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class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Joule expansion</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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href="https://ca.wikipedia.org/wiki/Expansi%C3%B3_lliure" title="Expansió lliure – Catalan" lang="ca" hreflang="ca" data-title="Expansió lliure" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Joule-ekspansion" title="Joule-ekspansion – Danish" lang="da" hreflang="da" data-title="Joule-ekspansion" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Gay-Lussac-Versuch" title="Gay-Lussac-Versuch – German" lang="de" hreflang="de" data-title="Gay-Lussac-Versuch" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Expansi%C3%B3n_libre" title="Expansión libre – Spanish" lang="es" hreflang="es" data-title="Expansión libre" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/D%C3%A9tente_de_Joule-Gay-Lussac" title="Détente de Joule-Gay-Lussac – French" lang="fr" hreflang="fr" data-title="Détente de Joule-Gay-Lussac" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9E%90%EC%9C%A0%ED%8C%BD%EC%B0%BD" title="자유팽창 – Korean" lang="ko" hreflang="ko" data-title="자유팽창" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Espansione_di_Joule" title="Espansione di Joule – Italian" lang="it" hreflang="it" data-title="Espansione di Joule" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%AA%D7%A4%D7%A9%D7%98%D7%95%D7%AA_%D7%92%27%D7%90%D7%95%D7%9C" title="התפשטות ג'אול – Hebrew" lang="he" hreflang="he" data-title="התפשטות ג'אול" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Joule-ekspansjon" title="Joule-ekspansjon – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Joule-ekspansjon" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Expans%C3%A3o_livre" title="Expansão livre – Portuguese" lang="pt" hreflang="pt" data-title="Expansão livre" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Destindere_Joule" title="Destindere Joule – Romanian" lang="ro" hreflang="ro" data-title="Destindere Joule" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Joule_geni%C5%9Flemesi" title="Joule genişlemesi – Turkish" lang="tr" hreflang="tr" data-title="Joule genişlemesi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Gi%C3%A3n_n%E1%BB%9F_t%E1%BB%B1_do" title="Giãn nở tự do – Vietnamese" lang="vi" hreflang="vi" data-title="Giãn nở tự do" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%87%AA%E7%94%B1%E6%93%B4%E6%95%A3" title="自由擴散 – Cantonese" lang="yue" hreflang="yue" data-title="自由擴散" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%87%AA%E7%94%B1%E8%86%A8%E8%84%B9" title="自由膨脹 – Chinese" lang="zh" hreflang="zh" data-title="自由膨脹" 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<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"><figure typeof="mw:File/Thumb"><a href="/wiki/File:JouleExpansion.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/JouleExpansion.svg/240px-JouleExpansion.svg.png" decoding="async" width="240" height="234" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/JouleExpansion.svg/360px-JouleExpansion.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/10/JouleExpansion.svg/480px-JouleExpansion.svg.png 2x" data-file-width="533" data-file-height="519" /></a><figcaption>The Joule expansion, in which a volume <span class="texhtml"><i>V</i><sub>i</sub> = <i>V</i><sub>0</sub></span> is expanded to a volume <span class="texhtml"><i>V</i><sub>f</sub> = 2<i>V</i><sub>0</sub></span> in a thermally isolated chamber.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Before_during_after_sudden_expansion.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Before_during_after_sudden_expansion.jpg/180px-Before_during_after_sudden_expansion.jpg" decoding="async" width="180" height="212" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Before_during_after_sudden_expansion.jpg/270px-Before_during_after_sudden_expansion.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Before_during_after_sudden_expansion.jpg/360px-Before_during_after_sudden_expansion.jpg 2x" data-file-width="597" data-file-height="702" /></a><figcaption>A free expansion of a gas can be achieved by moving the piston out faster than the fastest molecules in the gas.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Joule_expansion_quasi-static_but_irreversible.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Joule_expansion_quasi-static_but_irreversible.svg/180px-Joule_expansion_quasi-static_but_irreversible.svg.png" decoding="async" width="180" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Joule_expansion_quasi-static_but_irreversible.svg/270px-Joule_expansion_quasi-static_but_irreversible.