Adsorption-site grand partition function (${\Xi }_{\mathrm{site}}=1+z_s{e}^{\beta (\mu +ϵ)}$)

An independent site has one empty state and an occupied set with binding energy $-ϵ$ and vibrational partition function $z_s$. Its occupied probability is $\theta =z_s{e}^{\beta (\mu +ϵ)}/[1+z_s{e}^{\beta (\mu +ϵ)}]$. The denominator enforces the one-molecule-per-site constraint even though the vapour may obey classical statistics. Multiple occupancy or lateral interactions require a different model.

The occupied-to-vacant probability ratio of the adsorption-site grand partition function is $\theta /(1-\theta )=z_s{e}^{\beta ({\mu }_s+ϵ)}$. Taking its logarithm yields the displayed chemical potential. binding energy and vibrational energies must share a consistent reference so that zero-point energy is not counted twice.