© A.W.Marczewski 20022013
A Practical Guide to Isotherms of ADSORPTION on Heterogeneous Surfaces
Reload Adsorption Guide
Adsorption GLOSSARY
by
Adam W. Marczewski
See also: Adsorption References and Basics
Links:
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 A

Absorption
 See: Nernst law, see: Henry law.
Phenomenon of transfer of some species from one phase (volume/bulk or surface phase) into the volume/bulk of another phase. [Alternatively: partitioning between such 2 phases  see Nernst law]. (E.g. absorption of CO_{2} in cold water or desorption of CO_{2} from soda water into empty flask volume).We usually talk about absorption if some phase preferentially takes in some species.
The absorbed quantity (v_{abs}) depends linearly on the size of absorbing phase (i.e. volume, V_{bulk}) and in most cases depends linearly on concentration or pressure of absorbed species (see Henry law), i.e.
v_{abs} = K_{o} V_{bulk} p_{abs} (gas absorption)
where K_{o} is constant.
If the form of absorbed species is different in both phases (molecules/ ions, monomers/ dimers etc.), then the K_{o} may also change with absorbate pressure/concentration. Other factors are e.g. pH and ionic strength (for dissociating solutes), temperature etc. Yet other factors that change the simple way the system behaves are association and complexation by other molecules/ ions (absorption depends nonlinearly on equilibrium "analytical" concentration of adsorbate). This phenomenon often accompanies adsorption or is accompanied by adsorption (e.g.: in gas or liquid chromatography, where often some liquid stationary phase is immobilized on solid adsorbent or just column wall; in adsorption/ ion exchange on polymers, e.g. resins, where the bulk phase partially enters the volume of such solid).

Adsorption (1): phenomenon
 see: Isotherm (2) (Isotherm of adsorption)
Phenomenon of transfer of some species from one phase (usu. volume/bulk phase) into the socalled surface phase (in some cases in opposite direction). I.e. increase or decrease of concentration of some species when compared to the concentration in the neighbouring bulk phase. [Alternatively: partitioning between such 2 phases]. The surface phase lies on the boundary of two physically, chemically or structurally different phases and has usually limited thickness. Minimum thickness of surface phase is monolayer, however, it may extend into hundreds of nanometers from the phase boundary  depending on molecular and electrostatic interactions. Usually it is difficult to tell where exactly this "surface phase" ends (see electrical double layer or mulitilayer adsorption). The adsorbed quantity usually depends linearly on the size (i.e. surface area) of phase boundary.
Adsorption types:
 By strength / adsorption energy:

physical (physisorption) (relatively weak  adsorption energy comparable with gas liquefaction, reversible, strongly pressuredependent and temperaturedependent)  it is usually multilayer in character (in vapors at roughly p > 0.1p_{sat})

chemical (chemisorption) (strong  adsorption energy is comparable with typical energy of chemical reactions, practically irreversible, occurs easily at low pressures and high temperatures)  practically always monolayer; chemisorbed monolayer may be accompanied by a physically adsorbed multilayer

By mobility (with increasing temperature localized adsorption may become mobile ads.):
 localized  chemisorption and strong physical adsorption with higher energy barriers (ΔE > RT) between adjacent surface sites
 partially mobile  physical adsorption with moderate energy barriers (ΔE ~ RT) between adjacent surface sites
 mobile  usu. weak physical adsorption on surfaces with low energy barriers (ΔE < RT) between adjacent surface sites
 By adsorption system:
 gas (vapor) adsorption  all components in gas phase (in vapor adsorption, at p > ca. 0.1p_{sat} the multilayer forming becomes very important; at p = 0.4p_{sat} for nitrogen at 78 K and p = 0.175p_{sat} for benzene there is the onset of adsorbate condensation in mesopores; at ca. p = 0.98p_{sat} for nitrogen, the mesopores are completely filled and at p → 1 the macropores are filled and all external adsorbent planes should be covered by an infinite multilayer, i.e. liquefied adsorbate bulk)
 liquid adsorption  in solution or liquid mixture; some components may be solutes naturally occuring in gas or solid state, for some components solubility/miscibility limits may exist (may result in phase separation)
special cases:
 aqueous  from water solutions, often dilute solute adsorption; usu. solubility limits exist  similarily to vapor adsorption, multilayer may form; associating solutes may adsorb stronger; for dissociable solutes, their adsorption depends on ionic strength, I, solution pH and surface charge density  the solutes may be adsorbed in ionic and nonionic form; surface precipitation or ionic exchange may occur;
 nonelectrolyte  mixture of nonelectrolyte liquids with contact with solid, this may include also mixtures with water but without other dissociating components and/or salts (usu. excess adsorption of mixture components in the entire possible range of concentrations: from pure A to pure B), often binary (2comp), ternary (tertiary) (3comp), quarternary (4comp)).
 solute adsorption  usu. dilute solute adsorption (molecules or ions adsorbed from liquid phase)  see above
 By no. of components:
 singlecomponent adsorption (1 gas or vapor; solution of 1 solute, i.e. solvent + solute; mixture of 2 liquids  binary liquid system  strange, isn't it)
 multicomponent adsorption (mixture of gases; solution of more than 1 solute i.e. solvent + 2 or more solutes; mixture of 3 or more miscible liquids  strange, isn't it)

Adsorption (2): adsorbed amount
 Simplified: adsorbed amount. Usu. expressed as adsorbed amount per surface area or mass of adsorbing substance. May be expressed as "true"/absolute adsorption (i.e. total amount of a species in the surface phase) or excess adsorption (Gibbs adsorption, Γ). Adsorption units: [mol/g], [mmol/g], [µmol/m^{2}], [mg/g]; for vapor/gas adsorption also: [cm^{3} STP/g]  amount adsorbed is recalculated into volume at STP (STP = Standard Temperature and Pressure)

alphasplot, α_{s}plot method
 see Isotherm (2): porous: mesoporous & microporous
Method of analysis of vapor adsorption isotherm (usu. nitrogen at 77K or benzene at 293K) on porous solids. Adsorption on porous solid is compared with adsorption on some standard  usually nonporous material of the same chemical composition and chemical structure, possibly identical but nonporous material.
The isotherm data of the standard is converted into α_{s}(x) form, where
α_{s} = ads_{std}(x) / ads_{std}(x = x_{o}), x = p/p_{s}
Here, x_{o} = 0.40 for nitrogen at 7778K (closing point of isotherm hysteresis loop), for benzene at 293K by analogy x_{o} = 0.175
The idea behind choosing specific value of x_{o} is that at this point the "monolayer" part (including micropores)  should be filledup and no mesopore filling should occur. However, some authors use x_{o} = 0.40 for benzene too following the original alphas "recipe" for N_{2} vapors (I believe it is wrong).
The experimental isotherm is plotted in coordinates:
ads(x) vs. α_{s}(x)
Analogously to tplot method, the linear sections of the plot yield slopes (corresponding to available surface) and ordinates (related to the already filledup pores). However, in contrast to the tplot, it is a general method of analysis of pore structure, where no model is assumed a priori (though the data for specific standard may be needed, where in tplot one uses generally available equations). The micropore and mesopore volume and surface areas may be estimated as well as the external area (area of pores larger than mesopores, i.e. macropores).
see also: tplot method, t/Fplot method
 B

