Revision as of 19:46, 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 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.: ![{\displaystyle \left.N_{1}=N-\sum _{i=2}^{c}N_{i}\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95cdf8c9f8038dfe9347656698cb3328fdfeb343)
![{\displaystyle d\left(\beta A\right)=Ed\beta -(\beta p)dV+\beta \mu _{1}dN-\beta \mu _{1}\sum _{i=2}^{c}dN_{i}+\sum _{i=2}^{c}\beta \mu _{2}dN_{2};}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d1c3ca7e9b1a26ba7a3eeaf2e5c78869f5f8d55)
![{\displaystyle d\left(\beta A\right)=Ed\beta -(\beta p)dV+\beta \mu _{1}dN+\sum _{i=2}^{c}\beta (\mu _{2}-\mu _{i})dN_{i};}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11f0fbe4a2465124f5c0b2edb2390d0c6b97a0b8)
or,
![{\displaystyle d\left(\beta A\right)=Ed\beta -(\beta p)dV+\beta \mu _{1}dN+\sum _{i=2}^{c}\beta \mu _{i1}dN_{i};}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef366f7ee50df52c340d6fa4b627f4a4681fdd3d)
where
.
- Now considering the thermodynamical potential:
![{\displaystyle \beta A-\sum _{i=2}^{c}\left(N_{i}\beta \mu _{i1}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4583e2727b96dcc86e43559e2c25a52e76f79222)
![{\displaystyle d\left[\beta A-\sum _{i=2}^{c}(\beta \mu _{i1}N_{i})\right]=Ed\beta -\left(\beta p\right)dV+\beta \mu _{1}dN-N_{2}d\left(\beta \mu _{21}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1433629d7e037f8d3eeb8aa3a55db3e3a3085707)
Fixed pressure and temperature
In the Isothermal-Isobaric ensemble:
one can write:
![{\displaystyle d(\beta G)=Ed\beta +Vd(\beta p)+\sum _{i=1}^{c}\left(\beta \mu _{i}\right)dN_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/733632f3feb2c79cd0f3446f91accea2bb32466c)
where:
Fixed pressure and temperature: Semi-grand ensemble
Following the procedure described above one can write:
,
where the new thermodynamical Potential
is given by:
![{\displaystyle d(\beta \Phi )=d\left[\beta G-\sum _{i=2}^{c}(\beta \mu _{i1}N_{i})\right]=Ed\beta +Vd(\beta p)+\beta \mu _{1}dN-\sum _{i=2}^{c}N_{i}d(\beta \mu _{i1}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9ecb6a938d9b13ec7be5a28a8ecdf34dd065651)
Fixed pressure and temperature: Semi-grand ensemble: Partition function
TO BE CONTINUED SOON