Derivation of an Adsorption Isotherm, Chemical Potential, and Entropy for 2D Gas Adsorbates with Packing Exclusions and Attractive Interactions

Weakly bound adsorbates on surfaces may display 2D gas properties. In this work, we consider connections between chemical potential, partition functions, and equations of state and utilize these equations to derive isotherms for 2D gas adsorbates. After starting with equations for an ideal 2D gas, we extend the equations to the case of molecules approximated as hard discs (including the possibility of interadsorbate attractions) and show how the chemical potential and adsorption isotherm can be derived from either the equation of state or the partition function. An equation of state and adsorption isotherm are derived based on a generalized form of the van der Waals equation, accounting for both packing exclusion and the possibility of attractions between hard discs (gvdw-PEHD). We compare the derived isotherms to alternatives, including the Volmer equation. The gvdw-PEHD isotherm derived in this work uses a more realistic treatment of molecular properties and predicts a slower functional-form approach to saturation coverage for 2D gas adsorbates, when compared to the σ-Volmer equation. We also note that the absolute coverage form (the σ form) of equations is most convenient for use in thermodynamic analyses of 2D gas isotherms, as use of θ isotherms (such as the θ-Volmer equation) for 2D gas thermodynamic analyses can result in interpretation errors.