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:

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

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, $N$

$N=\sum _{i=1}^c{N}_i$;

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 $N_1\to N$ i.e.: $N_1=N-\sum _{i=2}^c{N}_i$

$d(\beta A)=Ed\beta -(\beta p)dV+\beta {\mu }_{1}dN-\beta {\mu }_1\sum _{i=2}^{c}d{N}_i+\sum _{i=2}^c\beta {\mu }_{2}d{N}_2;$

$d(\beta A)=Ed\beta -(\beta p)dV+\beta {\mu }_{1}dN+\sum _{i=2}^c\beta ({\mu }_2-{\mu }_i)d{N}_i;$

Or:

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

where ${\mu }_{i1}\equiv {\mu }_i-{\mu }_1$. Now considering the thermodynamical potential: $\beta A-\sum _{i=2}^c(N_i\beta {\mu }_{i1})$

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

Fixed pressure and temperature

In the Isothermal-Isobaric ensemble: $(N_1,N_2,\cdots ,N_c,p,T)$ ensemble we can write:

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

where:

• $G$ is the Gibbs energy function

Fixed pressure and temperature: Semigrand esemble

Following the procedure described above we can write:

$\beta G(\beta ,\beta p,N_1,N_2,\cdots N_c)\to \mathrm{\Phi }(\beta ,\beta p,N,\beta {\mu }_{21},\cdots ,\beta {\mu }_{c1})$, where the new thermodynamical Potential $\mathrm{\Phi }$ is given by:

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