BDDT isotherm [BDDT]
 see Isotherm (2): multilayer; 4parameter BrunauerDemingDemingTeller isotherm describes multilayer adsorption on energetically homogeneous solids

BET isotherm [BET] [nlayer BET]
 see BET surface area; see Isotherm (2): multilayer;
2parameter (∞layer) or 3parameter (nlayer) BrunauerEmmettTeller isotherm describes multilayer adsorption on energetically homogeneous solids

BET surface area
 see Specific surface area; the socalled BET surface area (denoted S_{BET} or A_{BET}) is estimated by using lowtemperature (7778K) nitrogen vapor adsorption isotherm data and linear form of BET equation called BETplot

BET plot, BET linear plot
 see Specific surface area; see isotherm plots.
A linear form of BET equation [BET]. It is used to estimate the socalled BET surface area with lowtemperature (7778K) nitrogen adsorption data.
The classic BET linear plot is: x/[a(1x)] vs. x
(where: x = p/p_{s})

BJH, BarretJoynerHalenda method
 see Isotherm (2): mesopores; a method of analysis of vapor adsorption data in mesopores (1950)  uses adsorption and/or desorption branch of adsorption isotherm starting data analysis from filled pores, requires the knowledge of adsorption isotherm on nonporous surface (e.g. experimental HJ or FHH)
 C

Capillary condensation, capillary evaporation
 See: Kelvin equation, hysteresis loop, adsorption in mesopores
The condensation/evaporation pressure, (p_{c} or p_{e}) over convex or concave surface is different than the condensation/evaporation pressure over flat surface (standard saturation pressure, p_{s}) (see Kelvin equation [Kelvin]. For convex surface (e.g. external surface of adsorbent granules) it is greater than the standard saturation pressure, whereas for concave surface (internal surface of adsorbent bead, e.g. pores, or spaces between adjacent beads) it is smaller. This phenomenon may be observed in pores of mesopore range (diameter ca. 2  50 nm). For larger pores (macropores) the condensation/evaporation pressure drop/increase is very small and hard to observe. In micropores this phenomenon changes character and cannot be treated as condensation/evaporation (single molecules or small clusters in pores  no condensed phase is present  see Theory of Micropore Volume Filling)

Competitive adsorption
 This term usually means that a molecule of component 1 is always replaced by a molecule of component 2 (if molecular sizes are the same). Such a property often makes it possible to consider adsorption of n1 components with nth a reference component (concentration of one of components may be calculated from concentrations of all other components). We always deal with competitive adsorption in solutions (solute  solvent), however, adsorption from gas mixtures may also be approximated in such a way at high surface coverages.
 D

DFT, Density Functional Theory
 used e.g. in analysis of adsorption isotherm data in order to find mesopore structure of adsorbent

DubininAstakhov (DA) [DA] isotherm
 see Isotherm (2): microporous
Isotherm equation describing adsorption in nonuniform micropores  may be considered a special solution of Stoeckli equation with the DR isotherm serving as a local isotherm. Special cases of DA equation are: Freundlich (F) and DubininRadushkevich (DR) isotherms

DubininRadushkevich (DR) [DR], biDR [biDR] (aka DubininIzotova) isotherms
 see Isotherm (2): microporous
Isotherms describing adsorption in micropores  DR for one type of pores and biDR for two types of micropores; biDR may be considered a special solution of the Stoeckli equation for bimodal pore distribution with the DR isotherm serving as a local isotherm

DubininStoeckli (DS) and DubininRaduskevichStoeckli (DRS) isotherms
 see Isotherm (2): microporous
Isotherms describing adsorption in micropores solutions of the Stoeckli equation with the DR isotherm serving as a local isotherm
 E

Electrical Double Layer (EDL)
 Due to the differences in the properties of two phases some electric charge may be created at the interface. E.g. surface functional groups (COOH, OH, NH_{2} etc.) may dissociate, some ionic species (e.g. RCOO^{}) may be adsorbed etc. The resulting surface charge is balanced by the diffuse layer of ions of opposite charge but equal in magnitude (Electroneutrality Condition) extending far from the surface. Original models were by Gouy, Chapman and Stern.

Elovich kinetic eq.
 Elovich rate equation  exponential (decaying) dependence of rate on adsorbed amount, attributed usually to chemisorption kinetics. Adsorbed amount (integrated equation) depends approx. linearly on log(time) without maximum (equilibrium) value. According to some analyses
may be used to describe middle part of IDM (intraparticle diffusion model) behavior.

Energy distribution function
 Adsorption energy distribution function χ(E) or F(E) describes the probability of finding sites of certain adsorption energy (see Isotherm (2): monolayer, heterogeneous). It is used in two forms:
 Integral form, F(E*) gives the probability of finding site characterized by an energy smaller than some threshhold, E ≤ E*. Normalized to 1 (values in the range 0 .. 1).
 Differential form, χ(E*) = (dF(E)/dE)_{(E*)}, gives the density of probability of finding site characterized by some energy E*. Normalized to 1 (∫χ(E)dE = 1)
For multicomponent systems the singlevariable energy distribution function is replaced by multidimensional distribution function, χ(E_{1} .. E_{n}). In specific cases, when the mixture components have similar properties such a multidimensional function (and correspondingly multidimensional General Integral Equation of Adsorption) may be replaced by singlevariable function.

Everett isotherm, Everett equation, Everett plot [Everett plot]
 Isotherm of adsorption (excess) from binary liquid mixtures of nonelectrolytes (no lateral interactions  or mutual compensation  in bulk or surface phase on solid surface (energetically homogeneous). A better model is regular solution theory (includes lateral interactions; based on lattice model of liquid and surface phase)

Excess, Ecess adsorption, Adsorption excess, Gibbs Adsorption
 The adsorption excess may be positive (species is preferentially adsorbed) or negative (species is repelled from surface phase) and is equal to the difference of total amount of a species close to the phase boundary (in the "whole" surface phase) minus the amount that would be present there, if the concentration would be the same as in the bulk phase, i.e. far away from the surface phase. Must be used in the description of adsorption systems where the adsorbate concentration in bulk phase is bigger than e.g. 0.02 to 0.05. Many isotherm equations require the absolute adsorption magnitude (adsorption 1). In such a case the main problem is finding the surface phase capacity (often monolayer capacity). It should be done by independent experimental methods, however if impossible  Everett equation (for system close to ideality) or Schay method may be used.

Experiment of adsorption  typical measurements
 The measured quantity is always adsorption excess.
 (almost) direct measurement of adsorbed amount (e.g. with McBain balance in gas/vapor adsorption; UV/VIS or IR reflection, ellipsometry, radiotracers)
 indirect measurement  by using "mass balance" of the system components prior to and after the experiment: pressure and volume (gas/vapor adsorption)  volumetric measurement, concentration and volume or mass (liquid asorption); various methods of composition determination, e.g. gas or liquid chromatography (liquids, organics), Atomic Absorption (metal ions), radiotracers etc.

External area, external surface area
 Basically the external area or adsorbent particles/granules (for spherical microporous granules the inner area is the micropore surface area and the granule spherical surface forms the external area). In practice it is the area of larger pores than the currently estimated ones. This "external" surface area may be determined from adsorption isotherm data by: (1) the alphas method (pores larger than mesopores) or by (2) tplot method (pores larger than micropores  it has some sense if the solid is microporous and contains little or no mesopores).
 F

FowlerGuggenheim (FG) isotherm [FG]
 see Isotherm (2): monolayer
FowlerGuggenheim (FG) (formally identical to Frumkin iso.) (nonspecific lateral interactions  meanfield aproximation), without interactions reduces to the Lagmuir isotherm. Adsorption calculated according to FG isotherm is always higher than for corresponding Langmuir equation characterized with the same adsorption equilibrium constant K. For higher values of interaction parameter α this isotherm develops phase transition.
See also a full Kiselev equation

Frumkin isotherm [Frumkin]
 see Isotherm (2): monolayer
Frumkin isotherm (formally identical to FowlerGuggenheim, FG) describes ion adsorption (e.g. on mercury, Hg).

FrenkelHalseyHill (FHH) [FHH] aka. Halsey isotherm
 see Isotherm (2): multilayer & porous: mesoporous; used also in tplot method

Freundlich (F) isotherm [F]
 see Isotherm (2): microporous;
see also: Generalized Freundlich (GF) aka. Sips (Isotherm (2): monolayer, heterogeneous)
 G

Generalized Freundlich (GF) [GF] aka. Sips isotherm (J.R. Sips 1948)
 see Isotherm (2): monolayer, heterogeneous;
Describes monolayer adsorption on energetically heterogeneous solids characterized by an asymmetrical energy distribution (extended towards high energies, has minimum energy). It is a special case of GL (Generalized Langmuir aka. MarczewskiJaroniec) isotherm. For energetically homogeneous solid it reduces to Langmuir isotherm. It may be also considered a monolayer modification of a simple Freundlich isotherm, which does not display monolayer behavior.
See also Freundlich (F) Isotherm (2): microporous

Generalized Langmuir (GL) [GL] (aka. MJ, MJ: MarczewskiJaroniec iso.)
(A.W. Marczewski & M. Jaroniec, 1983)
 see Isotherm (2): monolayer, heterogeneous;
Describes monolayer adsorption on energetically heterogeneous solids characterized by an asymmetrical or symmetrical energy distribution. For energetically homogeneous solid it reduces to Langmuir isotherm. Its special cases are: Generalized Langmuir, LF (symmetrical quasigaussian energy distribution), Generalized Freundlich, GF (asymmetrical, rightextended energy distribution) and Tóth, T (asymmetrical, left extended energy distribution) isotherms.
See also Langmuir (L) Isotherm (2): monolayer: homogeneous

Generalized Integral Equation of Adsorption (GIEA) [GIEA  gas]
 see Isotherm (2): monolayer, heterogeneous, GIEA;
This equation in a general form describes global/overall adsorption isotherm (observed isotherm) on energetically hetereogeneous solids in terms of local adsorption isotherm (adsorption on any specific site or specific surface patch depends on adsorption energy), energy distribution function (describes the variability of adsorption energy)
Ssee also Isotherm (2): monolayer: heterogeneous

Global isotherm
 see Isotherm (2): monolayer, heterogeneous;
Equation of isotherm of adsorption on energetically heterogeneous solid surface (e.g. LangmuirFreundlich, Generalized Freundlich, Tóth, Generalized Langmuir aka. MarczewskiJaroniec etc.); Also: any experimental isotherm on heterogeneous solid; summaric isotherm, result of calculation of General Integral Equation of Adsorption if the corresponding local isotherm and adsorption energy distribution function (and energetic surface topography if lateral interactions are included or molecular size is different than the size of surface site) is known (notion used also for adsorption in nonuniform micropores: Stoeckli integral equation)
 H

Halsey aka. FrenkelHalseyHill (FHH) isotherm [FHH]
 see Isotherm (2): multilayer & porous: mesoporous; used also in tplot method

HarkinsJura (HJ) isotherm [HJ]
 see Isotherm (2): multilayer & porous: mesoporous; used also in tplot method

Henry (H) isotherm
 see Isotherm (2): other equations (nonmonolayer)
Describes adsorption in gas or liquid phase at very low gas pressures (or solute concentration) where all types o adsorption isotherms reduce to the simple proportionality of adsorbent amount (a) and vs. adsorbate pressure/concentration. For gas adsorption (p  gas/vapor pressure):
a = K_{H} p
For solute adsorption (c  solute concentration):
a = K_{H} c

Henry law
 Describes gas absorption in liquid:
v_{gas} = K_{H} p_{gas}
where v is the volume of adsorbed gas per amount (mass or volume) of absorbing phase (e.g. [cm^{3} / 100cm^{3}]) and K_{H} is Henry equilibrium constant. See: Absorption; see also: Nernst law.
Henry law describes also adsorption on solid in gas and liquid phase at very low pressures or concentrations  see: Henry isotherm

Heterogeneity (here: energetic heterogeneity)
 Nonuniformity of adsorbent surface with respect to adsorption energy. Mor properly: nonuniformity of adsorption system with respect to adsorption energy  it includes possiblity that a nonuniform surface may display variuos levels of energetic heterogeneity or even behave energetically homogeneously vs. some adsorbates.

