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:

• 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
• $p$ is the pressure
• $\mu _{i}$ is the chemical potential of the species "i"
• 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_i } is the number of molecules of the species "i"