Maxwell’s thermodynamic surface
Maxwell's thermodynamic surface, at Cambridge University, with 2007 overlay annotation added on by American engineer Ronald Kriz. [5]
In thermodynamics, Maxwell’s thermodynamic surface is a famous three-dimensional plaster surface showing the various states of existence of a theoretical substance, modeled on water, namely gas, liquid, solid, states, and critical point, mapped on a Cartesian coordinates plot of the entropy (x), volume (y), and energy (z) of the body, made by Irish physicist James Maxwell, constructed over the course of about seven months, from November 1874 to July 1875, based on the descriptions of thermodynamics surfaces in American engineer Willard Gibbs' two 1873 papers on the graphical methods of thermodynamics. The "water sculpture" as it has been called was constructed in Cambridge and famously sent to Gibbs in 1875 as a gift of appreciation for his graphical work. [1] As summarized by American science historian Martin Klein in 1983: [20]

Maxwell expressed his appreciation of Gibbsthermodynamic surface by constructing a model of it for water and sending a cast of it to Gibbs and included a fourteen-page discussion of Gibbs surface in the 1875 edition of his textbook.”

Gibbs ‘graphical method’, as Maxwell called it, involved a three-dimensional mental model comparable to what we would now call a surface plot a kind of 3D boundary surface graph showing a number of points covering an undulating surface with rises and depressions like small mountains, hills and valleys. As each point on the surface (representing a particular conditions of temperature, volume, and pressure) moves over the surface plot, the conditions change (change of state) in certain predictable ways, giving a deep understanding of the behavior of the complex system based on being able to visualize the complex shape. [10]

Most refer to this sculpture made by Maxwell as Maxwell's thermodynamic surface. Maxwell called it "Gibbs' thermodynamics surface", among other names; others refer to it as a water statue, a water-substance surface, or thermodynamic surface for water, a "plaster cast showing the showing the thermodynamic properties of water" among other variants.

On his water 'statue', as some have called it, Maxwell located areas where the liquid, vapor, and solid phases coexist, areas where two phases can coexist, and a triangle representing coexistence of all three phases. On the surface of the model, Maxwell carved contours of constant pressure and temperature, as dictated by the equations of the first main principle and second main principle. [9] The surface embodies a number of complex core aspects of thermodynamics in respect to equilibrium. In the words of Gibbs biographer American engineer Lynde Wheeler: [1]

“It is impossible to describe this thermodynamic surface in detail without considerable prolixity and use of the notions and terms of higher geometry.”

Thomson thermodynamics surface (1871)
Scottish engineer James Thomson's 1871 thermodynamics surface model, on display at the Hunterian Museum and Art Gallery, University of Glasgow; possibly a precursor or influence to Maxwell's thermodynamic surface.

The molded shape depicts the geometry of the three-dimensional thermodynamic surface of the various states of existence of water: solid, liquid, or gas, shown on Cartesian coordinates of the entropy (x), volume (y), and energy (z) of the body. [2]

In total Maxwell made three surfaces, or possibly six (as discussed below), one he sent to Gibbs at Yale (where it resides currently on display in a glass case outside the Yale University physics department), the other two remained at Cambridge University, held presently by the Museum at the Cavendish Laboratory at Cambridge, one as pictured adjacent. [5] A step-by-step 2008 poster overview of the construction of Maxwell’s surface was compiled by American engineer Ronald Kriz. [12]

Thomson’s thermodynamic surface
In 1871, James Thomson constructed a plaster pressure-volume-temperature plot, based on data for carbon dioxide collected by Thomas Andrews. [22]

In July 1871, James Maxwell was writing his Theory of Heat and ruminating on a “question that had puzzled him”, as Carlo Cercignani reports, in respect to Thomas Andrews’ results on the continuity of liquid and gaseous states, and was corresponding with James Thomson, elder brother of William Thomson, and colleague of Andrews in Belfast, who suggested that the isothermal curves for a fluid below its critical temperature, for the transition from the gaseous to the fluid state to be possible—in the traditional pressure-volume diagram—should really show a minimum and maximum rather than a straight line segment parallel to the volume axis, as showed by Andrews’ data. In 1873, after reading Gibbs’ graphical thermodynamics papers, Maxwell found solution to his query, and thereafter made his thermodynamic surface, which he then mailed to Gibbs as a thank-you present, as discussed further below. [23]

Model substance: fictitious, imaginary, or water?
The question of what substance, whether water, fictitious, or imaginary, Maxwell sculpted and sent to Gibbs is a bit of a mystery. Historically, the surface has always been considered to have been a sculpture of the various states of water, as is exemplified by Gibbs’ biographer Muriel Rukeyser 1942 naming of it as a plaster surface a “statue of water” as well as Yale physicist and Gibbs' biographer Lynde Wheeler 1951 definition of it as a “thermodynamic surface for water”.

