Posts Tagged ‘Phase Rule’


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 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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Why don’t we use Gibbs’ original derivation of the Phase Rule?

This is a question for science historians with better information resources at their disposal than mine. But I can offer a couple of indicators.

The first point to make is that although Gibbs was undoubtedly the first to discover that a coexistent r-phase system containing n independent components has n + 2 – r degrees of freedom, he did not draw any attention to it; in fact his derivation is almost ‘hidden away’ in the text of his milestone monograph.

Nor did Gibbs apply the sobriquet ‘phase rule’; this seems to have originated in Europe. The Dutch chemist Hendrik Roozeboom, who in the 1880s began research into verifying Gibbs’ theoretical predictions of phase equilibria at the University of Amsterdam, certainly introduced the term “Phasenlehre”. But the earliest dated literature reference I can find is from 1893 when the German chemist Wilhelm Meyerhoffer, who was also working at the University of Amsterdam, published a paper entitled “Die Phasenregel”.

The second point concerns the two books which established chemical thermodynamics as a modern, practical science, and set the study curriculum at countless colleges around the world. One was American – the famous Thermodynamics by G.N. Lewis and Merle Randall, published in 1923. The other was European – Modern Thermodynamics by Edward Guggenheim, published in 1933.

The extraordinary thing about Lewis and Randall’s 600+ page book, the pivotal work which first made Gibbs’ powerful ideas accessible to students of physical science, is that it devotes barely a page to the Phase Rule and – crucially – does not even state the equation, let alone its derivation.

That job was left to Edward Guggenheim in Europe. In Chapter 1 of Modern Thermodynamics – Introduction and Fundamental Laws – he states the Phase Rule and gives the derivation.

But it is not Gibbs’ derivation based on a single intensive factor relation. Guggenheim’s method involves counting component concentrations, which are related by an equation of condition within each phase, and are also subject to individual constraints between phases since the chemical potential of any component is the same in all phases at equilibrium.

Two separate sets of constraints are thus imported into Guggenheim’s calculation of degrees of freedom


which to this writer’s eyes, lacks the simplicity of Gibbs’ approach.

Nevertheless, Guggenheim’s method is the one which has been taught to generations of students (including me), and will in all likelihood be taught for generations to come…

… unless CarnotCycle succeeds with this post in awakening interest in Gibbs’ own derivation, which is surely the original and arguably the best!

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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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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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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. Allgem. Physiol. 17, 81–124.

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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