Semi-grand ensembles

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:

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

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, $N$

$\left.N=N_{1}+N_{2}\right.$;

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 change $(N_{1},N_{2})\rightarrow (N,N_{2})$ i.e.: $\left.N_{1}=N-N_{2}\right.$

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

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

Or:

$d\left(\beta A\right)=Ed\beta -(\beta p)dV+\beta \mu _{1}dN+\beta \mu _{21}dN_{2};$

where $\mu _{21}=\mu _{2}-\mu _{1}$. Now considering the thermodynamical potentia: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \beta A-N_{2}\beta \mu _{21}}

$d\left(\beta A-\beta \mu _{21}N_{2}\right)=Ed\beta -\left(\beta p\right)dV+\beta \mu _{1}dN-N_{2}d\left(\beta \mu _{21}\right).$

Fixed pressure and temperature

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d (\beta G) = E d \beta + V d (\beta p) + \sum_{i=1}^c \mu_i d N_i }

TO BE CONTINUED ... SOON