Maxwell's first mention of his surface is found in a November 1874 letter to English chemist Thomas Andrews, concerning the experimental results of his noted article “On the Continuity of the Gaseous and Liquid States of Matter” (1869), an article also cited by Gibbs (1873), Maxwell gives indication that he has made a ‘clay model of a fancy surface’: [15]

“Are the numerical results of your former experiments on CO2 published anywhere except in the Philosophical Transactions [1869]? I have just finished a clay model of a fancy surface, showing the solid, liquid, and gaseous states, and the continuity of liquid and gaseous states. I am afraid that even CO2 would not make a very compact model if worked true to scale. But the data as to specific heat in the liquid and solid states are wanting, and also the latent heat of fusion and evaporation.”

In February 18th, 1875 letter to Gibbs, English mathematician Alexander Freeman wrote: [3]

“You will be gratified to hear that Prof. Maxwell has made a clay model of your thermodynamic surface wherein entropy, energy, and volume are the three coordinates, and is able to explain a great deal by it.”

In a July 8th, 1875 letter to Irish engineer James Thomson, Maxwell seems to have indicated that Thomson may have been in the process of making his own cast, suggesting that such models will be displayed in a special glass case at the Cavendish Laboratory: [18]

“It will give me great pleasure to receive on the part of the Cavendish Laboratory a cast of your thermodynamic model with lines marked on it. We know have got an excellent case with a glass front containing an thermometer by II Gonfio (before 1640, Wollastons optical and thermal apparatus, and others, and we shall have a special place for models such as yours. I enclosed a rough sketch of the lines of Gibbs’ surface, co-ordinates volume [x], entropy [y], energy [z], in an imaginary substance in which the principle features of the substance can be represented on a convenient scale.”

This sketch is depicted below, the coloring of which Maxwell goes into some detail explaining their mathematical and thermodynamic meanings: [14]

Maxwell surface (3D sketch 1875)Maxwell thermodynamic surface 400px
Maxwell's 3D Sketch of the lines on Gibbs’ thermodynamic surface (8 July 1875) [14]
Dark-scale photo of Maxwell's sculpture. [14]

In a follow-up July 15th, 1875 letter to Thomas Andrews, Maxwell comments: [6]

"I think you know Prof. J. Willard Gibbs' graphical methods in thermodynamics. Last winter I made several attempts to model the surface which he suggests, in which the three coordinates are volume, entropy and energy. The numerical data about entropy can only be obtained by integration from data which are for most bodies very insufficient, and besides it would require a very unwieldy model to get all the features, say of CO2, well represented, so I made no attempt at accuracy, but modeled a fictitious substance, in which the volume is greater when solid than when liquid; and in which, as in water, the saturated vapor becomes superheated by compression. When I had at last got a plaster cast I drew on it lines of equal pressure and temperature, so as to get a rough motion of their forms. This I did by placing the model in sunlight, and tracing the curve when the rays just grazed the surface... I send you a sketch of these lines..."
Maxwell's thermodynamic surface (National Museums Scotland)
One of Maxwell’s thermodynamic surfaces at the National Museums Scotland. (Ѻ)

As to the date and details of the statue that Gibbs actually received from Maxwell, the corroborating documentation seems to have disappeared, and moreover there seems to have been a certain amount of mystery surrounding the details of this package. In the 1909 “Josiah Willard Gibbs and His Relation to Modern Science”, for example, American science historian Fielding Garrison indicates that possibly up to six statues were made by Maxwell and that in his typical riddled correspondence style (see for example the origin of the symbol θ∆ics) that a certain amount of playful mystery accompanied the sending of each statue to its new host: [19]

“Copies of this model were distributed by Maxwell evidently with a certain amount of playful mystery, for each recipient thought that he was the happiest possessor of (at most) three. The writer knows of at six at least, and possibly there are more.”

