Grand canonical ensemble: Difference between revisions
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* <math>N</math> is the number of particles | * <math>N</math> 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 | * <math> \beta </math> is the [[inverse temperature]] | ||
* <math>U</math> is the potential energy, which depends on the coordinates of the particles (and on the [[models | interaction model]]) | * <math>U</math> 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> | ||
Revision as of 12:53, 31 August 2011
The grand-canonical ensemble is for "open" systems, where the number of particles, , can change. It can be viewed as an ensemble of canonical ensembles; there being a canonical ensemble for each value of , and the (weighted) sum over of these canonical ensembles constitutes the grand canonical ensemble. The weighting factor is and is known as the fugacity. The grand-canonical ensemble is particularly well suited to simulation studies of adsorption.
![{\displaystyle \exp \left[\beta \mu \right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ee894e58e3416f71930628e1b50c7ac1aa832e3)
Ensemble variables
- chemical potential,
- volume,
- temperature,
Grand canonical partition function
The grand canonical partition function for a one-component system in a three-dimensional space is given by:
![{\displaystyle \Xi _{\mu VT}=\sum _{N=0}^{\infty }\exp \left[\beta \mu N\right]Q_{NVT}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef904e42cf93b2f11bcec2757385884620f8c7a4)
i.e. for a classical system one has
![{\displaystyle \Xi _{\mu VT}=\sum _{N=0}^{\infty }\exp \left[\beta \mu N\right]{\frac {V^{N}}{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/2918a47c2ede5c1cc4f906029ca17adbff6daac9)
where:
- is the number of particles
- is the de Broglie thermal wavelength (which depends on the temperature)
- is the inverse temperature
- 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.:
See also
References
- Related reading