Grand canonical ensemble: Difference between revisions
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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: | ||
:<math> Q_{\mu VT} = \sum_{N=0}^{\infty} \frac{ \exp \left[ \beta \mu N \right] | :<math> Q_{\mu VT} = \sum_{N=0}^{\infty} \frac{ \exp \left[ \beta \mu N \right]}{N! \Lambda^{3N} } \int d (R^*)^{3N} \exp \left[ - \beta U \left( V, (R^*)^{3N} \right) \right] </math> | ||
where: | where: | ||
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* ''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>, | ||
Revision as of 12:08, 30 August 2011
The grand-canonical ensemble is particularly well suited to simulation studies of adsorption.
Ensemble variables
- Chemical potential,
- Volume,
- Temperature,
Grand canonical partition function
The classical grand canonical partition function for a one-component system in a three-dimensional space is given by:
![{\displaystyle Q_{\mu VT}=\sum _{N=0}^{\infty }{\frac {\exp \left[\beta \mu N\right]}{N!\Lambda ^{3N}}}\int d(R^{*})^{3N}\exp \left[-\beta U\left(V,(R^{*})^{3N}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc6c3dd6a6cec7a4333b65529f52958a37e718c1)
where:
- N is the number of particles
- is the de Broglie thermal wavelength (which depends on the temperature)
- , with being the Boltzmann constant
- U is the potential energy, which depends on the coordinates of the particles (and on the interaction model)
- represent the position coordinates of the particles (reduced with the system size): i.e.
Helmholtz energy and partition function
The corresponding thermodynamic potential, the grand potential, , for the aforementioned grand canonical partition function is:
- ,
where A is the Helmholtz energy function. Using the relation
one arrives at
i.e.:
