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* Page found: Reale Gase (eq 22) (force rerendering)

Occurences on the following pages:

   Reale Gase Eq: 22
TeX (as stored in database):
\begin{align}
 
& Y=\sum\limits_{N=0}^{\infty }{{}}{{\xi }^{N}}{{Z}_{N}}=\sum\limits_{N=0}^{\infty }{{}}\frac{1}{N!}{{\left( \xi V \right)}^{N}}\left[ 1+N(N-1)\frac{B}{V} \right] \\
 
& \xi :=\rho \Phi \left( \beta  \right) \\
 
& Y=\sum\limits_{N=0}^{\infty }{{}}{{\xi }^{N}}{{Z}_{N}}=\sum\limits_{N=0}^{\infty }{{}}\frac{1}{N!}{{\left( \xi V \right)}^{N}}\left[ 1+N(N-1)\frac{B}{V} \right]=\exp \left( \xi V \right)+\frac{B}{V}{{\left( \xi V \right)}^{2}}\frac{{{\partial }^{2}}}{\partial {{\left( \xi V \right)}^{2}}}\exp \left( \xi V \right) \\
 
& =\exp \left( \xi V \right)\left( 1+{{\xi }^{2}}BV \right) \\
 
\end{align}
MathML (33.33 KB / 4.341 KB) :
Y=N=0ξNZN=N=01N!(ξV)N[1+N(N-1)BV]YsuperscriptsubscriptN0superscriptξNsubscriptZNsuperscriptsubscriptN01NsuperscriptξVN1NN1BV\displaystyle Y=\sum\limits_{{N=0}}^{{\infty}}{{}}{{\xi}^{{N}}}{{Z}_{{N}}}=% \sum\limits_{{N=0}}^{{\infty}}{{}}\frac{1}{N!}{{\left(\xi V\right)}^{{N}}}% \left[1+N(N-1)\frac{B}{V}\right]
ξ:=ρΦ(β)assignξρnormal-Φβ\displaystyle\xi:=\rho\Phi\left(\beta\right)
Y=N=0ξNZN=N=01N!(ξV)N[1+N(N-1)BV]=exp(ξV)+BV(ξV)22(ξV)2exp(ξV)YsuperscriptsubscriptN0superscriptξNsubscriptZNsuperscriptsubscriptN01NsuperscriptξVN1NN1BVξVBVsuperscriptξV2superscript2fragmentssuperscriptfragmentsnormal-(ξVnormal-)2ξV\displaystyle Y=\sum\limits_{{N=0}}^{{\infty}}{{}}{{\xi}^{{N}}}{{Z}_{{N}}}=% \sum\limits_{{N=0}}^{{\infty}}{{}}\frac{1}{N!}{{\left(\xi V\right)}^{{N}}}% \left[1+N(N-1)\frac{B}{V}\right]=\exp\left(\xi V\right)+\frac{B}{V}{{\left(\xi V% \right)}^{{2}}}\frac{{{\partial}^{{2}}}}{\partial{{\left(\xi V\right)}^{{2}}}}% \exp\left(\xi V\right)
=exp(ξV)(1+ξ2BV)absentξV1superscriptξ2BV\displaystyle=\exp\left(\xi V\right)\left(1+{{\xi}^{{2}}}BV\right)

SVG (0 B / 8 B) :
PNG (0 B / 8 B) :
Hash : 9ed5ee27cdf1bd80417e3f86a3066e21

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Variables

24 results

  • mi exp (3/9/238)
  • mi ! (2/13/276)
  • mi ξ (11/35/685)
  • mo := (1/9/881)
  • mi Z (2/20/945)
  • mi (4/21/1025)
  • mi Y (2/24/1084)
  • mo [ (2/5/1298)
  • mo ] (2/5/1300)
  • mi Φ (1/11/1376)
  • mi β (1/39/1458)
  • mi ρ (1/7/1711)
  • mo (4/15/1888)
  • mi B (4/49/2737)
  • mi V (11/211/3347)
  • mi N (16/85/6684)
  • mo + (4/54/6970)
  • mo (2/133/7333)
  • mo (3/142/9643)
  • mo - (2/106/10795)
  • mo = (10/144/18270)
  • mo ( (11/169/18822)
  • mo ) (11/169/18905)
  • mo (24/419/59187)

