© A.W.Marczewski 2002

A Practical Guide to Isotherms of ADSORPTION on Heterogeneous Surfaces

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Graham plot Graham plot

Consult e.g. a monograph "Physical Adsorption on Solids" by Young and Crowell or other general literature on adsorption.
See also discussion on Henry constant and Global Heterogeneity page.

Graham [D.J. Graham, J.Phys.Chem. 57 (1953) 665] introduced a so called "Equilibrium function" defined as:
Graham function - definition for gas adsorption where   θ = a/am
For dilute solute adsorption replace Pressure with solute Concentration (competition of two components with constant concentration of component "2" - solvent):
Graham function - definition for dilute solute adsorption
For binary liquid adsorption replace: concentration, c, with ratio of molar fractions in liquid phase xl12 (competition of two components), and replace ratio of surface coverage θ to non-covered (1-θ) with ratio of molar fractions of component "1" and "2" in the surface phase, xs12:
Graham function - definition for binary liquid adsorption - function of surface molar fraction or surface coverage or Graham function - definition for binary liquid adsorption - function of molar fracction in the bulk

When plotted as KG vs. θ , Graham's Equilibrium function allows to study deviations of the adsorption system from ideality. This function should be constant for homogeneous monolayer (Langmuir), should increase for lateral interactons and decrease for energetic heterogeneity.
A minimum in this plot means a combination of heterogeneity (decrease of KG with the increasing θ - adsorption forces are stronger for low coverages, when highly energetic sites are being occupied) and lateral interactions (steady increase with growing coverage θ - usually this increase is of the order higher than 1 with respect to coverage).

Graham plot for weak heterogeneity (m=0.9) and weak interactions Graham plot for weak heterogeneity (m=0.9) and strong interactions Graham plot for moderate heterogeneity (m=0.7) and strong interactions Graham plot for weak heterogeneity (m=0.9) and weak interactions Graham plot for weak heterogeneity (m=0.9) and no interactions

Model Graham plot picture for isotherm equations: L (Langmuir), bi-Langmuir (bi-L), LF (Langmuir-Freudlich), T (Tóth), Radke-Prausnitz / Redlich-Peterson (RP), F-G (Fowler-Guggenheim), Kis (Kiselev), GF (Generalized Freudlich), GF-Kis (GF with associative interactions), full Kiselev (full Kis), full Kiselev with non-specific interactions (full Kis+FG), Jovanovic (Jov), Jovanovic-Freundlich ( Jovanovic + heterogeneity, JF/Jov-m) - dilute solute adsorption (for gas adsorption replace concentration, c, with pressure, p).

Parameters: am=1, K=1, Kn=1 (Kiselev), α=0.5 (FG), m=0.9 (GF,LF,Tóth). See next picture for stronger lateral interactions or below for higher heterogeneity. (legend)
Graham plot - model picture for weak heterogeneity (m=0.9) and weak interactions
legend

Parameters: am=1, K=1, Kn=3 (Kiselev), α=2 (FG), m=0.9 (GF,LF,Tóth). See above for weaker lateral interactions or below for higher heterogeneity. (legend)
Graham plot - model picture for weak heterogeneity (m=0.9) and strong interactions
legend

Parameters: am=1, K=1, Kn=3 (Kiselev), α=2 (FG), m=0.7 (GF,LF,Tóth). See previous for lower heterogeneity or above for weaker lateral interactions (Legend is at the top).
Graham plot - model picture for moderate heterogeneity (m=0.7) and strong interactions
legend

Parameters: am=1, K=1 (all except for bi-Langmuir), Kn=1 (Kiselev), Kn=0.5 (full Kiselev) (equiv. to Kn=1 for simplified Kiselev), α=0.5 (FG, full Kiselev+FG), m=0.9 (GF,LF,Tóth,JF/Jov-m). For bi-Langmuir (two sites): am=1 = am1 + am2 (am1=0.9, K1=1, am2=0.1, K2=10). See other picture for stronger lateral interactions or above for higher heterogeneity. (legend)
Graham plot - model picture for weak heterogeneity (m=0.9) and weak interactions
legend

Parameters: am=1, K=1 (all except for bi-Langmuir), m=0.9 (GF,LF,Tóth,RP,JF/Jov-m). For bi-Langmuir (two sites): am=1 = am1 + am2 (am1=0.9, K1=1, am2=0.1, K2=10). See other picture for stronger lateral interactions or above for higher heterogeneity. (legend)
Graham plot - model picture for weak heterogeneity (m=0.9) without interactions
legend

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