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# Difference between revisions of "Ideal gas Helmholtz energy function"

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

for the canonical ensemble partition function for an ideal gas, and

for the Helmholtz energy function, one has

using Stirling's approximation

one arrives at

where is the de Broglie thermal wavelength and is the Boltzmann constant.