Editing Ideal gas partition function

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for a system of ''N'' identical particles each of mass ''m'' is given by
for a system of ''N'' identical particles each of mass ''m'' is given by


:<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 dp^N dr^N \exp \left[ - \frac{H(p^N, r^N)}{k_B T}\right]</math>


where ''h'' is [[Planck constant |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]]
When the particles are distinguishable then the factor ''N!'' disappears. <math>H(p^N, r^N)</math> is the [[Hamiltonian]]
(Sir William Rowan Hamilton 1805-1865 Ireland)
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.
The Hamiltonian can be written as the sum of the kinetic and the potential energies of the system as follows
The Hamiltonian can be written as the sum of the kinetic and the potential energies of the system as follows


:<math>H({\mathbf p}^N, {\mathbf r}^N)= \sum_{i=1}^N \frac{|{\mathbf p}_i |^2}{2m} + {\mathcal V}({\mathbf r}^N)</math>
:<math>H(p^N, r^N)= \sum_{i=1}^N \frac{|p_i |^2}{2m} + V(r^N)</math>


Thus we have  
Thus we have  


:<math>Q_{NVT}=\frac{1}{N!}\frac{1}{h^{3N}}\int d{\mathbf p}^N \exp \left[ - \frac{|{\mathbf p}_i |^2}{2mk_B T}\right]
:<math>Q_{NVT}=\frac{1}{N!}\frac{1}{h^{3N}}\int dp^N \exp \left[ - \frac{|p_i |^2}{2mk_B T}\right]
\int  d{\mathbf r}^N  \exp \left[ - \frac{{\mathcal V}({\mathbf r}^N)} {k_B T}\right]</math>
\int  dr^N  \exp \left[ - \frac{V(r^N)} {k_B T}\right]</math>


This separation is only possible if <math>{\mathcal V}({\mathbf r}^N)</math> is independent of velocity (as is generally the case).
This separation is only possible if <math>V(r^N)</math> is independent of velocity (as is generally the case).
The momentum integral can be solved analytically:
The momentum integral can be solved analytically:


:<math>\int d{\mathbf p}^N \exp \left[ - \frac{|{\mathbf p} |^2}{2mk_B T}\right]=(2 \pi m k_B T)^{3N/2}</math>
:<math>\int dp^N \exp \left[ - \frac{|p |^2}{2mk_B T}\right]=(2 \pi m k_b T)^{3N/2}</math>


Thus we have  
Thus we have  


:<math>Q_{NVT}=\frac{1}{N!} \frac{1}{h^{3N}} \left( 2 \pi m k_B T\right)^{3N/2}
:<math>Q_{NVT}=\frac{1}{N!} \frac{1}{h^{3N}} \left( 2 \pi m k_B T\right)^{3N/2}
\int  d{\mathbf r}^N  \exp \left[ - \frac{{\mathcal V}({\mathbf r}^N)} {k_B T}\right]</math>
\int  dr^N  \exp \left[ - \frac{V(r^N)} {k_B T}\right]</math>




The integral over positions is known as the  
The integral over positions is known as the ''configuration integral'', <math>Z_{NVT}</math>
[[#configintegral|configuration integral]],  
<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  dr^N  \exp \left[ - \frac{V(r^N)} {k_B T}\right]</math>


In an [[ideal gas]] there are no interactions between particles so <math>{\mathcal V}({\mathbf r}^N)=0</math>.
In an [[ideal gas]] there are no interactions between particles so <math>V(r^N)=0</math>
Thus <math>\exp(-{\mathcal V}({\mathbf r}^N)/k_B T)=1</math> for every gas particle.
Thus <math>\exp(-V(r^N)/k_B T)=1</math> for every gas particle.
The integral of 1 over the coordinates of each atom is equal to the volume so for ''N'' particles
The integral of 1 over the coordinates of each atom is equal to the volume so for ''N'' particles
the ''configuration integral'' is given by <math>V^N</math> where ''V'' is the volume.
the ''configuration integral'' is given by <math>V^N</math> where ''V'' is the volume.
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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 (Eq. 4-12 in <ref>Terrell L. Hill "An Introduction to Statistical Thermodynamics" (1960) ISBN 0486652424 </ref>)
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= \frac{V}{\Lambda^{3}}</math>
:<math>q= \frac{V}{\Lambda^{3}}</math>
is the single particle translational partition function.
is the single particle translational partition function.


Thus one can now write the partition function for a real system can be built up from
Thus one can now write the partition function for a real system can be built up from
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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>
==References==
<references/>
==External links==
*<span id="configintegral"></span> [http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 Configuration integral page on VQWiki]
[[Category:Ideal gas]]
[[Category:Ideal gas]]
[[Category:Statistical mechanics]]
[[Category:Statistical mechanics]]
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