# 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 :

$S - * + M_{(g)} \rightleftharpoons S - M \label{Eq1}$

where :

• $$S - *$$ represents a vacant surface site

Assumption 1

In writing Equation $$\ref{Eq1}$$ we are making an inherent assumption that there are a fixed number of localized 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"

$K = \dfrac{[S-M]}{[S-*][M]}$

We may also note that :

• [ S - M ] is proportional to the surface coverage of adsorbed molecules, i.e. proportional to θ
• [ S - * ] is proportional to the number of vacant sites, i.e. proportional to (1-θ)
• [ M ] is proportional to the pressure of gas , P

Hence, it is also possible to define another equilibrium constant, b , as given below :

$b =\dfrac{\theta}{(1- \theta)P}$

Rearrangement then gives the following expression for the surface coverage

$\theta =\dfrac{b P}{1 + bP}$

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.

Assumption 2

$$b$$ is only a constant (independent of $$\theta$$) if the enthalpy of adsorption is independent of coverage. This is the second major assumption of the Langmuir Isotherm.

## Contributors

• Roger Nix (Queen Mary, University of London)