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* Page found: Reale Gase (eq 61) (force rerendering)

Occurences on the following pages:

   Reale Gase Eq: 61
TeX (as stored in database):
\begin{align}
 
& {{c}_{p}}-{{c}_{v}}=T{{\left( \frac{\partial p}{\partial T} \right)}_{v}}{{\left( \frac{\partial v}{\partial T} \right)}_{p}}=T\frac{R}{v-b}\frac{1}{{{\left( \frac{\partial T}{\partial v} \right)}_{p}}} \\
 
& RT=\left( p+\frac{a}{{{v}^{2}}} \right)\left( v-b \right) \\
 
& R{{\left( \frac{\partial T}{\partial v} \right)}_{p}}=\left( p+\frac{a}{{{v}^{2}}} \right)-\frac{2a}{{{v}^{3}}}\left( v-b \right) \\
 
& \left( p+\frac{a}{{{v}^{2}}} \right)=\frac{RT}{v-b} \\
 
& \Rightarrow {{c}_{p}}-{{c}_{v}}=T{{\left( \frac{\partial p}{\partial T} \right)}_{v}}{{\left( \frac{\partial v}{\partial T} \right)}_{p}}=T\frac{R}{v-b}\frac{1}{{{\left( \frac{\partial T}{\partial v} \right)}_{p}}}=\frac{R}{1-\frac{2a}{RT{{v}^{3}}}{{\left( v-b \right)}^{2}}} \\
 
\end{align}
MathML (37.607 KB / 4.813 KB) :
cp-cv=T(pT)v(vT)p=TRv-b1(Tv)psubscriptcpsubscriptcvTsubscriptpTvsubscriptvTpTRvb1subscriptTvp\displaystyle{{c}_{{p}}}-{{c}_{{v}}}=T{{\left(\frac{\partial p}{\partial T}% \right)}_{{v}}}{{\left(\frac{\partial v}{\partial T}\right)}_{{p}}}=T\frac{R}{% v-b}\frac{1}{{{\left(\frac{\partial T}{\partial v}\right)}_{{p}}}}
RT=(p+av2)(v-b)RTpasuperscriptv2vb\displaystyle RT=\left(p+\frac{a}{{{v}^{{2}}}}\right)\left(v-b\right)
R(Tv)p=(p+av2)-2av3(v-b)RsubscriptTvppasuperscriptv22asuperscriptv3vb\displaystyle R{{\left(\frac{\partial T}{\partial v}\right)}_{{p}}}=\left(p+% \frac{a}{{{v}^{{2}}}}\right)-\frac{2a}{{{v}^{{3}}}}\left(v-b\right)
(p+av2)=RTv-bpasuperscriptv2RTvb\displaystyle\left(p+\frac{a}{{{v}^{{2}}}}\right)=\frac{RT}{v-b}
cp-cv=T(pT)v(vT)p=TRv-b1(Tv)p=R1-2aRTv3(v-b)2fragmentsnormal-⇒subscriptcpsubscriptcvTsubscriptfragmentsnormal-(pTnormal-)vsubscriptfragmentsnormal-(vTnormal-)pTRvb1subscriptTvpR12aRTsuperscriptv3superscriptvb2\displaystyle\Rightarrow{{c}_{{p}}}-{{c}_{{v}}}=T{{\left(\frac{\partial p}{% \partial T}\right)}_{{v}}}{{\left(\frac{\partial v}{\partial T}\right)}_{{p}}}% =T\frac{R}{v-b}\frac{1}{{{\left(\frac{\partial T}{\partial v}\right)}_{{p}}}}=% \frac{R}{1-\frac{2a}{RT{{v}^{{3}}}}{{\left(v-b\right)}^{{2}}}}

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Hash : c5aa4e96cdf4f0fd54517c51155b94f1

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Variables

16 results

  • mo (1/22/1479)
  • mi b (6/49/2737)
  • mi c (4/37/2835)
  • mi v (20/211/3347)
  • mi p (12/77/4684)
  • mi a (5/37/5608)
  • mo + (3/54/6970)
  • mo (14/133/7333)
  • mo (14/142/9643)
  • mi T (14/142/10589)
  • mo - (10/106/10795)
  • mi R (7/86/14377)
  • mo = (8/144/18270)
  • mo ( (13/169/18822)
  • mo ) (13/169/18905)
  • mo (14/419/59187)

MathML


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\right)}_{{v}}}{{\left(\frac{\partial v}{\partial T}\right)}_{{p}}}=T\frac{R}{%
v-b}\frac{1}{{{\left(\frac{\partial T}{\partial v}\right)}_{{p}}}}</annotation></semantics></math></td>
<td class="eqpad"/></tr>
<tr id="S0.Ex2" class="equation baseline">
<td class="eqpad"/>
<td colspan="1" class="td right" style="text-align:right;"/>
<td colspan="1" class="td left" style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\displaystyle RT=\left(p+\frac{a}{{{v}^{{2}}}}\right)\left(v-b\right)" id="S0.Ex2.m2.1" display="inline" xref="S0.Ex2.m2.1.cmml"><semantics id="S0.Ex2.m2.1a" xref="S0.Ex2.m2.1.cmml"><mrow id="S0.Ex2.m2.1.14" xref="S0.Ex2.m2.1.14.cmml"><mrow id="S0.Ex2.m2.1.14.1" xref="S0.Ex2.m2.1.14.1.cmml"><mi id="S0.Ex2.m2.1.1" xref="S0.Ex2.m2.1.1.cmml">R</mi><mo id="S0.Ex2.m2.1.14.1.1" xref="S0.Ex2.m2.1.14.1.1.cmml"></mo><mi id="S0.Ex2.m2.1.2" xref="S0.Ex2.m2.1.2.cmml">T</mi></mrow><mo id="S0.Ex2.m2.1.3" xref="S0.Ex2.m2.1.3.cmml">=</mo><mrow id="S0.Ex2.m2.1.14.2" xref="S0.Ex2.m2.1.14.2.cmml"><mrow id="S0.Ex2.m2.1.14.2.2" xref="S0.Ex2.m2.1.14.2.2.cmml"><mo id="S0.Ex2.m2.1.14.2.2a" xref="S0.Ex2.m2.1.14.2.2.cmml">(</mo><mrow id="S0.Ex2.m2.1.14.2.2b" xref="S0.Ex2.m2.1.14.2.2.cmml"><mi id="S0.Ex2.m2.1.5" xref="S0.Ex2.m2.1.5.cmml">p</mi><mo id="S0.Ex2.m2.1.6" xref="S0.Ex2.m2.1.6.cmml">+</mo><mstyle displaystyle="true" id="S0.Ex2.m2.1.7" xref="S0.Ex2.m2.1.7.cmml"><mfrac id="S0.Ex2.m2.1.7a" xref="S0.Ex2.m2.1.7.cmml"><mi id="S0.Ex2.m2.1.7.2" xref="S0.Ex2.m2.1.7.2.cmml">a</mi><msup id="S0.Ex2.m2.1.7.3" xref="S0.Ex2.m2.1.7.3.cmml"><mi id="S0.Ex2.m2.1.7.3.1" xref="S0.Ex2.m2.1.7.3.1.cmml">v</mi><mn