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.

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