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| * <math> G </math> is the [[Gibbs energy function]] | | * <math> G </math> is the [[Gibbs energy function]] |
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| == Fixed pressure and temperature: Semigrand ensemble == | | == Fixed pressure and temperature: Semi-grand ensemble == |
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| Following the procedure described above we can write: | | Following the procedure described above we can write: |
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| == Fixed pressure and temperature: Semigrand ensemble: Partition function == | | == Fixed pressure and temperature: Semi-grand ensemble: Partition function == |
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| TO BE CONTINUED SOON | | TO BE CONTINUED SOON |
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