# Semi-grand ensembles

**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.

## Contents

- 1 Canonical ensemble: fixed volume, temperature and number(s) of molecules
- 2 Semi-grand ensemble at fixed volume and temperature
- 3 Fixed pressure and temperature
- 4 Fixed pressure and temperature: Semi-grand ensemble
- 5 Fixed pressure and temperature: Semi-grand ensemble: partition function
- 6 References

## Canonical ensemble: fixed volume, temperature and number(s) of molecules[edit]

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:

- 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
- is the number of molecules of the species

## Semi-grand ensemble at fixed volume and temperature[edit]

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 thermodynamic potential:

## Fixed pressure and temperature[edit]

In the isothermal-isobaric ensemble: one can write:

where:

- is the Gibbs energy function

## Fixed pressure and temperature: Semi-grand ensemble[edit]

Following the procedure described above one can write:

- ,

where the *new* thermodynamic potential is given by:

## Fixed pressure and temperature: Semi-grand ensemble: partition function[edit]

In the fixed composition ensemble one has:

## References[edit]

- Related reading