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.