Ideal gas Helmholtz energy function: Difference between revisions

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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>\left.A\right.=-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>
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:<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:Ideal gas]]
[[Category:Statistical mechanics]]
[[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.