© A.W.Marczewski 2002
A Practical Guide to Isotherms of ADSORPTION on Heterogeneous Surfaces
Reload Adsorption Guide
ADSORPTION: prediction
heterogeneity vs. molecular size (some ideas)
General Integral Equation /
GL (Generalized Langmuir) /
All equations (preview)
Adsorption type (
Linear Langmuir plot /
Graham plot /
Consistency /
Henry constant )
Popular isotherms
(
Mono-,
Multilayer,
Experimental,
Micro-,
Mesoporous
)
Data analysis:
LSq data fitting /
Heterogeneity: Global ,
σE /
Linear plots /
φ-function /
Pores
)
Prediction/Description of
Multicomponent adsorption /
Wastewater adsorption
Heterogeneity and Molecular Size ( Theory and Prediction / Simple binary isotherm )
Heterogeneity vs. molecular size and topography:
-
Theoretical approach based on simple statistics.
(Here are model pictures)
The proposed approach does not assume any particular adsorption isotherm, though localised physical adsorption in monolayer is implied. It is assumed, that the observed heterogeneity comes from the entire system:
- adsorbate with its specific properties (in the simplest case it is a polymer-like compound built of "mers" having the same size as surface sites)
- adsorption sites with their topography (neighbour-neighbour correlations - conditional probabilities) (e.g. pure random, patchwise, mixed, chessboard-like etc.)
- "effective" surface topography may also include some adsorbate-factor as not all possible combinations of adsorbate-mers - surface-sites are possible depending on the compound flexibility (polymer-like or stick-like etc.)
- as a heterogeneity measure, the adsorption energy dispersion is used - it is calculated by using all posssible energies of adsorption with their respective probabilities; in fact it is possible that in many situations (e.g. large molecules) not all possiblities may be observed due to site-screening effects (the weakest site-combinations may never be occupied), but here heterogeneity shows only a potential for differences in energies.
For more info consult this paper:
- "Energetic Heterogeneity and Molecular Size Effects in Physical Adsorption on Solid Surfaces", A.W.Marczewski, A.Derylo-Marczewska and M.Jaroniec, J.Colloid Interface Sci., 109, 310-324 (1986),
(doi).
(This paper is probably my most-often cited one and I must say it gave me a lot of fun to find all this!)
This approach explained several observed facts and helped to reject some wide-spread (at that time) myths.
- Adsorption energy is proportional to the "size" (observed e.g. for experimental adsorption data of n-alkanes) - always true! - does not depend on topography. It is true for:
- minimum adsorption energy
- average adsorption energy
- maximum adsorption energy
- Myth: random topography means larger heterogeneity - false! - for larger molecules surfaces with the same structural (i.e. "site") heterogeneity with patchwise topography show higher heterogeneity effects than those with random topography
- for patchwise topography the dispersion of adsorption energy is proportional to the molecular size
- for random topography the dispersion of adsorption energy is proportional to the square root of molecular size (energy variancy is proportional to the molecular size)
- for regular/ordered topographies (e.g. checkerboard topography) the observed heterogeneity will change periodically with molecular size
-
Model pictures for random topography and cut-Freundlich energy distribution function.
Cut-Freundlich distribution of adsorption energy of single mer on single site (black). Influence of r-mer molecule size on energy distribution function for random site topography.
1. Influence of molecule size (r-mer) on observed adsorption energy distribution (circles - average energy) (legend).
2. Influence of molecule size (r-mer) on observed adsorption energy distribution scaled to average mer adsorption energy of r-meric molecule (circles - average energy) (legend).
-
Simple equation of adsorption on heterogeneous solids from binary mixtures of gases or bi-component dilute solutions for components with different molecular sizes (gas adsorption: M.Jaroniec, Thin Solid Films, 81 L97 (1981)).
Read more in:
- "Competitive Adsorption of Binary Gas Mixtures on Energetically Heterogeneous Solids", M.Jaroniec and A.W.Marczewski, Thin Solid Films, 92, 385-391 (1982),
(doi).
- "An Equation for Multi-Solute Adsorption from Dilute Aqueous Solutions Involving Energetic Heterogeneity of the Solid and Differences in Molecular Sizes of the Solutes", M.Jaroniec, A.Derylo and A.W.Marczewski, Chem.Engng.Sci., 38, 307-311 (1983),
(doi).
For adsorption system characterised by quasi-gaussian energy distribution and heterogeneity coefficient m and components with size ratio r12 = r1 / r2 in conditions where adsorbed layer is almost filled-up :
K12 =
[a1 / a2 r12 ]1/m [c2 r12 / c1]
or in forms useful for data presentation:
[a1 / a2 r12 ] = { K12 [c1 / c2 r12] } m
[a1 / c1m ] = K12m [a2 / c2m]r12
In logarythmic form this equation is easier to use in data analysis and presentation:
-
ln(a1) - r12 ln(a2) = m ln(K12) + m [ln(c1) - r12 ln(c2)]
-
ln(a1) - m ln(c1) = m ln(K12) + r12 [ln(a2) - m ln(c2)]
The most useful forms are:
-
ln(a1 / a2r12) = m ln(K12) +
m ln(c1 / c2r12)
-
ln(a1 / c1m) = m ln(K12) + r12 ln(a2 / c2m)
This approach worked well for many gas mixtures and binary dilute solutions even for low concentrations/adsorptions, where the condition of almost complete monolayer filling cannot be met.
Certain weakness of this equation is fitting procedure - though there are apparently only 2 straight line parameters (m ln(K12) and m or m ln(K12) and r12) one of the remaing parameters must always be included in fitted variables (r12 or m, respectively) in fitted line x,y-data. So in fact raw adsorption data (a1, a2, c1, c2) should be optimised, with appropriate statistical weights. In such a case, the above linear forms may serve as a valuable illustration/presentation tool.
NOTE.
The meaning of heterogeneity parameter m (for 2 components - or even 3 for liquid adsorption - with different energy distributions - is it m1 m2 or may be m12 - relative heterogeneity of "1" and "2" - or sth. else?) or underlying surface topography may be explained within Theoretical Approach above.
Adsorption type (
Linear Langmuir plot /
Graham plot /
Consistency /
Henry constant )
Popular isotherms
(
Mono-,
Multilayer,
Experimental,
Micro-,
Mesoporous
)
Data analysis:
LSq data fitting /
Heterogeneity: Global ,
σE /
Linear plots /
φ-function /
Pores
)
Prediction/Description of
Multicomponent adsorption /
Wastewater adsorption
Heterogeneity and Molecular Size ( Theory and Prediction / Simple binary isotherm )
General Integral Equation /
GL (Generalized Langmuir) /
All equations (preview)
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