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* Page found: Entropie und Ökologie (eq math.2466.17)

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TeX (original user input): 

\begin{align}

& \Rightarrow {{S}_{mix}}={{S}_{nach}}-{{S}_{vor}}=-k\sum\limits_{i=1}^{3}{{}}{{N}_{i}}\ln \frac{{{N}_{i}}}{N}=-k\left( \sum\limits_{i=1}^{3}{{}}{{N}_{i}}\ln {{N}_{i}}-N\ln N \right)\approx -k\left( \sum\limits_{i=1}^{3}{{}}\ln {{N}_{i}}!-\ln N! \right) \\

& =k\ln \left( \frac{N!}{{{N}_{g}}!{{N}_{m}}!{{N}_{J}}!} \right) \\

& \frac{N!}{{{N}_{g}}!{{N}_{m}}!{{N}_{J}}!}=\Omega  \\

\end{align}

TeX (checked): 

{\begin{aligned}&\Rightarrow {{S}_{mix}}={{S}_{nach}}-{{S}_{vor}}=-k\sum \limits _{i=1}^{3}{}{{N}_{i}}\ln {\frac {{N}_{i}}{N}}=-k\left(\sum \limits _{i=1}^{3}{}{{N}_{i}}\ln {{N}_{i}}-N\ln N\right)\approx -k\left(\sum \limits _{i=1}^{3}{}\ln {{N}_{i}}!-\ln N!\right)\\&=k\ln \left({\frac {N!}{{{N}_{g}}!{{N}_{m}}!{{N}_{J}}!}}\right)\\&{\frac {N!}{{{N}_{g}}!{{N}_{m}}!{{N}_{J}}!}}=\Omega \\\end{aligned}}

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⇒Smix=Snach−Svor=−k∑i=13Niln⁡NiN=−k(∑i=13Niln⁡Ni−Nln⁡N)≈−k(∑i=13ln⁡Ni!−ln⁡N!)=kln⁡(N!Ng!Nm!NJ!)N!Ng!Nm!NJ!=Ω
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Calculated based on the variables occurring on the entire Entropie und Ökologie page

Identifiers

  • Smix
  • Snach
  • Svor
  • k
  • i
  • Ni
  • Ni
  • N
  • k
  • i
  • Ni
  • Ni
  • N
  • N
  • k
  • i
  • Ni
  • N
  • k
  • N
  • Ng
  • Nm
  • NJ
  • N
  • Ng
  • Nm
  • NJ
  • Ω

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Abgerufen von „https://wiki.physikerwelt.de/wiki/Spezial:Formelinformation“