Posts Tagged ‘James Clerk Maxwell’

From the perspective of classical thermodynamics, osmosis has a rather unclassical history. Part of the reason for this, I suspect, is that osmosis was originally categorised under the heading of biology. I can remember witnessing the first practical demonstration of osmosis in a biology class, the phenomenon being explained in terms of pores (think invisible holes) in the membrane that were big enough to let water molecules through, but not big enough to let sucrose molecules through. It was just like a kitchen sieve, we were told. It lets the fine flour pass through but not clumps. This was very much the method of biology in my day, explaining things in terms of imagined mechanism and analogy.

And it wasn’t just in my day. In 1883, JH van ‘t Hoff, an able theoretician and one of the founders of the new discipline of physical chemistry, became suddenly convinced that solutions and gases obeyed the same fundamental law, pv = RT. Imagined mechanism swiftly followed. In van ‘t Hoff’s interpretation, osmotic pressure depended on the impact of solute molecules against the semipermeable membrane because solvent molecules, being present on both sides of the membrane through which they could freely pass, did not enter into consideration.

It all seemed very plausible, especially when van ‘t Hoff used the osmotic pressure measurements of the German botanist Wilhelm Pfeffer to compute the value of R in what became known as the van ‘t Hoff equation


where Π is the osmotic pressure, and found that the calculated value for R was almost identical with the familiar gas constant. There really did seem to be a parallelism between the properties of solutions and gases.


JH van ‘t Hoff (1852-1911)

The first sign that there was anything amiss with the so-called gaseous theory of solutions came in 1891 when van ‘t Hoff’s close colleague Wilhelm Ostwald produced unassailable proof that osmotic pressure is independent of the nature of the membrane. This meant that hypothetical arguments as to the cause of osmotic pressure, such as van ‘t Hoff had used as the basis of his theory, were inadmissible.

A year later, in 1892, van ‘t Hoff changed his stance by declaring that the mechanism of osmosis was unimportant. But this did not affect the validity of his osmotic pressure equation ΠV = RT. After all, it had been shown to be in close agreement with experimental data for very dilute solutions.

It would be decades – the 1930s in fact – before the van ‘t Hoff equation’s formal identity with the ideal gas equation was shown to be coincidental, and that the proper thermodynamic explanation of osmotic pressure lay elsewhere.

But long before the 1930s, even before Wilhelm Pfeffer began his osmotic pressure experiments upon which van ‘t Hoff subsequently based his ideas, someone had already published a thermodynamically exact rationale for osmosis that did not rely on any hypothesis as to cause.

That someone was the American physicist Josiah Willard Gibbs. The year was 1875.


J. Willard Gibbs (1839-1903)

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Osmosis without mechanism

It is a remarkable feature of Gibbs’ On the Equilibrium of Heterogeneous Substances that having introduced the concept of chemical potential, he first considers osmotic forces before moving on to the fundamental equations for which the work is chiefly known. The reason is Gibbs’ insistence on logical order of presentation. The discussion of chemical potential immediately involves equations of condition, among whose different causes are what Gibbs calls a diaphragm, i.e. a semipermeable membrane. Hence the early appearance of the following section


In equation 77, Gibbs presents a new way of understanding osmotic pressure. He makes no hypotheses about how a semipermeable membrane might work, but simply states the equations of condition which follow from the presence of such a membrane in the kind of system he describes.

This frees osmosis from considerations of mechanism, and explains it solely in terms of differences in chemical potential in components which can pass the diaphragm while other components cannot.

In order to achieve equilibrium between say a solution and its solvent, where only the solvent can pass the diaphragm, the chemical potential of the solvent in the fluid on both sides of the membrane must be the same. This necessitates applying additional pressure to the solution to increase the chemical potential of the solvent in the solution so it equals that of the pure solvent, temperature remaining constant. At equilibrium, the resulting difference in pressure across the membrane is the osmotic pressure.

Note that increasing the pressure always increases the chemical potential since


is always positive (V1 is the partial molar volume of the solvent in the solution).

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Europe fails to notice (almost)

Gibbs published On the Equilibrium of Heterogeneous Substances in Transactions of the Connecticut Academy. Choosing such an obscure journal (seen from a European perspective) clearly would not attract much attention across the pond, but Gibbs had a secret weapon. He had a mailing list of the world’s greatest scientists to which he sent reprints of his papers.

