http://collections.peabody.yale.edu/search/Record/YPM-HSI-290012
Maxwellは, Gibbsの論文を読んでそのエネルギー，エントロ
ピー，体積の３次元モデルを石膏でつくって，その一つをGibbsに送ったそうです。 米国イエール大学にその石膏モデルが現存しております。3次元的なイメー
ジ(偏微分)は天才でもつきにく
いものなんですね。(垣内先生情報ありがとうございました。)
2018Jan17

Copy of plaster model of Gibbs' thermodynamic (energy entrophy surface) model
made by James Clerk Maxwell and sent to Josiah Willard Gibbs at Yale. Signed FEB
1909 (Frederick E Beach). Original model put on permanent display in Sloane
Physics Laboratory in 2006.
Catalog Number:

YPM HSI 290012

Department:

Historical Scientiﬁc Instruments

Summary Data:

Copy of plaster model of Gibbs' thermodynamic(energy entrophy surface) model made by James
Clerk Maxwell and sent to Josiah Willard Gibbs at
Yale. Signed FEB 1909 (Frederick E Beach).
Original model put on permanent display in Sloane
Physics Laboratory in 2006.

Description:

number 290012; lot count 1

Function:scientiﬁc instruments

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HSI.290012: Copy of plaster model of Gibbs'
thermodynamic (energy entrophy surface)model made by James Clerk Maxwell andsent to Josiah Willard Gibbs at Yale. Signed
FEB 1909 (Frederick E Beach). Originalmodel put on permanent display in Sloane
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https://en.wikipedia.org/wiki/Maxwell%27s̲thermodynamic̲surface

Maxwell's thermodynamic surface
Maxwell’s thermodynamic surface is an 1874
sculpture[2] made by Scottish physicist James Clerk
Maxwell (1831–1879). This model provides a three­dimensional plot of the various states of a fictitious
substance with water­like properties.[3] 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.[4][5]The model, in Maxwell's words, allowed "the principalfeatures of known substances [to] be represented on aconvenient scale."[6]Historic photograph of Maxwell’s plasterContents

model (taken by James Pickands II, and[1]

published in 1942).

Construction of the model
Uses of the model
Related models
See also
References
External links

Construction of the model
Gibbs' papers defined what Gibbs called the "thermodynamic surface," which expressed the relationshipbetween the volume, entropy, and energy of a substance at different temperatures and pressures. However,Gibbs did not include any diagrams of this surface.[4][7] After receiving reprints of Gibbs' papers, Maxwell
recognized the insight afforded by Gibbs' new point of view and set about constructing physical three­dimensional models of the surface.[8] This reflected Maxwell's talent as a strong visual thinker[9] andprefigured modern scientific visualization techniques.[4]Maxwell sculpted the original model in clay and made three plaster casts of the clay model, sending one to
Gibbs as a gift, keeping the other two in his laboratory at Cambridge University.[4] Maxwell's copy is on display
at the Cavendish Laboratory of Cambridge University,[4][10] while Gibbs' copy is on display at the Sloane
Physics Laboratory of Yale University,[11] where Gibbs held a professorship. 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[1] – and these photographs exposed a wider range of people to Maxwell's visualization
approach.

Uses of the model
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."[3] He sent sketches of theselines to a number of colleagues.[12] For example, his letter to Thomas
Andrews of 15 July 1875 included sketches of these lines.[3] Maxwell
provided a more detailed explanation and a clearer drawing of the lines in
the revised version of his book Theory of Heat,[13] and a version of thisdrawing appeared on a 2005 US postage stamp in honour of Gibbs.[7]As well as being on display in two countries, Maxwell's model lives on in theliterature of thermodynamics, and books on the subject often mention it,[14]Maxwell's sketch of theisothermals andisopiestics on his

though not always with complete historical accuracy. For example, thethermodynamic surface represented by the sculpture is often reported to bethat of water,[14] contrary to Maxwell's own statement.[3]surface (Plate IV of hiscollected letters).

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 pressure­volume­
temperature plot, based on data for carbon dioxide collected by Thomas Andrews.[15]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 plasterthermodynamic surface models.[16] These models were based on accurate experimental data obtained in theirlaboratory, and were accompanied by specialised tools for drawing the lines of equal pressure.[16]See also
History of thermodynamics
Scientiﬁc visualization

References
1. Muriel Rukeyser (1942), Willard Gibbs American Genius (reprinted by Ox Bow Press, ISBN 0‒918024‒57‒9), p. 203.
2. Maxwell, James Clerk (1990). The Scientiﬁc Letters and Papers of James Clerk Maxwell: 1874‒

1879 (https://books.google.com/books?id=JbNK9lRLHPEC&pg=PA148#v=onepage&q&f=false). 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)"

3. Maxwell, James Clerk (1995‒01‒01). Maxwell on Heat and Statistical Mechanics: On "Avoiding

All Personal Enquiries" of Molecules (https://books.google.com/books?id=hA‒oIDR0eXkC&pg=
PA248). 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 tomodel the surface which he suggests, in which the three coordinates are volume, entropy andenergy. The numerical data about entropy can only be obtained by integration from dataﬃ

which are for most bodies very insu

cient, 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, butmodelled 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 geta 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 toThomas Andrews, 15 July 1875)"
4. Thomas G.West (February 1999). "James Clerk Maxwell, Working in Wet Clay" (http://www.sigg
raph.org/publications/newsletter/v33n1/columns/west.html). SIGGRAPH Computer Graphics

Newsletter. 33 (1): 15–17. doi:10.1145/563666.563671 (https://doi.org/10.1145%2F563666.
563671).
5. Cropper, William H (2004). Great Physicists: The Life and Times of Leading Physicists from

Galileo to Hawking (https://books.google.com/books?id=UqbxZpELwHYC&pg=PA118). p. 118.
ISBN 9780195173246.
6. Maxwell and Harman, pp. 230‒231 (https://books.google.com/books?id=JbNK9lRLHPEC&pg=
PA230): "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)
7. Iowa State Chemical Engineer Drives Issue of New Stamp Honoring Father ofThermodynamics: Iowa State University – College of Engineering, 2004 (http://www.eng.iastat
e.edu/coe/feature/jolls.asp) Archived (https://web.archive.org/web/20121030193200/http://
www.eng.iastate.edu/coe/feature/jolls.asp) 2012‒10‒30 at the Wayback Machine..
8. Maxwell, Garber, Brush, and Everitt, p. 49 (https://books.google.com/books?id=hA‒oIDR0eXkC&pg=PA49).
9. 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(https://doi.org/10.1016%2F0168‒9002%2894%2901031‒5).
10. The Museum at the Cavendish Laboratory: Maxwell's Apparatus (http://www‒outreach.phy.cam.ac.uk/camphy/museum/area1/cabinet1.htm).
11. Kenneth R. Jolls (1990). "Gibbs and the art of thermodynamics". In D. G. Caldi; George D.
Mostow. Proceedings of the Gibbs Symposium, Yale University, May 15–17, 1989. American
Mathematical Society. p. 321. ISBN 978‒0‒8218‒0157‒4.
12. Maxwell, Garber, Brush, and Everitt, pp. 50 (https://books.google.com/books?id=hA‒oIDR0eXk
C&pg=PA50).
13. James Clerk Maxwell, Theory of Heat (https://archive.org/details/theoryofheat00maxwrich),revised in 1891 by John Strutt, 3rd Baron Rayleigh: the drawing of the lines appears as Figure
26d on page 207.
14. See, for example, Don S. Lemons, Mere Thermodynamics (https://books.google.com/books?id=WFZf2G80tNcC&pg=PA146), Johns Hopkins University Press, 2008, ISBN 0‒8018‒9015‒2, p.
146.

15. Johanna Levelt Sengers, How Fluids Unmix: Discoveries by the School of Van der Waals andKamerlingh Onnes (http://www.knaw.nl/waals/pdf/ﬂuids̲complete.pdf), Royal Netherlands
Academy of Arts and Sciences, 2002, pp. 56 & 104.
16. See the page 3D‒Models/Mixtures/Experiments: Kamerlingh Onnes (http://www.knaw.nl/waals/
experiments̲kamerlingh.html) from the Royal Netherlands Academy of Arts and Sciences.
Some of these models are on display at the Museum Boerhaave: Room 21 (http://www.museu
mboerhaave.nl/AAcollection/english/M21V01.html).

External links
Photograph of one of the two Cambridge copies (http://www‒outreach.phy.cam.ac.uk/camphys/museum/area1/images/cabinet1̲2.jpg) in the Museum at the Cavendish Laboratory; forbetter readable legends to go with the axes, see here (http://www.sv.vt.edu/classes/ESM471
4/methods/̲images/̲clayplaster.jpg)
Thermodynamic Case Study: Gibbs' Thermodynamic Graphical Method (https://web.archive.or
g/web/20140201163858/http://www.sv.vt.edu/classes/ESM4714/methods/Gibbs.html) at
Virginia Tech's Laboratory for Scientiﬁc Visual Analysis
Maxwell’s thermodynamic surface (http://www.eoht.info/page/Maxwell%E2%80%99s+thermod
ynamic+surface) at the "Encyclopedia of Human Thermodynamics"

Retrieved from "https://en.wikipedia.org/w/index.php?
title=Maxwell%27s̲thermodynamic̲surface&oldid=798940342"

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http://www.esm.rkriz.net/classes/ESM4714/methods/Gibbs.html

Thermodynamic Case Study:
Gibbs' Thermodynamic Graphical
MethodEnvisioning total derivatives of scalar functions with two
independent variables
as raised surfaces and tangent planes

ﬃ

Objective: To show the independence and su

ciency of a general graphicalmethod coextensive in its' application, that when applied to the thermodynamic
theory of state demonstrates the relationship between state variables of energy,
entropy, volume, temperature, and pressure with and without analytic expressions,where the emphasis will be not to use analytic experssions as recommended by
Gibbs and endorsed by Maxwell.
J. Willard Gibbs created the theory associated with thermodynamic state bydevelopment of a graphical method in his two historic publications, Part 1,"Graphical Methods in the Thermodynamics of Fluids" and Part 2,"A Method ofGeometrical Representation of the Thermodynamic Properties of Substances by
Means of Surfaces", Ref.[1]. In the ﬁrst publication, Part 1, as the title implies,
Gibbs developed a general graphical method that provided insight into
thermodynamic property relationships governed by the ﬁrst and second laws ofthermodynamics. In the ﬁrst paragraph Gibbs explains,"Although geometrical representations of propositions in thethermodynamics of ﬂuids are in general use, and have done good service indisseminating clear notions in this science, yet they have by no means
received the extension in respect to variety and generality of which they arecapable. So far as regards a general graphical method, which can exhibit atonce all the thermodynamic properties of a ﬂuid concerned in reversible
processes, and serve alike for the demonstration of general theorems andthe numerical solution of particular problems, it is the general if not theuniversal practice to use diagrams in which the rectilinear co‒ordinates
represent volume and pressure. The object of this article is to call attention

ﬀ

to certain diagrams of diﬀ

erent construction, which a

ord graphical methods

coextensive in their applications with that in ordinary use, and preferable to it
in many cases in respect of distinctness or of convenience."
Before developing this general graphical method Gibbs combined the equations of

the ﬁrst and second laws into a single equation which to this day is referred to as
the Gibbs' equation of state and can be found in any standard thermodynamic
textbook.

Figure 1. Gibbs' derivation of the equation of state prior to developing thegraphical method, Ref.[1].
The original brief derivation of the equation of state by Gibbs, equation (4), isshown above in Fig. 1. With this derivation Gibbs demonstrates why his "diagrams

ﬀ

of di

erent construction" is not just another interesting choice but fundamentallyﬀ

correct in the sense that the ﬁrst and second laws require that a diﬀchange in energy is related to a corresponding dierential

erential change in entropy andvolume, not "with that in ordinary use", e.g. pressure and volume. Fundamental
indeed. Later in his 1873 publication, Gibbs mentions Thomas Andrews discovery
of critical point but does not mention the P‒V isotherm diagram Andrew's used in
his discusion of critical point. Surely this must have been on Gibbs' mind, but it
was not Gibbs style to confront, but rather to note that his diagram was not "withthat in ordinary use". Having politely made this point Gibbs continues thedevelopment of his graphical method where the focus is now on creating asurface that must be fundmentally related to axes of energy, entropy, and volume,
not pressure, volume, and temperature which we continue to use to this day
contrary to Gibbs' insightful observation. This signiﬁcant point provides the insightnecessary to understanding Gibbs' lengthy development of his graphical method
where ironically only a few ﬁgures were actually drawn. The development of thegraphical method is accomplished with long descriptive sentences, which are atﬃ

times dicult to understand. But as we will see, this careful description wasnecessary if J.C. Maxwell was to reproduce and actually create a three‒dimensional (3D) model of Gibbs' fundamental surface in clay and plaster. It was
1873, 3D computer graphics did not exist, however visual thinking within ascientiﬁc context requires no such tools or data. After carefully describing various
reversible concepts graphically, pp. 310‒341, in the next to last paragraph of Ref.
[1], part 1, Gibbs concludes,"In the foregoing discussion, the equations which express the fundamental
principles of thermodynamics in an analytical form have been assumed, andthe aim has only been to show how the same relations may be expressedgeometrically. It would, however, be easy, starting from the ﬁrst and second
laws of thermodynamics as usually enunciated, to arrive at the same results
without the aid of analytical formulae, to arrive, for example, at theconception of energy, of entropy, of absolute temperature, in theconstruction of the diagram without the analytical deﬁnitions of thesequantities, and to obtain the various properties of the diagram without theanalytical expression of the thermodynamic properties which they involve.
Such a course would have been better ﬁtted to show the independence andﬃsu

