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 we can write:
,
where the new thermodynamical Potential is given by:
Fixed pressure and temperature: Semi-grand ensemble: Partition function
TO BE CONTINUED SOON