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| ==References== | | ==References== |
| | <references/> |
| | ;Related reading |
| | *[http://dx.doi.org/10.1063/1.3677193 Yiping Tang "A new method of semigrand canonical ensemble to calculate first-order phase transitions for binary mixtures", Journal of Chemical Physics '''136''' 034505 (2012)] |
| [[category: Statistical mechanics]] | | [[category: Statistical mechanics]] |
Latest revision as of 14:05, 20 January 2012
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[edit]
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[edit]
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 thermodynamic potential:
Fixed pressure and temperature[edit]
In the isothermal-isobaric ensemble: one can write:
where:
Fixed pressure and temperature: Semi-grand ensemble[edit]
Following the procedure described above one can write:
- ,
where the new thermodynamic potential is given by:
Fixed pressure and temperature: Semi-grand ensemble: partition function[edit]
In the fixed composition ensemble one has:
References[edit]
- Related reading