AWM's Adsorption page - Generalized Langmuir isotherm (GL)

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: 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, 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, M-J: Marczewski-Jaroniec 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), 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 → -∞):

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
Langmuir-Freundlich, 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:

Gaussian energy distribution (see Gauss iso.eqn.)
Rudzinski energy distribution, m>0 (see Rudzinski iso.eqn.):
(R. distr. is very similar to Langmuir-Freundlich for m<<1 - see above)
see more (section below)

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.Derylo-Marczewska and M.Jaroniec, Chemica Scripta, 28, 173-184 (1988) (pdf, hi-res pdf available upon e-mail request).
"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).
"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-E_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 finite-width 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 (1-F instead of F-0.5). 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)
Langmuir-Freundlich, 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) = 1-F(-z)
Generalized-Freundlich, 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:
1. m_R = n_LF (at θ→0 and θ→1 - very low and very high energies)
2. σ_R = σ_LF (at moderate coverages and energies)
and
Square (continuous, fixed width ΔE > 0) aka UNILAN

and
k-centered 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