One of the names on that list was James Clerk Maxwell, who instantly appreciated Gibbs’ work and began to promote it in Europe. On Wednesday 24 May 1876, the year that ‘Equilibrium’ was first published, Maxwell gave an address at the South Kensington Conferences in London on the subject of Gibbs’ development of the doctrine of available energy on the basis of his new concept of the chemical potentials of the constituent substances. But the audience did not share Maxwell’s enthusiasm, or in all likelihood share his grasp of Gibbs’ ideas. When Maxwell tragically died three years later, Gibbs’ powerful ideas lost their only real champion in Europe.

It was not until 1891 that interest in Gibbs masterwork would resurface through the agency of Wilhelm Ostwald, who together with van ‘t Hoff and Arrhenius were the founders of the modern school of physical chemistry.


Wilhelm Ostwald (1853-1932) He not only translated Gibbs’ masterwork into German, but also produced a profound proof – worthy of Sadi Carnot himself – that osmotic pressure must be independent of the nature of the semipermeable membrane.

Although perhaps overshadowed by his colleagues, Ostwald had a talent for sensing the direction that the future would take and was also a shrewd judge of intellect – he instinctively felt that there were hidden treasures in Gibbs’ magnum opus. After spending an entire year translating ‘Equilibrium’ into German, Ostwald wrote to Gibbs:

“The translation of your main work is nearly complete and I cannot resist repeating here my amazement. If you had published this work over a longer period of time in separate essays in an accessible journal, you would now be regarded as by far the greatest thermodynamicist since Clausius – not only in the small circle of those conversant with your work, but universally—and as one who frequently goes far beyond him in the certainty and scope of your physical judgment. The German translation, hopefully, will more secure for it the general recognition it deserves.”

The following year – 1892 – another respected scientist sent a letter to Gibbs regarding ‘Equilibrium’. This time it was the British physicist, Lord Rayleigh, who asked Gibbs:

“Have you ever thought of bringing out a new edition of, or a treatise founded upon, your “Equilibrium of Het. Substances.” The original version though now attracting the attention it deserves, is too condensed and too difficult for most, I might say all, readers. The result is that as has happened to myself, the idea is not grasped until the subject has come up in one’s own mind more or less independently.”

Rayleigh was probably just being diplomatic when he remarked that Gibbs’ treatise was ‘now attracting the attention it deserves’. The plain fact is that nobody gave it any attention at all. Gibbs and his explanation of osmosis in terms of chemical potential was passed over, while European and especially British theoretical work centered on the more familiar and more easily understood concept of vapor pressure.

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Gibbs tries again

Although van ‘t Hoff’s osmotic pressure equation ΠV = RT soon gained the status of a law, the gaseous theory that lay behind it remained clouded in controversy. In particular, van ‘t Hoff’s deduction of the proportionality between osmotic pressure and concentration was an analogy rather than a proof, since it made use of hypothetical considerations as to the cause of osmotic pressure. Following Ostwald’s proof that these were inadmissible, the gaseous theory began to look hollow. A better theory was needed.


Lord Kelvin (1824-1907) and Lord Rayleigh (1842-1919)

This was provided in 1896 by the British physicist, Lord Rayleigh, whose proof was free of hypothesis but did make use of Avogadro’s law, thereby continuing to assert a parallelism between the properties of solutions and gases. Heavyweight opposition to this soon materialized from the redoubtable Lord Kelvin. In a letter to Nature (21 January 1897) he charged that the application of Avogadro’s law to solutions had “manifestly no theoretical foundation at present” and further contended that

“No molecular theory can, for sugar or common salt or alcohol, dissolved in water, tell us what is the true osmotic pressure against a membrane permeable to water only, without taking into account laws quite unknown to us at present regarding the three sets of mutual attractions or repulsions: (1) between the molecules of the dissolved substance; (2) between the molecules of water; (3) between the molecules of the dissolved substance and the molecules of water.”

Lord Kelvin’s letter in Nature elicited a prompt response from none other than Josiah Willard Gibbs in America. Twenty-one years had now passed since James Clerk Maxwell first tried to interest Europe in the concept of chemical potentials. In Kelvin’s letter, with its feisty attack on the gaseous theory, Gibbs saw the opportunity to try again.

