Semi-grand ensembles

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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

In the fixed composition ensemble we will have:

TO BE CONTINUED SOON