© A.W.Marczewski 2002
A Practical Guide to Isotherms of ADSORPTION on Heterogeneous Surfaces
ADSORPTION: prediction
multicomponent systems (some ideas)
General Integral Equation / GL (Generalized Langmuir) / All equations (preview)
General | Prediction | References | Cases | Prediction steps | Examples
Chapter based on:
Unified description of physical adsorption
presented in references
and summarized in:
General Integral Equation of Adsorption in multicomponent systems
and Energy correlations
Data analysis - prediction of multi-component adsorption:
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 CCl_{4} 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/p_{s}) (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: |
Adsorbent Temp [K] |
Adsorbate | i | ln K_{x,i} ^{*} | m_{i} ^{*} | a_{m,i} ^{**} [mmol/g] |
---|---|---|---|---|---|
Aerosil 303K |
Benzene | 1 | 1.20 | 0.634 | 0.600 |
CCl_{4} | 2 | 0.54 | 0.769 | 0.536 | |
KSK silica gel 293K |
Benzene | 1 | 1.03 | 0.613 | 1.38 |
CCl_{4} | 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, a_{mono} = a (1-x), for LGD a_{mono} = a (1-x^{2}/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(a_{mono}).
^{**} a_{m,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) + CCl_{4}(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 | ||
---|---|---|---|---|
n^{s} ^{*} | m_{12} ^{**} | ln K_{12} ^{***} | ln K_{12} ^{****} | |
Aerosil | 0.567 | 0.938 | 0.61 | 1.45 |
KSK silica gel | 1.33 | 0.89 | 0.95 | 1.35 |
^{*} n^{s} = (a_{m,1} + a_{m,2})/2 (from vapor adsorption)
^{**} m_{12} are approximated as follows: from individual heterogeneity parameters m_{i} (from vapor adsorption), corresponding energy dispersions σ_{i} are calculated. From their positive difference σ_{12} (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 m_{12} is calculated.
^{***} ln K_{12} = ln K_{1} - ln K_{2} = ln (K_{1}/K_{2}), 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 K_{12} is obtained from fitting the model to the binary liquid adsorption data with n^{s} and m_{12} 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 ,
σ_{E} /
Linear plots /
φ-function /
Pores
)
Prediction/Description of
Multicomponent adsorption /
Wastewater adsorption
Heterogeneity and Molecular Size ( Theory and Prediction / Simple binary isotherm )
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