In his letter to Nature (18 March 1897), Gibbs opined that “Lord Kelvin’s very interesting problem concerning molecules which differ only in their power of passing a diaphragm, seems only to require for its solution the relation between density and pressure”, and highlighted the advantage of using his potentials to express van ‘t Hoff’s law:

“It will be convenient to use certain quantities which may be called the potentials of the solvent and of the solutum, the term being thus defined: – In any sensibly homogeneous mass, the potential of any independently variable component substance is the differential coefficient of the thermodynamic energy of the mass taken with respect to that component, the entropy and volume of the mass and the quantities of its other components remaining constant. The advantage of using such potentials in the theory of semi-permeable diaphragms consists partly in the convenient form of the condition of equilibrium, the potential for any substance to which a diaphragm is freely permeable having the same value on both sides of the diaphragm, and partly in our ability to express van’t Hoff law as a relation between the quantities characterizing the state of the solution, without reference to any experimental arrangement.”

But once again, Gibbs and his chemical potentials failed to garner interest in Europe. His timing was also unfortunate, since British experimental research into osmosis was soon to be stimulated by the aristocrat-turned-scientist Lord Berkeley, and this in turn would stimulate a new band of British theoreticians, including AW Porter and HL Callendar, who would base their theoretical efforts firmly on vapor pressure.

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Things Come Full Circle

As the new century dawned, van ‘t Hoff cemented his reputation with the award of the very first Nobel Prize for Chemistry “in recognition of the extraordinary services he has rendered by the discovery of the laws of chemical dynamics and osmotic pressure in solutions”.

The osmotic pressure law was held in high esteem, and despite Lord Kelvin’s protestations, Britain was well disposed towards the Gaseous Theory of Solutions. The idea circulating at the time was that the refinements of the ideal gas law that had been shown to apply to real gases, could equally well be applied to more concentrated solutions. As Lord Berkeley put it in the introduction to a paper communicated to the Royal Society in London in May 1904:

“The following work was undertaken with a view to obtaining data for the tentative application of van der Waals’ equation to concentrated solutions. It is evidently probable that if the ordinary gas equation be applicable to dilute solutions, then that of van der Waals, or one of analogous form, should apply to concentrated solutions – that is, to solutions having large osmotic pressures.”

Lord Berkeley’s landmark experimental studies on the osmotic pressure of concentrated solutions called renewed attention to the subject among theorists, who now had some fresh and very accurate data to work with. Alfred Porter at University College London attempted to make a more complete theory by considering the compressibility of a solution to which osmotic pressure was applied, while Hugh Callendar at Imperial College London combined the vapor pressure interpretation of osmosis with the hypothesis that osmosis could be described as vapor passing through a large number of fine capillaries in the semipermeable membrane. This was in 1908.


H L Callendar (1863-1930)

So seventeen years after Wilhelm Ostwald conclusively proved that hypothetical arguments as to the cause of osmotic pressure were inadmissible, things came full circle with hypothetical arguments once more being advanced as to the cause of osmotic pressure.

And as for Gibbs, his ideas were as far away as ever from British and European Science. The osmosis papers of both Porter (1907) and Callendar (1908) are substantial in referenced content, but nowhere do either of them make any mention of Gibbs or his explanation of osmosis on the basis of chemical potentials.

There is a special irony in this, since in Callendar’s case at least, the scientific papers of J Willard Gibbs were presumably close at hand. Perhaps even on his office bookshelf. Because that copy of Gibbs’ works shown in the header photo of this post – it’s a 1906 first edition – was Hugh Callendar’s personal copy, which he signed on the front endpaper.


Hugh Callendar’s signature on the endpaper of his personal copy of Gibbs’ Scientific Papers, Volume 1, Thermodynamics.

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Throughout this post, I have made repeated references to that inspired piece of thinking by Wilhelm Ostwald which conclusively demonstrated that osmotic pressure must be independent of the nature of the membrane.

Ostwald’s reasoning is so lucid and compelling, that one wonders why it didn’t put an end to speculation on osmotic mechanisms. But it didn’t, and hasn’t, and probably won’t.

Here is how Ostwald presented the argument in his own Lehrbuch der allgemeinen Chemie (1891). Enjoy.


