Ideal gas partition function: Difference between revisions

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[[Canonical ensemble]] partition function, ''Q'',
The [[canonical ensemble]] [[partition function]], ''Q'',
for a system of ''N'' identical particles each of mass ''m''
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 dp^N dr^N \exp \left[ - \frac{H(p^N, 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>


When the particles are distinguishable then the factor ''N!'' disappears. <math>H(p^N, r^N)</math> is the [[Hamiltonian]]
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]]
(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(p^N, r^N)= \sum_{i=1}^N \frac{|p_i |^2}{2m} + V(r^N)</math>
:<math>H({\mathbf p}^N, {\mathbf r}^N)= \sum_{i=1}^N \frac{|{\mathbf p}_i |^2}{2m} + {\mathcal V}({\mathbf r}^N)</math>


Thus we have  
Thus we have  


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


This separation is only possible if <math>V(r^N)</math> is independent of velocity (as is generally the case).
This separation is only possible if <math>{\mathcal V}({\mathbf 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 dp^N \exp \left[ - \frac{|p |^2}{2mk_B T}\right]=(2 \pi m k_b T)^{3N/2}</math>
:<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>


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  dr^N  \exp \left[ - \frac{V(r^N)} {k_B T}\right]</math>
\int  d{\mathbf r}^N  \exp \left[ - \frac{{\mathcal V}({\mathbf r}^N)} {k_B T}\right]</math>




The integral over positions is known as the ''configuration integral'', <math>Z_{NVT}</math>
The integral over positions is known as the  
[[#configintegral|configuration integral]],  
<math>Z_{NVT}</math> (from the German ''Zustandssumme'' meaning "sum over states")


:<math>Z_{NVT}= \int  dr^N  \exp \left[ - \frac{V(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>


In an ideal gas there are no interactions between particles so <math>V(r^N)=0</math>
In an [[ideal gas]] there are no interactions between particles so <math>{\mathcal V}({\mathbf r}^N)=0</math>.
Thus <math>\exp(-V(r^N)/k_B T)=1</math> for every gas particle.
Thus <math>\exp(-{\mathcal V}({\mathbf 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
one arrives at (Eq. 4-12 in <ref>Terrell L. Hill "An Introduction to Statistical Thermodynamics" (1960) ISBN 0486652424 </ref>)
 
:<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 = \frac{q^N}{N!}</math>
where
:<math>q= \frac{V}{\Lambda^{3}}</math>
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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particle interactions, ''i.e.''
particle interactions, ''i.e.''


:<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]]

Latest revision as of 12:13, 1 September 2011

The canonical ensemble partition function, Q, for a system of N identical particles each of mass m is given by

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 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]}

where h is Planck's constant, T is the temperature 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. When the particles are distinguishable then the factor N! disappears. 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 H(p^N, r^N)} is the Hamiltonian 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. The Hamiltonian can be written as the sum of the kinetic and the potential energies of the system as follows

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

Thus we have

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 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] \int d{\mathbf r}^N \exp \left[ - \frac{{\mathcal V}({\mathbf r}^N)} {k_B T}\right]}

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

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

Thus we have

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 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]}


The integral over positions is known as the configuration integral, 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 Z_{NVT}} (from the German Zustandssumme meaning "sum over states")

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 Z_{NVT}= \int d{\mathbf r}^N \exp \left[ - \frac{{\mathcal V}({\mathbf r}^N)} {k_B T}\right]}

In an ideal gas there are no interactions between particles so 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 {\mathcal V}({\mathbf r}^N)=0} . Thus 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 \exp(-{\mathcal V}({\mathbf r}^N)/k_B T)=1} for every gas particle. 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 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 V^N} where V is the volume. Thus we have

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 Q_{NVT}=\frac{V^N}{N!}\left( \frac{2 \pi m k_B T}{h^2}\right)^{3N/2}}

If we define the de Broglie thermal wavelength as 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} 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 = \sqrt{h^2 / 2 \pi m k_B T}}

one arrives at (Eq. 4-12 in [1])

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 Q_{NVT}=\frac{1}{N!} \left( \frac{V}{\Lambda^{3}}\right)^N = \frac{q^N}{N!}}

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 q= \frac{V}{\Lambda^{3}}}

is the single particle translational partition function.

Thus one can now write the partition function for a real system can be built up from the contribution of the ideal system (the momenta) and a contribution due to particle interactions, i.e.

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 Q_{NVT}=Q_{NVT}^{\rm ideal} ~Q_{NVT}^{\rm excess}}

References[edit]

  1. ↑ Terrell L. Hill "An Introduction to Statistical Thermodynamics" (1960) ISBN 0486652424

External links[edit]

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