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   Reale Gase Eq: 39
TeX (as stored in database):
\begin{align}
 
& f={{e}^{-\beta \phi }}-1\approx \left\{ \begin{matrix}
 
-1\quad f\ddot{u}r\,r<d  \\
 
-\beta \phi \quad f\ddot{u}r\,r>d  \\
 
\end{matrix} \right. \\
 
& \Rightarrow B=-\frac{2\pi }{3}{{d}^{3}}-\frac{2\pi }{kT}\int_{d}^{\infty }{{}}dr{{r}^{2}}\phi  \\
 
& \phi <0 \\
 
& \Rightarrow B=-\frac{2\pi }{3}{{d}^{3}}-\frac{2\pi }{kT}\int_{d}^{\infty }{{}}dr{{r}^{2}}\phi =-\frac{b}{{{N}_{A}}}+\frac{a}{kT{{N}_{A}}^{2}} \\
 
\end{align}
MathML (20.732 KB / 3.248 KB) :
f=e-βϕ-1{-1fu¨rr<d-βϕfu¨rr>dfragmentsfsuperscripteβϕ1fragmentsnormal-{<-1⁢f¨urrd>-⁢βϕ⁢f¨urrd\displaystyle f={{e}^{{-\beta\phi}}}-1\approx\left\{\begin{matrix}\par -1\quad f% \ddot{u}r\,r<d\\ \par -\beta\phi\quad f\ddot{u}r\,r>d\\ \par \end{matrix}\right.
B=-2π3d3-2πkTddrr2ϕfragmentsnormal-⇒B2π3superscriptd32πkTsuperscriptsubscriptddrsuperscriptr2ϕ\displaystyle\Rightarrow B=-\frac{2\pi}{3}{{d}^{{3}}}-\frac{2\pi}{kT}\int_{{d}% }^{{\infty}}{{}}dr{{r}^{{2}}}\phi
ϕ<0ϕ0\displaystyle\phi<0
B=-2π3d3-2πkTddrr2ϕ=-bNA+akTNA2fragmentsnormal-⇒B2π3superscriptd32πkTsuperscriptsubscriptddrsuperscriptr2ϕbsubscriptNAakTsuperscriptsubscriptNA2\displaystyle\Rightarrow B=-\frac{2\pi}{3}{{d}^{{3}}}-\frac{2\pi}{kT}\int_{{d}% }^{{\infty}}{{}}dr{{r}^{{2}}}\phi=-\frac{b}{{{N}_{{A}}}}+\frac{a}{kT{{N}_{{A}}% }^{{2}}}

SVG (0 B / 8 B) :
PNG (0 B / 8 B) :
Hash : 948ac4b2e0710af7e3961ffef5a22ca6

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  • mo ¨ (2/6/301)
  • mo (1/9/456)
  • mo (2/4/661)
  • mo { (1/2/716)
  • mi (2/21/1025)
  • mo < (2/19/1435)
  • mi π (4/16/1435)
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  • mi d (8/65/6342)
  • mi N (2/85/6684)
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  • mo = (4/144/18270)
  • mo (16/419/59187)

MathML


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<td colspan="1" class="td left" style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\displaystyle f={{e}^{{-\beta\phi}}}-1\approx\left\{\begin{matrix}\par&#10;-1\quad&#10;f%&#10;\ddot{u}r\,r&lt;d\\&#10;\par&#10;-\beta\phi\quad f\ddot{u}r\,r&gt;d\\&#10;\par&#10;\end{matrix}\right." id="S0.Ex1.m2.1" display="inline" xref="S0.Ex1.m2.1.cmml"><semantics id="S0.Ex1.m2.1a" xref="S0.Ex1.m2.1.cmml"><mrow id="S0.Ex1.m2.1b" xref="S0.Ex1.m2.1.cmml"><mi id="S0.Ex1.m2.1.1">f</mi><mo id="S0.Ex1.m2.1.2" xref="S0.Ex1.m2.1.2.cmml">=</mo><msup id="S0.Ex1.m2.1.11" xref="S0.Ex1.m2.1.11.cmml"><mi id="S0.Ex1.m2.1.3" xref="S0.Ex1.m2.1.3.cmml">e</mi><mrow id="S0.Ex1.m2.1.4.1" xref="S0.Ex1.m2.1.4.1.cmml"><mo id="S0.Ex1.m2.1.4.1.1" xref="S0.Ex1.m2.1.4.1.1.cmml">-</mo><mrow id="S0.Ex1.m2.1.4.1.4" xref="S0.Ex1.m2.1.4.1.4.cmml"><mi id="S0.Ex1.m2.1.4.1.2" xref="S0.Ex1.m2.1.4.1.2.cmml">β</mi><mo id="S0.Ex1.m2.1.4.1.4.1" xref="S0.Ex1.m2.1.4.1.4.1.cmml"></mo><mi id="S0.Ex1.m2.1.4.1.3" 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<td class="eqpad"/></tr>
<tr id="S0.Ex2" class="equation baseline">
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xref="S0.Ex2.m2.1">d</csymbol><csymbol cd="unknown" id="S0.Ex2.m2.1e.cmml" xref="S0.Ex2.m2.1">r</csymbol><apply id="S0.Ex2.m2.1.20.cmml" xref="S0.Ex2.m2.1.20"><csymbol cd="ambiguous" id="S0.Ex2.m2.1.20.1.cmml">superscript</csymbol><ci id="S0.Ex2.m2.1.15.cmml" xref="S0.Ex2.m2.1.15">r</ci><cn type="integer" id="S0.Ex2.m2.1.16.1.cmml" xref="S0.Ex2.m2.1.16.1">2</cn></apply><csymbol cd="unknown" id="S0.Ex2.m2.1f.cmml" xref="S0.Ex2.m2.1">ϕ</csymbol></cerror></annotation-xml><annotation id="S0.Ex2.m2.1c" encoding="application/x-tex" xref="S0.Ex2.m2.1.cmml">\displaystyle\Rightarrow B=-\frac{2\pi}{3}{{d}^{{3}}}-\frac{2\pi}{kT}\int_{{d}%
}^{{\infty}}{{}}dr{{r}^{{2}}}\phi</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\phi&lt;0" 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.m2.1.4" xref="S0.Ex3.m2.1.4.cmml"><mi id="S0.Ex3.m2.1.1" xref="S0.Ex3.m2.1.1.cmml">ϕ</mi><mo id="S0.Ex3.m2.1.2" xref="S0.Ex3.m2.1.2.cmml">&lt;</mo><mn id="S0.Ex3.m2.1.3" xref="S0.Ex3.m2.1.3.cmml">0</mn></mrow><annotation-xml id="S0.Ex3.m2.1.cmml" encoding="MathML-Content" xref="S0.Ex3.m2.1"><apply id="S0.Ex3.m2.1.4.cmml" xref="S0.Ex3.m2.1.4"><lt id="S0.Ex3.m2.1.2.cmml" xref="S0.Ex3.m2.1.2"/><ci id="S0.Ex3.m2.1.1.cmml" xref="S0.Ex3.m2.1.1">ϕ</ci><cn type="integer" id="S0.Ex3.m2.1.3.cmml" xref="S0.Ex3.m2.1.3">0</cn></apply></annotation-xml><annotation id="S0.Ex3.m2.1b" encoding="application/x-tex" xref="S0.Ex3.m2.1.cmml">\displaystyle\phi&lt;0</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\Rightarrow B=-\frac{2\pi}{3}{{d}^{{3}}}-\frac{2\pi}{kT}\int_{{d}%&#10;}^{{\infty}}{{}}dr{{r}^{{2}}}\phi=-\frac{b}{{{N}_{{A}}}}+\frac{a}{kT{{N}_{{A}}%&#10;}^{{2}}}" 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.m2.1b" xref="S0.Ex4.m2.1.cmml"><mo id="S0.Ex4.m2.1.1" xref="S0.Ex4.m2.1.1.cmml"></mo><mi id="S0.Ex4.m2.1.2">B</mi><mo id="S0.Ex4.m2.1.3" xref="S0.Ex4.m2.1.3.cmml">=</mo><mo id="S0.Ex4.m2.1.4" xref="S0.Ex4.m2.1.4.cmml">-</mo><mstyle displaystyle="true" id="S0.Ex4.m2.1.5" xref="S0.Ex4.m2.1.5.cmml"><mfrac