© 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 ,
σ_{E} /
Linear plots /
φfunction /
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 polymerlike compound built of "mers" having the same size as surface sites)
 adsorption sites with their topography (neighbourneighbour correlations  conditional probabilities) (e.g. pure random, patchwise, mixed, chessboardlike etc.)
 "effective" surface topography may also include some adsorbatefactor as not all possible combinations of adsorbatemers  surfacesites are possible depending on the compound flexibility (polymerlike or sticklike 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 sitescreening effects (the weakest sitecombinations 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.DeryloMarczewska and M.Jaroniec, J.Colloid Interface Sci., 109, 310324 (1986),
(doi).
(This paper is probably my mostoften 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 widespread (at that time) myths.
 Adsorption energy is proportional to the "size" (observed e.g. for experimental adsorption data of nalkanes)  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 cutFreundlich energy distribution function.
CutFreundlich distribution of adsorption energy of single mer on single site (black). Influence of rmer molecule size on energy distribution function for random site topography.
1. Influence of molecule size (rmer) on observed adsorption energy distribution (circles  average energy) (legend).
2. Influence of molecule size (rmer) on observed adsorption energy distribution scaled to average mer adsorption energy of rmeric molecule (circles  average energy) (legend).

Simple equation of adsorption on heterogeneous solids from binary mixtures of gases or bicomponent 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, 385391 (1982),
(doi).
 "An Equation for MultiSolute 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, 307311 (1983),
(doi).
For adsorption system characterised by quasigaussian energy distribution and heterogeneity coefficient m and components with size ratio r_{12} = r_{1} / r_{2} in conditions where adsorbed layer is almost filledup :
K_{12} =
[a_{1} / a_{2} ^{r12} ]^{1/m} [c_{2} ^{r12} / c_{1}]
or in forms useful for data presentation:
[a_{1} / a_{2} ^{r12} ] = { K_{12} [c_{1} / c_{2} ^{r12}] } ^{m}
[a_{1} / c_{1}^{m} ] = K_{12}^{m} [a_{2} / c_{2}^{m}]^{r12}
In logarythmic form this equation is easier to use in data analysis and presentation:

ln(a_{1})  r_{12} ln(a_{2}) = m ln(K_{12}) + m [ln(c_{1})  r_{12} ln(c_{2})]

ln(a_{1})  m ln(c_{1}) = m ln(K_{12}) + r_{12} [ln(a_{2})  m ln(c_{2})]
The most useful forms are:

ln(a_{1} / a_{2}^{r12}) = m ln(K_{12}) +
m ln(c_{1} / c_{2}^{r12})

ln(a_{1} / c_{1}^{m}) = m ln(K_{12}) + r_{12} ln(a_{2} / c_{2}^{m})
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(K_{12}) and m or m ln(K_{12}) and r_{12}) one of the remaing parameters must always be included in fitted variables (r_{12} or m, respectively) in fitted line x,ydata. So in fact raw adsorption data (a_{1}, a_{2}, c_{1}, c_{2}) 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 m_{1} m_{2} or may be m_{12}  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 ,
σ_{E} /
Linear plots /
φfunction /
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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