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Joule_expansion_quasi-static_but_irreversible.svg/360px-Joule_expansion_quasi-static_but_irreversible.svg.png 2x" data-file-width="1600" data-file-height="900" /></a><figcaption>This free expansion is irreversible but can be quasi-static for each chamber: quasi-<a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">equilibrium</a> is retained for each part but not for the whole system</figcaption></figure> <p>The <b>Joule expansion</b> (a subset of <b>free expansion</b>) is an <a href="/wiki/Irreversible_process_(thermodynamics)" class="mw-redirect" title="Irreversible process (thermodynamics)">irreversible process</a> in <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a> in which a volume of gas is kept in one side of a thermally isolated container (via a small partition), with the other side of the container being evacuated. The partition between the two parts of the container is then opened, and the gas fills the whole container. </p><p>The Joule expansion, treated as a thought experiment involving <a href="/wiki/Ideal_gas" title="Ideal gas">ideal gases</a>, is a useful exercise in classical thermodynamics. It provides a convenient example for calculating changes in thermodynamic quantities, including the resulting increase in <a href="/wiki/Entropy" title="Entropy">entropy</a> of the universe (<a href="/wiki/Entropy_production" title="Entropy production">entropy production</a>) that results from this inherently irreversible process. An actual Joule expansion experiment necessarily involves <a href="/wiki/Real_gas" title="Real gas">real gases</a>; the temperature change in such a process provides a measure of <a href="/wiki/Intermolecular_force" title="Intermolecular force">intermolecular forces</a>. </p><p>This type of expansion is named after <a href="/wiki/James_Prescott_Joule" title="James Prescott Joule">James Prescott Joule</a> who used this expansion, in 1845, in his study for the mechanical equivalent of heat, but this expansion was known long before Joule e.g. by <a href="/wiki/John_Leslie_(physicist)" title="John Leslie (physicist)">John Leslie</a>, in the beginning of the 19th century, and studied by <a href="/wiki/Joseph_Louis_Gay-Lussac" title="Joseph Louis Gay-Lussac">Joseph Louis Gay-Lussac</a> in 1807 with similar results as obtained by Joule.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>The Joule expansion should not be confused with the <a href="/wiki/Joule%E2%80%93Thomson_expansion" class="mw-redirect" title="Joule–Thomson expansion">Joule–Thomson expansion</a> or <i>throttling process</i> which refers to the steady flow of a gas from a region of higher pressure to one of lower pressure via a valve or porous plug. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Description">Description</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Joule_expansion&action=edit&section=1" title="Edit section: Description"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The process begins with gas under some pressure, <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 P_{\mathrm {i} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\mathrm {i} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0df8af1c605b5d39a766a26a778832cdbce5b515" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.182ex; height:2.509ex;" alt="{\displaystyle P_{\mathrm {i} }}"></span>, at temperature <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 T_{\mathrm {i} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mathrm {i} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66e56b3c534b48fc9795abb0b1e915dca96c180f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.047ex; height:2.509ex;" alt="{\displaystyle T_{\mathrm {i} }}"></span>, confined to one half of a <a href="/wiki/Thermal_contact" title="Thermal contact">thermally isolated</a> container (see the top part of the drawing at the beginning of this article). The gas occupies an initial volume <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 V_{\mathrm {i} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {i} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03290d0d6d1de16e268f03d63caddee91a89af66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.045ex; height:2.509ex;" alt="{\displaystyle V_{\mathrm {i} }}"></span>, mechanically separated from the other part of the container, which has a volume <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 V_{\mathrm {0} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {0} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bbb094a29067bfafb4f0540778c7de3ae0b9b30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle V_{\mathrm {0} }}"></span>, and is under near zero pressure. The tap (solid line) between the two halves of the container is then suddenly opened, and the gas expands to fill the entire container, which has a total volume of <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 V_{\mathrm {f} }=V_{\mathrm {i} }+V_{\mathrm {0} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {f} }=V_{\mathrm {i} }+V_{\mathrm {0} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c59ffe2dbc1fb4bdaa5a918bb6ec23b30e90e34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.592ex; height:2.509ex;" alt="{\displaystyle V_{\mathrm {f} }=V_{\mathrm {i} }+V_{\mathrm {0} }}"></span> (see the bottom part of the drawing). A thermometer inserted into the compartment on the left (not shown in the drawing) measures the <a href="/wiki/Thermodynamic_temperature" title="Thermodynamic temperature">temperature</a> of the gas before and after the expansion. </p><p>The <a href="/wiki/Thermodynamic_system" title="Thermodynamic system">system</a> in this experiment consists of both compartments; that is, the entire region occupied by the gas at the end of the experiment. Because this system is thermally isolated, it cannot exchange heat with its surroundings. Also, since the system's total volume is kept constant, the system cannot perform work on its surroundings.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> As a result, the change in <a href="/wiki/Internal_energy" title="Internal energy">internal energy</a>, <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 \Delta U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4716a2c49bbbe155e8b399117ca78342e802cbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.718ex; height:2.176ex;" alt="{\displaystyle \Delta U}"></span>, is zero. Internal energy consists of internal kinetic energy (due to the motion of the molecules) and internal potential energy (due to <a href="/wiki/Intermolecular_force" title="Intermolecular force">intermolecular forces</a>). When the molecular motion is random, temperature is the measure of the internal kinetic energy. In this case, the internal kinetic energy is called heat. If the chambers have not reached equilibrium, there will be some kinetic energy of flow, which is not detectable by a thermometer (and therefore is not a component of heat). Thus, a change in temperature indicates a change in kinetic energy, and some of this change will not appear as heat until and unless thermal equilibrium is reestablished. When heat is transferred into kinetic energy of flow, this causes a decrease in temperature.<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> In practice, the simple two-chamber free expansion experiment often incorporates a 'porous plug' through which the expanding air must flow to reach the lower pressure chamber. The purpose of this plug is to inhibit directional flow, thereby quickening the reestablishment of thermal equilibrium. Since the total internal energy does not change, the stagnation of flow in the receiving chamber converts kinetic energy of flow back into random motion (heat) so that the temperature climbs to its predicted value. If the initial air temperature is low enough that non-ideal gas properties cause condensation, some internal energy is converted into latent heat (an offsetting change in potential energy) in the liquid products. Thus, at low temperatures the Joule expansion process provides information on intermolecular forces. </p> <div class="mw-heading mw-heading3"><h3 id="Ideal_gases">Ideal gases</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Joule_expansion&action=edit&section=2" title="Edit section: Ideal gases"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the gas is ideal, both the initial (<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 T_{\mathrm {i} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mathrm {i} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66e56b3c534b48fc9795abb0b1e915dca96c180f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.047ex; height:2.509ex;" alt="{\displaystyle T_{\mathrm {i} }}"></span>, <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 P_{\mathrm {i} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\mathrm {i} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0df8af1c605b5d39a766a26a778832cdbce5b515" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.182ex; height:2.509ex;" alt="{\displaystyle P_{\mathrm {i} }}"></span>, <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 V_{\mathrm {i} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {i} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03290d0d6d1de16e268f03d63caddee91a89af66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.045ex; height:2.509ex;" alt="{\displaystyle V_{\mathrm {i} }}"></span>) and final (<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 T_{\mathrm {f} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mathrm {f} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/283b1d3199025c522dd46bbba1985fa82f946e39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.202ex; height:2.509ex;" alt="{\displaystyle T_{\mathrm {f} }}"></span>, <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 P_{\mathrm {f} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\mathrm {f} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f5567a673ec9d7f4a362ffeba793344a3addf56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.336ex; height:2.509ex;" alt="{\displaystyle P_{\mathrm {f} }}"></span>, <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 V_{\mathrm {f} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {f} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f311e5bbb8c57863a57700f3112d5d6edd958259" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.199ex; height:2.509ex;" alt="{\displaystyle V_{\mathrm {f} }}"></span>) conditions follow the <a href="/wiki/Ideal_Gas_Law" class="mw-redirect" title="Ideal Gas Law">Ideal Gas Law</a>, so that initially <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\mathrm {i} }V_{\mathrm {i} }=nRT_{\mathrm {i} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>n</mi> <mi>R</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\mathrm {i} }V_{\mathrm {i} }=nRT_{\mathrm {i} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f04159ae840488ea06d5d286482979910157bc0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.531ex; height:2.509ex;" alt="{\displaystyle P_{\mathrm {i} }V_{\mathrm {i} }=nRT_{\mathrm {i} }}"></span> and then, after the tap is opened, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\mathrm {f} }V_{\mathrm {f} }=nRT_{\mathrm {f} }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> </mrow> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>n</mi> <mi>R</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> </mrow> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\mathrm {f} }V_{\mathrm {f} }=nRT_{\mathrm {f} }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cdb4c106d81857de5001aa062f1b631f3f5ff2b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.641ex; height:2.509ex;" alt="{\displaystyle P_{\mathrm {f} }V_{\mathrm {f} }=nRT_{\mathrm {f} }.}"></span> </p><p>Here <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> is the number of moles of gas and <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> is the molar <a href="/wiki/Ideal_gas_constant" class="mw-redirect" title="Ideal gas constant">ideal gas constant</a>. Because the internal energy does not change and the <a href="/wiki/Internal_energy#Internal_energy_of_the_ideal_gas" title="Internal energy">internal energy of an ideal gas</a> is solely a function of temperature, the temperature of the gas does not change; therefore <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 T_{\mathrm {i} }=T_{\mathrm {f} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mathrm {i} }=T_{\mathrm {f} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c11cb01a39c906013ef4fc3d0e0f23bb1ffa8596" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.347ex; height:2.509ex;" alt="{\displaystyle T_{\mathrm {i} }=T_{\mathrm {f} }}"></span>. This implies that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\mathrm {i} }V_{\mathrm {i} }=P_{\mathrm {f} }V_{\mathrm {f} }=nRT_{\mathrm {i} }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> </mrow> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>n</mi> <mi>R</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\mathrm {i} }V_{\mathrm {i} }=P_{\mathrm {f} }V_{\mathrm {f} }=nRT_{\mathrm {i} }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c0e2c3e3055c5125adcf625a5f75d66399403e8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.812ex; height:2.509ex;" alt="{\displaystyle P_{\mathrm {i} }V_{\mathrm {i} }=P_{\mathrm {f} }V_{\mathrm {f} }=nRT_{\mathrm {i} }.}"></span> </p><p>Therefore if the volume doubles, the pressure halves. </p><p>The fact that the temperature does not change makes it easy to compute the change in entropy of the universe for this process. </p> <div class="mw-heading mw-heading3"><h3 id="Real_gases">Real gases</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Joule_expansion&action=edit&section=3" title="Edit section: Real gases"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Unlike ideal gases, the temperature of a real gas will change during a Joule expansion. At temperatures below their <b>inversion temperature</b> gases will cool during Joule expansion, while at higher temperatures they will heat up.<sup id="cite_ref-:0_5-0" class="reference"><a href="#cite_note-:0-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_6-0" class="reference"><a href="#cite_note-:1-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The inversion temperature of a gas is typically much higher than room temperature; exceptions are helium, with an inversion temperature of about 40 K, and hydrogen, with an inversion temperature of about 200 K. Since the internal energy of the gas during Joule expansion is constant, cooling must be due to the conversion of internal kinetic energy to internal potential energy, with the opposite being the case for warming. </p><p>Intermolecular forces are repulsive at short range and attractive at long range (for example, see the <a href="/wiki/Lennard-Jones_potential" title="Lennard-Jones potential">Lennard-Jones potential</a>). Since distances between gas molecules are large compared to molecular diameters, the energy of a gas is usually influenced mainly by the attractive part of the potential. As a result, expanding a gas usually increases the potential energy associated with intermolecular forces. Some textbooks say that for gases this must always be the case and that a Joule expansion must always produce cooling.<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><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> When molecules are close together, however, repulsive interactions are much more important and it is thus possible to get an increase in temperature during a Joule expansion.<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> </p><p>It is theoretically predicted that, at sufficiently high temperature, all gases will warm during a Joule expansion<sup id="cite_ref-:0_5-1" class="reference"><a href="#cite_note-:0-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The reason is that at any moment, a very small number of molecules will be undergoing collisions; for those few molecules, repulsive forces will dominate and the potential energy will be positive. As the temperature rises, both the frequency of collisions and the energy involved in the collisions increase, so the positive potential energy associated with collisions increases strongly. If the temperature is high enough, that can make the total potential energy positive, in spite of the much larger number of molecules experiencing weak attractive interactions. When the potential energy is positive, a constant energy expansion reduces potential energy and increases kinetic energy, resulting in an increase in temperature. This behavior has only been observed for hydrogen and helium; which have very weak attractive interactions. For other gases this "Joule inversion temperature" appears to be extremely high.<sup id="cite_ref-:1_6-1" class="reference"><a href="#cite_note-:1-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Entropy_production">Entropy production</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Joule_expansion&action=edit&section=4" title="Edit section: Entropy production"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Entropy is a <a href="/wiki/Function_of_state" class="mw-redirect" title="Function of state">function of state</a>, and therefore the entropy change can be computed directly from the knowledge of the final and initial equilibrium states. For an ideal gas, the change in entropy<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> is the same as for <a href="/wiki/Isothermal_process" title="Isothermal process">isothermal expansion</a> where all heat is converted to work: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta S=\int _{\text{i}}^{\text{f}}dS=\int _{V_{\text{i}}}^{V_{\text{f}}}{\frac {P\,dV}{T}}=\int _{V_{\text{i}}}^{V_{\text{f}}}{\frac {nR\,dV}{V}}=nR\ln {\frac {V_{\text{f}}}{V_{\text{i}}}}=Nk_{\text{B}}\ln {\frac {V_{\text{f}}}{V_{\text{i}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>S</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>f</mtext> </mrow> </msubsup> <mi>d</mi> <mi>S</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>f</mtext> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> </mrow> <mi>T</mi> </mfrac> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>f</mtext> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi>R</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> </mrow> <mi>V</mi> </mfrac> </mrow> <mo>=</mo> <mi>n</mi> <mi>R</mi> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>f</mtext> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mi>N</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>f</mtext> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta S=\int _{\text{i}}^{\text{f}}dS=\int _{V_{\text{i}}}^{V_{\text{f}}}{\frac {P\,dV}{T}}=\int _{V_{\text{i}}}^{V_{\text{f}}}{\frac {nR\,dV}{V}}=nR\ln {\frac {V_{\text{f}}}{V_{\text{i}}}}=Nk_{\text{B}}\ln {\frac {V_{\text{f}}}{V_{\text{i}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/511014f6ed457542574f466a15157d88e495500f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:68.074ex; height:6.676ex;" alt="{\displaystyle \Delta S=\int _{\text{i}}^{\text{f}}dS=\int _{V_{\text{i}}}^{V_{\text{f}}}{\frac {P\,dV}{T}}=\int _{V_{\text{i}}}^{V_{\text{f}}}{\frac {nR\,dV}{V}}=nR\ln {\frac {V_{\text{f}}}{V_{\text{i}}}}=Nk_{\text{B}}\ln {\frac {V_{\text{f}}}{V_{\text{i}}}}.}"></span> </p><p>For an ideal <a href="/wiki/Monatomic_gas" title="Monatomic gas">monatomic gas</a>, the entropy as a function of the internal energy <span class="texhtml mvar" style="font-style:italic;">U</span>, volume <span class="texhtml mvar" style="font-style:italic;">V</span>, and number of moles <span class="texhtml mvar" style="font-style:italic;">n</span> is given by the <a href="/wiki/Sackur%E2%80%93Tetrode_equation" title="Sackur–Tetrode equation">Sackur–Tetrode equation</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> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=nR\ln \left[{\frac {V}{N}}\left({\frac {4\pi m}{3h^{2}}}{\frac {U}{N}}\right)^{3/2}\right]+{\frac {5}{2}}nR.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mi>n</mi> <mi>R</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>V</mi> <mi>N</mi> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> <mi>m</mi> </mrow> <mrow> <mn>3</mn> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>U</mi> <mi>N</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mrow> <mi>n</mi> <mi>R</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=nR\ln \left[{\frac {V}{N}}\left({\frac {4\pi m}{3h^{2}}}{\frac {U}{N}}\right)^{3/2}\right]+{\frac {5}{2}}nR.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a4cdb095fde330d4b96da3ef46b381600489026" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:38.728ex; height:7.509ex;" alt="{\displaystyle S=nR\ln \left[{\frac {V}{N}}\left({\frac {4\pi m}{3h^{2}}}{\frac {U}{N}}\right)^{3/2}\right]+{\frac {5}{2}}nR.}"></span> </p><p>In this expression <span class="texhtml mvar" style="font-style:italic;">m</span> is the particle mass and <span class="texhtml"><i>h</i></span> is the Planck constant. For a monatomic ideal gas <span class="texhtml"><i>U</i> = <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span><i>nRT</i> = <i>nC</i><sub>V</sub><i>T</i></span>, with <span class="texhtml"><i>C</i><sub>V</sub></span> the molar heat capacity at constant volume. </p><p>A second way to evaluate the entropy change is to choose a route from the initial state to the final state where all the intermediate states are in equilibrium. Such a route can only be realized in the limit where the changes happen infinitely slowly. Such routes are also referred to as quasistatic routes. In some books one demands that a quasistatic route has to be reversible, here we don't add this extra condition. The net entropy change from the initial state to the final state is independent of the particular choice of the quasistatic route, as the entropy is a function of state. </p><p>Here is how we can effect the quasistatic route. Instead of letting the gas undergo a free expansion in which the volume is doubled, a free expansion is allowed in which the volume expands by a very small amount <span class="texhtml"><i>δV</i></span>. After thermal equilibrium is reached, we then let the gas undergo another free expansion by <span class="texhtml"><i>δV</i></span> and wait until thermal equilibrium is reached. We repeat this until the volume has been doubled. In the limit <span class="texhtml"><i>δV</i></span> to zero, this becomes an ideal quasistatic process, albeit an irreversible one. Now, according to the <a href="/wiki/Fundamental_thermodynamic_relation" title="Fundamental thermodynamic relation">fundamental thermodynamic relation</a>, we have: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} U=T\,\mathrm {d} S-P\,\mathrm {d} V.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>U</mi> <mo>=</mo> <mi>T</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> <mo>−</mo> <mi>P</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} U=T\,\mathrm {d} S-P\,\mathrm {d} V.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bedb28bf714d1055421d3c5002d581ad6d85fab9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.688ex; height:2.343ex;" alt="{\displaystyle \mathrm {d} U=T\,\mathrm {d} S-P\,\mathrm {d} V.