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<div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Sculpture of a substance's thermodynamic properties</div> <p class="mw-empty-elt"> </p> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti 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(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:404px;max-width:404px"><div class="trow"><div class="tsingle" style="width:402px;max-width:402px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Maxwell%27s_thermodynamic_surface,_commentary_book_figures_1,2.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/Maxwell%27s_thermodynamic_surface%2C_commentary_book_figures_1%2C2.jpg/400px-Maxwell%27s_thermodynamic_surface%2C_commentary_book_figures_1%2C2.jpg" decoding="async" width="400" height="177" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/Maxwell%27s_thermodynamic_surface%2C_commentary_book_figures_1%2C2.jpg/600px-Maxwell%27s_thermodynamic_surface%2C_commentary_book_figures_1%2C2.jpg 1.5x, 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srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Maxwell%27s_thermodynamic_surface%2C_commentary_book_figures_3%2C4.jpg/600px-Maxwell%27s_thermodynamic_surface%2C_commentary_book_figures_3%2C4.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Maxwell%27s_thermodynamic_surface%2C_commentary_book_figures_3%2C4.jpg/800px-Maxwell%27s_thermodynamic_surface%2C_commentary_book_figures_3%2C4.jpg 2x" data-file-width="1592" data-file-height="694" /></a></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Photographs of Maxwell's plaster model from different angles.</div></div></div></div> <p><b>Maxwell’s thermodynamic surface</b> is an 1874 sculpture<sup id="cite_ref-letters_1-0" class="reference"><a href="#cite_note-letters-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> made by Scottish physicist <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a> (1831–1879). This model provides a <a href="/wiki/Three-dimensional_space" title="Three-dimensional space">three-dimensional space</a> of the various states of a fictitious substance with water-like properties.<sup id="cite_ref-onheat_2-0" class="reference"><a href="#cite_note-onheat-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> This plot has coordinates <a href="/wiki/Volume" title="Volume">volume</a> (x), <a href="/wiki/Entropy" title="Entropy">entropy</a> (y), and <a href="/wiki/Energy" title="Energy">energy</a> (z). It was based on the American scientist <a href="/wiki/Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Josiah Willard Gibbs</a>’ graphical <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a> papers of 1873.<sup id="cite_ref-siggraph_3-0" class="reference"><a href="#cite_note-siggraph-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The model, in Maxwell's words, allowed "the principal features of known substances [to] be represented on a convenient scale."<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Construction_of_the_model">Construction of the model</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Maxwell%27s_thermodynamic_surface&action=edit&section=1" title="Edit section: Construction of the model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>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.<sup id="cite_ref-siggraph_3-1" class="reference"><a href="#cite_note-siggraph-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Iowa_6-0" class="reference"><a href="#cite_note-Iowa-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> 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.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> This reflected Maxwell's talent as a strong visual thinker<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> and prefigured modern <a href="/wiki/Scientific_visualization" title="Scientific visualization">scientific visualization</a> techniques.<sup id="cite_ref-siggraph_3-2" class="reference"><a href="#cite_note-siggraph-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Maxwell sculpted the original model in clay and made several <a href="/wiki/Plaster_cast" title="Plaster cast">plaster casts</a> of the clay model, sending one to Gibbs as a gift, keeping two in his laboratory at <a href="/wiki/Cambridge_University" class="mw-redirect" title="Cambridge University">Cambridge University</a>.<sup id="cite_ref-siggraph_3-3" class="reference"><a href="#cite_note-siggraph-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Maxwell's copy is on display at the <a href="/wiki/Cavendish_Laboratory" title="Cavendish Laboratory">Cavendish Laboratory</a> of Cambridge University,<sup id="cite_ref-siggraph_3-4" class="reference"><a href="#cite_note-siggraph-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> while Gibbs' copy is on display at the Sloane Physics Laboratory of <a href="/wiki/Yale_University" title="Yale University">Yale University</a>,<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> where Gibbs held a professorship. Two copies reside at the <a href="/wiki/National_Museum_of_Scotland" title="National Museum of Scotland">National Museum of Scotland</a>, one via <a href="/wiki/Peter_Tait_(physicist)" class="mw-redirect" title="Peter Tait (physicist)">Peter Tait</a> and the other via <a href="/wiki/George_Chrystal" title="George Chrystal">George Chrystal</a>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-nms_13-0" class="reference"><a href="#cite_note-nms-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Another was sent to <a href="/wiki/Thomas_Andrews_(scientist)" title="Thomas Andrews (scientist)">Thomas Andrews</a>.<sup id="cite_ref-nms_13-1" class="reference"><a href="#cite_note-nms-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> 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<sup id="cite_ref-Rukeyser_14-0" class="reference"><a href="#cite_note-Rukeyser-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> – and these photographs exposed a wider range of people to Maxwell's visualization approach. </p> <div class="mw-heading mw-heading2"><h2 id="Uses_of_the_model">Uses of the model</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Maxwell%27s_thermodynamic_surface&action=edit&section=2" title="Edit section: Uses of the model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:James_Clerk_Maxwell_-_Thermodynamic_surface_figure_from_book.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/James_Clerk_Maxwell_-_Thermodynamic_surface_figure_from_book.jpg/300px-James_Clerk_Maxwell_-_Thermodynamic_surface_figure_from_book.jpg" decoding="async" width="300" height="451" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/James_Clerk_Maxwell_-_Thermodynamic_surface_figure_from_book.jpg/450px-James_Clerk_Maxwell_-_Thermodynamic_surface_figure_from_book.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5b/James_Clerk_Maxwell_-_Thermodynamic_surface_figure_from_book.jpg/600px-James_Clerk_Maxwell_-_Thermodynamic_surface_figure_from_book.jpg 2x" data-file-width="1078" data-file-height="1620" /></a><figcaption>Diagram of thermodynamic surface from Maxwell's book <i>Theory of Heat</i>. The diagram is drawn roughly from the same angle as the upper left photo above, and shows the 3D axes <i>e</i> (energy, increasing downwards), <i>ϕ</i> (entropy, increasing to the lower right and out-of-plane), and <i>v</i> (volume, increasing to the upper right and into-plane).</figcaption></figure> <p>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 (<a href="/wiki/Phase_diagram#3-dimensional_diagrams" title="Phase diagram">pressure-volume-temperature</a>) surface was that it allowed to geometrically explain sharp, discontinuous <a href="/wiki/Phase_transition" title="Phase transition">phase transitions</a> as emerging from a purely continuous and smooth state function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(V,S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(V,S)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f19746cd4e12ddb4aa1684e905465c82b03390e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.912ex; height:2.843ex;" alt="{\displaystyle U(V,S)}"></span>; 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 <a href="/wiki/Tangent_plane" class="mw-redirect" title="Tangent plane">tangent plane</a> (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 <a href="/wiki/Triple_point" title="Triple point">triple point</a>, to identify the boundary between absolutely stable and metastable phases (e.g., <a href="/wiki/Superheating" title="Superheating">superheating</a> and <a href="/wiki/Supercooling" title="Supercooling">supercooling</a>), the <a href="/wiki/Spinodal" title="Spinodal">spinodal</a> boundary between metastable and unstable phases, and to illustrate the <a href="/wiki/Critical_point_(thermodynamics)" title="Critical point (thermodynamics)">critical point</a>.<sup id="cite_ref-theoryofheat_15-0" class="reference"><a href="#cite_note-theoryofheat-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>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."<sup id="cite_ref-onheat_2-1" class="reference"><a href="#cite_note-onheat-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> He sent sketches of these lines to a number of colleagues.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> For example, his letter to <a href="/wiki/Thomas_Andrews_(scientist)" title="Thomas Andrews (scientist)">Thomas Andrews</a> of 15 July 1875 included sketches of these lines.<sup id="cite_ref-onheat_2-2" class="reference"><a href="#cite_note-onheat-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Maxwell provided a more detailed explanation and a clearer drawing of the lines (pictured) in the revised version of his book <i>Theory of Heat</i>,<sup id="cite_ref-theoryofheat_15-1" class="reference"><a href="#cite_note-theoryofheat-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> and a version of this drawing appeared on a 2005 US postage stamp in honour of Gibbs.<sup id="cite_ref-Iowa_6-1" class="reference"><a href="#cite_note-Iowa-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>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,<sup id="cite_ref-mt_17-0" class="reference"><a href="#cite_note-mt-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> though not always with complete historical accuracy. For example, the thermodynamic surface represented by the sculpture is often reported to be that of water,<sup id="cite_ref-mt_17-1" class="reference"><a href="#cite_note-mt-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> contrary to Maxwell's own statement.<sup id="cite_ref-onheat_2-3" class="reference"><a href="#cite_note-onheat-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Related_models">Related models</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Maxwell%27s_thermodynamic_surface&action=edit&section=3" title="Edit section: Related models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Maxwell's model was not the first plaster model of a thermodynamic surface: in 1871, even before Gibbs' papers, <a href="/wiki/James_Thomson_(engineer)" title="James Thomson (engineer)">James Thomson</a> had constructed a plaster <a href="/wiki/Pressure" title="Pressure">pressure</a>-<a href="/wiki/Volume" title="Volume">volume</a>-<a href="/wiki/Temperature" title="Temperature">temperature</a> plot, based on data for <a href="/wiki/Carbon_dioxide" title="Carbon dioxide">carbon dioxide</a> collected by <a href="/wiki/Thomas_Andrews_(scientist)" title="Thomas Andrews (scientist)">Thomas Andrews</a>.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>Around 1900, the Dutch scientist <a href="/wiki/Heike_Kamerlingh_Onnes" title="Heike Kamerlingh Onnes">Heike Kamerlingh Onnes</a>, together with his student <a href="/wiki/Johannes_Petrus_Kuenen" class="mw-redirect" title="Johannes Petrus Kuenen">Johannes Petrus Kuenen</a> and his assistant Zaalberg van Zelst, continued Maxwell's work by constructing their own plaster thermodynamic surface models.<sup id="cite_ref-knaw_19-0" class="reference"><a href="#cite_note-knaw-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> 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.<sup id="cite_ref-knaw_19-1" class="reference"><a href="#cite_note-knaw-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Maxwell%27s_thermodynamic_surface&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/History_of_thermodynamics" title="History of thermodynamics">History of thermodynamics</a></li> <li><a rel="nofollow" class="external text" href="https://hmolpedia.com/images/9/99/Maxwell_thermodynamic_surface_construction.jpg">Process leading from Gibbs' drawings to Maxwell's thermodynamic surface</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Maxwell%27s_thermodynamic_surface&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-letters-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-letters_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFMaxwell1990" class="citation book cs1">Maxwell, James Clerk (1990). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JbNK9lRLHPEC&pg=PA148"><i>The Scientific Letters and Papers of James Clerk Maxwell: 1874-1879</i></a>. Cambridge University Press. p. 148. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780521256278" title="Special:BookSources/9780521256278"><bdi>9780521256278</bdi></a>. <q>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 <a href="/wiki/Thomas_Andrews_(scientist)" title="Thomas Andrews (scientist)">Thomas Andrews</a>, November, 1874)</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Scientific+Letters+and+Papers+of+James+Clerk+Maxwell%3A+1874-1879&rft.pages=148&rft.pub=Cambridge+University+Press&rft.date=1990&rft.isbn=9780521256278&rft.aulast=Maxwell&rft.aufirst=James+Clerk&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJbNK9lRLHPEC%26pg%3DPA148&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMaxwell%27s+thermodynamic+surface" class="Z3988"></span></span> </li> <li id="cite_note-onheat-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-onheat_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-onheat_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-onheat_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-onheat_2-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMaxwell1995" class="citation book cs1">Maxwell, James Clerk (1 January 1995). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hA-oIDR0eXkC&pg=PA248"><i>Maxwell on Heat and Statistical Mechanics: On "Avoiding All Personal Enquiries" of Molecules</i></a>. Lehigh University Press. p. 248. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780934223348" title="Special:BookSources/9780934223348"><bdi>9780934223348</bdi></a>. <q>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 CO<sub>2</sub>, 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 <a href="/wiki/Thomas_Andrews_(scientist)" title="Thomas Andrews (scientist)">Thomas Andrews</a>, 15 July 1875)</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Maxwell+on+Heat+and+Statistical+Mechanics%3A+On+%22Avoiding+All+Personal+Enquiries%22+of+Molecules&rft.pages=248&rft.pub=Lehigh+University+Press&rft.date=1995-01-01&rft.isbn=9780934223348&rft.aulast=Maxwell&rft.aufirst=James+Clerk&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DhA-oIDR0eXkC%26pg%3DPA248&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMaxwell%27s+thermodynamic+surface" class="Z3988"></span></span> </li> <li id="cite_note-siggraph-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-siggraph_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-siggraph_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-siggraph_3-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-siggraph_3-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-siggraph_3-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThomas_G.West1999" class="citation journal cs1">Thomas G.West (February 1999). <a rel="nofollow" class="external text" href="http://www.siggraph.org/publications/newsletter/v33n1/columns/west.html">"Images and reversals: James Clerk Maxwell, working in wet clay"</a>. <i>ACM SIGGRAPH Computer Graphics</i>. <b>33</b> (1): 15–17. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F563666.563671">10.1145/563666.563671</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:13968486">13968486</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=ACM+SIGGRAPH+Computer+Graphics&rft.atitle=Images+and+reversals%3A+James+Clerk+Maxwell%2C+working+in+wet+clay&rft.volume=33&rft.issue=1&rft.pages=15-17&rft.date=1999-02&rft_id=info%3Adoi%2F10.1145%2F563666.563671&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A13968486%23id-name%3DS2CID&rft.au=Thomas+G.West&rft_id=http%3A%2F%2Fwww.siggraph.org%2Fpublications%2Fnewsletter%2Fv33n1%2Fcolumns%2Fwest.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMaxwell%27s+thermodynamic+surface" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCropper2004" class="citation book cs1">Cropper, William H (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=UqbxZpELwHYC&pg=PA118"><i>Great Physicists: The Life and Times of Leading Physicists from Galileo to Hawking</i></a>. Oxford University Press. p. 118. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780195173246" title="Special:BookSources/9780195173246"><bdi>9780195173246</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Great+Physicists%3A+The+Life+and+Times+of+Leading+Physicists+from+Galileo+to+Hawking&rft.pages=118&rft.pub=Oxford+University+Press&rft.date=2004&rft.isbn=9780195173246&rft.aulast=Cropper&rft.aufirst=William+H&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DUqbxZpELwHYC%26pg%3DPA118&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMaxwell%27s+thermodynamic+surface" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Maxwell and Harman, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JbNK9lRLHPEC&pg=PA230">pp. 230-231</a>: "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 <a href="/wiki/James_Thomson_(engineer)" title="James Thomson (engineer)">James Thomson</a>, 8 July 1875)</span> </li> <li id="cite_note-Iowa-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-Iowa_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Iowa_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.eng.iastate.edu/coe/feature/jolls.asp">Iowa State Chemical Engineer Drives Issue of New Stamp Honoring Father of Thermodynamics: Iowa State University – College of Engineering, 2004</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20121030193200/http://www.eng.iastate.edu/coe/feature/jolls.asp">Archived</a> 2012-10-30 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Maxwell, Garber, Brush, and Everitt, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hA-oIDR0eXkC&pg=PA49">p. 49</a>.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Ken Brodlie, "Scientific visualization — past, present and future," Proceedings of the Third Workshop on Neutron Scattering Data Analysis, <i>Nuclear Instruments and Methods in Physics Research, Section A: Accelerators, Spectrometers, Detectors and Associated Equipment</i>, Volume 354, Issue 1, 15 January 1995, Pages 104-111, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0168-9002%2894%2901031-5">10.1016/0168-9002(94)01031-5</a>.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">The Museum at the <a href="/wiki/Cavendish_Laboratory" title="Cavendish Laboratory">Cavendish Laboratory</a>: <a rel="nofollow" class="external text" href="http://www-outreach.phy.cam.ac.uk/camphy/museum/area1/cabinet1.htm">Maxwell's Apparatus</a>.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKenneth_R._Jolls1990" class="citation book cs1">Kenneth R. Jolls (1990). "Gibbs and the art of thermodynamics". In D. G. Caldi; George D. Mostow (eds.). <i>Proceedings of the Gibbs Symposium, Yale University, May 15–17, 1989</i>. American Mathematical Society. p. 321. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-0157-4" title="Special:BookSources/978-0-8218-0157-4"><bdi>978-0-8218-0157-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Gibbs+and+the+art+of+thermodynamics&rft.btitle=Proceedings+of+the+Gibbs+Symposium%2C+Yale+University%2C+May+15%E2%80%9317%2C+1989&rft.pages=321&rft.pub=American+Mathematical+Society&rft.date=1990&rft.isbn=978-0-8218-0157-4&rft.au=Kenneth+R.+Jolls&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMaxwell%27s+thermodynamic+surface" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.nms.ac.uk/explore-our-collections/collection-search-results/thermodynamic-model/417665">"Thermodynamic model"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Thermodynamic+model&rft_id=https%3A%2F%2Fwww.nms.ac.uk%2Fexplore-our-collections%2Fcollection-search-results%2Fthermodynamic-model%2F417665&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMaxwell%27s+thermodynamic+surface" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.nms.ac.uk/explore-our-collections/collection-search-results/thermodynamic-model/425122">"Thermodynamic model"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Thermodynamic+model&rft_id=https%3A%2F%2Fwww.nms.ac.uk%2Fexplore-our-collections%2Fcollection-search-results%2Fthermodynamic-model%2F425122&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMaxwell%27s+thermodynamic+surface" class="Z3988"></span></span> </li> <li id="cite_note-nms-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-nms_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-nms_13-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.nms.ac.uk/explore-our-collections/stories/science-and-technology/james-clerk-maxwell-inventions/james-clerk-maxwell/thermodynamic-surface">"Thermodynamic surface"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Thermodynamic+surface&rft_id=https%3A%2F%2Fwww.nms.ac.uk%2Fexplore-our-collections%2Fstories%2Fscience-and-technology%2Fjames-clerk-maxwell-inventions%2Fjames-clerk-maxwell%2Fthermodynamic-surface&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMaxwell%27s+thermodynamic+surface" class="Z3988"></span></span> </li> <li id="cite_note-Rukeyser-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-Rukeyser_14-0">^</a></b></span> <span class="reference-text"><a href="/wiki/Muriel_Rukeyser" title="Muriel Rukeyser">Muriel Rukeyser</a> (1942), <i>Willard Gibbs American Genius</i> (reprinted by Ox Bow Press, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-918024-57-9" title="Special:BookSources/0-918024-57-9">0-918024-57-9</a>), p. 203.</span> </li> <li id="cite_note-theoryofheat-15"><span class="mw-cite-backlink">^ <a href="#cite_ref-theoryofheat_15-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-theoryofheat_15-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a>, <i><a rel="nofollow" class="external text" href="https://archive.org/details/theoryofheat00maxwrich">Theory of Heat</a></i>, revised in 1891 by <a href="/wiki/John_Strutt,_3rd_Baron_Rayleigh" class="mw-redirect" title="John Strutt, 3rd Baron Rayleigh">John Strutt, 3rd Baron Rayleigh</a>: the drawing of the lines appears as Figure 26d on page 207.</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">Maxwell, Garber, Brush, and Everitt, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hA-oIDR0eXkC&pg=PA50">pp. 50</a>.</span> </li> <li id="cite_note-mt-17"><span class="mw-cite-backlink">^ <a href="#cite_ref-mt_17-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-mt_17-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">See, for example, Don S. Lemons, <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=WFZf2G80tNcC&pg=PA146">Mere Thermodynamics</a></i>, Johns Hopkins University Press, 2008, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8018-9015-2" title="Special:BookSources/0-8018-9015-2">0-8018-9015-2</a>, p. 146.