id="S0.Ex2.m2.1.7.3.2.1" xref="S0.Ex2.m2.1.7.3.2.1.cmml">2</mn></msup></mfrac></mstyle></mrow><mo id="S0.Ex2.m2.1.14.2.2c" xref="S0.Ex2.m2.1.14.2.2.cmml">)</mo></mrow><mo id="S0.Ex2.m2.1.14.2.1" xref="S0.Ex2.m2.1.14.2.1.cmml"></mo><mrow id="S0.Ex2.m2.1.14.2.3" xref="S0.Ex2.m2.1.14.2.3.cmml"><mo id="S0.Ex2.m2.1.14.2.3a" xref="S0.Ex2.m2.1.14.2.3.cmml">(</mo><mrow id="S0.Ex2.m2.1.14.2.3b" xref="S0.Ex2.m2.1.14.2.3.cmml"><mi id="S0.Ex2.m2.1.10" xref="S0.Ex2.m2.1.10.cmml">v</mi><mo id="S0.Ex2.m2.1.11" xref="S0.Ex2.m2.1.11.cmml">-</mo><mi id="S0.Ex2.m2.1.12" xref="S0.Ex2.m2.1.12.cmml">b</mi></mrow><mo id="S0.Ex2.m2.1.14.2.3c" xref="S0.Ex2.m2.1.14.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml id="S0.Ex2.m2.1.cmml" encoding="MathML-Content" xref="S0.Ex2.m2.1"><apply id="S0.Ex2.m2.1.14.cmml" xref="S0.Ex2.m2.1.14"><eq id="S0.Ex2.m2.1.3.cmml" xref="S0.Ex2.m2.1.3"/><apply id="S0.Ex2.m2.1.14.1.cmml" xref="S0.Ex2.m2.1.14.1"><times id="S0.Ex2.m2.1.14.1.1.cmml" xref="S0.Ex2.m2.1.14.1.1"/><ci id="S0.Ex2.m2.1.1.cmml" xref="S0.Ex2.m2.1.1">R</ci><ci id="S0.Ex2.m2.1.2.cmml" xref="S0.Ex2.m2.1.2">T</ci></apply><apply id="S0.Ex2.m2.1.14.2.cmml" xref="S0.Ex2.m2.1.14.2"><times id="S0.Ex2.m2.1.14.2.1.cmml" xref="S0.Ex2.m2.1.14.2.1"/><apply id="S0.Ex2.m2.1.14.2.2.cmml" xref="S0.Ex2.m2.1.14.2.2"><plus id="S0.Ex2.m2.1.6.cmml" xref="S0.Ex2.m2.1.6"/><ci id="S0.Ex2.m2.1.5.cmml" xref="S0.Ex2.m2.1.5">p</ci><apply id="S0.Ex2.m2.1.7.cmml" xref="S0.Ex2.m2.1.7"><divide id="S0.Ex2.m2.1.7.1.cmml"/><ci id="S0.Ex2.m2.1.7.2.cmml" xref="S0.Ex2.m2.1.7.2">a</ci><apply id="S0.Ex2.m2.1.7.3.cmml" xref="S0.Ex2.m2.1.7.3"><csymbol cd="ambiguous" id="S0.Ex2.m2.1.7.3.3.cmml">superscript</csymbol><ci id="S0.Ex2.m2.1.7.3.1.cmml" xref="S0.Ex2.m2.1.7.3.1">v</ci><cn type="integer" id="S0.Ex2.m2.1.7.3.2.1.cmml" xref="S0.Ex2.m2.1.7.3.2.1">2</cn></apply></apply></apply><apply id="S0.Ex2.m2.1.14.2.3.cmml" xref="S0.Ex2.m2.1.14.2.3"><minus id="S0.Ex2.m2.1.11.cmml" xref="S0.Ex2.m2.1.11"/><ci id="S0.Ex2.m2.1.10.cmml" xref="S0.Ex2.m2.1.10">v</ci><ci id="S0.Ex2.m2.1.12.cmml" xref="S0.Ex2.m2.1.12">b</ci></apply></apply></apply></annotation-xml><annotation id="S0.Ex2.m2.1b" encoding="application/x-tex" xref="S0.Ex2.m2.1.cmml">\displaystyle RT=\left(p+\frac{a}{{{v}^{{2}}}}\right)\left(v-b\right)</annotation></semantics></math></td>
