Grand canonical ensemble

The grand-canonical ensemble is particularly well suited to simulation studies of adsorption.

Ensemble variables

• Chemical potential, $\left.\mu \right.$

• Volume, $\left.V\right.$

• Temperature, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.T\right.}

Grand canonical partition function

The classical grand canonical partition function for a one-component system in a three-dimensional space is given by:

$Q_{\mu VT}=\sum _{N=0}^{\infty }{\frac {\exp \left[\beta \mu N\right]V^{N}}{N!\Lambda ^{3N}}}\int d(R^{*})^{3N}\exp \left[-\beta U\left(V,(R^{*})^{3N}\right)\right]$

where:

• N is the number of particles

• Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.\Lambda \right.} is the de Broglie thermal wavelength (which depends on the temperature)

• $\beta ={\frac {1}{k_{B}T}}$, with $k_{B}$ being the Boltzmann constant

• U is the potential energy, which depends on the coordinates of the particles (and on the interaction model)

• Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left(R^{*}\right)^{3N}} represent the Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 3N} position coordinates of the particles (reduced with the system size): i.e. Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int d(R^{*})^{3N}=1}

Helmholtz energy and partition function

The corresponding thermodynamic potential, the grand potential, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Omega } , for the aforementioned grand canonical partition function is:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Omega =\left.A-\mu N\right.} ,

where A is the Helmholtz energy function. Using the relation

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.U\right.=TS-PV+\mu N}

one arrives at

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.\Omega \right.=-PV}

i.e.:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.pV=k_{B}T\log Q_{\mu VT}\right.}