Semi-grand ensembles: Difference between revisions

From SklogWiki
Jump to navigation Jump to search
mNo edit summary
Line 1: Line 1:
== 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.



Revision as of 17:41, 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 will consider a system with "c" components;. In the Canonical Ensemble, the differential equation energy for the Helmholtz energy function can be written as:

,

where:

  • is the Helmholtz energy function
  • is the Boltzmann constant
  • is the absolute temperature
  • is the internal energy
  • is the pressure
  • is the chemical potential of the species "i"
  • is the number of molecules of the species "i"

Semi-grand ensemble at fixed volume and temperature

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

;

but the composition can change, from the thermodynamics we can apply a Legendre's 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: ensemble we can write:

where:

Fixed pressure and temperature: Semi-grand ensemble

Following the procedure described above we can write:

, where the new thermodynamical Potential is given by:


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

TO BE CONTINUED SOON