3.2 Langmuir Isotherm - derivation from equilibrium considerations
We may derive the Langmuir isotherm by treating the adsorption process as we would any other equilibrium process - except in this case the equilibrium is between the gas phase molecules (M), together with vacant surface sites, and the species adsorbed on the surface. Thus, for a non-dissociative (molecular) adsorption process we consider the adsorption to be represented by the following chemical equation :
S - * + M (g) = S - M
where : S - * , represents a vacant surface site
Note - in writing this equation we are making an inherent assumption that there are a fixed number of localised surface sites present on the surface. This is the first major assumption of the Langmuir isotherm.
We may now define an equilibrium constant ( K ) in terms of the concentrations of "reactants" and "products"
We may also note that :
Hence, it is also possible to define another equilibrium constant, b , as given below :
Rearrangement then gives the following expression for the surface coverage
which is the usual form of expressing the Langmuir Isotherm.
As with all chemical reactions, the equilibrium constant, b , is both temperature-dependent and related to the Gibbs free energy and hence to the enthalpy change for the process.
Note - b is only a constant (independent of θ ) if the enthalpy of adsorption is independent of coverage - this is the second major assumption of the Langmuir Isotherm.