Vibrational factor in a Langmuir equilibrium constant ($b_s(T)=z_s(T){\lambda }^3{e}^{ϵ/(k_{B}T)}/(k_{B}T)$)

Equating the site chemical potential with that of an internally structureless classical vapour gives $\theta =b_{s}p/(1+b_{s}p)$. The coefficient $b_s$ includes the bound vibrational partition factor, the translational thermal de Broglie wavelength, and the positive binding Boltzmann factor. It has inverse-pressure units. Half coverage occurs at $p=1/b_s$; an unspecified $z_s(T)$ prevents a numerical pressure scale, but not the functional isotherm.