HilldeBoer isotherm (HdB) [HdB]
 see Isotherm (2): monolayer (mobile)
mobile adsorption with lateral interactions, its special case is Volmer isotherm [Volmer]

Hill (linear) plot
 See Isotherm plots.
A linear plot for Langmuir and LangmuirFreundlich isotherms (log[θ/(1θ)] vs. log(p) for gas adsorption and vs. log(c) for dilute solute adsorption); see also Isotherm (2): monolayer;

Homogeneity (here: energetic homogeneity)
 Uniformity of adsorbent surface with respect to adsorption energy. Mor properly: uniformity of adsorption system with respect to adsorption energy  it includes possiblity that a nonuniform surface may behave energetically homogeneously vs. some adsorbates.

"Hydraulic" pore size
 Calculated from the amount of adsorbate (e.g. nitrogen at 7778 K) adsorbed at specific range of relative pressures x = p/p_{s} (p_{s} is saturation pressure) and corresponding surface area. Then by assuming some specific pore shape the "hydraulic" pore size may be calculated. E.g. for slitshaped pores the average slit width x is given by x = 2V/S, for cylindrical capillaries the pore width is d = 4V/S and for spherical pores we have d = 6V/S (V is adsorbate volume, e.g. total adsorbed amount at x = 0.98; S is pore surface area, e.g. BET area or mesopore area claculated according to the BJH method)

Hysteresis, hysteresis loop (of adsorption isotherm)
 Characteristic behavior of gas/vapor adsorption/desorption isotherms measured on porous solids, where the course of adsorption and desorption isotherm branches is different over some range of pressures (or relative pressures, x = p/p_{s}). The branches follow the same course for low pressures below the socalled hysteresis loop closing point x < x_{h} (e.g. for N_{2} adsorption at 77K x_{h} = 0.39 .. 0.46 and for benzene adsorption at 20°C
x_{h} = 0.175) and for high pressures close to saturation (x → 1 for vapor adsorption and p >> p_{h} for gas adsorption). The differences in adsorption and desorption result from the mechanism of adsorption in mesopores. The vapor over concave surface (internal pore surface) may condense at pressure lower than the condensation pressure for flat surface (curvature radius, r → ∞) and at higher pressure over convex surface (e.g. external surface of the adsorbent bead). This condensation pressure drop or increase depends also on the shape of surface (sphere, cylinder, slit, saddle etc.). This behavior is described by the Kelvin equation [Kelvin]. The actual condensation pressure depends on the shape and radius of adsorbate meniscus and not only on the adsorbent surface. E.g. if cylindrical pore in adsorption is covered by liquid film of the adsorbate, this forms a cylindrical meniscus. After fillingup of pore system (adsorption branch is completed) the decrease of pressure starts the desorption branch. However, this cylindrical pore is filledup by adsorbate and the only pore part available for desorption (i.e. in contsant/equilibrium with vapor) is now only the filledpore opening with spherical meniscus. This meniscus corresponds to smaller condensation/evaporation pressure than the original cylindrical meniscus (of the same radius). This difference of condensation and evaporation pressure for some pore (corresponding to the same adsorbed amounts!) results in isotherm hysteresis. Another example may be pore with narrow opening, e.g. bottleshaped pore. Both adsorption (large radius of the pore inside) and desorption (small radius of bottleneck) menisci have the same general shape (spherical), however, the evaporation radius for desorption (small pore) will be lower that that for adsorption.
If the pores are uniform, the adsorption and desporption branches will have parallel vertical sections, which make it easy to determine pore radiuses and volumes (if their shape is known)  see: BJH method.
 I

IAS theory
 Ideal Adsorbed Solution (IAS) theory (Myers & Prausnitz)  thermodynamic theory (an extension of Raoult law to adsorption system) allowing for prediction of adsorption (from 1component to multicomponent adsorption). Does not base on any particular adsorption isotherm  uses experimental isotherm data.

Isobare: type of relation of physicochemical quantities
 Dependence of some physicochemical quantity (e.g. adsorption, enthalpy) on some other variable (e.g. temperature) obtained for a fixed system pressure (e.g. open container at atmospheric pressure).

Isochore: type of relation of physicochemical quantities
 Dependence of some physicochemical quantity (e.g. pressure, enthalpy) on some other variable (e.g. temperature) obtained for a fixed system size (volume).

Isostere: type of relation of physicochemical quantities
 Dependence of some physicochemical quantity (e.g. pressure, enthalpy) on some other variable (e.g. temperature) obtained for a fixed adsorbed amount. E.g. isosteric heat of adsorption.

Isotherm (1): type of relation of physicochemical quantities
 Dependence of some physicochemical quantity (e.g. adsorbed amount) on some other variable (e.g. presssure) for a fixed temperature. Here, isotherm of adsorption  the dependence of adsorption (adsorbed amount) on equilibrium pressure (gases) or concentration (liquids).

Isotherm (2): equation of adsorption isotherm
 Often short for "isotherm equation" or more specifically "equation of adsorption isotherm" (see also a general Isotherm (1)).
Most important adsorption isotherms are:

For nonporous or macroporous solids,

Monolayer (localized adsorption): Homogeneous and heterogeneous nonporous solids without lateral interactions or multilayer effects (lateral interactions and multilayer effects may be easily incorporated):
 General Integral Equation of Adsorption [GIEA]; requires equation of socalled "local isotherm" for homogeneous solid (e.g. Langmuir for monolayer or BET for multilayer adsorption) and  if lateral interactions are considered (local isotherm e.g. FG or Kiselev)  information on energetic topography of solid surface
 Isotherm equations of adsorption on heterogeneous solids
 Langmuir (L) [L] (energetically homogeneous surface).
 Generalized Langmuir (GL) [GL] (aka. MJ, MJ: MarczewskiJaroniec iso.) (energetically heterogeneous) (A.W. Marczewski & Jaroniec 1983) with its specific cases:
 Langmuir [L] (GL limiting case without heterogeneity).
 LangmuirFreundlich (LF) [LF](symmetrical adsorption energy distribution function)
 Generalized Freundlich aka. Sips (GF) [GF] (J.R. Sips 1948)(energy distribution extended towards high energies)
 Tóth (T) [Toth] (energy distribution extended towards low energies)
 "Square" (Sq) aka. UNILAN [Sq] (continuous/constant energy distribution function) (related to Tiemkin)
 "Gauss" (G) [Gauss] (gaussian energy distribution function)(nonanalytical isotherm)
 Rudzinski (R) [R] (quasigaussian energy distribution function)(nonanalytical isotherm but very similar to LF)

Monolayer (localized adsorption): Homogeneous surface and lateral interactions:
 Langmuir (L) [L] (no interactions).
 FowlerGuggenheim (FG) [FG] (formally identical to Frumkin iso.) (nonspecific lateral interactions  meanfield aproximation), without interactions reduces to Lagmuir.
 Kiselev (Kis) [Kis] (specific associative lateral interactions; in full form [fullKis] incorporates nonspecific FGlike interactions), without interactions reduces to Lagmuir.

Monolayer (mobile adsorption): Homogeneous surface:
 Volmer (no lateral interactions) (does not reduce to Langmuir) [Volmer].
 Hillde Boer (HB) (nonspecific lateral interactions  meanfield aproximation), [HdB];without lateral interactions reduces to Volmer.

Other equations:
 Henry (H): a = Kc (or a = Kp) (nonmonolayer); all isotherms of adsorption should give this behavior for c → 0 (or c → 0). Many equations (F,RP,Jos,Tiemkin,LF,GF,GL,R) do not conform to this requirement and may be used in a limited adsorption range.
 Jovanovic (Jov) [Jov] (monolayer with interactions; homogeneous surface  may be easily extended on heterogeneous surfaces: JovanovicFreundlich [JF/Jovm])
 RadkePrausnitz aka. RedlichPeterson (RP) [RP](nonmonolayer; heterogeneous surface)
 Jossens (Jos) [Jos] (nonmonolayer; heterogeneous surface)
 Tiemkin (nonmonolayer; heterogeneous surface) [Tiemkin] (related to Sq/UNILAN iso. with continuous energy distribution); used in catalysis

Multilayer: Homogeneous surface (heterogeneity may easily be introduced).
 BrunauerEmmettTeller (BET) [BET] (1938) (2parms. standard BET eq. assumes infinite no. of adsorbate layer; for macro and mesoporous solids nlayer limited 3parms. version [nlayer BET] is appropriate).
 Hüttig [Hüttig] (2parms).
 BrunauerDemingDemingTeller (BDDT) [BDDT](4parms. equation).
 Sircar (3parms) [Sircar].
 HarkinsJura (HJ) [HJ], FrenkelHalseyHill aka. Halsey (FHH) [FHH]  used only for surfaces already completely covered by adsorbate  describe continuous liquid film (in contrast to discrete layers in BET etc.)  often used for porous solids.

Porous solids

Microporous solids
 Freundlich (F) [F], DubininRadushkevich (DR) [DR], DubininAstakhov (DA) [DA], biDR [biDR] aka DubininIzotova, DubininStoeckli (DS), DubininRadushkevichStoeckli (DRS), Stoeckli [Stoeckli]
 Stoeckli integral equation of adsorption on nonuniform microporous solids [Stoeckli] (DubininRadushkevich isotherm is "local isotherm" for homogeneous micropores; distribution of micropore sizes must be known in advance in order to calculate global adsorption isotherm, or it may be found from experimental data i.e. global adsorption isotherm)

Mesoporous solids
 HarkinsJura (HJ) [HJ], FrenkelHalseyHill aka. Halsey (FHH) [FHH] isotherms (may be used for analysis of mesopore structure by using e.g. nitrogen adsorption data and BJH method)

IUPAC pore size classification
 See also Pore size distribution (PSD)
According to IUPAC (International Union of Pure and Applied Chemistry) the pores are divided according to their size into 3 main groups:

IUPAC isotherm classification
 The IUPAC (International Union of Pure and Applied Chemistry) classification of gas/vapor adsorption is based on BDDT classification (Brunauer and coworkers). The isotherms are divided into 5 types:
 Type I with 2 subtypes: Ia and Ib
Isotherms show long plateau (for approx. x = p/p_{s} ca. 0.1 .. 0.9) with possible small rise at x → 1.
 Ia  typical for microporous solids having small external surface (e.g. zeolites or activated carbons) (typically DR or DA isotherm)
 Ib  typical for microporous solids having larger micropores, the shape is closer to Langmuir isotherm (monolayer adsorption on energetically homogeneous surface) than for type Ia
 Type II  (reversible, physical) mono and multilayer adsorption on "flat" (i.e. having no micro or mesopores) surfaces  typical for BETtype isotherms of N_{2}; isotherm has linear section in ca. 0.1  0.5 range of x; Bpoint method of surface area determination may be used
 Type III  typical for water adsorption (weak adsorption forces, strong adsorbateadsorbate interactions); low equilibrium constant in fitted BET equation, low accuracy of BET surface area determination
 Type IV  typical for mesoporous solids, displays hysteresis loop due to capillary condensation. At lower pressures it is similar to type II, for high pressures x → 1 displays maximum adsorption (adsorption limit related to total capacity of pore system)
 Type V  like type III but with maximum adsorption at x → 1
 J

Jossens (Jos) isotherm [Joss]
 see Isotherm (2): other equations: nonmonolayer (heterogeneous);

Jovanovic (Jov) isotherm [Jov]
 see Isotherm (2): other equations: monolayer (homogeneous and heterogeneous);
 K

Kelvin equation [Kelvin]
 The vapor over concave surface (internal pore surface) may condense/evaporate at pressure lower than the condensation/evaporation pressure for flat surface (curvature radius, r → ∞) and at higher pressure over convex surface (e.g. external surface of the adsorbent bead). This condensation/evaporation pressure drop or increase depends also on the shape of surface (sphere, cylinder, slit, saddle etc.). This behavior is described by the Kelvin equation. The actual condensation pressure depends on the shape and radius of adsorbate meniscus and not only on the adsorbent surface. Other factors are temperature and adsorbate surface tension. In more advanced theories, the influence of the curvature of the liquid on the surface tension is also taken into account.
See also: BJH method, Hysteresis loop.

Kinetics of adsorption, adsorption rate
 The rate of adsorption (da/dt)  equivalent for batch conditions is (dc/dt). Various physical models are used. Simple equations include:
 Adsorption kinetics corresponding to Langmuir isotherm (and its derivatives including energetic heterogeneity and lateral interactions:
 Langmuir rate equation (Langmuir kinetic equation). Introduced by I. Langmuir (series of papers: 19161918) as an explanation of Langmuir isotherm:
 Analytical solutions for homogeneous surface and absence of lateral interactions (A.W. Marczewski 2010, 2012): IKL (integrated kinetic Langmuir eq.) and gIKL (generalized IKL).
 Nonanalytical solutions for nonideal systems (A.W. Marczewski 2011): RSK (regular solution kinetic model: nonspecific interactions), mRSK (modified regular solution kinetic model: nonspecific and specific/assciative Kiselev interactions) and LFmRSK (mRSK with LF heterogeneity).
 Statistical rate theory (SRT). Introduced by Ward et. al (1982). Corresponds to the same equilibrium model as Langmuir rate equation (i.e. Langmuir isotherm) and its derivatives for nonideal systems (energetic heterogeneity: LF isotherm, lateral interactions: FG isotherm), but rate equation has completely different properities (e.g. infinite initial rate)
 Kinetics of diffusion into porous particles:
 Intraparticle diffusion model (IDM), Crank et al.  kinetics of occlusion by pure diffusion into porous particles; initial rate is infinite
 Pore diffusion model (PDM), McKay et al. 1996  simple shrinking core model with external film transfer resistance  initial rate is finite, time to equilibrium is finite (artifact caused by simplified solution).
 Empirical equations
 First order kinetics (FOE), pseudofirst order kinetics (PFOE)  empirical equations, special cases of IKL/gIKL and MOE equations
 Second order kinetics (SOE), pseudosecond order kinetics (PFOE)  empirical equations, special cases of IKL/gIKL and MOE equations
 Mixed 1,2order kinetics (MOE)  the same mathematical form as IKL/gIKL  agreement with equilibrium Langmuir isotherm is not required, may be used for adsorption and desorption
 Nth order kinetics (NOE), pseudonth order kinetics (PNOE)  empirical equations, 1st and 2nd order equations are its special cases
 Inifinite series eq. (ISE), pseudoinifinite series eq. (PISE)  empirical equations, FOE/PFOE, SOE/PSOE, NOE/PNOE and MOE equations are its special cases
 Multiexponential equation (mexp)  approximation of adsorption by a sum of parallel first order terms
 Elovich eq.  Elovichian kinetics  kinetics of chemisorption

Kiselev isotherm (Kis) [Kis]
 see Isotherm (2): monolayer: homogeneous
Kiselev isotherm (Kis) includes specific associative (for molecule A associates AA, AAA, AAAA ...) lateral interactions; in full form [fullKis] it also incorporates nonspecific FGlike (FowlerGuggenheim, [FG]) interactions. This equation without lateral interactions reduces to Lagmuir isotherm [L]. Adsorption calculated according to the Kiselev equation is always higher than for corresponding Langmuir isotherm characterized by the same value of adsorption equilibrium constant K.
 L

Lagregren eq.; Lagergren linear plot
 Lagergren equation (Lagergren 1898)  linear form of the integrated pseudofirst order equation (PFOE).

Langmuir (L) [L]
 see Isotherm (2): monolayer: homogeneous
see also: Langmuir (L), Generalized Langmuir (GL) and LangmuirFreundlich (LF) (Isotherm (2): monolayer: heterogeneous);

Langmuir linear plot
 see Specific surface area; see isotherm plots.
A linear form of Langmuir equation [L]. It is used to estimate the socalled Langmuir surface area with lowtemperature (7778K) nitrogen adsorption data.
The classic Langmuir linear plot used to determine the surface area is:
x/a vs. x (where: x = p/p_{s})

Lagmuir surface area
 see Specific surface area; the socalled Langmuir surface area (denoted S_{L} or A_{L}) is estimated by using lowtemperature (7778K) nitrogen vapor adsorption isotherm data and linear form of Langmuir equation called Langmuir linear plot.
CAUTION. This estimate should be used with caution  only for isotherms that show plateau at relative pressures (x = p/p_{s}) 0.1 .. 0.3, i.e. isotherms of IUPAC type Ia or Ib. Even then, the isotherms of type Ia (for microporous solids with small external surace like zolites or activated carbons) cannot be described by Langmuir isotherm (monolayer adsorption on homogeneous solids) and Langmuir linear plot may serve only as an approximation of quasiasymptotic behavior of such experimental isotherms. So, the estimated equilibrium constant has no physical meaning and the estimated "monolayer capacity" is approximation of micropore "adsorption capacity".

LangmuirFreundlich (LF) [LF]
 see Isotherm (2): monolayer, heterogeneous;
see also Langmuir (L) Isotherm (2): monolayer: homogeneous, and Langmuir (L) and Generalized Langmuir (GL) (Isotherm (2): monolayer, heterogeneous)

Lateral interactions
 see Isotherm (2): monolayer: homogeneous
Interactions between adsorbed molecules. For nonspecific interactions see FowlerGuggenheim (FG) isotherm (localized physical adsorption) and mobile physical adsorption). For specific (associative) interactions see Kiselev isotherm (localized physical adsorption). Lateral interactions increase adsorbed amount when compared to the isotherm with the same adsorption forces (adsorbateadsorbent, i.e. "vertical" forces) but lacking lateral interactions (e.g. Langmuir or Volmerisotherms)

Local isotherm
 see Isotherm (2): monolayer, heterogeneous.
Isotherm equation on energetically homogeneous solid surface (e.g. Langmuir, FG, Kiselev, BET) used in General Integral Equation of Adsorption [GIEA] describing adsorption on energetically heterogeneous solids (see: global isotherm). This notion is also used for adsorption in nonuniform micropores (e.g. DR) in a structural version of GIEA equation, the Stoeckli integral equation [Stoeckli])

Localized physical adsorption
 See Isotherm (2): monolayer: localized; See also Adsorption: localized
Physical adsorption in which the energetic bareer between adjacent adsorption sites V is large when compared to kinetic energy of molecules, kT: V/kT >> 1. It means that the molecule has to be desorbed and then adsorbed again in order to move from one surface site to another (at least statistically). If this barreer is of the same order as kT: V/kT ~ 1 then molecules become partially mobile (see also partially mobile physical adsorption), when kinetic energy is even higher then molecules become practically completely mobile (see also isotherms and mobile physical adsorption)
 M

Macropores
 Pores with diameter larger than that of mesopores i.e. 50 nm according to IUPAC pore classification

Mesopores
 Pores with diameter smaller in the range 2  50 nm (IUPAC pore classification), in some papers the range 2  100 nm is still used

Micropores
 Pores with diameter smaller than 2 nm according to IUPAC pore classification

Mobile physical adsorption
 See Isotherm (2): monolayer: mobile; See also Adsorption: mobile
Physical adsorption in which the energetic bareer between adjacent adsorption sites V is small when compared to kinetic energy of molecules, kT: V/kT < 1. It means that the adsorbed molecule may easily move from one surface site to another (at least statistically). If this barreer is of the same order as kT: V/kT ~ 1 then molecules become partially mobile (see also partially mobile physical adsorption), when the energy bareer is large when compared to kinetic energy then molecules become practically localized (see also isotherms and localized physical adsorption)

