A Statistical Derivation of Langmuir's Adsorption Isotherm

The commonest form of Langmuir's adsorption isotherm iswhere θ is the fraction of the surface of the solid covered by adsorbed molecules, p the gas pressure in equilibrium with the adsorbed layer and A = A (T) a function of the temperature alone. This formula is usually derived by a kinetic argument which balances the rates of deposition and re-evaporation. It is perhaps not without interest to show that formula (1) and similar formulae can be obtained directly by the usual statistical methods which evaluate all the properties of the equilibrium state of any assembly. The ordinary derivation is apt to obscure the essentially thermodynamic character of (1) and to lead one to think that its form depends on the precise mechanisms of deposition and re-evaporation, whereas in fact it depends only on the whole set of states, adsorbed and free, accessible to the molecules in question. By suitable use of the usual technique for handling assemblies obeying the Fermi-Dirac statistics the saturation effect can be naturally incorporated in the theory ab initio.