<td class="eqpad"/></tr>
<tr id="S0.Ex3" class="equation baseline">
<td class="eqpad"/>
<td colspan="1" class="td right" style="text-align:right;"/>
<td colspan="1" class="td left" style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\displaystyle R{{\left(\frac{\partial T}{\partial v}\right)}_{{p}}}=\left(p+%&#10;\frac{a}{{{v}^{{2}}}}\right)-\frac{2a}{{{v}^{{3}}}}\left(v-b\right)" id="S0.Ex3.m2.1" display="inline" xref="S0.Ex3.m2.1.cmml"><semantics id="S0.Ex3.m2.1a" xref="S0.Ex3.m2.1.cmml"><mrow id="S0.Ex3.m2.1.19" xref="S0.Ex3.m2.1.19.cmml"><mrow id="S0.Ex3.m2.1.19.1" xref="S0.Ex3.m2.1.19.1.cmml"><mi id="S0.Ex3.m2.1.1" xref="S0.Ex3.m2.1.1.cmml">R</mi><mo id="S0.Ex3.m2.1.19.1.1" xref="S0.Ex3.m2.1.19.1.1.cmml"></mo><msub id="S0.Ex3.m2.1.19.1.2" xref="S0.Ex3.m2.1.19.1.2.cmml"><mrow id="S0.Ex3.m2.1.3" xref="S0.Ex3.m2.1.3.cmml"><mo id="S0.Ex3.m2.1.3a" xref="S0.Ex3.m2.1.3.cmml">(</mo><mstyle displaystyle="true" id="S0.Ex3.m2.1.3b" xref="S0.Ex3.m2.1.3.cmml"><mfrac id="S0.Ex3.m2.1.3c" xref="S0.Ex3.m2.1.3.cmml"><mrow id="S0.Ex3.m2.1.3.2" xref="S0.Ex3.m2.1.3.2.cmml"><mo id="S0.Ex3.m2.1.3.2.1" xref="S0.Ex3.m2.1.3.2.1.cmml"></mo><mo id="S0.Ex3.m2.1.3.2a" xref="S0.Ex3.m2.1.3.2.cmml"></mo><mi id="S0.Ex3.m2.1.3.2.2" xref="S0.Ex3.m2.1.3.2.2.cmml">T</mi></mrow><mrow id="S0.Ex3.m2.1.3.3" xref="S0.Ex3.m2.1.3.3.cmml"><mo id="S0.Ex3.m2.1.3.3.1" xref="S0.Ex3.m2.1.3.3.1.cmml"></mo><mo id="S0.Ex3.m2.1.3.3a" xref="S0.Ex3.m2.1.3.3.cmml"></mo><mi id="S0.Ex3.m2.1.3.3.2" xref="S0.Ex3.m2.1.3.3.2.cmml">v</mi></mrow></mfrac></mstyle><mo id="S0.Ex3.m2.1.3d" xref="S0.Ex3.m2.1.3.cmml">)</mo></mrow><mi id="S0.Ex3.m2.1.5.1" xref="S0.Ex3.m2.1.5.1.cmml">p</mi></msub></mrow><mo id="S0.Ex3.m2.1.6" xref="S0.Ex3.m2.1.6.cmml">=</mo><mrow id="S0.Ex3.m2.1.19.2" xref="S0.Ex3.m2.1.19.2.cmml"><mrow id="S0.Ex3.m2.1.19.2.1" xref="S0.Ex3.m2.1.19.2.1.cmml"><mo id="S0.Ex3.m2.1.19.2.1a" xref="S0.Ex3.m2.1.19.2.1.cmml">(</mo><mrow id="S0.Ex3.m2.1.19.2.1b" xref="S0.Ex3.m2.1.19.2.1.cmml"><mi id="S0.Ex3.m2.1.8" xref="S0.Ex3.m2.1.8.cmml">p</mi><mo id="S0.Ex3.m2.1.9" xref="S0.Ex3.m2.1.9.cmml">+</mo><mstyle