Monolayer
 Surface phase with thickness not exceeding 1 molecule size. For nonspherical molecules it may have different thickness depending on orientation of adsorbed molecules. Often used to describe a class of adsorption isotherms that assume that no multilayer effects exist, or sometimes describes a behavior of a part of multilayer closest to the adsorbent surface. Mostly true for chemisorption

Monolayer isotherm (see: Isotherm (2): monolayer)
 Often used to describe a class of adsorption isotherms that assume that no multilayer effects exist, or sometimes describes a behavior of a part of multilayer closest to the adsorbent surface. May be used esp. for gas adsorption far from vapor saturation point and for dilute solute adsorption far from solubility limit. If multilayer effects are considered, the monolayer part of adsorption may be analysed in order to describe adsorbatesurface interactions and surface heterogeneity. If a monolayer is chemisorbed (strong adsorption) then following adsorption layers (aka. adlayers) are adsorbed physically (weaker adsorption)

Multilayer isotherm (see: Isotherm: multilayer)
 Assumes that the adsorbed species is ordered in layers forming monolayer in contact with the surface and subsequently adsorbed layers of adsorbate not in contact with surface (usu. it is assumed that the presence of surface does not affect or affects weakly layers no. 2,3...).

Multicomponent adsorption
 Adsorption from gas mixtures (more than 1 component); adsorption from dilute solutions containing solvent and more than 1 dissolved component; adsorption from liquid mixtures containing more than 2 components (adsorption in binary mixture may be reduced to a singlevariable isotherm).
Most often used equation: multiLangmuir aka. MarkhamBenton equation (homogeneous surface, no lateral interaction). This equation may be used as a "local isotherm" in the multicomponent version of General Integral Equation of Adsorption  (multidimensional) energy distribution function and possibly topography of surface sites must be known.
 N

Nernst law, Nernst partition coefficient
 (see also: Absorption; see: Henry law)
The Nernst law describes partition of substances between 2 immiscible (nonmiscible) or partially miscible fluids A and B (fluid = gas/vapor or liquid), i.e. gas  liquid or liquid  liquid (and also for any combination of such phases separated by a suitable semipermeable membrane, A:B, or separated by another phase C, ACB). Such partition may also occur between some solids (e.g. resins).
Partition coefficient K_{N} is defined as a ratio of equilibrium concentrations in both phases,
K_{N} = [x]_{phase1} / [x]_{phase2}
As a rule as a phase1 is chosen either (i) phase with lower density (i.e. "upper" phase  e.g. gas over liquid, organic liquid over water) (ii) organic phase1 over water phase2 (many organic solvents have low densities). If the value of partition coefficient is unknown for partition of a substance between 2 liquids, approximate value may be obtained by using solubility data:
K_{N,approx} = [x_{sat}]_{phase1} / [x_{sat}]_{phase2}.
However, one has to be cautious with systems where the substance partitioning between two fluids is present in different forms  this law is valid for identical form in both bulk phases only. If a substance dissociates, associates etc. (e.g. benzoic acid in benzene/water system  depending on solution pH benzoate anions appear in water, dimers appear in benzene) then the classic form of partion coefficient calculated from "analytical" concentrations in both phases will give value which will be concentration and pH dependent. In such a case the real concentrations of partitioned species present in both phases should be taken into account (e.g. molecular form of beznoic acid, BA, for partition between benzene and water solution: in benzene the BA dimers are formed, whereas in water most inportant factor is dissociation which depends on pH and ionic strength  both dimers and benzoate anions do not take part directly in partitioning).
In general in the formulation of partition coefficient any kind of variable describing concentration may be used (e.g. for gas phase, pressure: p_{x} = c_{x}RT). Henry law of gas absorption is a special case of Nernst law.
 O

Overall coverage  global coverage
 Adsorption coverage averaged all over the surface (various sites/surface patches having different local coverage)  see: Isotherm (2): General Integral Equation of adsorption
 P

Partition coefficient, partition law
 See: Nernst partition law, Henry law, Absorption
Describes in what proportion some species (substance) is (spontaneously) partitioned (i.e. divided) between two phases separated by a phase boundary, semipermeable membrane or another phase.

Pore size, Average pore size
 see also: Hydraulic pore size, Pore size distribution (PSD), IUPAC pore classification
Usually calculated from the amount of adsorbate (e.g. nitrogen at 7778 K) adsorbed at specific range of relative pressures x = p/p_{s} (p_{s} is saturation pressure) and averaged in various ways. One has to be cautious as to the definition of "size"  it may be diameter, radius, width or even halfwidth!

Pore size distribution function (PSD)
 Usu. mesopore size distribution function (see: BJH method). Usu. gives the volume of pores as a function of their size (e.g. diameter D) e.g. in integral form the volume of pores of diameter D ≤ D*  V(D*). Most often is is used in a diferential form dV/dD. Very often a semilogarithimic function dV/dlog(D) is used  it is a good choice if pores have wide distribution (e.g. from 2  100 nm). However, for simplicity, both types of pore distribution functions are usu. presented as the f(D) vs. log(D). Related functions are distributions of pore surface dS/dD and dS/dlog(D). The relation obtained from the hydraulic model of pores depends on the pore shape  for slit shaped pores we have dS/dD = (2/D) dV/dD, for cylindical pores dS/dD = (4/D) dV/dD and for spherical pores dS/dD = (6/D) dV/dD. If the pore model is different from the actual one, PSDfunctions obtained from adsorption and desorption data are different. See also: hysteresis loop of adsorption data and IUPAC pore classification.

Pore volume, Total pore volume
 Usually calculated from the adsorbate (e.g. nitrogen at 7778 K) volume at relative pressure x = p/p_{s} ca. 0.98 (p_{s} is saturation pressure), where micropores and mesopores (but not the macropores) are already filledup
 Q
 R

RadkePrausnitz aka. RedlichPeterson (RP) isotherm [RP]
 see Isotherm (2): other equations (nonmonolayer)
 S

Scatchard plot (an analog of linear Langmuir plot)
 See Isotherm plots.
A linear plot for Langmuir isotherm (for gas adsorption: a/p vs. a; for solute adsorption: a/c vs. a)  an analog of linear Langmuir plot (a vs a/p); see also Isotherm (2): monolayer;

Sircar isotherm [Sircar]
 see Isotherm (2): multilayer;

Sorption
 The mechanism of sorption is not specified. May include sorption by surface i.e. adsorption (chemisorption and physisorption), ionic exchange, surface precipitation (insoluble compounds formed near or at the surface), electrosorption (adsorption due to electrical potential), absorption (sorption by volume) etc.