ciency of a graphical method, but perhaps less suitable for an

examination of the comparative advantages or disadvantages of di

ﬀ

erent

graphical methods."
Evidently, according to Gibbs, the equation of state brieﬂy derived at the beginingof his ﬁrst publication was not as insightful as his graphical method. Interesting.
To understand how the thermodynamic relationship of properties can be better
understood graphically "without the analytic expressions of the various properties"
it is necessary to actually read and study Gibbs original 1873 publications, part 1
and 2, Ref.[1]. This emphasis on the graphical method in the ﬁrst publication (part1) was further developed in Gibbs' second publication (part 2) as a surface, "A
Method of Geometrical Representation of the Thermodynamic Properties of
Substances by Means of Surfaces". Using only visual thinking, the relationshipbetween energy, entropy, and volume was envisioned as a surface. After studying
Gibbs' publications, James Clerk Maxwell reproduced, in clay and plaster, asculptured surface of the thermodynamic property relationship that Gibbs
envisioned and described in detail but did not draw.

Figure 2. J. Clerk Maxwell sculpture: display case Cavendish Laboratory,
Cambridge, England.
Download: http://www‒outreach.phy.cam.ac.uk/camphys/museum/area1/images/cabinet1̲2.jpg
Maxwell sent one these sculptures to Gibbs (1874), which now gathers dust in adisplay case at Yale. Another of Maxwell's sculptures can be viewed in a display
case at Cavendish Laboratory at Cambridge, Fig. 2. Unlike contemporay ancetodal

ﬃ

reports on visual thinking by scientists, Gibbs su

ciently described his visual

thinking as a reproducible graphical method. It is important to note that thedevelopment of the graphical method has nothing to do with the use of graphicaltools. In 1874 there were no graphical tools except for clay and plaster and the
sun's grazing rays to locate lines of constant pressure and temperature mappedon the energy‒entropy‒volume sculptured surface, Refs.[2 and 3]. Maxwell further
developed and described how lines of constant temperature and pressure can be
scribed on the energy‒entropy‒volume diagram in his textbook, Theory of Heat,
pp. 195‒208, Ref.[2]. It is interesting that no equations are used by Maxwell to
describe stable and unstable conditions associated with solid, liquid, and gaseousstates as well as other state variable relationships. According to Maxwell, the statevariable relationships are better understood by using the graphical method asorginially proposed by Gibbs. When the word relationship is used today, it means amathematical relationship ("formulae"), not in the sense Gibbs and Maxwell
envisioned as a graphical method. Because todays reader expects thedevelopment of a mathematical derivation not a graphical method, Gibbs' 1873

ﬃ

publication is dicult to understand. This observation is complicated by the fact

that Maxwell's diagrams are diﬃ

cult to envision in 3D. The diagram created by

Maxwell, Fig. 3, closely resembles the lines seen on the clay and plaster sculpturein Figure 2. These same lines appear on a postage stamp in honor of Josiah
Willard Gibbs, Fig. 4. In fact it is diﬃ

cult to understand how these lines mapped

onto the energy‒entropy‒volume 3D surface without a picture of the clay andpaster model, which was not published with an explanation until 2002, Plates II and
III Ref.[3].

Figure 3. Maxwell's graphical model based on Gibbs' original graphical method.
Information highlighted in light blue shows a connection to the equation of state.

Figure 4. Stamp honoring J.W. Gibbs. NOTE: lines of pressure and temperature.
In a letter to Thomas Andrew, 15 July 1875, Ref.[3], Maxwell describes a cleverway to draw these lines onto the surface by using parallel rays of light: "When I
had at last got a plaster cast I drew on it lines of equal pressure and equal
temperature, so as to get a rough notion of their forms. This I did by placing themodel in the sun light, and tracing the curve when the rays just grazed thesurface. For it v(olume),φ(entropy), and e(nergy) are the co‒ordinates, P(ressure)φ= de/dv, and T(emperature) = de/d()." Of course this is a simple task using

current graphical software tools. However the graphical method was developed
independent of the tools, e.g. either contemporary computer graphic software orclay and plaster in 1874. Tools are still important, however, the thought behinddevelopment of the graphical method occurs ﬁrst by how the scientist
understands the science. It is the intent here to show how Gibbs' energy‒entropy‒
volume sculpture with lines of constant pressure and temperature scribed on the
sculptured surface, can be implemented using a general graphical method
developed to envision total derivatives of any arbitrary scalar function with two
independent variables. Such a general graphical method, as it relates to existing
analytic methods, is summarized below.

Figure 5. Developement of the graphical method used to understand the total
derivative of any arbitrary scalar function of two indenpendent variables.
The ﬁgure above demonstrates that envisioning the total derivative of a scalarfunction, f(x,y), is an extension of envisioning the gradient of the same scalarfunction f(x,y) by combining this vector gradient with the iniﬁnitesimal position
vector that deﬁnes two points, A and B. Mathematically, points A and B coexist in
the x‒y plane as a vector. However for the thermodynamicists Gibbs and Maxwell,
who ﬁrst envisioned thermodynamic state as a graphical method, each of thesepoints represented a unique thermodynamic state that exist on a curved surface
that deﬁne three thermodynamic state variables, energy, entropy and volume.
Although points A and B coexist in the entropy‒volume plane in a mathematicalsense, to Gibbs and Maxwell these points are projected upwards onto the energy‒entropy‒volume surface in a graphical sense. Gibbs discusses this idea in the lastparagraph of Ref.[1], part 1,"The possibility of treating the thermodynamics of ﬂuids by such graphicalmethods as have been described evidently arises from the fact that the stateof the body considered, like the position of a point in a plane, is capable of
two and only two independent variations. It is, perhaps, worthy of notice, thatwhen the diagram is only used to demonstrate or illustrate general theorems,
it is not necessary, although it may be convenient, to assume any particularmethod of forming the diagram; it is enough to suppose the diﬀ

erent statesof the body to be presented continuously by points upon a sheet."
Gibbs and Maxwell continued to use the graphical method to develop thethermodynamic theory of state by asking the question, what thermodynamic
processes exist when moving from point A to point B on the energy‒entropy‒
volume diagram, Fig. 2. The projection of points A and B from the entropy‒volume
plane onto the curved two‒dimensional energy surface above is the visualcongnitive process that allows the scientist to "see" and hypothesize thethermodynamic relationships associated with points A and B. The graphicalmethod orginally developed by Gibbs to understand the relationship ofthermodynamic state variables is further developed by Maxwell, who provides an
excellent description of numerous thermodynamic processes associated withpoints A and B using Fig. 3, which was taken from his textbook, "Theory of Heat",Ref.[2]. To appreciate how the thermodynamic theory of state was developed
using only a graphical method it is necessary to read and study pp. 195‒208 in
Maxwell's "Theory of Heat" together with the ﬁg.26d on page 207, Ref.[2]. Lines inthis ﬁgure are combined here with color originally deﬁned by Maxwell in Ref.[3]and mapped onto the surface in Fig. 6 , to help the reader make the cognitive linkto the mathematical concepts of a gradient in Fig. 5. It is interesting that theoriginal ﬁgure in Maxwell's textbook avoided equations to describe numerous
thermodynamic processes that occur between any two arbitrary points A and B.
This is consistent with Gibbs' conclusion page 342 Ref.[1], ".. , to arrive at thesame results without the aid of analytical formulae, ..". However, since Gibbs ﬁrst
published his idea in 1873, the basis for a fundamental understanding of thethermodynamic theory of state has become mathematical not graphical,
consequently it is necessary to provide a mathematical link back to Gibbs' originalgraphical method.

Figure 6. Figures 1, 2, and 3 combined with color as described by Maxwell in Ref.
[3].
A poster‒sized graphical summary of Gibbs' and Maxwell's ﬁgures combined
provides a link to the original graphical thinking developed by Gibbs in Ref.[1].
Although Maxwell carefully deﬁned and constructed lines of constant pressure
and temperature, see Fig. 3, there was confusion in understanding how these lines
mapped onto the energy‒entropy‒volume surface. Professor W.P. Boynton and twoof his students at the University of California, Berkely, unsuccessfully attempted
to recreate Maxwell's sculpture, Ref.[4], from the ﬁg.26d on page 207, Ref.[2].
After reading Boynton's paper it is interesting to observe ﬁrst hand how important
it is to at least have a photograph of Maxwell's sculptured surface. Unless you've
visited the display case at the Cavendish Physics Laboratory, only recently (2002)
a photograph of Maxwell's clay and plaster model was published, Plates II and III in
Ref.[3], as well as posted on the web, see Fig. 2. Understanding that Gibbs original
Fig. 1 is three dimensional is also diﬃ

cult to see, as shown below, where theentropy and volume axis is not perpendicular. A simple modiﬁcation is ﬁrst
necessary before projecting this ﬁgure into Maxwell's three‒dimensional energy,
entropy, volume coordinate space. With this modiﬁed ﬁgure using only visual
thinking in a scientiﬁc context, pp. 385‒388, Gibbs derives a well known
relationship called the Clausius‒Clapeyron equation by envisioning "... successivepositions of the rolling plane ..." that come in contact with the "L" (liquid) and "V"(vapor) tangent points on the "primitive" ("thermodynamic") energy surface where

Δη Δυ
Δυ

small angle rotation about the SV line is used to show dp/dt= BL/CV=
Gibbs recallsΔη= [heat of vaporization, r] / t, hence dp/dt = r / [tthe well know Clausius‒Clayperon equation, where

Δυ

/

.