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><a href="/wiki/Anneke_Levelt_Sengers" title="Anneke Levelt Sengers">Johanna Levelt Sengers</a>, <i><a rel="nofollow" class="external text" href="http://www.knaw.nl/waals/pdf/fluids_complete.pdf">How Fluids Unmix: Discoveries by the School of Van der Waals and Kamerlingh Onnes</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070530063315/http://www.knaw.nl/waals/pdf/fluids_complete.pdf">Archived</a> 2007-05-30 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></i>, <a href="/wiki/Royal_Netherlands_Academy_of_Arts_and_Sciences" title="Royal Netherlands Academy of Arts and Sciences">Royal Netherlands Academy of Arts and Sciences</a>, 2002, pp. 56 & 104.</span> </li> <li id="cite_note-knaw-19"><span class="mw-cite-backlink">^ <a href="#cite_ref-knaw_19-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-knaw_19-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">See the page <a rel="nofollow" class="external text" href="http://www.knaw.nl/waals/experiments_kamerlingh.html">3D-Models/Mixtures/Experiments: Kamerlingh Onnes</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20041210065914/http://www.knaw.nl/waals/experiments_kamerlingh.html">Archived</a> 2004-12-10 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> from the <a href="/wiki/Royal_Netherlands_Academy_of_Arts_and_Sciences" title="Royal Netherlands Academy of Arts and Sciences">Royal Netherlands Academy of Arts and Sciences</a>. Some of these models are on display at the <a href="/wiki/Museum_Boerhaave" title="Museum Boerhaave">Museum Boerhaave</a>: <a rel="nofollow" class="external text" href="http://www.museumboerhaave.nl/AAcollection/english/M21V01.html">Room 21</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110607145133/http://www.museumboerhaave.nl/AAcollection/english/M21V01.html">Archived</a> 2011-06-07 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Maxwell%27s_thermodynamic_surface&action=edit&section=6" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www-outreach.phy.cam.ac.uk/camphys/museum/area1/images/cabinet1_2.jpg">Photograph of one of the two Cambridge copies</a> in the Museum at the <a href="/wiki/Cavendish_Laboratory" title="Cavendish Laboratory">Cavendish Laboratory</a>; for better readable legends to go with the axes, see <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100610182147/http://www.sv.vt.edu/classes/ESM4714/methods/_images/_clayplaster.jpg">here</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20140201163858/http://www.sv.vt.edu/classes/ESM4714/methods/Gibbs.html">Thermodynamic Case Study: Gibbs' Thermodynamic Graphical Method</a> at Virginia Tech's Laboratory for Scientific Visual Analysis</li> <li><a rel="nofollow" class="external text" href="http://www.eoht.info/page/Maxwell%E2%80%99s+thermodynamic+surface">Maxwell’s thermodynamic surface</a> at the "Encyclopedia of Human Thermodynamics"</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Maxwell%27s_thermodynamic_surface&oldid=1173129017">https://en.wikipedia.org/w/index.php?title=Maxwell%27s_thermodynamic_surface&oldid=1173129017</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:1874_in_science" title="Category:1874 in science">1874 in science</a></li><li><a href="/wiki/Category:1874_sculptures" title="Category:1874 sculptures">1874 sculptures</a></li><li><a href="/wiki/Category:Historical_scientific_instruments" title="Category:Historical scientific instruments">Historical scientific instruments</a></li><li><a href="/wiki/Category:Sculptures_in_England" title="Category:Sculptures in England">Sculptures in England</a></li><li><a href="/wiki/Category:Sculptures_in_the_United_States" title="Category:Sculptures in the United States">Sculptures in the United States</a></li><li><a href="/wiki/Category:Scottish_sculpture" title="Category:Scottish sculpture">Scottish sculpture</a></li><li><a href="/wiki/Category:Numerical_function_drawing" title="Category:Numerical function drawing">Numerical function drawing</a></li><li><a href="/wiki/Category:Scientific_models" title="Category:Scientific models">Scientific models</a></li><li><a href="/wiki/Category:Thermodynamics" title="Category:Thermodynamics">Thermodynamics</a></li><li><a href="/wiki/Category:Works_by_James_Clerk_Maxwell" title="Category:Works by James Clerk Maxwell">Works by James Clerk Maxwell</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Use_dmy_dates_from_April_2022" title="Category:Use dmy dates from April 2022">Use dmy dates from April 2022</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 31 August 2023, at 12:59<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; 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