Semi-grand ensembles: Difference between revisions

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to the differential equation written above in terms of <math> A (T,V,N_1,N_2) </math>.  
to the differential equation written above in terms of <math> A (T,V,N_1,N_2) </math>.  


# Consider the change <math> (N_1,N_2) \rightarrow (N,N_2) </math>
# Consider the change <math> (N_1,N_2) \rightarrow (N,N_2) </math> i.e.: <math> \left. N_1 = N-N_2  \right. </math>
: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \sum_{i=1}^2 (\beta \mu_i) d N_i </math
 
 
: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V +  \beta \mu_1 d N - \beta \mu_1 d N_2 + \beta \mu_2 d N_2; </math>
 
 
: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V +  \beta \mu_1 d N + \beta (\mu_2-\mu_1) d N_2. </math>
 
TO BE CONTINUED

Revision as of 13:14, 5 March 2007

General Features

Semi-grand ensembles are used in Monte Carlo simulation of mixtures.

In this ensembles the total number of molecules is fixed, but the composition can change.

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

We will consider a binary system;. In the Canonical Ensemble, the differential equation energy for the Helmholtz energy function can be written as:

,

where:

  • is the Helmholtz energy function
  • is the Boltzmann constant
  • is the absolute temperature
  • is the internal energy
  • is the pressure
  • is the chemical potential of the species "i"
  • is the number of molecules of the species "i"

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 .

  1. Consider the change i.e.:



TO BE CONTINUED