Semi-grand ensembles: Difference between revisions

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: <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_1 \sum_{i=2}^c d N_i + \sum_{i=2}^c \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>
: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V +  \beta \mu_1 d N + \sum_{i=2}^c \beta (\mu_2-\mu_i) d N_i; </math>


Or:
Or:


: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V +  \beta \mu_1 d N + \beta \mu_{21} d N_2; </math>
: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V +  \beta \mu_1 d N + \sum_{i=2}^c \beta \mu_{i1} d N_i; </math>


where <math> \mu_{21} = \mu_2 - \mu_1 </math>. Now considering the thermodynamical potentia: <math> \beta A - N_2 \beta \mu_{21} </math>
where <math> \left. \mu_{i1} \equiv  \mu_i - \mu_1 \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 - \beta \mu_{21} N_2 \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 - N_2 d \left( \beta \mu_{21} \right).
</math>
</math>



Revision as of 15:52, 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 system with "c" components;. 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 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: