If there are n independent component substances, the phase has a total of n+2 variables

These quantities are not all independently variable however, because they are related by the Gibbs-Duhem equation. If all but one of the quantities are varied, the variation of the last is given by the equation. A single-phase system is thus capable of (n+2) – 1 independent variations.

Now suppose we have two phases, each containing the same n components, in coexistent equilibrium with each other. Signifying one phase by a single prime and the other by a double prime we may write

since this is the definition of equilibrium between phases. So in the two-phase system the total number of variables remains unchanged at n+2, but there are now two Gibbs-Duhem equations, one for each phase. It follows that if all but two of the quantities are varied, the variations of the last two are given by the two equations. A two-phase system is thus capable of (n+2) – 2 independent variations.

It is evident from the foregoing that regardless of the number of coexistent phases in equilibrium, the total number of variables will still be n+2 while the number subtracted (called the number of constraints) will be equal to the number of Gibbs-Duhem equations i.e. one for each phase.

A system of r coexistent phases is thus capable of n+2 – r independent variations, which are also called degrees of freedom (F). Therefore

This is the Phase Rule as derived by Gibbs himself. In contemporary textbooks it is usually written

where C is the number of independent components and P is the number of phases in coexistent equilibrium.

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gibbs twitter

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P Mander February 2015

photo credits: see below

For a binary solution containing a solute in equilibrium with its vapour, the Phase Rule tells us that the system possesses two degrees  of freedom. So if temperature is held constant, there will be a relation at equilibrium between pressure and concentration. Applying this to an atmospheric gas in contact with water, the Phase Rule predicts an equilibrium relation between the concentration of the gas in the water and its partial pressure in the atmosphere.

This indeed proves to be the case. The relation was first discovered in Manchester, England, by William Henry (1774-1836), possibly in connexion with his mineral water manufacturing activities in that city. William Henry was also a fellow of the Royal Society; in 1803 he published his findings in the society’s Philosophical Transactions, thereby ensuring that his discovery would reach the wider world of science.

William Henry was fortunate in having another citizen of Manchester – John Dalton, pioneer of the atomic theory – as a neighbour and co-worker, since the latter had just recently (1801) discovered the law of partial pressures which conceptually underpins what we now call Henry’s law. Both laws are strictly applicable only to ideal gases, but provided that real gases are not too near their critical temperatures and pressures, and are only moderately soluble, deviations from these laws are not large.

With temperature held constant, Henry’s law implies that if disequilibrium arises in the partial pressure of a gas phase component, the system will react to restore equilibrium by either increasing or decreasing the concentration of that component in the liquid phase. This is of central significance to what follows.

The plaque commemorating William Henry’s birthplace in Manchester, England. Some of the dates are a bit astray; Dalton’s law predates Henry’s law, since the latter is conceptually dependent on the former.

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The Air Bubble

Over a century after William Henry’s discovery, with the Great War having recently broken out and military minds planning slaughter in the fields of Flanders, an experimental  biologist in neutral Denmark had his mind on something much more peaceful. Richard Ege was studying the habits of water beetles, in particular the Corixidae, Dytiscidae and Notonecta. These beetles, which favour well-aerated aquatic environments, carry around with them a bubble of air which they periodically replenish. Now at the time it was not known whether the air bubble had a purely hydrostatic function, or whether it might be used for respiratory purposes.

Richard Ege decided to investigate this by altering the composition of the gas in the bubble, confining the beetles under water, and seeing how long they survived. He reasoned that if the bubble was carried only for buoyancy, then replacing the air with nitrogen would make no difference to survival time. But if the bubble had a respiratory function, then replacing the air with nitrogen would shorten the survival time.

Ege duly conducted the experiments, and found that beetles with bubbles containing air survived for up to six hours, while those with bubbles containing only nitrogen survived no longer than five minutes. Conclusion – the bubble acted as a source of oxygen and therefore had a respiratory function. Case closed.

Or it would have been, had not Richard Ege conducted a further experiment, which produced a completely unexpected result.

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The Problem

Ege replaced the air in the bubble with pure oxygen, expecting to see the submerged survival time extended considerably beyond six hours. But a surprise was in store. When he conducted the experiment, he found the exact opposite. With a bubble of oxygen, survival time was reduced to a mere 35 minutes.

I can picture Richard Ege in his laboratory, scratching his head and puzzling over this result. No doubt he carefully checked the purity of his oxygen supply and re-ran the experiment. Same result. Survival time 35 minutes. And I can imagine him pacing up and down, turning over the problem in his mind until – aha! the truth begins to dawn on him. But to return from fancy to fact.

