Semi-grand ensembles: Difference between revisions

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== General features ==
== General features ==
Semi-grand ensembles are used in Monte Carlo simulation of mixtures.
Semi-grand ensembles are used in Monte Carlo simulation of mixtures. In these ensembles the total number of molecules is fixed, but the composition can change.
 
In these ensembles the total number of molecules is fixed, but the composition can change.


== Canonical ensemble: fixed volume, temperature and number(s) of molecules ==
== Canonical ensemble: fixed volume, temperature and number(s) of molecules ==

Revision as of 17:49, 5 March 2007

General features

Semi-grand ensembles are used in Monte Carlo simulation of mixtures. In these ensembles the total number of molecules is fixed, but the composition can change.

Canonical ensemble: fixed volume, temperature and number(s) of molecules

We shall consider a system consisting of c components;. In the canonical ensemble, the differential equation energy for the Helmholtz energy function can be written as:

,

where:

Semi-grand ensemble at fixed volume and temperature

Consider now that we wish to consider a system with fixed total number of particles,

;

but the composition can change, from thermodynamic considerations one can apply a Legendre transform [HAVE TO CHECK ACCURACY] to the differential equation written above in terms of .

  • Consider the variable change i.e.:



or,

where .

  • Now considering the thermodynamical potential:

Fixed pressure and temperature

In the Isothermal-Isobaric ensemble: one can write:

where:

Fixed pressure and temperature: Semi-grand ensemble

Following the procedure described above one can write:

,

where the new thermodynamical Potential is given by:

Fixed pressure and temperature: Semi-grand ensemble: Partition function

TO BE CONTINUED SOON