A Practical Guide to Isotherms of ADSORPTION on Heterogeneous Surfaces Generalized Langmuir (GL) isotherm:
[Marczewski-Jaroniec iso.]
GL energy distribution
and
Other energy distributions

GL Equation | GL special forms | GL distributions ( graphs | functions ) | Other distr.
Alternative formulation of distributions ( GL | Other distr. )

GL: First introduced by myself (AWM) and M. Jaroniec

"A New Isotherm Equation for Single-Solute Adsorption from Dilute Solutions on Energetically Heterogeneous Solids", A.W.Marczewski and M.Jaroniec, Mh.Chem., 114, 711-715 (1983), (doi).
Later discussed and used to analyse adsorption data in a series of papers:
(theory) Mh.Chem., 115, 997-1012 (1984), (doi).
(gas adsorption) Mh.Chem., 115, 1013-1038 (1984), (doi).
(binary liquid adsorption) Mh.Chem., 115, 541-550 (1984)), (doi).
and so on ...

NOTE
Isotherm equations below are good e.g. for dilute solutions of organics. For gas adsorption replace concentration, c, with pressure, p. If you deal with weakly soluble solutes (or your concentrations are close to solubility limit, cs) or vapours (pressures close to saturation pressure, ps), you should take it into account, e.g. by using "multilayer correction"

GL - Generalized Langmuir (aka. MJ, M-J: Marczewski-Jaroniec iso.): where: θt is global (overall) adsorption isotherm (overall coverage) obtained from General Integral Equation of Adsorption for Langmuir local equation and 0<m,n≤1 .

GL isotherm for specific values of parameters reduces to the simpler well known monolayer isotherm equations:
Langmuir (L) (m=n=1), Langmuir-Freundlich (LF) (0 < m=n ≤ 1), Generalized Freundlich (GF) aka. Sips eq. (n=1, 0 < m ≤ 1) and Toth (T) (m=1, 0 < n ≤ 1)).
Generally parameter m is responsible for the isotherm behaviour at c → 0 (and ΔE → +∞) whereas n for the course at c → ∞ (and ΔE → -∞). This equation may be extended to lateral interactions and multilayer.

Model energy distribution functions for GL and its special cases (generally parameter m is responsible for the behaviour at ΔE → +∞ and n for ΔE → -∞):  Energy distribution functions and their asymptotes (generally parameter m is responsible for the behaviour at ΔE → +∞ and n for ΔE → -∞):

In the above equations:

Other energy distribution functions and their asymptotes:

Alternative formulations of energy distribution functions
See General Integral Equation of Adsorption

Based on:
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) (pdf, hi-res pdf available upon e-mail request).
2. "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).
3. "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).

The same distributions may be expressed through the z = E-Eo (ΔE in the formulas above), where Eo is characteristic energy, and the integral (cumulative) energy distribution function F(E) or F(z) (see references above and General Integral Equation of Adsorption).

In the following formulas:
• ΔE = Emax - Emin   (for finite-width distributions)
• z = E - Eo - where Eo is characteristic energy (see above);
Unless specified otherwise:
Eo is average energy for symmetric distributions;
Eo is Emin or Emax for asymmetric energy distributions with minimum and maximum energy, respectively.
• F(E), F(z) - integral (cumulative) energy distribution function, 0≤F≤1
• χ(z) = dF(z)/dz, χ(E) = dF(E)/dE - differential energy distribution function
(For a homogeneous surface - e.g. Langmuir eq. - χ(z) is defined as the Dirac's delta, δD(z), which is equal to 0 everywhere but in some fixed point - in this case z=0 - and is normalized to 1; F(z) is a simple Heaviside (step) function, F(z) = 0 for z<0 and F(z) = 1 for z≥0 - may be also defined alternately)
Energy distributions and energy dispersions:
CAUTION!
Over a period of 3 days (Nov.5 - Nov.7, 2003) there were some erroneous formulas for energy dispersion for LF, GF and Tóth (opposite sign of expression under square root) and z(F) for Sq (1-F instead of F-0.5). There were also some minor inconsistencies in parameter naming schemes (m or n).
And also:

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