], which is

associated with thevolumetric expansion as the liquid expands to vapor is the same as the line length
CV.

Figure 7. Gibbs' original Figure 1 modiﬁed to show 3D axes.
Creating 3D ﬁgures was a challenge in 1873. Perhaps this partially explains why
the graphical method was not fostered, but the analytical model continued to be
developed over the last 134 years. When information in Figures 1, 2, and 3, part 2of Ref.[1]. is combined with using color originally proposed by Maxwell in Plates IV
and V, Ref.[3], it is much easier to understand Gibbs' graphical method. Maxwell
explains that the sculpture is a hypothetical model (".. not to scale ..") that
communicates the idea of thermodynamic state as a surface that was describedby Gibbs in Ref.[1] but not drawn by Gibbs. This emphasis on using the geometric
thinking to develop thermodynamic state theory was discussed in Ref.[5], "It is thegeometry that led Maxwell into the discussion of the physics, rather than thephysics leading to a sense of the surfaces that may represent those properties.
Clearly geometry allowed Maxwell to penetrate the physics of thermodynamics in
new ways." Although the discussion and development of new physics motivated bygeometric thinking is extremely rare, Gibbs' and Maxwell's published results
demonstrated it is possible ‒‒ and reproducible.
Using the best available computer graphical software tools (1991‒1993), D.C. Coy

in his dissertation developed a reproducible method to create Gibbs' surfaces for
actual (to scale) materials, Ref.[6]. Unfortunately because computer graphics is a
new fast evolving technology, the graphic software Application Programming
Interface (API ‒ an interface programmers use to build applications) used by Coy
no longer works on current computer operating systems. However Coy carefully
outlined the development of the surfaces for actual materials which can berevived using current graphical APIs. To avoid a similar situation in the future it isrecommended that future diagrams be created using a ISO standard format, e.g.
VRML‒2 or X3D. These standard ﬁle formats can be loaded into future graphicalsoftware APIs as they evolve with the demands of ever changing graphic
hardware and operating systems. This should extend the usefullness of diagrams
over time, however this will inevitably change as well. To archive results for futureapplications it is recommended to store a list (ASCII human readable text format)
of point‒coordinates (vertices) that deﬁne points on the energy‒entropy‒volume
surface and the corresponding list of polygons (connectivity) that connect
vertices into polygon shapes. A list of colors assigned to vertices and polygons
can also be used to map properties, e.g. pressure and temperature, onto theenergy‒entropy‒volume surface. These archived ASCII text ﬁle lists will not changewith ever changing graphic software APIs. Using this archived format only minimalchanges will be necessary to reconstruct Gibbs' surfaces 134 years from now.
Specialized computer graphical programs were developed by Coy to extend Gibbs'
graphical method to include Legendre transformations in Dr. Coy's dissertation,"Visualizing thermodynamic stability and phase‒equilibrium through computer
graphics", Ref.[6]. After 120 years Dr. Coy and his professor Ken Jolls were theﬁrst to provide a detailed explanation on how to calculate and create Gibbs'
surfaces to scale and created a web site that exempliﬁes how others can share in

ﬀ

this insightful experience. Except for Coy and Jolls e

ort in 1993 little has been

done to promote the graphical method. Instead analytic methods based on Gibbs
equation of state have been developed over the last 134 years. As insightful as
the graphical methods described by Gibbs and Maxwell may be, students have notbeen taught to think visually, at least in the context of science and math. This iswhat Dr. Coy describes as Gibbs' geometric thinking. Unfortuantely 134 years ofmomentum has been devoted to developing analytic methods. Perhaps we havebecome analytically QWERTY (comtemporary computer keyboard conﬁguration)even though we are aware there are more e

ﬃ

cient keyboards and graphicalmethods that can be more insightful. With this observation the challenge andopportunity exists to teach analytic and graphical methods together and showthat both methods complement each other. This can be accomplished whengraphical methods are ﬁrst developed in a general sense and then applied to a
speciﬁc problem. At the end of his ﬁrst publication Gibbs supported the idea ofdeveloping a general graphical method independent of the application or graphicaltools, "Such a course would have been better ﬁtted to show the independence andﬃsu

ciency of a graphical method, ....", pg. 342 Ref.[1]. Perhaps the lack ofgraphical tools in 1873 encouraged geometric thinking.
Independent of using a speciﬁc graphical tool, a general graphical method (" ...
coextensive in their applications ... ") was developed above as a surface in Fig. 5,that can be used to describe Gibbs' equation of state. The objective here is tocombine the graphical and analytic methods to facilitate an understanding of therelationship between the various thermodynamic state variables.

