Semi-grand ensembles: Difference between revisions

From SklogWiki
Jump to navigation Jump to search
m (corrections)
m (still correcting errors)
Line 44: Line 44:
* Now considering the thermodynamical potential: <math> \beta A - \sum_{i=2}^c \left( N_i \beta \mu_{i1} \right) </math>
* Now considering the thermodynamical potential: <math> \beta A - \sum_{i=2}^c \left( N_i \beta \mu_{i1} \right) </math>


:<math> d \left[ \beta A - \sum_{i=2}^c ( \beta \mu_{i1} N_i ) \right] = E d \beta - \left( \beta p \right) d V + \beta \mu_{1} d N - N_2 d \left( \beta \mu_{21} \right).
:<math> d \left[ \beta A - \sum_{i=2}^c ( \beta \mu_{i1} N_i ) \right] = E d \beta - \left( \beta p \right) d V + \beta \mu_{1} d N - i
\sum_{i=2}^c N_i d \left( \beta \mu_{i1} \right).
</math>
</math>



Revision as of 10:14, 7 September 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:

References