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, $\left.T\right.$

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:

• $\left.N\right.$ is the number of particles

• 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 \left. \Lambda \right. } is the de Broglie thermal wavelength (which depends on the temperature)

• 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 \beta = \frac{1}{k_B T} } , with 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 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 (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 \left( R^*\right)^{3N} } represent the 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 3N} position coordinates of the particles (reduced with the system size): i.e. $\int d(R^{*})^{3N}=1$

Helmholtz energy and partition function

The corresponding thermodynamic potential, the grand potential, 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 \Omega} , for the aforementioned grand canonical partition function is:

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 \Omega = \left. A - \mu N \right. } ,

where A is the Helmholtz energy function. Using the relation

$\left.U\right.=TS-PV+\mu N$

one arrives at

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 \left. \Omega \right.= -PV}

i.e.:

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 \left. p V = k_B T \log Q_{\mu V T } \right. }