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* Page found: Reale Gase (eq 13) (force rerendering)

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   Reale Gase Eq: 13
TeX (as stored in database):
\begin{align}
 
& \int_{{{R}^{3}}}^{{}}{{}}{{f}_{ij}}(r){{d}^{3}}r=\int_{0}^{\infty }{{}}\left( {{e}^{-\beta \phi (r)}}-1 \right)4\pi {{r}^{2}}dr \\
 
& =\left. \left( {{e}^{-\beta \phi (r)}}-1 \right)\frac{4\pi }{3}{{r}^{3}} \right|_{0}^{\infty }+\frac{4\pi \beta }{3}\int_{0}^{\infty }{{}}\left( \frac{\partial \phi }{\partial r}r \right){{e}^{-\beta \phi (r)}}{{r}^{2}}dr \\
 
\end{align}
MathML (21.719 KB / 3.076 KB) :
R3fij(r)d3r=0(e-βϕ(r)-1)4πr2drsubscriptsuperscriptR3subscriptfijrsuperscriptd3rsuperscriptsubscript0superscripteβϕr14πsuperscriptr2dr\displaystyle\int_{{{{R}^{{3}}}}}{{}}{{f}_{{ij}}}(r){{d}^{{3}}}r=\int_{{0}}^{{% \infty}}{{}}\left({{e}^{{-\beta\phi(r)}}}-1\right)4\pi{{r}^{{2}}}dr
=(e-βϕ(r)-1)4π3r3|0+4πβ30(ϕrr)e-βϕ(r)r2drabsentevaluated-atsuperscripteβϕr14π3superscriptr304πβ3superscriptsubscript0ϕrrsuperscripteβϕrsuperscriptr2dr\displaystyle=\left.\left({{e}^{{-\beta\phi(r)}}}-1\right)\frac{4\pi}{3}{{r}^{% {3}}}\right|_{{0}}^{{\infty}}+\frac{4\pi\beta}{3}\int_{{0}}^{{\infty}}{{}}% \left(\frac{\partial\phi}{\partial r}r\right){{e}^{{-\beta\phi(r)}}}{{r}^{{2}}% }dr

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Hash : c2dd502a3a6ecd1c91eeda642fc98d60

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Variables

20 results

  • mi (3/21/1025)
  • mi π (3/16/1435)
  • mi β (4/39/1458)
  • mi ϕ (4/26/1589)
  • mo (3/29/2089)
  • mi f (1/22/3433)
  • mi j (1/28/3458)
  • mi d (3/65/6342)
  • mo + (1/54/6970)
  • mi e (3/15/7131)
  • mo (2/133/7333)
  • mo | (1/4/7788)
  • mi i (1/38/7943)
  • mo (2/142/9643)
  • mo - (5/106/10795)
  • mi R (13/86/14377)
  • mo = (2/144/18270)
  • mo ( (7/169/18822)
  • mo ) (7/169/18905)
  • mo (26/419/59187)

MathML


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id="S0.Ex2.m2.1.24.cmml" xref="S0.Ex2.m2.1.24">e</ci><apply id="S0.Ex2.m2.1.25.1.cmml" xref="S0.Ex2.m2.1.25.1"><minus id="S0.Ex2.m2.1.25.1.1.cmml" xref="S0.Ex2.m2.1.25.1.1"/><apply id="S0.Ex2.m2.1.25.1.7.cmml" xref="S0.Ex2.m2.1.25.1.7"><times id="S0.Ex2.m2.1.25.1.7.1.cmml" xref="S0.Ex2.m2.1.25.1.7.1"/><ci id="S0.Ex2.m2.1.25.1.2.cmml" xref="S0.Ex2.m2.1.25.1.2">β</ci><ci id="S0.Ex2.m2.1.25.1.3.cmml" xref="S0.Ex2.m2.1.25.1.3">ϕ</ci><ci id="S0.Ex2.m2.1.25.1.5.cmml" xref="S0.Ex2.m2.1.25.1.5">r</ci></apply></apply></apply><apply id="S0.Ex2.m2.1.30.2.2.2.2.4.cmml" xref="S0.Ex2.m2.1.30.2.2.2.2.4"><csymbol cd="ambiguous" id="S0.Ex2.m2.1.30.2.2.2.2.4.1.cmml">superscript</csymbol><ci id="S0.Ex2.m2.1.26.cmml" xref="S0.Ex2.m2.1.26">r</ci><cn type="integer" id="S0.Ex2.m2.1.27.1.cmml" xref="S0.Ex2.m2.1.27.1">2</cn></apply><ci id="S0.Ex2.m2.1.28.cmml" xref="S0.Ex2.m2.1.28">d</ci><ci id="S0.Ex2.m2.1.29.cmml" xref="S0.Ex2.m2.1.29">r</ci></apply></apply></apply></apply></apply></annotation-xml><annotation id="S0.Ex2.m2.1b" encoding="application/x-tex" xref="S0.Ex2.m2.1.cmml">\displaystyle=\left.\left({{e}^{{-\beta\phi(r)}}}-1\right)\frac{4\pi}{3}{{r}^{%
{3}}}\right|_{{0}}^{{\infty}}+\frac{4\pi\beta}{3}\int_{{0}}^{{\infty}}{{}}%
\left(\frac{\partial\phi}{\partial r}r\right){{e}^{{-\beta\phi(r)}}}{{r}^{{2}}%
}dr</annotation></semantics></math></td>
<td class="eqpad"/></tr>
</table>