}"></span> </p><p>As this equation relates changes in thermodynamic state variables, it is valid for any quasistatic change, regardless of whether it is irreversible or reversible. For the above defined path we have that <span class="texhtml">d<i>U</i> = 0</span> and thus <span class="texhtml"><i>T</i> d<i>S</i> = <i>P</i> d<i>V</i></span>, and hence the increase in entropy for the Joule expansion is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta S=\int _{i}^{f}\mathrm {d} S=\int _{V_{0}}^{2V_{0}}{\frac {P\,\mathrm {d} V}{T}}=\int _{V_{0}}^{2V_{0}}{\frac {nR\,\mathrm {d} V}{V}}=nR\ln 2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>S</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> </mrow> <mi>T</mi> </mfrac> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi>R</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> </mrow> <mi>V</mi> </mfrac> </mrow> <mo>=</mo> <mi>n</mi> <mi>R</mi> <mi>ln</mi> <mo>⁡</mo> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta S=\int _{i}^{f}\mathrm {d} S=\int _{V_{0}}^{2V_{0}}{\frac {P\,\mathrm {d} V}{T}}=\int _{V_{0}}^{2V_{0}}{\frac {nR\,\mathrm {d} V}{V}}=nR\ln 2.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/373420b31bf297b374cc7065904142a423e8aec5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:55.191ex; height:6.676ex;" alt="{\displaystyle \Delta S=\int _{i}^{f}\mathrm {d} S=\int _{V_{0}}^{2V_{0}}{\frac {P\,\mathrm {d} V}{T}}=\int _{V_{0}}^{2V_{0}}{\frac {nR\,\mathrm {d} V}{V}}=nR\ln 2.}"></span> </p><p>A third way to compute the entropy change involves a route consisting of reversible adiabatic expansion followed by heating. We first let the system undergo a reversible adiabatic expansion in which the volume is doubled. During the expansion, the system performs work and the gas temperature goes down, so we have to supply heat to the system equal to the work performed to bring it to the same final state as in case of Joule expansion. </p><p>During the reversible <a href="/wiki/Adiabatic_expansion" class="mw-redirect" title="Adiabatic expansion">adiabatic expansion</a>, we have <span class="texhtml">d<i>S</i> = 0</span>. From the classical expression for the entropy it can be derived that the temperature after the doubling of the volume at constant entropy is given as: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=T_{i}2^{-R/C_{V}}=T_{i}2^{-2/3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mrow> </msup> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=T_{i}2^{-R/C_{V}}=T_{i}2^{-2/3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77862ed61b90e421fc1c062beb1ebdaf6218b9f4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.395ex; height:3.176ex;" alt="{\displaystyle T=T_{i}2^{-R/C_{V}}=T_{i}2^{-2/3}}"></span> for the monoatomic ideal gas. Heating the gas up to the initial temperature <span class="texhtml"><i>T</i><sub>i</sub></span> increases the entropy by the amount <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta S=n\int _{T}^{T_{i}}C_{\mathrm {V} }{\frac {\mathrm {d} T'}{T'}}=nR\ln 2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>S</mi> <mo>=</mo> <mi>n</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msubsup> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>T</mi> <mo>′</mo> </msup> </mrow> <msup> <mi>T</mi> <mo>′</mo> </msup> </mfrac> </mrow> <mo>=</mo> <mi>n</mi> <mi>R</mi> <mi>ln</mi> <mo>⁡</mo> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta S=n\int _{T}^{T_{i}}C_{\mathrm {V} }{\frac {\mathrm {d} T'}{T'}}=nR\ln 2.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f3b0dd452ccc858f752d24e75d59b2f6cf1354c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.424ex; height:6.176ex;" alt="{\displaystyle \Delta S=n\int _{T}^{T_{i}}C_{\mathrm {V} }{\frac {\mathrm {d} T'}{T'}}=nR\ln 2.}"></span> </p><p>We might ask what the work would be if, once the Joule expansion has occurred, the gas is put back into the left-hand side by compressing it. The best method (i.e. the method involving the least work) is that of a reversible isothermal compression, which would take work <span class="texhtml mvar" style="font-style:italic;">W</span> given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=-\int _{2V_{0}}^{V_{0}}P\,\mathrm {d} V=-\int _{2V_{0}}^{V_{0}}{\frac {nRT}{V}}\mathrm {d} V=nRT\ln 2=T\Delta S_{\text{gas}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msubsup> <mi>P</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> <mo>=</mo> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi>R</mi> <mi>T</mi> </mrow> <mi>V</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> <mo>=</mo> <mi>n</mi> <mi>R</mi> <mi>T</mi> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> <mo>=</mo> <mi>T</mi> <mi mathvariant="normal">Δ</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>gas</mtext> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=-\int _{2V_{0}}^{V_{0}}P\,\mathrm {d} V=-\int _{2V_{0}}^{V_{0}}{\frac {nRT}{V}}\mathrm {d} V=nRT\ln 2=T\Delta S_{\text{gas}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dac8f4625bb778ee0599b3b4a3c03c80743bacc1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:59.728ex; height:6.509ex;" alt="{\displaystyle W=-\int _{2V_{0}}^{V_{0}}P\,\mathrm {d} V=-\int _{2V_{0}}^{V_{0}}{\frac {nRT}{V}}\mathrm {d} V=nRT\ln 2=T\Delta S_{\text{gas}}.