Semi-grand ensembles

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 will consider a system with "c" components;. In the Canonical Ensemble, the differential equation energy for the Helmholtz energy function can be written as:

$d\left(\beta A\right)=Ed\beta -(\beta p)dV+\sum _{i=1}^{c}(\beta \mu _{i})dN_{i}$,

where:

• $A$ is the Helmholtz energy function
• $\beta \equiv 1/k_{B}T$
• $k_{B}$ is the Boltzmann constant
• $T$ is the absolute temperature
• $E$ is the internal energy
• $p$ is the pressure
• $\mu _{i}$ is the chemical potential of the species "i"
• $N_{i}$ 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, $N$

$\left.N=\sum _{i=1}^{c}N_{i}\right.$;

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 $A(T,V,N_{1},N_{2})$.

1. Consider the variable change $N_{1}\rightarrow N$ i.e.: $\left.N_{1}=N-\sum _{i=2}^{c}N_{i}\right.$

$d\left(\beta A\right)=Ed\beta -(\beta p)dV+\beta \mu _{1}dN-\beta \mu _{1}\sum _{i=2}^{c}dN_{i}+\sum _{i=2}^{c}\beta \mu _{2}dN_{2};$

$d\left(\beta A\right)=Ed\beta -(\beta p)dV+\beta \mu _{1}dN+\sum _{i=2}^{c}\beta (\mu _{2}-\mu _{i})dN_{i};$

Or:

$d\left(\beta A\right)=Ed\beta -(\beta p)dV+\beta \mu _{1}dN+\sum _{i=2}^{c}\beta \mu _{i1}dN_{i};$

where $\left.\mu _{i1}\equiv \mu _{i}-\mu _{1}\right.$. Now considering the thermodynamical potential: $\beta A-\sum _{i=2}^{c}\left(N_{i}\beta \mu _{i1}\right)$

$d\left[\beta A-\sum _{i=2}^{c}(\beta \mu _{i1}N_{i})\right]=Ed\beta -\left(\beta p\right)dV+\beta \mu _{1}dN-N_{2}d\left(\beta \mu _{21}\right).$

Fixed pressure and temperature

In the Isothermal-Isobaric ensemble: $(N_{1},N_{2},\cdots ,N_{c},p,T)$ ensemble we can write:

$d(\beta G)=Ed\beta +Vd(\beta p)+\sum _{i=1}^{c}\left(\beta \mu _{i}\right)dN_{i}$

where:

• $G$ is the Gibbs energy function

Fixed pressure and temperature: Semigrand esemble

Following the procedure described above we can write:

$\beta G(\beta ,\beta p,N_{1},N_{2},\cdots N_{c})\rightarrow \Phi (\beta ,\beta p,N,\beta \mu _{21},\cdots ,\beta \mu _{c1})$, where the new thermodynamical Potential $\Phi$ is given by:

$d\Phi =d\left[\beta G-\sum _{i=2}^{c}(\beta \mu _{i1}N_{i})\right]=Ed\beta +Vd(\beta p)+\beta \mu _{1}dN-\sum _{i=2}^{c}N_{i}d(\beta \mu _{i1}).$