ﬃ

Figure 8. Gibbs' energy function of entropy and volume. Coe

cients highlighted

green and blue show
the connection

to the components of the vector gradient of the energy function.
The ﬁnal equation is

known as Gibbs equation of state. This equation is the total derivative of theenergy function.
According to the general graphical method developed to envision total derivatives,
all ﬁve thermodynamic state variables (e,η, v, T, P), related in the equation ofstate above, coexist at any arbitrary point A, at x=x1 y=y1 in Fig. 5. Hence any
point on the energy‒entropy‒volume surface uniquely describes a thermodynamic
state. In Gibbs equation of state the vector components of a gradient of thescalar energy function are the deﬁnitions for temperature and pressure. Hencelines of constant temperature and pressure can be mapped as a collection ofconnected points on the curved surface with the same slopes where the surface
is deﬁned by the scalar two‒dimensional function of energy as a function ofentropy and volume. This is why Maxwell could use the sun's "grazing" rays toidentify points on the surface with similar slopes. The vector components of thegradient that deﬁne the slopes of the tangent plane are highlighted using green
and blue colors. Maxwell also used color to highlight lines of constant pressure
and temperature scribed onto his sculptured surface, which has since faded inFig. 2 but reconstructed in Fig. 6. In Coy's dissertation this tangent plane wasdrawn in special cases to understand properties of interest. Using the graphicalmethod for envisioning total derivatives, a glyph representing the total derivative
also uses the idea of a tangent plane to show the pressure, temperature, position
vector, and the corresponding total derivative. This glyph could be drawn

interactively in a highlighted window next to the surface as the mouse pointermoves from point to point (A to B) along the scalar two‒dimensional functionenergy‒entropy‒volume surface. If the mouse pointer is static then no glyph isdrawn. This is an essential requirement because the total derivative only exists
with respect to a position vector that deﬁnes two points A and B. From athermodynamic view point this corresponds to the underlying question whenworking with thermodynamic theory of state, that is, what thermodynamic
processes exist when moving from point to point along the energy‒entropy‒
volume surface. Using this idea Maxwell demonstrated how to determine whether

ﬀ

a substance, composed of a mixture of dierent states that coexist at points A

and B, e.g. solid, liquid, or gas (triple point), "will tend of itself to pass from one ofthese states to the other". This is determined by by the amount of energy freely
available (Gibbs' free Energy) which is associated with the curvature of the"primitive" surface and orientation of the triple point tangent plane which Gibbs
labels as the "dissipated energy surface". Maxwell lists triple point thermodynamic
processes associated with Gibbs' free energy next to ﬁg.26d on page 206 of Ref.
[2]. Hence implementation of the glyph at points A and B would ﬁrst require anunderstanding of the underlying thermodynamic concepts associated with the liston page 206 of Ref[2].

Figure 9. Total derivative glyph for any arbitrary scalar function of two
independent
variables ( x, y ).

NOTE:

Points A and B must be deﬁned if glyph is to be drawn.

ηHere the scalar function, f(x,y), is Gibbs energy, e(,v), that is a function ofentropy, &eta, and volume, v. The gradient terms highlighted in green and blue arethemselves thermodynamic state variables: temperature, T, and negative pressure,‒P. It is fortunate that the gradient terms are additional thermodynamic statevariables that coexist at point A along with energy, entropy and volume.
Regardless if the gradient terms highlighted with green and blue colors are
additional properties or not, the total derivative can always be calculated andenvisioned using the glyph. In this case the total derivative is a diﬀ

erential changeof energy, de, associated with moving from point A to point B on the energy‒
entropy‒volume diagram.

It is now possible to explore other total derivative equations that could be used toﬀ

