A Practical Guide to Isotherms of ADSORPTION on Heterogeneous Surfaces

multicomponent systems
(some ideas)

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 )
Heterogeneity and Molecular Size ( Theory and Prediction / Simple binary isotherm )

Chapter based on:
presented in references and summarized in:
General Integral Equation of Adsorption in multicomponent systems
and Energy correlations

Data analysis - prediction of multi-component adsorption:

• Prediction of multicomponent adsorption for components with non-linearly correlated adsorption energy distributions. (Here are model pictures and examples)

Necessary theoretical background is given in:
multi-component GIEA and Energy correlation

Applicability - method requirements:
Proposed method assumes that a certain correlation (generally non-linear) exists between adsorption energies of various adsorbates on a solid surface. This energy correlation assumes that at least the order of site energies is the same for all components, i.e. if the site energy (it means of course adsorption energy of component on a site) order for a component "i" is:
EiA ≤ EiD ≤ EiC ≤ EiB
then for any other component "j" we must have at least:
EjA ≤ EjD ≤ EjC ≤ EjB
where A - D denote site types (a stronger condition where "≤" is replaced by "<" is easier to analyse).

Generally no limit of no. of components/active site exists. Energy distribution function may be discrete (like in the example above) or continuous. Prediction requires data for simple (single-component or binary) systems or at least some estimated values of heterogeneity parameters or energy dispersion σE values. However, this method will succeed only if its main assumption about sequence of adsorption energy magnitudes (or sequence of adsorption energy differences, e.g. Ein like Ejn and Ekn) being common for all single or binary systems is true. In solutions it is always reasonable to choose a "reference adsorbate" ("n") which should be as different from all other adsorbates as possible (e.g. organic solutes "1","2","3" vs. water reference, "n").
If adsorbates showing very specific interactions with the surface are used, the prediction may not be satisfactory if those interactions are different for those adsorbates (e.g. acid R-COOH and amine R-NH2 adsorbed on the surface having acidic and basic functional groups. In such a case the parts of adsorption energy distributions related to acid-base interaction cannot be predicted properly (This does not include adsorbate-adsorbate interaction, that should be taken into account by other methods: formation of ion pairs, association etc.). However, the non-specific part of adsorption energy - if it is large comparing to the specific one - may still lead to reasonable prediction of at least some of the parameters.

References
1. "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.Derylo-Marczewska and M.Jaroniec, Chemica Scripta, 28, 173-184 (1988).
2. "Correlations of Heterogeneity Parameters for Single-Solute and Multi-Solute Adsorption from Dilute Solutions", A.W.Marczewski, A.Derylo-Marczewska and M.Jaroniec, J.Chem.Soc.Faraday Trans.I, 84, 2951-2957 (1988), (doi).
3. "A Simplified Integral Equation for Adsorption of Gas Mixtures on Heterogeneous Surfaces", A.W.Marczewski, A.Derylo-Marczewska and M.Jaroniec, Mh.Chem., 120, 225-230 (1989), (doi).
4. "Prediction of the Heterogeneity Parameters for Adsorption of Multicomponent Liquid Mixtures on Solids", A.W.Marczewski, A.Derylo-Marczewska, M.Jaroniec and J.Oscik, Z.phys.Chem., 270(4), 834-838 (1989) (pdf, hi-res pdf available upon e-mail request).
5. "Analysis of Experimental Data for Adsorption of Organic Substances from Dilute Aqueous Solutions on Activated Carbon", A.Derylo-Marczewska and A.W.Marczewski, Polish J. Chem., 71, 618-629 (1997)
6. "A General Model for Adsorption of Organic Solutes from Dilute Aqueous Solutions on Heterogeneous Solids: Application for Prediction of Multisolute Adsorption", A.Derylo-Marczewska and A.W.Marczewski, Langmuir, 13, 1245-1250 (1997), (doi).

Prediction cases:
This method, described in details and tested on experimental data, may reasonably 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 tertiary-mixture adsorption is wanted; quality of prediction - very good).
• Prediction of adsorption in multi-component 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 tertiary-mixture adsorption is wanted; quality: heterogeneity parameters are nicely estimated, other parameters may require correction) (for the prediction in e.g. tertiary systems, it is much better if binary liquid mixtures are available)

