Semi-grand ensembles: Difference between revisions

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== Canonical Ensemble: fixed volume, temperature and number(s) of molecules ==
== Canonical Ensemble: fixed volume, temperature and number(s) of molecules ==


We will consider a system with "c" components;.  
We shall consider a system consisting of ''c'' components;.  
In the Canonical Ensemble, the differential
In the [[Canonical ensemble|canonical ensemble]], the differential
equation energy for the [[Helmholtz energy function]] can be written as:
equation energy for the [[Helmholtz energy function]] can be written as:


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*<math> \beta \equiv 1/k_B T </math>
*<math> \beta \equiv 1/k_B T </math>
*<math> k_B</math> is the [[Boltzmann constant]]
*<math> k_B</math> is the [[Boltzmann constant]]
*<math> T </math> is the absolute temperature
*<math> T </math> is the [[absolute temperature]]
*<math> E </math> is the internal energy
*<math> E </math> is the [[internal energy]]
*<math>  p </math> is the pressure
*<math>  p </math> is the [[pressure]]
*<math> \mu_i </math> is the chemical potential of the species "i"
*<math> \mu_i </math> is the [[Chemical potential|chemical potential]] of the species <math>i</math>
*<math> N_i </math> is the number of molecules of the species "i"
*<math> N_i </math> is the number of molecules of the species <math>i</math>


== Semi-grand ensemble at fixed volume and temperature ==
== Semi-grand ensemble at fixed volume and temperature ==

Revision as of 17:44, 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 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