© A.W.Marczewski 2002
A Practical Guide to Isotherms of ADSORPTION on Heterogeneous Surfaces
Reload Adsorption Guide
ADSORPTION: popular isotherms
General Integral Equation /
GL (Generalized Langmuir) /
All equations (preview)
Adsorption type (
Linear Langmuir plot /
Graham plot /
Consistency /
Henry constant )
Popular isotherms
(
Mono,
Multilayer,
Experimental,
Micro,
Mesoporous
)
Data analysis:
LSq data fitting /
Global heterogeneity /
Linear plots /
φfunction /
Pores
)
Prediction/Description of
Multicomponent adsorption /
Wastewater adsorption
Heterogeneity and Molecular Size ( Theory and Prediction / Simple binary isotherm )
Monolayer: Heterogeneous  Homogeneous ( localized  mobile )
Multilayer 
Experimental 
Microporous 
Mesoporous
Popular isotherm types
(see data analysis and their linear forms) :

Monolayer (flat surface) localized adsorption on energetically heterogeneous surfaces (local isotherm  Langmuir):
All those equations are based on so called General Integral Equation and may include lateral interactions (of FG, simplified Kiselev and full Kiselev type  see local isotherms and derivation) or multilayer formation (BET  see other isotherms below or general formulation).

GL  Generalized Langmuir (aka. MJ, MJ: MarczewskiJaroniec) is a combination of Langmuir local isotherm with surface heterogeneity controlled by 2 parameters (0 ≤ m,n ≤ 1), for specific parameter values this equation reduces to Langmuir (L)(homogeneous), Generalized Freundlich (GF)/Sips, LangmuirFreundlich (LF) and Tóth (T) isotherms.

GF  Generalized Freudlich (aka. Sips eq.) (GL with n=1)  exponentially decreasing distribution of ads. energy, but in contrast to the "true Freundlich"  monolayer; reduces to Freundlich eq. for low adsorption values  does not show Henry behaviour at low conc.

LF  LangmuirFreundlich (GL with m=n)  symmetrical quasigaussian energy distribution (reduces to Freundlich eq. for low adsorption values; does not show Henry behaviour for low conc.)

T  Tóth (GL with m=1)  asymmetrical quasigaussian energy distribution with strong tail for low adsorption energies (reduces to Henry isotherm (a=Kc) for low adsorption values)

Sq  "Square" (aka. UNILAN)  monolayer physical adsorption with continuous (constant) energy distribution of ΔE width. Displays Henry behaviour (a=Kc) for low conc. (or pressures); for coverages close to 0.5 similar to LF and Gauss; related to a nonmonolayer Tiemkin eq. ( below ).

G  "Gauss"  monolayer physical adsorption with (true) Gaussian energy distribution of σ dispersion; no analytical isotherm equation exists. Displays Henry behaviour (a=Kc) for low very conc. (or pressures); for coverages close to 0.5 similar to LF, Sq etc. with similar energy dispersion.

R  Rudzinski  monolayer physical adsorption with simple symmetrical quasigaussian energy distribution; no analytical isotherm equation exists but this isotherm is very similar to LF. Does not display Henry behaviour at low conc.
 Monolayer isotherms for homogeneous adsorption:
All isotherms follow Henry behaviour at very low concentrations or pressures, like other isotherms without heterogeneity effects.
 Localized adsorption:

L  Langmuir eq.  monolayer localized physical adsorption on homogeneous surface; may be extended with heterogeneity effects, lateral interactions and multilayer effects

FG  FowlerGuggenheim eq.  localized physical adsorption with nonspecific lateral interactions (meanfield approx.) on homogeneous surface (gas phase), may be extended with heterogeneity effects. It reduces to Langmuir isotherm in absence of lateral interactions.
Identical in form equation is known as Frumkin isotherm and used in electrochemistry (e.g. adsorption of ions on mercury etc.)

Kis  simplified Kiselev eq.  localized physical adsorption with specific lateral interactions (associative) on homogeneous surface (gas phase), in full form it combines FG nonspecific (meanfield) lateral interactions with specific (associative) ones. It may be extended with heterogeneity effects. Reduces to Langmuir isotherm in absence of lateral interactions.

Jov  Jovanovic eq.  unlike L, FG or Kiselev it considers vertical interactions bulk/surface phase (derived from kinetic considerations, however, no proper statisticalmechanics derivation exists); homogeneous but may be extended with heterogeneity effects (see e.g.
JF/Jovm). Does not reduce to Langmuir isotherm.
 Mobile adsorption:
(Partially mobile models are known but eqns. are much more complicated.)
Should be used for gases and vapors not for liquids.

