Maxwell's Thermodynamic Surface
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Maxwell's Thermodynamic Surface
Photographs of Maxwell's plaster model from different angles.

Maxwell's thermodynamic surface is an 1874 sculpture[1] made by Scottish physicist James Clerk Maxwell (1831-1879). This model provides a three-dimensional space of the various states of a fictitious substance with water-like properties.[2] This plot has coordinates volume (x), entropy (y), and energy (z). It was based on the American scientist Josiah Willard Gibbs' graphical thermodynamics papers of 1873.[3][4] The model, in Maxwell's words, allowed "the principal features of known substances [to] be represented on a convenient scale."[5]

## Construction of the model

Gibbs' papers defined what Gibbs called the "thermodynamic surface," which expressed the relationship between the volume, entropy, and energy of a substance at different temperatures and pressures. However, Gibbs did not include any diagrams of this surface.[3][6] After receiving reprints of Gibbs' papers, Maxwell recognized the insight afforded by Gibbs' new point of view and set about constructing physical three-dimensional models of the surface.[7] This reflected Maxwell's talent as a strong visual thinker[8] and prefigured modern scientific visualization techniques.[3]

Maxwell sculpted the original model in clay and made several plaster casts of the clay model, sending one to Gibbs as a gift, keeping two in his laboratory at Cambridge University.[3] Maxwell's copy is on display at the Cavendish Laboratory of Cambridge University,[3][9] while Gibbs' copy is on display at the Sloane Physics Laboratory of Yale University,[10] where Gibbs held a professorship. Two copies reside at the National Museum of Scotland, one via Peter Tait and the other via George Chrystal.[11][12][13] Another was sent to Thomas Andrews.[13] A number of historic photographs were taken of these plaster casts during the middle of the twentieth century - including one by James Pickands II, published in 1942[14] - and these photographs exposed a wider range of people to Maxwell's visualization approach.

## Uses of the model

Diagram of thermodynamic surface from Maxwell's book Theory of Heat. The diagram is drawn roughly from the same angle as the upper left photo above, and shows the 3D axes e (energy, increasing downwards), ? (entropy, increasing to the lower right and out-of-plane), and v (volume, increasing to the upper right and into-plane).

As explained by Gibbs and appreciated by Maxwell, the advantage of a U-V-S (energy-volume-entropy) surface over the usual P-V-T (pressure-volume-temperature) surface was that it allowed to geometrically explain sharp, discontinuous phase transitions as emerging from a purely continuous and smooth state function ${\displaystyle U(V,S)}$; Maxwell's surface demonstrated the generic behaviour for a substance that can exist in solid, liquid, and gaseous phases. The basic geometrical operation involved simply placing a tangent plane (such as a flat sheet of glass) on the surface and rolling it around, observing where it touches the surface. Using this operation, it was possible to explain phase coexistence, the triple point, to identify the boundary between absolutely stable and metastable phases (e.g., superheating and supercooling), the spinodal boundary between metastable and unstable phases, and to illustrate the critical point.[15]

Maxwell drew lines of equal pressure (isopiestics) and of equal temperature (isothermals) on his plaster cast by placing it in the sunlight, and "tracing the curve when the rays just grazed the surface."[2] He sent sketches of these lines to a number of colleagues.[16] For example, his letter to Thomas Andrews of 15 July 1875 included sketches of these lines.[2] Maxwell provided a more detailed explanation and a clearer drawing of the lines (pictured) in the revised version of his book Theory of Heat,[15] and a version of this drawing appeared on a 2005 US postage stamp in honour of Gibbs.[6]

As well as being on display in two countries, Maxwell's model lives on in the literature of thermodynamics, and books on the subject often mention it,[17] though not always with complete historical accuracy. For example, the thermodynamic surface represented by the sculpture is often reported to be that of water,[17] contrary to Maxwell's own statement.[2]

## Related models

Maxwell's model was not the first plaster model of a thermodynamic surface: in 1871, even before Gibbs' papers, James Thomson had constructed a plaster pressure-volume-temperature plot, based on data for carbon dioxide collected by Thomas Andrews.[18]