Necessary prediction steps:
• Prediction of adsorption in gas mixtures:
Measure single-component isotherms - calculate isotherm parameters - get energy distributions χi(Ei) - convert to Ei(F) and zi(F). Use General Integral Equation for mixtures, calculate isotherm data (if possible measure several isotherm points for mixed isotherm and correct parameters - mainly equilibrium constants). Another possibility is fitting all data together (see LSQ for multiple data sets) - it may require more points measured for mixture and/or additional weighting, however, it may result in a solution allowing for a wider range of prediction (though fitting quality for individual isotherms may be worse). We (me and my wife) used this method for fitting several hundreds of isotherm data points of solute and multi-solute adsorption on activated carbons (see below).
• Prediction of adsorption in multi-component dilute solutions:
see prediction steps for gas mixtures (above)
• Prediction of adsorption in liquid mixtures:
1. From single vapor isotherm data - calculate energy distributions χi(Ei) - convert to Ei(F) and zi(F). Use General Integral Equation for mixtures:
• Calculate energy distributions of energy differences for suitable component pairs:
zin(F) = zi(F)-zn(F)
(determine "reference component" by choosing one with the smallest or one with the highest heterogeneity effects), then use calculate isotherm data by using suitable "competitive" form of 1-dimensional multi-GIEA i.e. the one for (n-1) component "in" pairs (if possible measure several isotherm points for mixed isotherm and correct parameters - mainly equilibrium constants).
• Use original energy distributions zi(F) in a suitable form of multi-GIEA (for n individual components)
2. From binary liquid adsorption data - calculate energy distributions χij(Eij) - convert to Eij(F) and zij(F). Then use General Integral Equation for mixtures (its competitive form - for component pairs). All parameters of competitive adsorption (binary mixtures) should be given with respect to the same "reference component", i.e. (n-1) "ij" component pairs
Note.This method is easiest to use if single-component (prediction of multicomponent gas, solute or liquid mixture adsorption) or binary (prediction of multicomponent mixture adsorption) isotherm data are fitted with some theoretical equation (e.g. isotherms corresponding to Gaussian or LF energy distribution), then analytical equations of energy distributions zi(F) are used in the prediction.
E.g.:
For 2 components and LF isotherm, LF isotherm is fitted -> common monolayer capacity, am, heterogeneity parameters mLF,1, mLF,2 and equilibrium constants K1 and K2 are obtained.
Energy functions ELF,1(F), ELF,2(F) and zLF,1(F) and zLF,2(F) are calculated from theoretical energy distributions χLF,i(E), then used in multi-GIEA.
By comparing predicted adsorption values with experimental ones one can slightly correct predicted theoretical isotherm(s) (e.g. by correcting individual Ki or competitive Kij equilibrium constants - energy distributions/energy dispersions/heterogeneity parameters should be OK).

Model pictures will be here
Necessary theoretical background and model pictures are given in:
multi-component GIEA and Energy correlations

Examples of analysis/prediction (see the references):
Hand drawing after my paper Ref. 1 above.

 Step 1 - single-component adsorption experiment (data from literature, see ref.1 above): Single-component adsorption of pure benzene and CCl4 vapors on wide porous silica gel KSK and aerosil (experimental data): Step 2 - fitting (see ref.1 above): The isotherms are shown in log-log reduced scale (x=p/ps) (left and top scales). Points are experimental data (raw or transformed), red lines are fitted theoretical GF isotherms (General Freundlich) with LGD (Lopez-Gonzalez-Dietz) multilayer. Lower lines represent calculated monolayer part of adsorption as calculated according to BET (blue) and LGD isotherm (bottom and right scales) - better fitting to LGD is obvious: Table 1. Single-component data fitting: General Freundlich isotherm (GF) with multilayer factor (BET, LGD).
Temp [K]
Adsorbate i ln Kx,i * mi * am,i **
[mmol/g]
Aerosil
303K
Benzene 1 1.20 0.634 0.600
CCl4 2 0.54 0.769 0.536
KSK silica gel
293K
Benzene 1 1.03 0.613 1.38
CCl4 2 0.08 0.79 1.28

* GF isotherm with multilayer BET correction and LGD (Lopez-Gonzalez-Dietz) correction (see multilayer isotherms). The original adsorption data was reduced to its monolayer part (e.g. for BET, amono = a (1-x), for LGD amono = a (1-x2/2)/(1-x) ); for both multilayer isotherms the pressure, p, in original GF equation was replaced by [x/(1-x)]. The LSQ fitting was performed for log(amono).

** am,i values presented in the table have been obtained as average of 3 values: (1) obtained from GF-LGD fitting, (2) obtained by the B-point method, (3) obtained from adsorption of benzene vapours and corrected with respect to adsorbate molar volume.

 Step 3 - prediction, comparison with multicomponent experiment (see ref.1 above): Experimetal isotherms of binary adsorption in liquid mixtures of benzene(1) + CCl4(2) are compared with predicted/corrected theoretical lines. The heterogeneity parameters have been used as obtained by the prediction method shown above, adsorption capacities are average values as obtained for individual components in adsorption from gas phase. The equilibrium constants are corrected by comparing the experimental data with theoretical lines in log-log plot of GF isotherm: Table 2. Predicted and corrected parameters of binary liquid adsorption.

Adsorbent Predicted from vapor adsorption of pure components Corrected - fitted to the experimental data
ns * m12 ** ln K12 *** ln K12 ****
Aerosil 0.567 0.938 0.61 1.45
KSK silica gel 1.33 0.89 0.95 1.35

* ns = (am,1 + am,2)/2 (from vapor adsorption)

** m12 are approximated as follows: from individual heterogeneity parameters mi (from vapor adsorption), corresponding energy dispersions σi are calculated. From their positive difference σ12 (corresponds to the distribution of differences of adsorption energy for molecules 1-2 or 2-1, or in other words their competition to the surface sites) the heterogeneity parameter m12 is calculated.

*** ln K12 = ln K1 - ln K2 = ln (K1/K2), corresponds to the difference of average adsorption energies of 1 and 2 (from vapor phase). It original values are properly determined (usually LSQ data fitting is not very sensitive to the K values; the fitted model - energy distribution, multilayer, lateral interactions etc. - may also be an approximation) and at the same time the competition in binary mixture is not altered by the change of state (vapor to liquid) or mixing, then such value may describe adsorption competition from liquid mixture.

**** this value of ln K12 is obtained from fitting the model to the binary liquid adsorption data with ns and m12 fixed to the predicted values.

Conclusions (see references above):
The prediction method worked very well with respect to the heterogeneity effects - the relative differences in adsorption energies of components are well preserved even after change of state of matter. The adsorption equilibrium constants depend to higher degree on the adsorbate state or medium type (gas or liquid, mixing effects in liquid) in which or from which molecules are adsorbed.

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 )
Heterogeneity and Molecular Size ( Theory and Prediction / Simple binary isotherm )

General Integral Equation / GL (Generalized Langmuir) / All equations (preview)

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