Ideal gas partition function - Revision history

Carl McBride: Added a reference

2011-09-01T11:13:52Z

Added a reference

← Older revision		Revision as of 13:13, 1 September 2011
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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 = \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>
	==References==		==References==
			<references/>
	==External links==		==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]		*<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]]

Nice and Tidy: Placed external links at foot of page.

2008-12-11T18:02:02Z

Placed external links at foot of page.

← Older revision		Revision as of 20:02, 11 December 2008
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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==
	==External links==		==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]		*<span id="configintegral"></span> [http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 Configuration integral page on VQWiki]
	~~==References==~~
	[[Category:Ideal gas]]		[[Category:Ideal gas]]
	[[Category:Statistical mechanics]]		[[Category:Statistical mechanics]]

Carl McBride at 15:10, 12 February 2008

2008-02-12T15:10:21Z

← Older revision		Revision as of 17:10, 12 February 2008
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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 [[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]]		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]]
	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.

Carl McBride: Slight tidy up

2008-01-23T09:34:43Z

Slight tidy up

← Older revision		Revision as of 11:34, 23 January 2008
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	The integral over positions is known as the		The integral over positions is known as the
	[[#configintegral\|''configuration integral'']],		[[#configintegral\|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")

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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>

			==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]
	==References==		==References==
	* <span id="configintegral"></span> [http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 Configuration integral (statistical mechanics)]

	[[Category:Ideal gas]]		[[Category:Ideal gas]]
	[[Category:Statistical mechanics]]		[[Category:Statistical mechanics]]

Vql at 17:22, 22 January 2008

2008-01-22T17:22:16Z

← Older revision		Revision as of 19:22, 22 January 2008
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	The integral over positions is known as the ''configuration integral'', <math>Z_{NVT}</math> (from the German ''Zustandssumme'' meaning "sum over states")		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 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>Q_{NVT}=Q_{NVT}^{\rm ideal} ~Q_{NVT}^{\rm excess}</math>		:<math>Q_{NVT}=Q_{NVT}^{\rm ideal} ~Q_{NVT}^{\rm excess}</math>

			==References==
			* <span id="configintegral"></span> [http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 Configuration integral (statistical mechanics)]

	[[Category:Ideal gas]]		[[Category:Ideal gas]]
	[[Category:Statistical mechanics]]		[[Category:Statistical mechanics]]

Nice and Tidy at 14:12, 10 July 2007

2007-07-10T14:12:21Z

← Older revision		Revision as of 16:12, 10 July 2007
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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 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>

	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]]		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]]
Line 9:		Line 9:
	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} + \~~Phi~~(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{\~~Phi~~(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{\~~Phi~~(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> (from the German ''Zustandssumme'' meaning "sum over states")		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}= \int dr^N \exp \left[ - \frac{\~~Phi~~(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>\~~Phi~~(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(-\~~Phi~~(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.

Carl McBride at 15:10, 26 June 2007

2007-06-26T15:10:17Z

← Older revision		Revision as of 17:10, 26 June 2007
Line 27:		Line 27:


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

Carl McBride at 15:05, 26 June 2007

2007-06-26T15:05:06Z

← Older revision		Revision as of 17:05, 26 June 2007
Line 4:		Line 4:
	:<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 dp^N dr^N \exp \left[ - \frac{H(p^N, 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'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.

Carl McBride at 14:50, 26 June 2007

2007-06-26T14:50:09Z

← Older revision		Revision as of 16:50, 26 June 2007
Line 1:		Line 1:
	The canonical [[partition function]], ''Q'',		The [[canonical ensemble]] [[partition function]], ''Q'',
	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

Carl McBride at 14:45, 26 June 2007

2007-06-26T14:45:08Z

← Older revision		Revision as of 16:45, 26 June 2007
Line 10:		Line 10:
	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(p^N, r^N)= \sum_{i=1}^N \frac{\|p_i \|^2}{2m} + \Phi(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 dp^N \exp \left[ - \frac{\|p_i \|^2}{2mk_B T}\right]
	\int dr^N \exp \left[ - \frac{V(r^N)} {k_B T}\right]</math>		\int dr^N \exp \left[ - \frac{\Phi(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>V(r^N)</math> is independent of velocity (as is generally the case).
Line 25:		Line 25:

	:<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 dr^N \exp \left[ - \frac{\Phi(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 ''configuration integral'', <math>Z_{NVT}</math>

	:<math>Z_{NVT}= \int dr^N \exp \left[ - \frac{V(r^N)} {k_B T}\right]</math>		:<math>Z_{NVT}= \int dr^N \exp \left[ - \frac{\Phi(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>\Phi(r^N)=0</math>
	Thus <math>\exp(-V(r^N)/k_B T)=1</math> for every gas particle.		Thus <math>\exp(-\Phi(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.