Gibbs’ 1942 biographer Muriel Rukeyser, following her prolonged study of the Gibbs archives material at Yale, specifically states:

“The statue which Clerk Maxwell sent to Willard Gibbs was a statue of water.”

In 1951, however, Gibbs biographer, Yale physics professor Lynde Wheeler, specifically states:

“There is no record of Maxwell’s having corresponded with Gibbs on this (or any other) occasion, although one would expect that his gift of the model of the thermodynamic surface would have been accompanied by a letter of transmittal. This was either lost in transit or mislaid by Gibbs, as no such letter is included in the volume of his correspondence assembled by his brother-in-law nor did Gibbs ever allude to it as far as is known.”

From these various letters and opinions, which outline the view that: Maxwell had originally had "made several attempts to model the surface"; that he finished making a (likely first-draft) clay model surface in November (1874); that he was communicating news of the sculpture to Freeman (and thus to Gibbs) in February (1875); that he made detailed color-coded three-dimensional sketch of the model by July 8th (1875); that he had added grooved isothermal and isochoric lines to the model by July 15th (1875); and that he made three physical models in total, if not six or more.

Free energy | available energy
The idea of Gibbs' "free" energy, according to the 2009 views of American engineer Ronald Kriz, is often used in equations but not fundamentally understood as originally developed by Gibbs as a geometric property used to describe triple point thermodynamic processes. Understanding is often disguised by demonstrating use of equations. To understand Gibbs free energy, according to Kriz, it is first necessary to understand Gibbs' original thermodynamic (3D) U-S-V surface. After studying how Maxwell created this surface, the relationship of triple point thermodynamic processes associated with Gibbs' free energy can be fundamentally understood as original proposed by Gibbs without equations. [11]

One should also note that Gibbs never used the term “free energy” but instead used the term “available energy” (graphically signified by section AB, figure 3, below); the former term arising from the 1882 paper “The Thermodynamics of Chemical Processes” by German physicist Hermann Helmholtz, who commented on the earlier graphical work of Gibbs.
Gibbs (figure 1)
Gibbs 1873 figure 1.

Overview | short history
In 1873, American engineer Willard Gibbs published his first thermodynamics paper, “Graphical Methods in the Thermodynamics of Fluids”, in which Gibbs uses the two coordinates of the entropy and volume to represent the state of the body. [3]

In his second follow-up paper, “A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces”, published later that year, Gibbs added in the third coordinate of the energy of the body, as shown in the following three diagrams, albeit mostly described via equations and verbally. [4]

To go though one example, of the difficulty involved in conceptualizing, three-dimensionally, Gibbs' logic, figure 3 (below middle), according to Gibbs, shows a plane perpendicular to the axis of v (volume) and passing through point A, which represents the initial state of the body. MN is the section of the surface of dissipated energy. Qε and Qη are sections of the planes η = 0 and ε = 0, and therefore parallel to the axes of ε (internal energy) and η (entropy), respectively. AD and AE are the energy and entropy of the body in its initial state, AB and AC its available energy (Gibbs free energy) and its capacity for entropy (the amount by which the entropy of the body can be increased without changing the energy of the body or increasing its volume) respectively:

Maxwell shape (construction)

In short, above (left) we have American engineer Willard Gibbs' 1873 figures two and three (above left and middle) used by Scottish physicist James Maxwell in 1875 to create a three-dimensional entropy (x), volume (y), energy (z) thermodynamic surface diagram for water, transposed the two figures of Gibbs (above right) onto the volume-entropy coordinates (transposed to bottom of cube) and energy-entropy coordinates (flipped upside down and transposed to back of cube), respectively, of a three-dimensional Cartesian coordinates; the region AB being the first three-dimensional representation of Gibbs free energy, or what Gibbs called "available energy"; the region AC being its capacity for entropy, what Gibbs defined as "the amount by which the entropy of the body can be increased without changing the energy of the body or increasing its volume.

