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3.2: Langmuir Isotherm - derivation from equilibrium considerations

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    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:

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


    • \(S - *\) 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.

    3.2: Langmuir Isotherm - derivation from equilibrium considerations is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Roger Nix.

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