Chemical potential of an adsorbed species on equivalent sites (${\mu }_s=-ϵ+k_{B}T\ln [\theta /((1-\theta )z_s)]$)

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.