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 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:

• Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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
• Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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.: $\left.N_{1}=N-\sum _{i=2}^{c}N_{i}\right.$

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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle d\left(\beta A\right)=Ed\beta -(\beta p)dV+\beta \mu _{1}dN+\beta (\mu _{2}-\mu _{1})dN_{2};}

Or:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle d\left(\beta A\right)=Ed\beta -(\beta p)dV+\beta \mu _{1}dN+\beta \mu _{21}dN_{2};}

where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \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}}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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: 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 (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 \left( \beta \mu_i \right) d N_i }

where:

• 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 G } is the Gibbs energy function