“… it may be stated with certainty that the amount of pressure is independent of the nature of the membrane, provided that the membrane is not permeable by the dissolved substance. To understand this, let it be supposed that two separating partitions, A and B, formed of different membranes, are placed in a cylinder (fig. 17). Let the space between the membranes contain a solution and let there be pure water in the space at the ends of the cylinder. Let the membrane A show a higher pressure, P, and the membrane B show a smaller pressure, p. At the outset, water will pass through both membranes into the inner space until the pressure p is attained, when the passage of water through B will cease, but the passage through A will continue. As soon as the pressure in the inner space has been thus increased above p, water will be pressed out through B. The pressure can never reach the value P; water must enter continuously through A, while a finite difference of pressures is maintained. If this were realised we should have a machine capable of performing infinite work, which is impossible. A similar demonstration holds good if p>P ; it is, therefore, necessary that P=p; in other words, it follows necessarily that osmotic pressure is independent of the nature of the membrane.”

(English translation by Matthew Pattison Muir)

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P Mander July 2015


James Clerk Maxwell and the geometrical figure with which he proved his famous thermodynamic relations

Historical background

Every student of thermodynamics sooner or later encounters the Maxwell relations – an extremely useful set of statements of equality among partial derivatives, principally involving the state variables P, V, T and S. They are general thermodynamic relations valid for all systems.

The four relations originally stated by Maxwell are easily derived from the (exact) differential relations of the thermodynamic potentials:

dU = TdS – PdV   ⇒   (∂T/∂V)S = –(∂P/∂S)V
dH = TdS + VdP   ⇒   (∂T/∂P)S = (∂V/∂S)P
dG = –SdT + VdP   ⇒   –(∂S/∂P)T = (∂V/∂T)P
dA = –SdT – PdV   ⇒   (∂S/∂V)T = (∂P/∂T)V

This is how we obtain these Maxwell relations today, but it disguises the history of their discovery. The thermodynamic state functions H, G and A were yet to be created when Maxwell published the above relations in his 1871 textbook Theory of Heat. The startling fact is that Maxwell navigated his way to these relations using nothing more than a diagram of the Carnot cycle, allied to an ingenious exercise in plane geometry.

Another historical truth that modern derivations conceal is that entropy did not feature as the conjugate variable to temperature (θ) in Maxwell’s original relations; instead Maxwell used Rankine’s thermodynamic function (Φ) which is identical with – and predates – the state function entropy (S) introduced by Clausius in 1865.

Maxwell’s use of Φ instead of S was not a matter of personal preference. It could not have been otherwise, because Maxwell misunderstood the term entropy at the time when he wrote his book (1871), believing it to represent the available energy of a system. From a dimensional perspective – and one must remember that Maxwell was one of the founders of dimensional analysis – it was impossible for entropy as he understood it to be the conjugate variable to temperature. By contrast, it was clear to Maxwell that Rankine’s Φ had the requisite dimensions of ML2T-2θ-1.

Two years later, in an 1873 publication entitled A method of geometrical representation of the thermodynamic properties of substances by means of surfaces, the American physicist Josiah Willard Gibbs politely pointed out Maxwell’s error in regard to the units of measurement of entropy:


Maxwell responded in a subsequent edition of Theory of Heat with a contrite apology for misleading his readers:


– – – –

Carnot Cycle revisited

The centrepiece of the geometrical construction with which Maxwell proves his thermodynamic relations is a quadrilateral drawn 37 years earlier by Émile Clapeyron in his 1834 paper Mémoire sur la Puissance Motrice de la Chaleur (Memoir on the motive power of heat).


When Émile Clapeyron drew this PV-plane representation of the Carnot cycle in 1834, heat was believed to be a conserved quantity. By the time Maxwell used the diagram in 1871, heat and work were understood to be interconvertible forms of energy, the latter being the conserved quantity.

This is the first analytical representation of the Carnot cycle, shown as a closed curve on a pressure-volume indicator diagram. The sides ab and cd represent isothermal lines, the sides ad and bc adiabatic lines. By assigning infinitely small values to the variations of volume and pressure during the successive operations of the cycle, Clapeyron renders this quadrilateral a parallelogram.

The area enclosed by the curve equates to the work done in a complete cycle, and Maxwell uses the following contrivance to set this area equal to unity.