displaystyle="true" id="S0.Ex3.m2.1.10" xref="S0.Ex3.m2.1.10.cmml"><mfrac id="S0.Ex3.m2.1.10a" xref="S0.Ex3.m2.1.10.cmml"><mi id="S0.Ex3.m2.1.10.2" xref="S0.Ex3.m2.1.10.2.cmml">a</mi><msup id="S0.Ex3.m2.1.10.3" xref="S0.Ex3.m2.1.10.3.cmml"><mi id="S0.Ex3.m2.1.10.3.1" xref="S0.Ex3.m2.1.10.3.1.cmml">v</mi><mn id="S0.Ex3.m2.1.10.3.2.1" xref="S0.Ex3.m2.1.10.3.2.1.cmml">2</mn></msup></mfrac></mstyle></mrow><mo id="S0.Ex3.m2.1.19.2.1c" xref="S0.Ex3.m2.1.19.2.1.cmml">)</mo></mrow><mo id="S0.Ex3.m2.1.12" xref="S0.Ex3.m2.1.12.cmml">-</mo><mrow id="S0.Ex3.m2.1.19.2.2" xref="S0.Ex3.m2.1.19.2.2.cmml"><mstyle displaystyle="true" id="S0.Ex3.m2.1.13" xref="S0.Ex3.m2.1.13.cmml"><mfrac id="S0.Ex3.m2.1.13a" xref="S0.Ex3.m2.1.13.cmml"><mrow id="S0.Ex3.m2.1.13.2" xref="S0.Ex3.m2.1.13.2.cmml"><mn id="S0.Ex3.m2.1.13.2.1" xref="S0.Ex3.m2.1.13.2.1.cmml">2</mn><mo id="S0.Ex3.m2.1.13.2.3" xref="S0.Ex3.m2.1.13.2.3.cmml"></mo><mi id="S0.Ex3.m2.1.13.2.2" xref="S0.Ex3.m2.1.13.2.2.cmml">a</mi></mrow><msup id="S0.Ex3.m2.1.13.3" xref="S0.Ex3.m2.1.13.3.cmml"><mi id="S0.Ex3.m2.1.13.3.1" xref="S0.Ex3.m2.1.13.3.1.cmml">v</mi><mn id="S0.Ex3.m2.1.13.3.2.1" xref="S0.Ex3.m2.1.13.3.2.1.cmml">3</mn></msup></mfrac></mstyle><mo id="S0.Ex3.m2.1.19.2.2.1" xref="S0.Ex3.m2.1.19.2.2.1.cmml"></mo><mrow id="S0.Ex3.m2.1.19.2.2.2" xref="S0.Ex3.m2.1.19.2.2.2.cmml"><mo id="S0.Ex3.m2.1.19.2.2.2a" xref="S0.Ex3.m2.1.19.2.2.2.cmml">(</mo><mrow id="S0.Ex3.m2.1.19.2.2.2b" xref="S0.Ex3.m2.1.19.2.2.2.cmml"><mi id="S0.Ex3.m2.1.15" xref="S0.Ex3.m2.1.15.cmml">v</mi><mo id="S0.Ex3.m2.1.16" xref="S0.Ex3.m2.1.16.cmml">-</mo><mi id="S0.Ex3.m2.1.17" xref="S0.Ex3.m2.1.17.cmml">b</mi></mrow><mo id="S0.Ex3.m2.1.19.2.2.2c" xref="S0.Ex3.m2.1.19.2.2.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml id="S0.Ex3.m2.1.cmml" encoding="MathML-Content" xref="S0.Ex3.m2.1"><apply id="S0.Ex3.m2.1.19.cmml" xref="S0.Ex3.m2.1.19"><eq id="S0.Ex3.m2.1.6.cmml" xref="S0.Ex3.m2.1.6"/><apply id="S0.Ex3.m2.1.19.1.cmml" xref="S0.Ex3.m2.1.19.1"><times id="S0.Ex3.m2.1.19.1.1.cmml" xref="S0.Ex3.m2.1.19.1.1"/><ci id="S0.Ex3.m2.1.1.cmml" xref="S0.Ex3.m2.1.1">R</ci><apply id="S0.Ex3.m2.1.19.1.2.cmml" xref="S0.Ex3.m2.1.19.1.2"><csymbol cd="ambiguous" id="S0.Ex3.m2.1.19.1.2.1.cmml">subscript</csymbol><apply