MathML


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xref="S0.Ex1.m2.1.29.3">V</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation id="S0.Ex1.m2.1b" encoding="application/x-tex" xref="S0.Ex1.m2.1.cmml">\displaystyle Y=\sum\limits_{{N=0}}^{{\infty}}{{}}{{\xi}^{{N}}}{{Z}_{{N}}}=%
\sum\limits_{{N=0}}^{{\infty}}{{}}\frac{1}{N!}{{\left(\xi V\right)}^{{N}}}%
\left[1+N(N-1)\frac{B}{V}\right]</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\xi:=\rho\Phi\left(\beta\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.m2.1.9" xref="S0.Ex2.m2.1.9.cmml"><mi id="S0.Ex2.m2.1.1" xref="S0.Ex2.m2.1.1.cmml">ξ</mi><mo id="S0.Ex2.m2.1.3" xref="S0.Ex2.m2.1.3.cmml">:=</mo><mrow id="S0.Ex2.m2.1.9.1" xref="S0.Ex2.m2.1.9.1.cmml"><mi id="S0.Ex2.m2.1.4" xref="S0.Ex2.m2.1.4.cmml">ρ</mi><mo id="S0.Ex2.m2.1.9.1.1" xref="S0.Ex2.m2.1.9.1.1.cmml"></mo><mi mathvariant="normal" id="S0.Ex2.m2.1.5" xref="S0.Ex2.m2.1.5.cmml">Φ</mi><mo id="S0.Ex2.m2.1.9.1.1a" xref="S0.Ex2.m2.1.9.1.1.cmml"></mo><mrow id="S0.Ex2.m2.1.7" xref="S0.Ex2.m2.1.7.cmml"><mo id="S0.Ex2.m2.1.7a" xref="S0.Ex2.m2.1.7.cmml">(</mo><mi id="S0.Ex2.m2.1.7b" xref="S0.Ex2.m2.1.7.cmml">β</mi><mo id="S0.Ex2.m2.1.7c" xref="S0.Ex2.m2.1.7.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.m2.1.9.cmml" xref="S0.Ex2.m2.1.9"><csymbol cd="latexml" id="S0.Ex2.m2.1.3.cmml" xref="S0.Ex2.m2.1.3">assign</csymbol><ci id="S0.Ex2.m2.1.1.cmml" xref="S0.Ex2.m2.1.1">ξ</ci><apply id="S0.Ex2.m2.1.9.1.cmml" xref="S0.Ex2.m2.1.9.1"><times id="S0.Ex2.m2.1.9.1.1.cmml" xref="S0.Ex2.m2.1.9.1.1"/><ci id="S0.Ex2.m2.1.4.cmml" xref="S0.Ex2.m2.1.4">ρ</ci><ci id="S0.Ex2.m2.1.5.cmml" xref="S0.Ex2.m2.1.5">normal-Φ</ci><ci id="S0.Ex2.m2.1.7.cmml" xref="S0.Ex2.m2.1.7">β</ci></apply></apply></annotation-xml><annotation id="S0.Ex2.m2.1b" encoding="application/x-tex" xref="S0.Ex2.m2.1.cmml">\displaystyle\xi:=\rho\Phi\left(\beta\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 Y=\sum\limits_{{N=0}}^{{\infty}}{{}}{{\xi}^{{N}}}{{Z}_{{N}}}=%&#10;\sum\limits_{{N=0}}^{{\infty}}{{}}\frac{1}{N!}{{\left(\xi V\right)}^{{N}}}%&#10;\left[1+N(N-1)\frac{B}{V}\right]=\exp\left(\xi V\right)+\frac{B}{V}{{\left(\xi&#10;V%&#10;\right)}^{{2}}}\frac{{{\partial}^{{2}}}}{\partial{{\left(\xi V\right)}^{{2}}}}%&#10;\exp\left(\xi V\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.m2.1.50" xref="S0.Ex3.m2.1.50.cmml"><mi id="S0.Ex3.m2.1.1" xref="S0.Ex3.m2.1.1.cmml">Y</mi><mo id="S0.Ex3.m2.1.2" xref="S0.Ex3.m2.1.2.cmml">=</mo><mrow id="S0.Ex3.m2.1.50.2" xref="S0.Ex3.m2.1.50.2.cmml"><mover id="S0.Ex3.m2.1.50.2.1" xref="S0.Ex3.m2.1.50.2.1.cmml"><mstyle displaystyle="true" id="S0.Ex3.m2.1.50.2.1.2" xref="S0.Ex3.m2.1.50.2.1.2.cmml"><munder id="S0.Ex3.m2.1.50.2.1.2a" 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xref="S0.Ex3.m2.1.50.4.2.3.1"><times id="S0.Ex3.m2.1.50.4.2.3.1.1.cmml" xref="S0.Ex3.m2.1.50.4.2.3.1.1"/><ci id="S0.Ex3.m2.1.47.cmml" xref="S0.Ex3.m2.1.47">ξ</ci><ci id="S0.Ex3.m2.1.48.cmml" xref="S0.Ex3.m2.1.48">V</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation id="S0.Ex3.m2.1b" encoding="application/x-tex" xref="S0.Ex3.m2.1.cmml">\displaystyle Y=\sum\limits_{{N=0}}^{{\infty}}{{}}{{\xi}^{{N}}}{{Z}_{{N}}}=%
\sum\limits_{{N=0}}^{{\infty}}{{}}\frac{1}{N!}{{\left(\xi V\right)}^{{N}}}%
\left[1+N(N-1)\frac{B}{V}\right]=\exp\left(\xi V\right)+\frac{B}{V}{{\left(\xi
V%
\right)}^{{2}}}\frac{{{\partial}^{{2}}}}{\partial{{\left(\xi V\right)}^{{2}}}}%
\exp\left(\xi V\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;"/>
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<td class="eqpad"/></tr>
</table>