Posts Tagged ‘intensive variable’


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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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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P Mander February 2015



Imagine a perfect gas contained by a rigid-walled cylinder equipped with a frictionless piston held in position by a removable external agency such as a magnet. There are finite differences in the pressure (P1>P2) and volume (V2>V1) of the gas in the two compartments, while the temperature can be regarded as constant.

If the constraint on the piston is removed, will the piston move? And if so, in which direction?

Common sense, otherwise known as dimensional analysis, tells us that differences in volume (dimensions L3) cannot give rise to a force. But differences in pressure (dimensions ML-1T-2) certainly can. There will be a net force of P1–P2 per unit area of piston, driving it to the right.

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The driving force

In thermodynamics, there exists a set of variables which act as “generalized forces” driving a system from one state to another. Pressure is one such variable, temperature is another and chemical potential is yet a third. What they all have in common is that they are intensive variables – those which are independent of the quantity of a phase.

Each intensive variable has a conjugate extensive variable – those which are dependent on the quantity of a phase – and together they form a generalized force-displacement pair which has the dimensions of work (= energy ML2T-2). Examples include pressure × volume, temperature × entropy, and voltage × charge.

But back to those intensive variables. Experience confirms that it is these generalized forces which are the agents of change. And spontaneous change results when there are finite differences in intensive variables between one system and another – the direction of change being determined by their relative values.

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The result is work

In the illustrated example above, the freed piston moves spontaneously to the right because of finite differences in pressure (P1>P2). As a result, the system consisting of the left hand compartment does PV work on the system consisting of the right hand compartment. Likewise, finite differences in other intensive variables such as temperature and chemical potential will act through their respective force-displacement pairs to perform work.

We can thus conclude that finite differences in intensive variables drive spontaneous change, and that due to the dimensionality of their respective conjugate extensive variables, spontaneous change results in the performance of work. It should however be noted that this work is not always useful work. The spontaneous diffusion of two gases into each other is a classic case, where it is difficult to imagine how the work could be usefully obtained.

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P Mander March 2015