© A.W.Marczewski 2002
A Practical Guide to Isotherms of ADSORPTION on Heterogeneous Surfaces
Reload Adsorption Guide
Generalized Langmuir (GL) isotherm:
[MarczewskiJaroniec iso.]
GL energy distribution
and
Other energy distributions
see also: General Integral Equation of Adsorption and GL isotherm
see also Energy distribution and Calculation of energy distribution
see also: energy dispersion and Global Heterogeneity concept
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 SingleSolute Adsorption from Dilute Solutions on Energetically Heterogeneous Solids", A.W.Marczewski and M.Jaroniec, Mh.Chem.,
114, 711715 (1983),
(
doi).
Later discussed and used to analyse adsorption data in a series of papers:
(theory) Mh.Chem.,
115, 9971012 (1984),
(
doi).
(gas adsorption) Mh.Chem.,
115, 10131038 (1984),
(
doi).
(binary liquid adsorption) Mh.Chem.,
115, 541550 (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, c_{s}) or vapours (pressures close to saturation pressure, p_{s}), you should take it into account, e.g. by using "multilayer correction"
GL  Generalized Langmuir (aka. MJ, MJ: MarczewskiJaroniec iso.):
where: θ_{t} 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),
LangmuirFreundlich (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 → ∞):

Langmuir, L (m=n=1) isotherm (homogeneous)
Energy distribution function χ(E) is defined by Dirac's delta (impulse) function, δ_{D}(E)

Generalized Langmuir, GL (m≠n, m≠1, n≠1) isotherm

LangmuirFreundlich, LF (m=n, n≠1) isotherm

Tóth (m=1, n≠1) isotherm

Generalized Freundlich, GF (n=1, m≠1) isotherm (ΔE > 0  see below)
In the above equations:
 ΔE = E  E_{o}
 E = ε/RT is reduced energy
 E_{o} is a characteristic energy.
for symmetrical LF: E_{o} = E_{avg} (average energy)
for asymmetrical GF: E_{o} = E_{min} (minimum energy)


or
Other energy distribution functions and their asymptotes:
Alternative formulations of energy distribution functions
See General Integral Equation of Adsorption
Based on:
 "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).
 "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).
The same distributions may be expressed through the z = EE_{o} (ΔE in the formulas above), where E_{o} 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 = E_{max}  E_{min} (for finitewidth distributions)

z = E  E_{o}  where E_{o} is characteristic energy (see above);
Unless specified otherwise:
E_{o} is average energy for symmetric distributions;
E_{o} is E_{min} or E_{max} 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, δ_{D}(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 (1F instead of F0.5). There were also some minor inconsistencies in parameter naming schemes (m or n).

Langmuir, L (m=n=1) isotherm (homogeneous)
Differential energy distribution function χ(z) is defined by Dirac's delta (impulse) function, δ_{D}(z) and the integral (cumulative) energy distribution function is defined by the simple Heaviside (step) function (Heaviside function defined alternately)

LangmuirFreundlich, LF (m=n), ∞<z<+∞
where
(direct formula for z≤0 and F≤0.5)
For z>0 and F>0.5 use symmetry: F(z) = 1F(z)

GeneralizedFreundlich, GF (n=1), z>0

Tóth isotherm (m=1), ∞<z<+∞
E_{o} is characteristic energy (it is not average energy here)
where
and
And also:

Gauss distribution (σ > 0), ∞<z<+∞
and

Rudzinski, R (0 < m < ∞), ∞<z<+∞
This equation will behave similarly to LF if:

m_{R} = n_{LF} (at θ→0 and θ→1  very low and very high energies)

σ_{R} = σ_{LF} (at moderate coverages and energies)
and

Square (continuous, fixed width ΔE > 0) aka UNILAN
and

kcentered discrete (fixed width ΔE > 0, k ≥ 2)
and
and

cut Freundlich, cF (decreasing exponential with minimum energy E_{min}, m > 0)
Here: characteristic energy, E_{o} = E_{min} + 1/m = E_{av} (average energy)
and
GL Equation 
GL special forms 
GL distributions (
graphs 
functions )  Other distr.
Alternative formulation of distributions (
GL 
Other distr. )
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