Grand canonical ensemble: Difference between revisions
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The '''grand-canonical ensemble''' is particularly well suited to simulation studies of adsorption. | The '''grand-canonical ensemble''' is particularly well suited to simulation studies of adsorption. | ||
== Ensemble variables == | == Ensemble variables == | ||
* [[Chemical potential]], <math> \left. \mu \right. </math> | * [[Chemical potential]], <math> \left. \mu \right. </math> | ||
* Volume, <math> \left. V \right. </math> | * Volume, <math> \left. V \right. </math> | ||
* [[Temperature]], <math> \left. T \right. </math> | |||
* Temperature, <math> \left. T \right. </math> | |||
== Grand canonical partition function == | == Grand canonical partition function == | ||
The classical grand canonical partition function for a one-component system in a three-dimensional space is given by: | The classical grand canonical partition function for a one-component system in a three-dimensional space is given by: | ||
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* ''N'' is the number of particles | * ''N'' is the number of particles | ||
* <math> \left. \Lambda \right. </math> is the [[de Broglie thermal wavelength]] (which depends on the temperature) | * <math> \left. \Lambda \right. </math> is the [[de Broglie thermal wavelength]] (which depends on the temperature) | ||
* <math> \beta = \frac{1}{k_B T} </math>, with <math> k_B </math> being the [[Boltzmann constant]] | * <math> \beta = \frac{1}{k_B T} </math>, with <math> k_B </math> being the [[Boltzmann constant]] | ||
* ''U'' is the potential energy, which depends on the coordinates of the particles (and on the [[models | interaction model]]) | * ''U'' is the potential energy, which depends on the coordinates of the particles (and on the [[models | interaction model]]) | ||
* <math> \left( R^*\right)^{3N} </math> represent the <math>3N</math> position coordinates of the particles (reduced with the system size): i.e. <math> \int d (R^*)^{3N} = 1 </math> | * <math> \left( R^*\right)^{3N} </math> represent the <math>3N</math> position coordinates of the particles (reduced with the system size): i.e. <math> \int d (R^*)^{3N} = 1 </math> | ||
== Helmholtz energy and partition function == | == Helmholtz energy and partition function == | ||
The corresponding thermodynamic potential, the '''grand potential''', <math>\Omega</math>, | The corresponding thermodynamic potential, the '''grand potential''', <math>\Omega</math>, | ||
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:<math> \left. p V = k_B T \log Q_{\mu V T } \right. </math> | :<math> \left. p V = k_B T \log Q_{\mu V T } \right. </math> | ||
==See also== | |||
*[[Monte Carlo in the grand-canonical ensemble]] | |||
==References== | |||
[[Category:Statistical mechanics]] | [[Category:Statistical mechanics]] | ||
Revision as of 13:44, 1 April 2008
The grand-canonical ensemble is particularly well suited to simulation studies of adsorption.
Ensemble variables
- Chemical potential, 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 \left. \mu \right. }
- Volume,
- Temperature, 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 \left. T \right. }
Grand canonical partition function
The classical grand canonical partition function for a one-component system in a three-dimensional space 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_{\mu VT} = \sum_{N=0}^{\infty} \frac{ \exp \left[ \beta \mu N \right] V^N}{N! \Lambda^{3N} } \int d (R^*)^{3N} \exp \left[ - \beta U \left( V, (R^*)^{3N} \right) \right] }
where:
- N is the number of particles
- 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 \left. \Lambda \right. } is the de Broglie thermal wavelength (which depends on the temperature)
- 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 \beta = \frac{1}{k_B T} } , with 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 } being the Boltzmann constant
- U is the potential energy, which depends on the coordinates of the particles (and on the interaction model)
- 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 \left( R^*\right)^{3N} } represent the 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 3N} position coordinates of the particles (reduced with the system size): 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 \int d (R^*)^{3N} = 1 }
Helmholtz energy and partition function
The corresponding thermodynamic potential, the grand potential, 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 \Omega} , for the aforementioned grand canonical partition function is:
- 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 \Omega = \left. A - \mu N \right. } ,
where A is the Helmholtz energy function. Using the relation
- 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 \left.U\right.=TS -PV + \mu N}
one arrives at
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 \left. p V = k_B T \log Q_{\mu V T } \right. }