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Maxwell’s thermodynamic surface is an 1874 sculpture made by Scottish physicist
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish physicist and mathematician who was responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism an ...
(1831–1879). This model provides a
three-dimensional space In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values ('' coordinates'') are required to determine the position of a point. Most commonly, it is the three- ...
of the various states of a fictitious substance with water-like properties. This plot has coordinates
volume Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...
(x),
entropy Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...
(y), and
energy Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...
(z). It was based on the American scientist
Josiah Willard Gibbs Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American mechanical engineer and scientist who made fundamental theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynami ...
’ graphical
thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
papers of 1873. In Maxwell's words, the model allowed "the principal features of known substances obe represented on a convenient scale."


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.Iowa State Chemical Engineer Drives Issue of New Stamp Honoring Father of Thermodynamics: Iowa State University – College of Engineering, 2004
.
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. This reflected Maxwell's talent as a strong visual thinker and prefigured modern
scientific visualization Scientific visualization ( also spelled scientific visualisation) is an interdisciplinary branch of science concerned with the visualization of scientific phenomena. Michael Friendly (2008)"Milestones in the history of thematic cartography, st ...
techniques. Maxwell sculpted the original model in clay and made several
plaster cast A plaster cast is a copy made in plaster of another 3-dimensional form. The original from which the cast is taken may be a sculpture, building, a face, a pregnant belly, a fossil or other remains such as fresh or fossilised footprints – ...
s of the clay model, sending one to Gibbs as a gift, keeping two in his laboratory at
Cambridge University The University of Cambridge is a Public university, public collegiate university, collegiate research university in Cambridge, England. Founded in 1209, the University of Cambridge is the List of oldest universities in continuous operation, wo ...
. Maxwell's copy is on display at the
Cavendish Laboratory The Cavendish Laboratory is the Department of Physics at the University of Cambridge, and is part of the School of Physical Sciences. The laboratory was opened in 1874 on the New Museums Site as a laboratory for experimental physics and is named ...
of Cambridge University, while Gibbs' copy is on display at the Sloane Physics Laboratory of
Yale University Yale University is a Private university, private Ivy League research university in New Haven, Connecticut, United States. Founded in 1701, Yale is the List of Colonial Colleges, third-oldest institution of higher education in the United Stat ...
, where Gibbs held a professorship. Two copies reside at the
National Museum of Scotland The National Museum of Scotland in Edinburgh, Scotland, is a museum of Scottish history and culture. It was formed in 2006 with the merger of the new Museum of Scotland, with collections relating to Scottish antiquities, culture and history, ...
, one via Peter Tait and the other via
George Chrystal George Chrystal FRSE FRS (8 March 1851 – 3 November 1911) was a Scottish mathematician. He is primarily known for his books on algebra and his studies of seiches (wave patterns in large inland bodies of water) which earned him a Gold Meda ...
. Another was sent to Thomas Andrews. 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 Muriel Rukeyser (1942), ''Willard Gibbs American Genius'' (reprinted by Ox Bow Press, ), p. 203. – and these photographs exposed a wider range of people to Maxwell's visualization approach.


Uses of the model

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 transition In physics, chemistry, and other related fields like biology, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic Sta ...
s as emerging from a purely continuous and smooth state function 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 In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
(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 In thermodynamics, the triple point of a substance is the temperature and pressure at which the three Phase (matter), phases (gas, liquid, and solid) of that substance coexist in thermodynamic equilibrium.. It is that temperature and pressure at ...
, to identify the boundary between absolutely stable and metastable phases (e.g.,
superheating In thermodynamics, superheating (sometimes referred to as boiling retardation, or boiling delay) is the phenomenon in which a liquid is heated to a temperature higher than its boiling point, without boiling. This is a so-called ''metastable state ...
and
supercooling Supercooling, also known as undercooling, is the process of lowering the temperature of a liquid below its freezing point without it becoming a solid. Per the established international definition, supercooling means ''‘cooling a substance be ...
), the spinodal boundary between metastable and unstable phases, and to illustrate the critical point. 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." He sent sketches of these lines to a number of colleagues. For example, his letter to Thomas Andrews of 15 July 1875 included sketches of these lines. Maxwell provided a more detailed explanation and a clearer drawing of the lines (pictured) in the revised version of his book ''Theory of Heat'',
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish physicist and mathematician who was responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism an ...
,
Theory of Heat
', revised in 1891 by John Strutt, 3rd Baron Rayleigh: the drawing of the lines appears as Figure 26d on page 207.
and a version of this drawing appeared on a 2005 US postage stamp in honour of Gibbs. 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,See, for example, Don S. Lemons,
Mere Thermodynamics
', Johns Hopkins University Press, 2008, , p. 146.
though not always with complete historical accuracy. For example, the thermodynamic surface represented by the sculpture is often reported to be that of water, contrary to Maxwell's own statement.


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 Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and eve ...
-
volume Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...
-
temperature Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
plot, based on data for
carbon dioxide Carbon dioxide is a chemical compound with the chemical formula . It is made up of molecules that each have one carbon atom covalent bond, covalently double bonded to two oxygen atoms. It is found in a gas state at room temperature and at norma ...
collected by Thomas Andrews. Johanna Levelt Sengers,
How Fluids Unmix: Discoveries by the School of Van der Waals and Kamerlingh Onnes
'',
Royal Netherlands Academy of Arts and Sciences The Royal Netherlands Academy of Arts and Sciences (, KNAW) is an organization dedicated to the advancement of science and literature in the Netherlands. The academy is housed in the Trippenhuis in Amsterdam. In addition to various advisory a ...
, 2002, pp. 56 & 104.
Around 1900, the Dutch scientist
Heike Kamerlingh Onnes Heike Kamerlingh Onnes (; 21 September 1853 – 21 February 1926) was a Dutch Experimental physics, experimental physicist. After studying in Groningen and Heidelberg, he became Professor of Experimental Physics at Leiden University, where he tau ...
, 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.See the pag
3D-Models/Mixtures/Experiments: Kamerlingh Onnes
from the
Royal Netherlands Academy of Arts and Sciences The Royal Netherlands Academy of Arts and Sciences (, KNAW) is an organization dedicated to the advancement of science and literature in the Netherlands. The academy is housed in the Trippenhuis in Amsterdam. In addition to various advisory a ...
. Some of these models are on display at the Museum Boerhaave
Room 21
.
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.


See also

*
History of thermodynamics The history of thermodynamics is a fundamental strand in the history of physics, the history of chemistry, and the history of science in general. Due to the relevance of thermodynamics in much of science and technology, its history is finely wov ...


References

{{Reflist


External links


Photograph of one of the two Cambridge copies
in the Museum at the
Cavendish Laboratory The Cavendish Laboratory is the Department of Physics at the University of Cambridge, and is part of the School of Physical Sciences. The laboratory was opened in 1874 on the New Museums Site as a laboratory for experimental physics and is named ...
; for better readable legends to go with the axes, se
here


at Virginia Tech's Laboratory for Scientific Visual Analysis
Maxwell’s thermodynamic surface
at the "Encyclopedia of Human Thermodynamics" 1874 in science 1874 sculptures Historical scientific instruments Sculptures in England Sculptures in the United States Scottish sculpture Numerical function drawing Scientific models Thermodynamics Works by James Clerk Maxwell