© A.W.Marczewski 2002

A Practical Guide to Isotherms of ADSORPTION on Heterogeneous Surfaces

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:

where *θ = a/a _{m}*

For dilute solute adsorption replace Pressure with solute Concentration (competition of two components with constant concentration of component "2" - solvent):

For binary liquid adsorption replace: concentration, c, with ratio of molar fractions in liquid phase x

or

When plotted as * K_{G} 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 K

**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:**
a_{m}=1, K=1, K_{n}=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)

legend

**Parameters:**
a_{m}=1, K=1, K_{n}=3 (Kiselev), α=2 (FG), m=0.9 (GF,LF,Tóth). See above for weaker lateral interactions or below for higher heterogeneity. (legend)

legend

**Parameters:**
a_{m}=1, K=1, K_{n}=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).

legend

**Parameters:**
a_{m}=1, K=1 (all except for bi-Langmuir), K_{n}=1 (Kiselev), K_{n}=0.5 (full Kiselev) (equiv. to K_{n}=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): a_{m}=1 = a_{m1} + a_{m2} (a_{m1}=0.9, K_{1}=1, a_{m2}=0.1, K_{2}=10).
See other picture for stronger lateral interactions or above for higher heterogeneity. (legend)

legend

**Parameters:**
a_{m}=1, K=1 (all except for bi-Langmuir), m=0.9 (GF,LF,Tóth,RP,JF/Jov-m).
For bi-Langmuir (two sites): a_{m}=1 = a_{m1} + a_{m2} (a_{m1}=0.9, K_{1}=1, a_{m2}=0.1, K_{2}=10).
See other picture for stronger lateral interactions or above for higher heterogeneity. (legend)

legend

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