}"></span> </p><p>During the Joule expansion the surroundings do not change, i.e. the entropy of the surroundings is constant. Therefore the entropy change of the so-called "universe" is equal to the entropy change of the gas which is <span class="texhtml"><i>nR</i> ln 2</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Real-gas_effect">Real-gas effect</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Joule_expansion&action=edit&section=5" title="Edit section: Real-gas effect"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Joule performed his experiment with air at room temperature which was expanded from a pressure of about 22 bar. Air, under these conditions, is almost an ideal gas, but not quite. As a result the real temperature change will not be exactly zero. With our present knowledge of the thermodynamic properties of air <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> we can calculate that the temperature of the air should drop by about 3 degrees Celsius when the volume is doubled under adiabatic conditions. However, due to the low heat capacity of the air and the high heat capacity of the strong copper containers and the water of the calorimeter, the observed temperature drop is much smaller, so Joule found that the temperature change was zero within his measuring accuracy. </p> <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=Joule_expansion&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Szilard%27s_engine" class="mw-redirect" title="Szilard's engine">Szilard's engine</a>, a reverse thought experiment</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=Joule_expansion&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Joule/Gay-Lussac_expansions" class="extiw" title="commons:Category:Joule/Gay-Lussac expansions">Joule/Gay-Lussac expansions</a></span>.</div></div> </div> <p>The majority of good undergraduate textbooks deal with this expansion in great depth; see e.g. <i>Concepts in Thermal Physics</i>, Blundell & Blundell, OUP <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><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-856770-7" title="Special:BookSources/0-19-856770-7">0-19-856770-7</a> </p> <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-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">D.S.L. Cardwell, From Watt to Clausius, Heinemann, London (1957)</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">M.J. Klein, Principles of the theory of heat, D. Reidel Pub.Cy., Dordrecht (1986)</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Note that the fact that the gas expands in a vacuum and thus against zero pressure is irrelevant. The work done by the system would also be zero if the right hand side of the chamber were not evacuated, but is instead filled with a gas at a lower pressure. While the expanding gas would then do work against the gas in the right-hand side of the container, the whole system doesn't do any work against the environment.</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">V.A. Kirillin, et al, Engineering Thermodynamics,(1981) Mir Publishers, Chapter 7.7 p.265</span> </li> <li id="cite_note-:0-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_5-1"><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 id="CITEREFGoussardRoulet1993" class="citation journal cs1">Goussard, J.-O.; Roulet, B. (1993). "Free expansion for real gases". <i>Am. J. Phys</i>. <b>61</b> (9): 845–848. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1993AmJPh..61..845G">1993AmJPh..61..845G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.17417">10.1119/1.17417</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Am.+J.+Phys.&rft.atitle=Free+expansion+for+real+gases&rft.volume=61&rft.issue=9&rft.pages=845-848&rft.date=1993&rft_id=info%3Adoi%2F10.1119%2F1.17417&rft_id=info%3Abibcode%2F1993AmJPh..61..845G&rft.aulast=Goussard&rft.aufirst=J.-O.&rft.au=Roulet%2C+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJoule+expansion" class="Z3988"></span></span> </li> <li id="cite_note-:1-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_6-1"><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 id="CITEREFAlbarrán-ZavalaEspinoza-ElizarrarazAngulo-Brown2009" class="citation journal cs1">Albarrán-Zavala, E.; Espinoza-Elizarraraz, B.A.; Angulo-Brown, F. 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