© A.W.Marczewski 2002
A Practical Guide to Isotherms of ADSORPTION on Heterogeneous Surfaces
Reload Adsorption Guide
ADSORPTION:
Prediction of multicomponent adsorption and
Heterogeneity and 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 )
Data analysis  prediction of multicomponent adsorption:

Prediction of multicomponent adsorption for components with nonlinearly correlated adsorption energy distributions.
(Here are model pictures)
Proposed method assumes that a certain correlation (generally nonlinear) exists between adsorption energies of various adsorbates on a solid surface. Generally no limit of no. of components exists. (Necessary theoretical background is given in multicomponent GIEA and Energy correlation sections). Prediction requires data for simple (singlecomponent or binary) systems or at least estimated values of heterogeneity parameters or energy dispersion σ_{E} values.
For more info consult these papers:
 "Unified Theoretical Description of Physical Adsorption from Gaseous and Liquid Phases on Heterogeneous Solid Surfaces and Its Application for Predicting Multicomponent Adsorption Equilibria", A.W.Marczewski, A.DeryloMarczewska and M.Jaroniec, Chemica Scripta, 28, 173184 (1988) (pdf, hires pdf available upon email request).
 "Correlations of Heterogeneity Parameters for SingleSolute and MultiSolute Adsorption from Dilute Solutions", A.W.Marczewski, A.DeryloMarczewska and M.Jaroniec, J.Chem.Soc.Faraday Trans.I, 84, 29512957 (1988),
(doi).
 "A Simplified Integral Equation for Adsorption of Gas Mixtures on Heterogeneous Surfaces", A.W.Marczewski, A.DeryloMarczewska and M.Jaroniec, Mh.Chem., 120, 225230 (1989),
(doi).
 "Prediction of the Heterogeneity Parameters for Adsorption of Multicomponent Liquid Mixtures on Solids", A.W.Marczewski, A.DeryloMarczewska, M.Jaroniec and J.Oscik, Z.phys.Chem., 270(4), 834838 (1989) (pdf, hires pdf available upon email request).
 "Analysis of Experimental Data for Adsorption of Organic Substances from Dilute Aqueous Solutions on Activated Carbon", A.DeryloMarczewska and A.W.Marczewski, Polish J. Chem., 71, 618629 (1997)
 "A General Model for Adsorption of Organic Solutes from Dilute Aqueous Solutions on Heterogeneous Solids: Application for Prediction of Multisolute Adsorption", A.DeryloMarczewska and A.W.Marczewski, Langmuir, 13, 12451250 (1997),
(doi).
This method may predict adsorption (or at least is able to estimate the magnitude of heterogeneity effects) in many situations:
 Prediction of adsorption in gas mixtures (required: single gas adsorption data for all components; useful: binary gas mixture data  if e.g. prediction of tertiarymixture adsorption is wanted; quality of prediction  very good).
 Prediction of adsorption in multicomponent dilute solutions (required: single solute adsorption data for all components; useful: binary dilute solute adsorption data  if e.g. prediction of tertiary mixture adsorption is wanted; quality of prediction  good or very good).
 Prediction of adsorption in liquid mixtures (required: single gas/vapour adsorption data for all components; useful: binary gas mixture data  if e.g. prediction of tertiarymixture adsorption is wanted; quality: heterogeneity parameters are nicely estimated, other parameters require correction) (for the prediction in e.g. tertiary systems, it is much better if binary liquid mixtures are available).
See prediction steps

Prediction of adsorption in wastewater  unlimited no. of components and variable composition.
(Here are model pictures)
Proposed method assumes that all adsorbates are similar in chemical properties and their distributions af adsorption energies have the same shape but different position on energy axis (same heterogeneities, different adsorption equilibrium constants). Generally no limit of no. of components exists. Prediction requires data for simple (singlecomponent or binary) systems.
In the light of the above presented prediction method as well as a heterogeneity vs. molecular size and heterogeneity study it is crude oversimplification. However, the usefulness of this approach may overcome its limits.
E.g. for wastewater treatment, where usually a large and only partially known no. and kind of pollutants is present. In such a case, if only an approximate mixtureprofile is known, the adsorption of the entire mixture may be quite well predicted. In this way the right amount of adsorbent (e.g. active carbon) may be determined with a narrow margin error.
For more info consult these papers:
 "A Method of Investigation of Adsorption Equilibria on Active Carbons",
A.W.Marczewski, A.DeryloMarczewska and M.Jaroniec, Ochrona Srodowiska, 521/23(3233), pp. 3742, PZITS, Wroclaw (1987) (paper and oral presentation in Polish),

"A Simple Method for Describing MultiSolute Adsorption Equilibria on Activated Carbons", A.W.Marczewski, A.DeryloMarczewska and M.Jaroniec, Chem.Engng.Sci., 45(1), 143149 (1990),
(doi).
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

Simple equation of adsorption on heterogeneous solids from binary mixtures of gases or bicomponent dilute solutions for components with different molecular sizes (for 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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