understand di

erent complex scientiﬁc phenomena. This requires that the two‒

dimensional function be plotted as a raised curved surface where the idea of a
total derivative glyph can be invoked. Because of the availability of popular
computer graphic tools, two‒dimensional functions are readily plotted as raised
curved surfaces simply because "it looks interesting". With these raised surface
plots, there is now a conceptual link to the total derivative and the same visual
thinking ﬁrst implemented by Gibbs and Maxwell can be realized coextensive to its
application, that is, if the user appreciates the signiﬁcance of a total derivative andits' associated properties. Other concepts related to the total derivative can be
explored. For example the idea of the comoving derivative, used in continuum
mechanics, is an extension to the total derivative in the same sense that the ideaof the total derivative is an extension of the gradient. Similarly other examples will
come to mind depending on the experience of the scientist. Create the graphicalmethod ‒‒ discover, or in this case, create the science. How to design the tools
follow. For Maxwell it was clay and plaster. Unfortunately because of the wide‒
spread use of computers and "easy‒to‒use" graphical software, we are tool rich
but methodology poor. This emphasis on just using existing tools without visual
thinking does not encourage scientists to develop new graphical methods.
Summary
The development of the thermodynamic theory of state is a rare but excellent
example that demonstrates how scientists combine analytic and graphicalmethods together with how they understand science. How scientists combine
analytical and graphical models into new knowledge exempliﬁes a cognitive
processes that includes visual thinking or what Dr. Coy describes as "geometricreasoning", Ref.[6]. This new knowledge was reported and documented by Gibbs
as a graphical method, so that others could reproduce and build on that
understanding. As the graphical method was being developed by Gibbs the intent
was not to use graphics for presentation but rather to develop the theory. This iscontrary to the popular belief that imaging in science is used for presentation
which can at times be insightful. For example if scientists do experience any
insight when working with diagrams or imaging, this experience is understood tobe unique to the researcher as a subjective experience, which may haveeducational value by reporting this experience as a scientiﬁc visual thinking
anecdote: a short tale narrating an interesting or amusing biographical incident
that may be useful for educational or outreach purposes to the general public
who do not understand the science. At best the result becomes an intriguing
picture on a stamp or the cover of a scientiﬁc text book. After reading and
STUDYING Gibbs and Maxwell, perhaps the reader would agree that neither Gibbs

nor Maxwell developed their graphical method for presentation, a metaphor, or as
an intriguing ancetodal experience that could not be scientiﬁcally reproduced.
Rather the graphical method was su

ﬃ

ciently developed and described by Gibbs to

be inclusive with developing the thermodynamic theory of state, which was
reproduced and further developed graphically by Maxwell. Recall in summary
Gibbs states,"In the foregoing discussion, the equations which express the fundamental
principles of thermodynamics in an analytical form have been assumed, andthe aim has only been to show how the same relations may be expressedgeometrically. It would, however, be easy, starting from the ﬁrst and second
laws of thermodynamics as usually enunciated, to arrive at the same results
without the aid of analytical formulae, to arrive, for example, at theconception of energy, of entropy, of absolute temperature, in theconstruction of the diagram without the analytical deﬁnitions of thesequantities, and to obtain the various properties of the diagram without theanalytical expression of the thermodynamic properties which they involve."
This is not a subjective process, e.g. what visual tools were used, how were theyused, or how were the tools designed. The integrity of Gibbs' and Maxwell'sgraphical method is a well established, scientiﬁc, objective, and a reproducible
process that has nothing to do with the subjective use of tools. This graphicalmethod is inclusive with the developement of the thermodynamic theory of state
where Gibbs demonstrates that understanding this theory can be accomplished"...without the aid of analytic formulae", e.g. his equation of state. In fact Gibbs
thought his graphical method was so important that,"Such a course would have been better ﬁtted to show the independence andﬃsu

ciency of a graphical method, but perhaps less suitable for an

ﬀ

examination of the comparative advantages or disadvantages of dierentgraphical methods."
Hopefully the independence and su

ﬃ

ciency of a graphical method, as proposedby Gibbs, was developed and demonstrated here by envisioning energy as asurface deﬁned as a scalar function of two independent variables, e.g. entropy
and volume, where the gradient of the scalar function are slopes tangent to thissurface and equal to temperature and negative pressure, as deﬁned in Figs. 5 and8. However since neither this surface nor the gradient lines tangent to thissurface are not associated with a speciﬁc set of physical properties, this generalgraphical method is indeed coextensive in its application.

References:
1. J.W. Gibbs, Part 1,"Graphical Methods in the Thermodynamics of Fluids" andPart 2, "A Method of Geometrical Representation of the Thermodynamic
Properties of Substances by Means of Surfaces", Transactions of theConnecticut Academy, Vol. II, Part 1, pp. 309‒342 and Part 2, pp. 382‒404",
1873.
2. J. Clerk Maxwell, Theory of Heat, Longmans, Green, and Co, pp. 195‒208,
1904.
3. The Scientiﬁc Letters and Papers of James Clerk Maxwell, Vol. III, 1874‒1879,
Edited by P.M. Harman, Cambridge University Press, 2002.
4. W.P. Boynton, "Gibbs' Thermodynamic Model", Phys. Rev., Series 1, Vol.10, pp.
228‒233, 1900.
5. Maxwell on heat and statistical mechanics: on "avoiding all personal enquiries"
of molecules, edited by E. Gaber, S.G. Brush, and C.W.F. Everitt, Lehigh
University Press, pp. 49‒50, 1995.
6. Daniel Charles Coy, "Visualizing thermodynamic stability and phase‒
equilibrium through computer graphics", Ph.D. dissertation, Iowa State
University, 1993.

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Ronald D. Kriz, Short Bio
Engineering Science and Mechanics
College of Engineering, Virginia TechBlacksburg, Virginia 24061
Created September, 2007
Revised July, 2011
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