Posts Tagged ‘J. Willard Gibbs’

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CarnotCycle has been fortunate to acquire the original two-volume set of Gibbs’ scientific papers published by Longmans, Green and Co. in 1906, three years after Gibbs’ death. Volume I is devoted to thermodynamics, while Volume II covers dynamics, vector analysis and multiple algebra, the electromagnetic theory of light, and other miscellaneous topics. Considering that these first edition books are well over a century old, they are in remarkably fine condition.

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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

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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

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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

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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

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

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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

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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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Volume One of the Scientific Papers of J. Willard Gibbs, published posthumously in 1906, is devoted to Thermodynamics. Chief among its content is the hugely long and desperately difficult “On the equilibrium of heterogeneous substances (1876, 1878)”, with which Gibbs single-handedly laid the theoretical foundations of chemical thermodynamics.

In contrast to James Clerk Maxwell’s textbook Theory of Heat (1871), which uses no calculus at all and hardly any algebra, preferring geometry as the means of demonstrating relationships between quantities, Gibbs’ magnum opus is stuffed with differential equations. Turning the pages of this calculus-laden work, one could easily be drawn to the conclusion that the writer was not a visual thinker.

But in Gibbs’ case, this is far from the truth.

The first two papers on thermodynamics that Gibbs published, in 1873, were in fact visually-led. Paper I deals with indicator diagrams and their comparative properties, while Paper II shows how the relations between the state variables V, P, T, U, S, given in analytical form by dU=TdS – PdV, may be expressed geometrically by means of a surface.

Indeed Gibbs propels the visual argument further by pointing out that analytical formulae are strictly unnecessary for comprehending relationships between thermodynamic state variables, since they can just as easily be understood by applying graphical methods.

Gibbs’ advocacy of the visual approach found instant favor with Maxwell, who in the fourth edition of Theory of Heat devoted no less than 12 pages to an illustrated discussion of Gibbs’ thermodynamic surface, including the wild diagram shown at the head of this post. Maxwell’s enthusiasm was such that he sculpted a clay model of the surface, from which he made a plaster cast and sent it to Gibbs at Yale in 1874.

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Besides his passion for using geometrical constructions to demonstrate connexions between quantities, Maxwell had an influential voice in the scientific world, and it is almost certain that he would have used it to promulgate the geometrical approach to understanding thermodynamic relationships that Gibbs had pioneered. But Maxwell’s death in 1879 at the early age of 48 brought such initiatives to a premature end.

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James Clerk Maxwell (1831-1879) striking a pose. The studio backdrop and furnishings confirm that he was quite short in stature, but had large hands with a broad palm and relatively short fingers – the strong, practical hands of a sculptor.

The exposure that Gibbs’ thermodynamic surface gained through the agency of Maxwell proved to be short-lived; no other contemporary scientist followed Maxwell’s lead. One explanation could be that Gibbs’ visual approach lacked appeal because — for reasons best known to himself — he described it in words, not pictures. Another could be that Maxwell’s illustrations of the surface were found too difficult: a joke reportedly circulated at the time that “only one man lived who could understand Gibbs’ papers. That was Maxwell, and now he is dead.”

Whatever the actual truth, the fact remains that none of the milestone literature in the post-Maxwell period took up Gibbs’ visual approach to understanding relationships between thermodynamic properties. Instead, the approach taken in textbooks by Max Planck (1879), GH Bryan (1909), JR Partington (1913) and most importantly by Lewis & Randall (1923) and Guggenheim (1933), was analytical.

Writing in 1936, the American mathematician Edwin Wilson (who had attended Gibbs’ lectures at Yale in 1901-2) argued that Gibbs’ entropy-temperature diagram in Paper I and the thermodynamic surface in Paper II were both victims of the inevitable choices that science makes as it evolves.

He commented: “Science goes on its way, picking and choosing and modifying. The trend of the last fifty years is not towards Papers I and II. Interesting as they are historically, and important because of the preparation they afforded Willard Gibbs for writing his great memoir III [On the Equilibrium of Heterogeneous Substances], there is no present indication that they are in themselves significant for present or future science.”

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JR Partington’s fascinating Text-book of Thermodynamics (with Special Reference to Chemistry) of 1913, although presenting the subject analytically, nonetheless points out the graphical origins of Gibbs’ early discoveries.

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James Riddick Partington (1886-1965), whose Text-book of Thermodynamics was published just before the outbreak of the Great War. It provides a detailed and historically fascinating view of the subject in the decade before Lewis & Randall produced their watershed work.