Thus, not only does Gibbs only provide the reader with a two-dimensional picture for a three-dimensional discussion, but he uses the obscure Greek notation for thermodynamic quantities of French engineer Francois Massieu. [13]

Gibbs thermodynamic surface (Plate IV)
July 8th, 1875 sketches by Maxwell of "Gibbs' thermodynamics surface", as he called it, on a volume, entropy, energy plot. [12]
In any event, with this addition, Gibbs found it possible to cover in great generality the determination of the “conditions for equilibrium” for any body of variable composition, and in particular to take into account of those states of the body which are not homogeneous (i.e. those consisting of a mixture of more than one state), although still in equilibrium. The difficulty at this point in the work of Gibbs is that it is difficult to visualize the three-dimensional thermodynamic picture that he describes in his 1873 papers. As computer graphics designer Thomas West summarizes: [10]

“Gibbs wrote about his new method and described it mathematically, yet he made no effort to make a diagram of what was, apparently, clearly seen in his own mind’s eye. The new method and the difficulty in having to visualize such complex material resulted in little attention from Gibbs’ scientific colleagues, especially in the United States. Yet, when Maxwell read Gibbs’ papers in Britain, he immediately saw the power and the potential of the new graphical method and would spend an entire winter constructing a 3D clay model of a surface using Gibbs’ data.”

Maxwell received Gibbs two graphical papers in circa 1873-74. As commented by Gibbs stamp designer American chemical engineer Kenneth Jolls: “Maxwell was so attracted by Gibbs geometrical approach that he spent an entire winter building the model for the energy-entropy-volume surface (USV surface) for water.” [8]

By July, Maxwell was into the final stages of preparation for the publication for the updated edition of his Theory of Heat (1875), in which he devoted a fourteen page section to thermodynamic surfaces, entitled "Representation of the Properties of a Substance by Means of a Surface", of which figure 26d "thermodynamic surface", in the section, ended up being the backdrop design on the 2005 commemorative Gibbs stamp. [7]

In conclusion the effect of these models in science, already by 1909, as American science historian Fielding Garrison summarizes: [19]

“These solid diagrams have played a great part in the elaborate studies of the continuity of gaseous and liquid states by Van der Waals and his pupils, of which we have recently witnessed the final triumph in the liquefaction of helium.”

See main: Thermodynamics anecdotes
Gibbs was said to have been flattered and pleased with the gift of the water sculpture from Maxwell. Owing to his typical modesty, however, when students asked about it is said that Gibbs told them it came from a “friend in England”, rather than make mention that it came from the famous James Maxwell. [9] The anecdote seems to have originated from American writer Muriel Rukeyser's 1942 biography on Gibbs, where she states that the student was the father of a "Leonard Bacon". [16]

Into the turn of the 20th-century, college students, such as at the University of California, Berkeley (1900), were attempting to make or reconstruct models of Gibbs surface, similar to what Maxwell did. [27]

See also
Maxwell’s demon

Gibbs stamp (no s)
Thermodynamics surface (Maxwell 1875)
2005 commemorative Gibbs stamp, showing an overlay of figure 26d "thermodynamic surface" of Scottish physisict James Maxwell's 1875 Theory of Heat, based on Gibbs' two 1873 graphical thermodynamics papers.