id="S0.Ex3.m2.1.3.cmml" xref="S0.Ex3.m2.1.3"><divide id="S0.Ex3.m2.1.3.1.cmml"/><apply id="S0.Ex3.m2.1.3.2.cmml" xref="S0.Ex3.m2.1.3.2"><partialdiff id="S0.Ex3.m2.1.3.2.1.cmml" xref="S0.Ex3.m2.1.3.2.1"/><ci id="S0.Ex3.m2.1.3.2.2.cmml" xref="S0.Ex3.m2.1.3.2.2">T</ci></apply><apply id="S0.Ex3.m2.1.3.3.cmml" xref="S0.Ex3.m2.1.3.3"><partialdiff id="S0.Ex3.m2.1.3.3.1.cmml" xref="S0.Ex3.m2.1.3.3.1"/><ci id="S0.Ex3.m2.1.3.3.2.cmml" xref="S0.Ex3.m2.1.3.3.2">v</ci></apply></apply><ci id="S0.Ex3.m2.1.5.1.cmml" xref="S0.Ex3.m2.1.5.1">p</ci></apply></apply><apply id="S0.Ex3.m2.1.19.2.cmml" xref="S0.Ex3.m2.1.19.2"><minus id="S0.Ex3.m2.1.12.cmml" xref="S0.Ex3.m2.1.12"/><apply id="S0.Ex3.m2.1.19.2.1.cmml" xref="S0.Ex3.m2.1.19.2.1"><plus id="S0.Ex3.m2.1.9.cmml" xref="S0.Ex3.m2.1.9"/><ci id="S0.Ex3.m2.1.8.cmml" xref="S0.Ex3.m2.1.8">p</ci><apply id="S0.Ex3.m2.1.10.cmml" xref="S0.Ex3.m2.1.10"><divide id="S0.Ex3.m2.1.10.1.cmml"/><ci id="S0.Ex3.m2.1.10.2.cmml" xref="S0.Ex3.m2.1.10.2">a</ci><apply id="S0.Ex3.m2.1.10.3.cmml" xref="S0.Ex3.m2.1.10.3"><csymbol cd="ambiguous" id="S0.Ex3.m2.1.10.3.3.cmml">superscript</csymbol><ci id="S0.Ex3.m2.1.10.3.1.cmml" xref="S0.Ex3.m2.1.10.3.1">v</ci><cn type="integer" id="S0.Ex3.m2.1.10.3.2.1.cmml" xref="S0.Ex3.m2.1.10.3.2.1">2</cn></apply></apply></apply><apply id="S0.Ex3.m2.1.19.2.2.cmml" xref="S0.Ex3.m2.1.19.2.2"><times id="S0.Ex3.m2.1.19.2.2.1.cmml" xref="S0.Ex3.m2.1.19.2.2.1"/><apply id="S0.Ex3.m2.1.13.cmml" xref="S0.Ex3.m2.1.13"><divide id="S0.Ex3.m2.1.13.1.cmml"/><apply id="S0.Ex3.m2.1.13.2.cmml" xref="S0.Ex3.m2.1.13.2"><times id="S0.Ex3.m2.1.13.2.3.cmml" xref="S0.Ex3.m2.1.13.2.3"/><cn type="integer" id="S0.Ex3.m2.1.13.2.1.cmml" xref="S0.Ex3.m2.1.13.2.1">2</cn><ci id="S0.Ex3.m2.1.13.2.2.cmml" xref="S0.Ex3.m2.1.13.2.2">a</ci></apply><apply id="S0.Ex3.m2.1.13.3.cmml" xref="S0.Ex3.m2.1.13.3"><csymbol cd="ambiguous" id="S0.Ex3.m2.1.13.3.3.cmml">superscript</csymbol><ci id="S0.Ex3.m2.1.13.3.1.cmml" xref="S0.Ex3.m2.1.13.3.1">v</ci><cn type="integer" id="S0.Ex3.m2.1.13.3.2.1.cmml" xref="S0.Ex3.m2.1.13.3.2.1">3</cn></apply></apply><apply id="S0.Ex3.m2.1.19.2.2.2.cmml" xref="S0.Ex3.m2.1.19.2.2.2"><minus id="S0.Ex3.m2.1.16.cmml" xref="S0.Ex3.m2.1.16"/><ci id="S0.Ex3.m2.1.15.cmml" xref="S0.Ex3.m2.1.15">v</ci><ci id="S0.Ex3.m2.1.17.cmml" xref="S0.Ex3.m2.1.17">b</ci></apply></apply></apply></apply></annotation-xml><annotation id="S0.Ex3.m2.1b" encoding="application/x-tex" xref="S0.Ex3.m2.1.cmml">\displaystyle R{{\left(\frac{\partial T}{\partial v}\right)}_{{p}}}=\left(p+%