Volmer  Volmer eq.  mobile physical gas adsorption on homogeneous surface (gas phase), may be extended with heterogeneity effects.

HB  Hillde Boer eq.  mobile physical gas adsorption with FGlike (meanfield) lateral interactions on homogeneous surface (gas phase), may be extended with heterogeneity effects.

Multilayer isotherms (reduced concentration: x = c/c_{s} or reduced pressure: x = p/p_{s})  most based on BET equation (all reduce to Langmuir isotherm at lower relative concentrations x=c/c_{s} or pressures x = p/p_{s} and display Henry behaviour for very low x → 0; all those equations may also serve as local isotherms in General Integral Equation of Adsorption involving energetic heterogeneity effects):

BrunauerEmmettTeller (BET)  2parameter equation for infinite no. of layers (on flat surface only)  predicts always too big multilayer adsorption. For a finite no. of layers (e.g. in a pore) a 3parameter nlayer BET equation is obtained (see its classic form) (Note: actual max. statistical multilayer thickness for nlayer BET is not n but: 1+(n1)/2). Another nlayer isotherm is a 4parameter BDDT (BrunauerDemingDemingTeller) equation.

Hüttig  2parameter equation  predicted adsorption is always smaller than for BET

Sircar  3parameter equation

LopezGonzalez & Dietz (LGD)  2parameter equation  average of BET and Hüttig, may be also approximated by a simpler LGDa equation given by AWM.

Other experimental isotherms:
Not limited by monolayer capacity; may show maximum adsorption for dilute solutions of weakly soluble substances or vapours because of concentration/pressure limit

F  Freundlich equation (see also below).
GL equation may reduce to F for low adsorption values for specific values of heterogeneity parameters (e.g. specifically LF,GF). Very often used (with success) to describe adsorption in a narrow or moderately large adsorption range (for strongly heterogeneous or microporous solids, e.g. active carbons, it may translate into a quite large concentration range)

RP  RedlichPeterson (invented) a.k.a. RadkePrasunitz (popularized):
for low adsorptions similar to Henry (a=Kc) isotherm, for medium and high adsorptions behaves like Freundlich eq. (a=(Kc)^{m}) (it may be considered as a harmonic combination of Henry and Freundlich equations).
This equation is even more successful than Freudlich (above) in practical description of experimental data.

Jos  Jossens eq. (experimental; exponential dependence of isosteric heat of adsorption on adsorption)

Tiemkin (Temkin, Tiomkin)  Tiemkin eq.  experimental  used predominantly in gas catalysis; strong heterogeneity with constant energy distribution (continuous from ∞ to +∞; may be treated as a simplification of ( above ) Sq aka. UNILAN isotherm, but only far from monolayer filling region).

Adsorption in micropores (maximum adsorption capacity is analogous to monolayer capacity):
Based on the Theory of Pore Volume Filling by Dubinin and Radushkevich

Adsorption in mesopores (usually N_{2} adsorption):
Adsorption is a combination of adsorption on mesopore walls with condensation of adsorbate in pores, where meniscus reached a critical radius. Generally such isotherms display hysteresis loops (adsorption and desorption go along different paths) in the mesopore region (x = p/p_{s} ≥ 0.40 for N_{2} and x ≥ 0.175  0.98 for benzene). It is assumed that in this "mesopore" range all micropores are filled and monolayer is filled, too.
Methods of data analysis use meniscus radius calculated from Kelvin equation and statistical layer thickness of adsorbate (often given by HarkinsJura (HJ) or by Halsey / FrenkelHalseyHill (FHH) eq. with specific expressions for N_{2} adsorption: t_{HJ} and t_{Halsey}, respectively  see also above) in order to find pore diameter.
 Calculations depend on a mesopore shape:
 Mesopore filling model / pore distribution calculation scheme:
 BarretJoynerHalenda (BJH)
 simplified BJH
 Dubinin (like BJH, but takes into account the dependence of N_{2} surface tension on meniscus radius)
 DensityFunctional Theory (DFT)
Monolayer: Heterogeneous  Homogeneous ( localized  mobile )
Multilayer 
Experimental 
Microporous 
Mesoporous
Adsorption type (
Linear Langmuir plot /
Graham plot /
Consistency /
Henry constant )
Popular isotherms
(
Mono,
Multilayer,
Experimental,
Micro,
Mesoporous
)
Data analysis:
LSq data fitting /
Global heterogeneity /
Linear plots /
φfunction /
Pores
)
Prediction/Description of
Multicomponent adsorption /
Wastewater adsorption
Heterogeneity and Molecular Size (
Theory and Prediction /
Simple binary isotherm )
General Integral Equation /
GL (Generalized Langmuir) /
All equations (preview)
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