1. Wheeler, Lynde Phelps. (1951). Josiah Willard Gibbs: the History of a Great Mind (thermodynamic surface, pgs. 73-74; plaster picture, pg. 86). Ox Bow Press.
2. Maxwell plaster picture (image) – Cambridge University.
3. Gibbs, J. Willard. (1873). "Graphical Methods in the Thermodynamics of Fluids", Transactions of the Connecticut Academy, I. pp. 309-342, April-May.
4. Gibbs, J. Willard. (1873). "A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces", Transactions of the Connecticut Academy, II. pp.382-404, Dec.
5. Kriz, Ronald D. (2007). “Thermodynamic Case Study: Gibbs’ Thermodynamic Graphical Method: Envisioning Total Derivatives of a Scalar Function with Two Independent Variables as Raised Surfaces and Tangent Planes.”
6. Maxwell, James. (1875). “Letter to Thomas Andrews”, Jul 15; in The Scientific Letters and Papers of James Clerk Maxwell (pgs. 236-37). Volume 3; 1874-1879.
7. Maxwell, James. (1875). Theory of Heat (Fig. 26d. “Thermodynamic Surface”, pg. 207). Dover.
8. Caldi, D. G. and Mostow, George D. (1989). Proceedings of the Gibbs Symposium, May 15-17 (Kenneth Jolls, plaster model, pg. 296, 309). American Mathematical Society.
Maxwell surface (Yale)
Maxwell's thermodynamic surface on display at the physics department at Yale University.
9. Cropper, William H. (2004). Great Physicists: the Life and Times of Leading Physicists from Galileo to Hawking (section II: Thermodynamics, pgs. 41-134; ch. 9: “The Greatest Simplicity: Willard Gibbs”, pgs 106-23; water statue, pg. 118, 309). Oxford University Press.
10. (a) Email communication from Ronald Kriz to Libb Thims (27 Jul 2010).
(b) West, Thomas G. (1999). “James Clerk Maxwell, Working in Wet Clay”, SIGGraph Computer Graphics Newsletter: Images and Reversals, 32(1), Feb.
11. Kriz, Ronald D. (2009). “Triple Point Processes Associated with Gibbs’ Free Energy”, College of Engineering, Virginia Tech,
12. Kriz, Ronald D. (2008). “Maxwell’s thermodynamic surface (poster overview)” (4848x3131px),
13. (a) Massieu, Francois. (1869). “Sur les Fonctions Caracteristiques des Divers Fluids (On the Various Functions Characteristic of Fluids)”, Comptes Rendus 69, pgs. 858-62, 1057-61.
(b) Gibbs refers to Massieu’s notation in 1875 and 1876 (See: Sci. Pap. 1, pg. 86-87, 358).
14. Maxwell, James. (1875). “Letter to James Thomson: Diagram of Gibbs’ Thermodynamic Surface”, circa July 8th, James Thomson Papers, MS 13/22i, Library of the Queens University Belfast; in: The Scientific Letters and Papers of James Maxwell, Volume 3 (1874-1879) (pgs. 232-33). Cambridge University Press, 2002.
15. Maxwell, James. (1874). “Letter to Thomas Andrews”, Nov; In: P.G. Tait and Alexander Crum-Brown, “Memoir of Dr. Thomas Andrews,” The Scientific Papers of the late Thomas Andrews with a Memoir. London: Macmillan and Co, 1889: ix-lxii, liv-lv; In: Maxwell, James. (1855-78). “Documents on Thermodynamics (47 documents)”, in: Maxwell on Heat and Statistical Mechanics: On Avoiding all Personal Enquires of Molecules (pgs. 171-288), by Elizabeth Garber, Stephen Brush, and C. W. Everitt, Lehigh University Press, 1995.
16. Rukeyser, Muriel. (1942). Willard Gibbs: American Genius (The Surface, pg. 202-03). Garden City, New York: Doubleday.
17. Freeman, Alexander. (1875). “Letter to J. W. Gibbs”, Feb 18; in J.W. Gibbs, Scientific Correspondence, 86. Beinecke Rare book and manuscript Library, Yale University Library.
18. Maxwell, James. (1875). “Letter to James Thomson”, July 8th; in The Scientific Letters and Papers of James Clerk Maxwell (pgs. 230-31). Volume 3; 1874-1879.
19. Garrison, Fielding H. (1909). “Josiah Willard Gibbs and His Relation to Modern Science” (note 41, pg. 481), The Popular Science Monthly, 74: 470-84.
20. Klein, Martin J. (1983). “The Scientific Style of Josiah Willard Gibbs” (Gibbs surface, pg. 158), in: Springs of Creativity (pgs. 141-), by Rutherford Aris, Howard Davis, and Roger Stuewer. U of Minnesota Press.
21. Boynton, W.P. (1900). “Gibbs’ Thermodynamic Model”, Physical Review, 10: 228-33.
22. Sengers, Johanna L. (2002). How Fluids Unmix: Discoveries by the School of Van der Waals and Kamerlingh Onnes (GB) (abs) (pgs. 56, 104). Royal Netherlands Academy of Arts and Sciences.
23. Cercignani, Carlo. (1998). Ludwig Boltzmann: the Man Who Trusted Atoms (foreward by Roger Penrose) (pgs. 136-37). Oxford University Press.

Further reading
● Porter, Spencer K. (1971). “The Volume-Entropy-Energy Surface of J. W. Gibbs” (abstract), J. Chem. Educ., 48(4):231.

External links
Maxwell’s thermodynamic surface – Wikipedia.

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