Specific surface area
 See also: BET surface area, Langmuir surface area
Adsorbent specific surface area (A or S [m^{2}/g]) usu. calculated from the adsorption data of nitrogen vapor (at 7778 K) by using the linear form of BET isotherm equation ( x/[a(1x)] vs. x where x = p/p_{s}) (for nonporous, macroporous and mesoporus solids) or Langmuir equation ( x/a vs. x or p/a vs. p) (for microporous solids) (see isotherm plots). Although the use of both equations has serious deficiencies, it is a kind of standard and useful in practice  especially if used with caution or for comparisons.
Generally the estimation of surface area requires the assumption about the area occupied by adsorbate molecules (crosssection area A or σ  for N_{2} usu. 0.162 nm^{2}) on the surface. If we may separate from adsorption data the monolayer part of adsorption isotherm (requires some isotherm equation) and this monolayer is filled completely, then the (no. of molecules) times their (crosssection area) gives (adsorbent surface area).

Stoeckli equation, Stoeckli integral equation
 Stoeckli integral equation is a general equation describing adsorption in micropores. (It is a structural counterpart of the General Integral Equation of Adsorption used for energetic heterogeneity). For adsorption in uniform micropores a DubininRadushkevich (DR) isotherm is a solution  the DR equation serves as a local isotherm. Depending on the micropore size distribution various adsorption isotherms are obtained, e.g. DubininAstakhov (DA) (with its special cases: Freundlich, F and DubininRadushkevich, DR), biDR, DubininStoeckli (DS) or DubininRadushkevichStoeckli (DRS).

STP, Standard Temperature and Pressure (gas conditions)
 STP: T = 273.15K and p = 1013.25 hPa
Currently two similar sets of conditions are used (see "Physical Chemistry" by P.W. Atkins):
Normal conditions (for gases, t = 0°C) equiv. to STP:
T_{norm} = 273.15K (t = 0°C) and p_{norm} = 1013.25 hPa
Standard conditions (thermodynamics, t = 25°C  room temperature):
T^{Θ} = T_{std} = 298.15K and p^{Θ} = p_{std} = 1 bar = 10^{5}Pa = 1000 hPa

Surface charge density
 The density of electric charge at (usu.) solid liquid interface. The surface functional groups (NH_{2}, COOH, =CO, SiOH, SiOSi etc.) may dissociate or accept protons, the adsorbed substances  the adsorbate proper and the ions of backgroud electrolyte  may possess its own charge. This charge is "countered" by a socalled diffuse layer containing higher concentration of ionbearing entities of opposite charge. The total charge of this EDL (see: Electrical Double Layer) is 0 according to the Electroneutrality Condition (the total charge of solution and adsorbent must be 0).
 T

tplot method (de Boer), t/Fplot method (Kadlec)
 see Isotherm (2): microporous: microporous;
Method of analysis of vapor isotherm (usu. nitrogen at 77K) on microporous solids.
Uses isotherm data over the range where the monolayer is filledup but before the condensation in classical mesopores starts (isotherm histeresis closing point, for nitrogen at 7778K is at approx. x = p/p_{s} = 0.40 .. 0.45). The course of isotherm plot
ads(x) vs. t(x)
where t(x) is statistical thickness of adsorbate is extrapolated towards t(x) = 0 (and x = 0). The intersection with adsorption axis gives micropore capacity and the slope yields socalles S_{ext} "external surface area" (nonmicropore area). The statistical thickness t(x) = t_{mono} θ(x) is often calculated with HarkinsJura (HJ) or FrenkelHalseyHill (FHH) aka. Halsey equations.
The t/Fplot includes analysis of micropore part of isotherm with the use of DR equation.
see also: α_{s}plot

Tiemkin isotherm [Tiemkin]
 see Isotherm (2): other equations: nonmonolayer (heterogeneous); corresponds to an infinite continuous and constant distribution of adsorption energies; used mainly in catalysis; it is a special form of a more general UNILAN aka Sq isotherm [Sq]

Topography, surface topography
 Energetic topography of surface sites must be known if lateral interactions on heterogeneous solid surface are to be taken into consideration (see also: Isotherm (2): monolayer, heterogeneous)
 Patchwise topography  sites of given energy have neighbors of the same energy (patches are so large that the sites on patch boundaries may be neglected  almost 100% energy correlation). E.g. crystalline solids, mixtures of energetically homogeneous solids, solids with partially covered chemically modified surface.
 Random topography  sites have neigbors of random energy (the probability of finding a given energy neighbor for any site is the same as probability for the entire surface  no energy correlation). E.g. amorphous solids, solids with large amounts of impurities, very well homogenized (finely ground) solid mixtures, solids with developed porous structure
 Mixed topography  probability of finding a given neighbor for some site is not equal to the surfaceaverage probability  partial energy correlation). E.g. partially crystalized solids (amorhous + crystalline), mixture of chemically different amorhous solids.
In most cases one of the boundary models (patchwise or random) is assumed as they are much easier to solve or isotherms are fitted and a better fitting one is assumed to reflect solid structure
Usually for continuous energy distribution function (e.g. amorphous solid with developed pore structure and impurities) swe use General Integral Equation of Adsorption which is easy to solve for random topography (which is also an obvious choice). However, for discrete energy distribution function (e.g. crystalline, well defined solid or mixture of such solids or such solid with chemically modified surface) we may use simple summation of individual adsorption isotherms on homogeneous patches (patchwise model).

Tóth (T) [Toth]
 see Isotherm (2): monolayer, heterogeneous;
see also Langmuir (L) Isotherm (2): monolayer: homogeneous and Langmuir (L) and Generalized Langmuir (GL) (Isotherm (2): monolayer, heterogeneous)
localized adsorption without lateral interactions on heterogeneous solid, corresponds to asymmetric quasigaussian energy distribution function (no min./max. energy, widened towards lower energies)
 U

UNILAN isotherm aka Sq [Sq]
 see Isotherm (2): monolayer (heterogeneous)
localized adsorption without lateral interactions, coresponds to finite (max. and min. energy) continuous energy distribution function (constant value)
 V

Volmer isotherm [Volmer]
 see Isotherm (2): monolayer (mobile)
mobile adsorption without lateral interactions; special case of HilldeBoer (HdB) isotherm
 W
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