On Friday 14 May 1915, at a meeting of the Danish Natural History Society in Copenhagen, Richard Ege presented his work, together with a well-argued scientific explanation. This is minuted in the proceedings of the society and can be seen here (in Danish) on Ege also published his work in English in a well-respected German journal*, which is how the wider world came to know of it. In entomological circles, it is a celebrated paper and is still regularly referenced, not least  as a reminder to experimenters that the truth is sometimes not as simple as it first appears.

*Ege, R., 1915. On the respiratory function of the air stores carried by some aquatic insects (Corixidae, Dytiscidae and Notonecta). Z. Allg. Physiol. 17, 81–124.

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The Solution

So what was the solution to the water beetle puzzle? Step forward Henry’s Law.

Ege realised that the respiratory activities of the submerged beetle would result in a disequilibrium between the partial pressures of the gases in bubble and those in the atmosphere, and that this would result in gaseous exchange between the bubble and the water in the direction of restored equilibrium. Below, I have summarised the analysis of his experiments:

Experiment 1: Bubble containing air (21% oxygen, 79% nitrogen)
Survival time: Up to six hours

As the submerged beetle moves around, it consumes oxygen from the bubble, lowering the oxygen partial pressure below the atmospheric partial pressure with which the oxygen concentration in the water is in equilibrium. This causes oxygen to diffuse out of the water and into the bubble. In other words the bubble acts as a kind of gill, enabling the beetle to replenish the oxygen it consumes from the bubble with oxygen from the water. By balancing supply and demand, the beetle can remain submerged for a lengthy period.

The beetle cannot remain submerged indefinitely however, because the lowered oxygen partial pressure in the bubble necessarily raises the nitrogen partial pressure. This disequilibrium causes nitrogen to diffuse into the water, reducing the volume of the bubble. As the surface area of the bubble shrinks, the rate of oxygen uptake from the water gradually diminishes to a point where oxygen demand can no longer be met and the beetle rises to the surface to renew the air bubble, after about six hours.

Note – CO2 produced by the submerged beetle does not accumulate in the bubble, since its partial pressure is immediately higher than the atmospheric partial pressure with which the CO2 concentration of the water is in equilibrium. The CO2 thus diffuses out into the water. This process occurs rapidly since CO2 has a much higher solubility in water than the other gases.

Experiment 2: Bubble containing pure nitrogen
Survival time: 5 minutes

Did you wonder how the beetles survived for even 5 minutes without any oxygen?

Without oxygen in the bubble, the submerged beetle is forced to rely solely on the partial pressure disequilibrium to obtain oxygen from the water. But even this supply is soon restricted as the bubble surface area gets smaller. This happens much more quickly than in the first experiment because the oxygen which diffuses into the bubble is immediately consumed and so the nitrogen partial pressure remains close to 100%. Due to the greater disequilibrium, the nitrogen diffuses more rapidly into the water, causing the bubble to shrink at a faster rate. Within a few minutes, the rate of oxygen uptake from the water diminishes to a point where oxygen demand can no longer be met and the beetle rises to the surface.

Experiment 3: Bubble containing pure oxygen
Survival time: 35 minutes

With a partial pressure of 100%, the oxygen in the bubble is depleted at a much faster rate than if it contained air, because in addition to oxygen consumption by the beetle, the large disequilibrium in partial pressure causes rapid diffusion of oxygen into the water. Even allowing for some inward diffusion of nitrogen, the oxygen partial pressure in the bubble remains considerably above that of the atmosphere. No uptake of oxygen from the water can occur, and so the bubble simply continues to shrink until the oxygen is exhausted – a process which takes about half an hour.

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At the time of the water beetle experiments, 24-year-old Richard Ege was living on Øster Farimagsgade in downtown Copenhagen. His apartment was on the third floor of this building.

Richard Ege (1891-1974) was born in Rønde near the Danish market town of Randers in Jutland. He studied zoology at the University of Copenhagen, and later became interested in physiological chemistry. He was appointed the university’s first professor of biochemistry in 1928, a post he held until 1962. During the second world war when Denmark was occupied by Hitler’s Germany, Richard Ege and his wife did much to help Danish Jews escape to Sweden. He died in Holte, a northern suburb of Copenhagen, and his ashes are interred in Mariebjerg cemetery in Gentofte.

I am indebted to Helle Nygaard for supplying the biographical details on Richard Ege, and for discovering the archived proceedings of the Danish Natural History Society for 1915 on

Photo Credits
Header photo (left to right): William Henry, Wikipedia; Water Beetles, NC State University; Richard Ege, Det Kongelige Bibliotek