Applying Carnot’s principle, Maxwell expresses the work W done as a function of the heat H supplied

W = H(T2 – T1)/T2

with T2 and T1 representing the absolute temperatures of the source and sink respectively.
Maxwell then defines

T2 – T1 = 1
H/T2 = 1

The conversion of heat into work is thus expressed as the product of a unit change in temperature T and a unit change in Rankine’s thermodynamic function Φ, equivalent to entropy S:

W = Δ1T . Δ1S = 1

Maxwell’s definitions also give the parallelogram the property that any line drawn from one isothermal line to the other, or from one adiabatic line to the other, is of unit length when reckoned in the respective dimensions of temperature or entropy. This is of central significance to what follows.

– – – –

Geometrical extensions

Maxwell’s geometric machinations consist in extending the isothermal (T1T2) and adiabatic lines (Φ1Φ2) of the original figure ABCD and adding vertical lines (pressure) and horizontal lines (volume) to create four further parallelograms with the aim of proving their areas also equal to unity, while at the same time enabling each of these areas to be expressed in terms of pressure and volume as a base-altitude product.


As the image from Theory of Heat shown at the head of this article reveals, Maxwell did not fully trace out the perimeters of three (!) of the four added parallelograms. I have extended four lines to the arbitrarily labelled points E, F and H in order to complete the figure.

– parallelogram AKQD stands on the same base AD as ABCD and lies between the same parallels T1T2 so its area is also unity, expressible in terms of volume and pressure as the base-altitude product AK.Ak

– parallelogram ABEL stands on the same base AB as ABCD and lies between the same parallels Φ1Φ2 so its area is also unity, expressible in terms of volume and pressure as the base-altitude product AL.Al

– parallelogram AMFD stands on the same base AD as ABCD and lies between the same parallels T1T2 so its area is also unity, expressible in terms of pressure and volume as the base-altitude product AM.Am

– parallelogram ABHN stands on the same base AB as ABCD and lies between the same parallels Φ1Φ2 so its area is also unity, expressible in terms of pressure and volume as the base-altitude product AN.An

– line AD, which represents a unit rise in entropy at constant temperature, resolves into the vertical (pressure) and horizontal (volume) components Ak and Am

– line AB, which represents a unit rise in temperature at constant entropy, resolves into the vertical (pressure) and horizontal (volume) components Al and An

– in summary: ABCD = AK.Ak = AL.Al = AM.Am = AN.An = 1 [dimensions ML2T-2]

– – – –

Maxwell’s thermodynamic relations

Maxwell’s next step is to interpret the physical meaning of these four pairs of lines.

AK is the volume increase per unit rise in temperature at constant pressure: (∂V/∂T)P
Ak is the pressure decrease per unit rise in entropy at constant temperature: –(∂P/∂S)T

Recalling the property of partial derivatives that given the implicit function f(x,y,z) = 0


Since AK = 1/Ak

(∂V/∂T)P = –(∂S/∂P)T

AL is the volume increase per unit rise in entropy at constant pressure: (∂V/∂S)P
Al is the pressure increase per unit rise in temperature at constant entropy: (∂P/∂T)S

Since AL = 1/Al

(∂V/∂S)P = (∂T/∂P)S

AM is the pressure increase per unit rise in temperature at constant volume: (∂P/∂T)V
Am is the volume increase per unit rise in entropy at constant temperature: (∂V/∂S)T

Since AM = 1/Am

(∂P/∂T)V = (∂S/∂V)T

AN is the pressure increase per unit rise in entropy at constant volume: (∂P/∂S)V
An is the volume decrease per unit rise in temperature at constant entropy: –(∂V/∂T)S

Since AN = 1/An

(∂P/∂S)V = –(∂T/∂V)S

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In his own words

I leave it to the man himself to conclude this post:

“We have thus obtained four relations among the physical properties of the substance. These four relations are not independent of each other, so as to rank as separate truths. Any one might be deduced from any other. The equality of the products AK, Ak &c., to the parallelogram ABCD and to each other is merely a geometrical truth, and does not depend on thermodynamic principles. What we learn from thermodynamics is that the parallelogram and the four products are each equal to unity, whatever be the nature of the substance or its condition as to pressure and temperature.”


– – – –

P Mander August 2014


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.


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.


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.

– – – –


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.


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:


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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P Mander May 2014