\frac{a}{{{v}^{{2}}}}\right)-\frac{2a}{{{v}^{{3}}}}\left(v-b\right)</annotation></semantics></math></td>
<td class="eqpad"/></tr>
<tr id="S0.Ex4" class="equation baseline">
<td class="eqpad"/>
<td colspan="1" class="td right" style="text-align:right;"/>
<td colspan="1" class="td left" style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\displaystyle\left(p+\frac{a}{{{v}^{{2}}}}\right)=\frac{RT}{v-b}" id="S0.Ex4.m2.1" display="inline" xref="S0.Ex4.m2.1.cmml"><semantics id="S0.Ex4.m2.1a" xref="S0.Ex4.m2.1.cmml"><mrow id="S0.Ex4.m2.1.8" xref="S0.Ex4.m2.1.8.cmml"><mrow id="S0.Ex4.m2.1.8.1" xref="S0.Ex4.m2.1.8.1.cmml"><mo id="S0.Ex4.m2.1.8.1a" xref="S0.Ex4.m2.1.8.1.cmml">(</mo><mrow id="S0.Ex4.m2.1.8.1b" xref="S0.Ex4.m2.1.8.1.cmml"><mi id="S0.Ex4.m2.1.2" xref="S0.Ex4.m2.1.2.cmml">p</mi><mo id="S0.Ex4.m2.1.3" xref="S0.Ex4.m2.1.3.cmml">+</mo><mstyle displaystyle="true" id="S0.Ex4.m2.1.4" xref="S0.Ex4.m2.1.4.cmml"><mfrac id="S0.Ex4.m2.1.4a" xref="S0.Ex4.m2.1.4.cmml"><mi id="S0.Ex4.m2.1.4.2" xref="S0.Ex4.m2.1.4.2.cmml">a</mi><msup id="S0.Ex4.m2.1.4.3" xref="S0.Ex4.m2.1.4.3.cmml"><mi id="S0.Ex4.m2.1.4.3.1" xref="S0.Ex4.m2.1.4.3.1.cmml">v</mi><mn id="S0.Ex4.m2.1.4.3.2.1" xref="S0.Ex4.m2.1.4.3.2.1.cmml">2</mn></msup></mfrac></mstyle></mrow><mo id="S0.Ex4.m2.1.8.1c" xref="S0.Ex4.m2.1.8.1.cmml">)</mo></mrow><mo id="S0.Ex4.m2.1.6" xref="S0.Ex4.m2.1.6.cmml">=</mo><mstyle displaystyle="true" id="S0.Ex4.m2.1.7" xref="S0.Ex4.m2.1.7.cmml"><mfrac id="S0.Ex4.m2.1.7a" xref="S0.Ex4.m2.1.7.cmml"><mrow id="S0.Ex4.m2.1.7.2" xref="S0.Ex4.m2.1.7.2.cmml"><mi id="S0.Ex4.m2.1.7.2.1" xref="S0.Ex4.m2.1.7.2.1.cmml">R</mi><mo id="S0.Ex4.m2.1.7.2.3" xref="S0.Ex4.m2.1.7.2.3.cmml"></mo><mi id="S0.Ex4.m2.1.7.2.2" xref="S0.Ex4.m2.1.7.2.2.cmml">T</mi></mrow><mrow id="S0.Ex4.m2.1.7.3" xref="S0.Ex4.m2.1.7.3.cmml"><mi id="S0.Ex4.m2.1.7.3.1" xref="S0.Ex4.m2.1.7.3.1.cmml">v</mi><mo