Maxwell's thermodynamic surface, at Cambridge University, with 2007 overlay annotation added on by American engineer Ronald Kriz. [5] |

“Maxwellexpressed his appreciation of Gibbs’thermodynamic surfaceby constructing a model of it forwaterand 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]

Synonyms

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.

Overview

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 thisthermodynamic surfacein detail without considerable prolixity and use of the notions and terms of higher geometry.”

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

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

In July 1871, James Maxwell was writing his

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

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 thePhilosophical Transactions[1869]? I have just finished aclay 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 tospecific heatin 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 aclay modelof yourthermodynamic surfacewherein 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 animaginary substancein 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'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 afictitious 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..."

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

“Copies of thismodelwere 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

“The statue which Clerk Maxwell sent to Willard Gibbs was astatue 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 themodelof thethermodynamic surfacewould 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 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

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)

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]

July 8th, 1875 sketches by Maxwell of "Gibbs' thermodynamics surface", as he called it, on a volume, entropy, energy plot. [12] |

“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

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

In conclusion the effect of these models in science, already by 1909, as American science historian

“Thesesolid diagramshave 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.”

Anecdote

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]See main: Thermodynamics anecdotes

Education

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

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. |

References

1. Wheeler, Lynde Phelps. (1951).

2. Maxwell plaster picture (image) – Cambridge University.

3. Gibbs, J. Willard. (1873). "Graphical Methods in the Thermodynamics of Fluids",

4. Gibbs, J. Willard. (1873). "A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces",

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.” Sv.Vt.edu.

6. Maxwell, James. (1875). “Letter to Thomas Andrews”, Jul 15; in

7. Maxwell, James. (1875).

8. Caldi,

Maxwell's thermodynamic surface on display at the physics department at Yale University. |

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”,

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), Sv.Vt.edu.

13. (a) Massieu, Francois. (1869). “Sur les Fonctions Caracteristiques des Divers Fluids (On the Various Functions Characteristic of Fluids)”,

(b) Gibbs refers to Massieu’s notation in 1875 and 1876 (See:

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:

15. Maxwell, James. (1874). “Letter to Thomas Andrews”, Nov; In: P.G. Tait and Alexander Crum-Brown, “Memoir of Dr. Thomas Andrews,”

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

19. Garrison, Fielding H. (1909). “Josiah Willard Gibbs and His Relation to Modern Science” (note 41, pg. 481),

20. Klein, Martin J. (1983). “The Scientific Style of Josiah Willard Gibbs” (Gibbs surface, pg. 158), in:

21. Boynton, W.P. (1900). “Gibbs’ Thermodynamic Model”,

22. Sengers, Johanna L. (2002).

23. Cercignani, Carlo. (1998).

Further reading

● Porter, Spencer K. (1971). “The Volume-Entropy-Energy Surface of J. W. Gibbs” (abstract),

External links

● Maxwell’s thermodynamic surface – Wikipedia.