Editing Ideal gas partition function
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:<math>Q_{NVT}=\frac{1}{N!}\frac{1}{h^{3N}}\int\int d{\mathbf p}^N d{\mathbf r}^N \exp \left[ - \frac{H({\mathbf p}^N,{\mathbf r}^N)}{k_B T}\right]</math> | :<math>Q_{NVT}=\frac{1}{N!}\frac{1}{h^{3N}}\int\int d{\mathbf p}^N d{\mathbf r}^N \exp \left[ - \frac{H({\mathbf p}^N,{\mathbf r}^N)}{k_B T}\right]</math> | ||
where ''h'' is [[ | where ''h'' is [[Planck's constant]], ''T'' is the [[temperature]] and <math>k_B</math> is the [[Boltzmann constant]]. When the particles are distinguishable then the factor ''N!'' disappears. <math>H(p^N, r^N)</math> is the [[Hamiltonian]] | ||
corresponding to the total energy of the system. | corresponding to the total energy of the system. | ||
''H'' is a function of the ''3N'' positions and ''3N'' momenta of the particles in the system. | ''H'' is a function of the ''3N'' positions and ''3N'' momenta of the particles in the system. | ||
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The integral over positions is known as the | The integral over positions is known as the ''configuration integral'', <math>Z_{NVT}</math> (from the German ''Zustandssumme'' meaning "sum over states") | ||
<math>Z_{NVT}</math> (from the German ''Zustandssumme'' meaning "sum over states") | |||
:<math>Z_{NVT}= \int d{\mathbf r}^N \exp \left[ - \frac{{\mathcal V}({\mathbf r}^N)} {k_B T}\right]</math> | :<math>Z_{NVT}= \int d{\mathbf r}^N \exp \left[ - \frac{{\mathcal V}({\mathbf r}^N)} {k_B T}\right]</math> | ||
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:<math>\Lambda = \sqrt{h^2 / 2 \pi m k_B T}</math> | :<math>\Lambda = \sqrt{h^2 / 2 \pi m k_B T}</math> | ||
one arrives at | one arrives at | ||
:<math>Q_{NVT}=\frac{1}{N!} \left( \frac{V}{\Lambda^{3}}\right)^N = \frac{q^N}{N!}</math> | :<math>Q_{NVT}=\frac{1}{N!} \left( \frac{V}{\Lambda^{3}}\right)^N = \frac{q^N}{N!}</math> | ||
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:<math>Q_{NVT}=Q_{NVT}^{\rm ideal} ~Q_{NVT}^{\rm excess}</math> | :<math>Q_{NVT}=Q_{NVT}^{\rm ideal} ~Q_{NVT}^{\rm excess}</math> | ||
[[Category:Ideal gas]] | [[Category:Ideal gas]] | ||
[[Category:Statistical mechanics]] | [[Category:Statistical mechanics]] |