Ideal gas Helmholtz energy function: Difference between revisions
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Carl McBride (talk | contribs) m (New page: From equations :<math>Q_{NVT}=\frac{1}{N!} \left( \frac{V}{\Lambda^{3}}\right)^N</math> and :<math>A=-k_B T \ln Q_{NVT}</math> one has :<math>A=-k_BT\left(\ln \frac{1}{N!} + N\ln\frac{V}...) |
Carl McBride (talk | contribs) m (defined a couple of terms) |
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From equations | From equations | ||
:<math>Q_{NVT}=\frac{1}{N!} \left( \frac{V}{\Lambda^{3}}\right)^N</math> | :<math>Q_{NVT}=\frac{1}{N!} \left( \frac{V}{\Lambda^{3}}\right)^N</math> | ||
and | for the [[ Ideal gas partition function | canonical ensemble partition function for an ideal gas]], and | ||
:<math>A=-k_B T \ln Q_{NVT}</math> | :<math>\left.A\right.=-k_B T \ln Q_{NVT}</math> | ||
one has | for the [[Helmholtz energy function]], one has | ||
:<math>A=-k_BT\left(\ln \frac{1}{N!} + N\ln\frac{V}{\Lambda^{3}}\right)</math> | :<math>A=-k_BT\left(\ln \frac{1}{N!} + N\ln\frac{V}{\Lambda^{3}}\right)</math> | ||
::<math>=-k_BT\left(-\ln N! + N\ln\frac{VN}{\Lambda^3N}\right)</math> | ::<math>=-k_BT\left(-\ln N! + N\ln\frac{VN}{\Lambda^3N}\right)</math> | ||
| Line 11: | Line 11: | ||
one arrives at | one arrives at | ||
<math>A=Nk_BT\left(\ln \Lambda^3 \rho -1 \right)</math> | :<math>A=Nk_BT\left(\ln \Lambda^3 \rho -1 \right)</math> | ||
where <math>\Lambda</math>is the [[de Broglie thermal wavelength]] and <math>k_B</math> is the [[Boltzmann constant]]. | |||
[[Category:Ideal gas]] | |||
[[Category:Statistical mechanics]] | |||
Latest revision as of 11:19, 4 August 2008
From equations
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle Q_{NVT}={\frac {1}{N!}}\left({\frac {V}{\Lambda ^{3}}}\right)^{N}}
for the canonical ensemble partition function for an ideal gas, and
for the Helmholtz energy function, one has
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A=-k_{B}T\left(\ln {\frac {1}{N!}}+N\ln {\frac {V}{\Lambda ^{3}}}\right)}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =-k_BT\left(-\ln N! + N\ln\frac{VN}{\Lambda^3N}\right)}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =-k_BT\left(-\ln N! + N\ln\frac{N}{\Lambda^3 \rho}\right)}
using Stirling's approximation
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =-k_BT\left( -N\ln N +N + N\ln N - N\ln \Lambda^3 \rho \right)}
one arrives at
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=Nk_BT\left(\ln \Lambda^3 \rho -1 \right)}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Lambda} is the de Broglie thermal wavelength and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_B} is the Boltzmann constant.