Commenting on Paper II, Partington writes: “In this very important memoir Gibbs shows that the conditions of equilibrium of two parts of a substance in contact can be expressed geometrically in terms of the position of the tangent planes to the volume-entropy-energy surface of the substance, and he finds that the analytical expression of this property is that the value of this function (U–TS+PV) shall be the same for the two states at the same temperature and pressure.”

For those of us educated in the analytical age, it is indeed remarkable to discover that the free energy function was first obtained by Gibbs using purely graphical methods, and that the pressure-temperature equilibrium relation G(α)=G(β) between two phases of a substance in contact was originally derived from geometrical considerations.

In fact the volume-entropy-energy diagram enabled Gibbs to reach a further conclusion of great importance to his future work in thermodynamics: namely that the volume, entropy and energy of a mixture of portions of a substance in different states (whether in equilibrium or not) are the sums of the volumes, entropies and energies of the separate parts. This suggested to Gibbs that mixtures of substances differing in chemical composition, as well as physical state, might be treated in a similar manner.

It was this clue from Paper II that gave Gibbs the conceptual springboard he needed for investigating chemical equilibrium, the subject matter of Paper III – On the Equilibrium of Heterogeneous Substances.

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Ok, so let’s take a closer look at Gibbs’ thermodynamic surface of 1873, realized by Maxwell in 1874. Each point on this surface describes the state of a body (of invariable composition) in terms of its volume, entropy and energy.

Now if we were to slice vertical sections of this surface perpendicular to the energy-volume plane, the curve of section would represent the relation between energy and entropy when the volume is constant; the tangent of the angle of slope of this curve of section is therefore (dU/dS)V. By similar reasoning, the curve of section of the surface perpendicular to the energy-entropy plane represents the relation between energy and volume when the entropy is constant. The tangent of the angle of slope of this curve of section is therefore (dU/dV)S.

From the fundamental thermodynamic relation dU = TdS – PdV, we can identify (dU/dS)V as the absolute temperature T which reckoned from zero is essentially positive, and (dU/dV)S as the pressure P which is reckoned negative when the energy U increases as the volume V increases.

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The first appearance in print of the Gibbs free energy function, as the equation of the tangent plane at any point of the (v,η,ε) surface, in “A method of geometrical representation of the thermodynamic properties of substances by means of surfaces”

The tangent plane therefore represents the same temperature and pressure at all points. Gibbs used this geometrical property of the model to show that if two points in the surface (such as ε’ and ε”) have a common tangent plane, the states they represent can exist permanently in contact. He then gave the analytical expression of this condition – that what we now know as the Gibbs free energy of states ε’ and ε” are equal. But he did not show the geometrical reasoning by which he reached his conclusion.

Maybe he thought we could work it all out in our heads, who knows. Personally I much prefer to see these things drawn – and especially in this case, for it is a rewarding exercise in solid geometry to see how the answer emerges. CarnotCycle is indebted to Ronald Kriz for making available the following explanatory diagram:

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This diagram uses the Greek letters employed by Gibbs to denote internal energy (ε) and entropy (η). Source: Ronald Kriz, private communication

The common tangent plane through states ε’ and ε” cuts the axis of energy at a single point, marked ε. Beginning with the liquid state ε’, the length ε’ε on the axis of energy is the sum of Δε’η (due to the entropy change) and Δε’v (due to the volume change).

Since the tangent plane defines t’ = Δε’η/η’ and –p’ = Δε’v/v’ we have

ε = ε’ – t’η’ + p’v’

The right hand member of this equation is composed entirely of state variables, and thus denotes a state function associated with the point ε’ on the thermodynamic surface.

Turning to the gas state ε”, the length ε”ε on the axis of energy is the sum of Δε”η (due to the entropy change) and Δε”v (due to the volume change).

Since the tangent plane defines t” = Δε”η/η” and –p” = Δε”v/v” we have

ε = ε” – t”η” + p”v”

The right hand member of this equation is composed entirely of state variables, and thus denotes a state function associated with the point ε” on the thermodynamic surface.

An identical result will be obtained for all such pairs of points on the so-called node-couple curve, the branches of which unite at the isopycnic or critical point. Since the magnitude of the state function ε–tη+pv (in modern notation U–TS+PV=G) is the same for each pair, it is demonstrated that G(ε’)=G(ε”) is the analytical expression of the condition of coexistent equilibrium of separate states of a substance of invariable composition at the same temperature and pressure.

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