Around 1900, the Dutch scientist Heike Kamerlingh Onnes, together with his student Johannes Petrus Kuenen and his assistant Zaalberg van Zelst, continued Maxwell's work by constructing their own plaster thermodynamic surface models.[19] These models were based on accurate experimental data obtained in their laboratory, and were accompanied by specialised tools for drawing the lines of equal pressure.[19]

## References

1. ^ Maxwell, James Clerk (1990). The Scientific Letters and Papers of James Clerk Maxwell: 1874-1879. p. 148. ISBN 9780521256278. 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." (letter to Thomas Andrews, November, 1874)
2. ^ a b c d Maxwell, James Clerk (1995-01-01). Maxwell on Heat and Statistical Mechanics: On "Avoiding All Personal Enquiries" of Molecules. p. 248. ISBN 9780934223348. I think you know Prof. J. Willard Gibbs's (Yale College Connecticut) 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 modelled a fictitious substance, in which the volume is greater when solid than when liquid; and in which, as in water, the saturated vapour 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..." (letter to Thomas Andrews, 15 July 1875)
3. Thomas G.West (February 1999). "James Clerk Maxwell, Working in Wet Clay". SIGGRAPH Computer Graphics Newsletter. 33 (1): 15-17. doi:10.1145/563666.563671. S2CID 13968486.
4. ^ Cropper, William H (2004). Great Physicists: The Life and Times of Leading Physicists from Galileo to Hawking. p. 118. ISBN 9780195173246.
5. ^ Maxwell and Harman, pp. 230-231: "I enclose a rough sketch of the lines on Gibbs' surface, co-ordinates Volume Entropy Energy in an imaginary substance in which the principal features of known substances can be represented on a convenient scale." (letter to James Thomson, 8 July 1875)
6. ^ a b
7. ^ Maxwell, Garber, Brush, and Everitt, p. 49.
8. ^ Ken Brodlie, "Scientific visualization -- past, present and future," Proceedings of the Third Workshop on Neutron Scattering Data Analysis, Nuclear Instruments and Methods in Physics Research, Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, Volume 354, Issue 1, 15 January 1995, Pages 104-111, doi:10.1016/0168-9002(94)01031-5.
9. ^ The Museum at the Cavendish Laboratory: Maxwell's Apparatus.
10. ^ Kenneth R. Jolls (1990). "Gibbs and the art of thermodynamics". In D. G. Caldi; George D. Mostow (eds.). Proceedings of the Gibbs Symposium, Yale University, May 15-17, 1989. American Mathematical Society. p. 321. ISBN 978-0-8218-0157-4.
11. ^ https://www.nms.ac.uk/explore-our-collections/collection-search-results/thermodynamic-model/417665
12. ^ https://www.nms.ac.uk/explore-our-collections/collection-search-results/thermodynamic-model/425122
13. ^ a b https://www.nms.ac.uk/explore-our-collections/stories/science-and-technology/james-clerk-maxwell-inventions/james-clerk-maxwell/thermodynamic-surface
14. ^ Muriel Rukeyser (1942), Willard Gibbs American Genius (reprinted by Ox Bow Press, ISBN 0-918024-57-9), p. 203.
15. ^ a b James Clerk Maxwell, Theory of Heat, revised in 1891 by John Strutt, 3rd Baron Rayleigh: the drawing of the lines appears as Figure 26d on page 207.
16. ^ Maxwell, Garber, Brush, and Everitt, pp. 50.
17. ^ a b See, for example, Don S. Lemons, Mere Thermodynamics, Johns Hopkins University Press, 2008, ISBN 0-8018-9015-2, p. 146.
18. ^ Johanna Levelt Sengers, How Fluids Unmix: Discoveries by the School of Van der Waals and Kamerlingh Onnes Archived 2007-05-30 at the Wayback Machine, Royal Netherlands Academy of Arts and Sciences, 2002, pp. 56 & 104.
19. ^ a b See the page 3D-Models/Mixtures/Experiments: Kamerlingh Onnes Archived 2004-12-10 at the Wayback Machine from the Royal Netherlands Academy of Arts and Sciences. Some of these models are on display at the Museum Boerhaave: Room 21 Archived 2011-06-07 at the Wayback Machine.