id="S0.Ex4.m2.1.5a" xref="S0.Ex4.m2.1.5.cmml"><mrow id="S0.Ex4.m2.1.5.2" xref="S0.Ex4.m2.1.5.2.cmml"><mn id="S0.Ex4.m2.1.5.2.1" xref="S0.Ex4.m2.1.5.2.1.cmml">2</mn><mo id="S0.Ex4.m2.1.5.2.3" xref="S0.Ex4.m2.1.5.2.3.cmml"></mo><mi id="S0.Ex4.m2.1.5.2.2" xref="S0.Ex4.m2.1.5.2.2.cmml">π</mi></mrow><mn id="S0.Ex4.m2.1.5.3" xref="S0.Ex4.m2.1.5.3.cmml">3</mn></mfrac></mstyle><msup id="S0.Ex4.m2.1.23" xref="S0.Ex4.m2.1.23.cmml"><mi id="S0.Ex4.m2.1.6" xref="S0.Ex4.m2.1.6.cmml">d</mi><mn id="S0.Ex4.m2.1.7.1" xref="S0.Ex4.m2.1.7.1.cmml">3</mn></msup><mo id="S0.Ex4.m2.1.8" xref="S0.Ex4.m2.1.8.cmml">-</mo><mstyle displaystyle="true" id="S0.Ex4.m2.1.9" xref="S0.Ex4.m2.1.9.cmml"><mfrac id="S0.Ex4.m2.1.9a" xref="S0.Ex4.m2.1.9.cmml"><mrow id="S0.Ex4.m2.1.9.2" xref="S0.Ex4.m2.1.9.2.cmml"><mn id="S0.Ex4.m2.1.9.2.1" xref="S0.Ex4.m2.1.9.2.1.cmml">2</mn><mo id="S0.Ex4.m2.1.9.2.3" xref="S0.Ex4.m2.1.9.2.3.cmml"></mo><mi id="S0.Ex4.m2.1.9.2.2" xref="S0.Ex4.m2.1.9.2.2.cmml">π</mi></mrow><mrow id="S0.Ex4.m2.1.9.3" xref="S0.Ex4.m2.1.9.3.cmml"><mi id="S0.Ex4.m2.1.9.3.1" xref="S0.Ex4.m2.1.9.3.1.cmml">k</mi><mo id="S0.Ex4.m2.1.9.3.3" xref="S0.Ex4.m2.1.9.3.3.cmml"></mo><mi id="S0.Ex4.m2.1.9.3.2" xref="S0.Ex4.m2.1.9.3.2.cmml">T</mi></mrow></mfrac></mstyle><mstyle 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id="S0.Ex4.m2.1.22.3.7a" xref="S0.Ex4.m2.1.22.3.7.cmml"/><none id="S0.Ex4.m2.1.22.3.7b" xref="S0.Ex4.m2.1.22.3.7.cmml"/><mn id="S0.Ex4.m2.1.22.3.5.1" xref="S0.Ex4.m2.1.22.3.5.1.cmml">2</mn></mmultiscripts></mrow></mfrac></mstyle></mrow><annotation-xml id="S0.Ex4.m2.1.cmml" encoding="MathML-Content" xref="S0.Ex4.m2.1"><cerror id="S0.Ex4.m2.1a.cmml" xref="S0.Ex4.m2.1"><csymbol cd="ambiguous" id="S0.Ex4.m2.1b.cmml" xref="S0.Ex4.m2.1">fragments</csymbol><ci id="S0.Ex4.m2.1.1.cmml" xref="S0.Ex4.m2.1.1">normal-⇒</ci><csymbol cd="unknown" id="S0.Ex4.m2.1c.cmml" xref="S0.Ex4.m2.1">B</csymbol><eq id="S0.Ex4.m2.1.3.cmml" xref="S0.Ex4.m2.1.3"/><minus id="S0.Ex4.m2.1.4.cmml" xref="S0.Ex4.m2.1.4"/><apply id="S0.Ex4.m2.1.5.cmml" xref="S0.Ex4.m2.1.5"><divide id="S0.Ex4.m2.1.5.1.cmml"/><apply id="S0.Ex4.m2.1.5.2.cmml" xref="S0.Ex4.m2.1.5.2"><times id="S0.Ex4.m2.1.5.2.3.cmml" xref="S0.Ex4.m2.1.5.2.3"/><cn type="integer" id="S0.Ex4.m2.1.5.2.1.cmml" xref="S0.Ex4.m2.1.5.2.1">2</cn><ci 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xref="S0.Ex4.m2.1.22.3.2">T</ci><apply id="S0.Ex4.m2.1.22.3.7.cmml" xref="S0.Ex4.m2.1.22.3.7"><csymbol cd="ambiguous" id="S0.Ex4.m2.1.22.3.7.1.cmml">superscript</csymbol><apply id="S0.Ex4.m2.1.22.3.7.2.cmml"><csymbol cd="ambiguous" id="S0.Ex4.m2.1.22.3.7.2.1.cmml">subscript</csymbol><ci id="S0.Ex4.m2.1.22.3.3.cmml" xref="S0.Ex4.m2.1.22.3.3">N</ci><ci id="S0.Ex4.m2.1.22.3.4.1.cmml" xref="S0.Ex4.m2.1.22.3.4.1">A</ci></apply><cn type="integer" id="S0.Ex4.m2.1.22.3.5.1.cmml" xref="S0.Ex4.m2.1.22.3.5.1">2</cn></apply></apply></apply></cerror></annotation-xml><annotation id="S0.Ex4.m2.1c" encoding="application/x-tex" xref="S0.Ex4.m2.1.cmml">\displaystyle\Rightarrow B=-\frac{2\pi}{3}{{d}^{{3}}}-\frac{2\pi}{kT}\int_{{d}%
}^{{\infty}}{{}}dr{{r}^{{2}}}\phi=-\frac{b}{{{N}_{{A}}}}+\frac{a}{kT{{N}_{{A}}%
}^{{2}}}</annotation></semantics></math></td>
<td class="eqpad"/></tr>
</table>