A Practical Guide to Isotherms of ADSORPTION on Heterogeneous Surfaces

Global Heterogeneity, H: calculation plot: Jovanovic
(see local heterogeneity h(θ) function)

Global heterogeneity plot θ(1-θ) vs. ln(p) for Jovanovic and Jovanovic-Freundlich (JF aka Jov-m, i.e. Jovanovic + heterogeneity) compared with Langmuir, Generalized Freundlich (GF)/Sips and Freundlich (F) equations.

All equations (JF/Jov-m, GF and F) corresponding to heterogeneous surfaces and characterized by the same value of heterogeneity parameter m for low surface coverages θ behave in the same way. The same applies to local Jovanovic and Langmuir isotherms.
However, for medium and high coverages θ) the differences become clear. Freundlich quickly reaches saturation, whereas Jovanovic and JF/Jov-m tend much faster to θ → 1 in comparison to Langmuir and Langmuir-based GF isotherm of the same parameter m.
For other Langmuir-based isotherms see model picture for GL
Those differences result partially from their implied local isotherms, i.e. different adsorption/interaction models (Langmuir, Jovanovic, Freundlich), partially from the magnitude of heterogeneity effects (heterogeneous or homogeneous: JF (Jov-m)/ Jov; GF / L) or shape of energy distribution function corresponding to apparently identical heterogeneity parameters or other factors (e.g. Freundlich should be used for vapor adsorption with p ≤ ps or in solutions for weakly soluble substances with c ≤ cs).

For Freundlich equation point θ=1 corresponds to p=po , for other equations θ=1 is reached for p→∞ or [p/(1-x)]→∞ (for vapors x = p/ps and for weakly soluble solutes; x=c/cs).
NOTE. Replace p with c for dilute solute adsorption.

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