id="S0.Ex4.m2.1.7.3.2" xref="S0.Ex4.m2.1.7.3.2.cmml">-</mo><mi id="S0.Ex4.m2.1.7.3.3" xref="S0.Ex4.m2.1.7.3.3.cmml">b</mi></mrow></mfrac></mstyle></mrow><annotation-xml id="S0.Ex4.m2.1.cmml" encoding="MathML-Content" xref="S0.Ex4.m2.1"><apply id="S0.Ex4.m2.1.8.cmml" xref="S0.Ex4.m2.1.8"><eq id="S0.Ex4.m2.1.6.cmml" xref="S0.Ex4.m2.1.6"/><apply id="S0.Ex4.m2.1.8.1.cmml" xref="S0.Ex4.m2.1.8.1"><plus id="S0.Ex4.m2.1.3.cmml" xref="S0.Ex4.m2.1.3"/><ci id="S0.Ex4.m2.1.2.cmml" xref="S0.Ex4.m2.1.2">p</ci><apply id="S0.Ex4.m2.1.4.cmml" xref="S0.Ex4.m2.1.4"><divide id="S0.Ex4.m2.1.4.1.cmml"/><ci id="S0.Ex4.m2.1.4.2.cmml" xref="S0.Ex4.m2.1.4.2">a</ci><apply id="S0.Ex4.m2.1.4.3.cmml" xref="S0.Ex4.m2.1.4.3"><csymbol cd="ambiguous" id="S0.Ex4.m2.1.4.3.3.cmml">superscript</csymbol><ci id="S0.Ex4.m2.1.4.3.1.cmml" xref="S0.Ex4.m2.1.4.3.1">v</ci><cn type="integer" id="S0.Ex4.m2.1.4.3.2.1.cmml" xref="S0.Ex4.m2.1.4.3.2.1">2</cn></apply></apply></apply><apply id="S0.Ex4.m2.1.7.cmml" xref="S0.Ex4.m2.1.7"><divide id="S0.Ex4.m2.1.7.1.cmml"/><apply id="S0.Ex4.m2.1.7.2.cmml" xref="S0.Ex4.m2.1.7.2"><times id="S0.Ex4.m2.1.7.2.3.cmml" xref="S0.Ex4.m2.1.7.2.3"/><ci id="S0.Ex4.m2.1.7.2.1.cmml" xref="S0.Ex4.m2.1.7.2.1">R</ci><ci id="S0.Ex4.m2.1.7.2.2.cmml" xref="S0.Ex4.m2.1.7.2.2">T</ci></apply><apply id="S0.Ex4.m2.1.7.3.cmml" xref="S0.Ex4.m2.1.7.3"><minus id="S0.Ex4.m2.1.7.3.2.cmml" xref="S0.Ex4.m2.1.7.3.2"/><ci id="S0.Ex4.m2.1.7.3.1.cmml" xref="S0.Ex4.m2.1.7.3.1">v</ci><ci id="S0.Ex4.m2.1.7.3.3.cmml" xref="S0.Ex4.m2.1.7.3.3">b</ci></apply></apply></apply></annotation-xml><annotation id="S0.Ex4.m2.1b" encoding="application/x-tex" xref="S0.Ex4.m2.1.cmml">\displaystyle\left(p+\frac{a}{{{v}^{{2}}}}\right)=\frac{RT}{v-b}</annotation></semantics></math></td>
<td class="eqpad"/></tr>
<tr id="S0.Ex5" class="equation baseline">
<td class="eqpad"/>
<td colspan="1" class="td right" style="text-align:right;"/>
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\partial T}\right)}_{{v}}}{{\left(\frac{\partial v}{\partial T}\right)}_{{p}}}%
=T\frac{R}{v-b}\frac{1}{{{\left(\frac{\partial T}{\partial v}\right)}_{{p}}}}=%
\frac{R}{1-\frac{2a}{RT{{v}^{{3}}}}{{\left(v-b\right)}^{{2}}}}</annotation></semantics></math></td>
<td class="eqpad"/></tr>
</table>