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| == Fixed pressure and temperature == | | == Fixed pressure and temperature == |
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| In the [[Isothermal-Isobaric ensemble]]: <math> (N_1,N_2, \cdots, N_c, p, T) </math> one can write: | | In the [[Isothermal-isobaric ensemble]]: <math> (N_1,N_2, \cdots, N_c, p, T) </math> one can write: |
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| :<math> d (\beta G) = E d \beta + V d (\beta p) + \sum_{i=1}^c \left( \beta \mu_i \right) d N_i </math> | | :<math> d (\beta G) = E d \beta + V d (\beta p) + \sum_{i=1}^c \left( \beta \mu_i \right) d N_i </math> |
Revision as of 13:59, 21 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
In the fixed composition ensemble one has: