Water adsorption on carbons--critical review of the most popular analytical approaches.

Abstract

The purpose of the current study is to present the state of art in the field of analytical description of water sorption on carbons. We discuss the most important and promising models proposed recently (for example by Mahle; Talu and Meunier; and Malakhov and Volkov) as well as some older theoretical models inspired by the pioneering ideas proposed in the papers of Dubinin, Serpinsky, Barton, D'Arcy, Watt, Do and Do and others. The applicability, advantages, and defects of all these analytical formulas are pointed out and some new approaches in this field are presented. The special attention is paid to the finite adsorption space and the possible involvement of partial chemisorption, i.e. the existence of various types of the hydrophilic centres. Since the calculation of isosteric enthalpy from an adsorption equation, and the comparison of theoretical enthalpy plot with the values measured calorimetrically, is the fundamental condition for the verification of the correctness of an adsorption model, for all considered models we show the corresponding adsorption enthalpy equations. The validity of all mentioned above models is verified for the data measured for five water-activated carbon systems. Finally, a summary of obtained results and some perspectives and suggestions for the description of experimental data are presented. From the analysis of experimental data it is seen that developed recently- the heterogeneous Do and Do model is probably the most successful for the simultaneous description of water adsorption and enthalpy of adsorption results.

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Water adsorption on carbons — Critical review of the most
popular analytical approaches☆
Sylwester Furmaniak, Piotr A. Gauden, Artur P. Terzyk⁎, Gerhard Rychlicki
N. Copernicus University, Department of Chemistry, Physicochemistry of Carbon Materials Research Group, Gagarin St. 7, 87-100 Toruń, Poland
Available online 30 August 2007
Abstract
The purpose of the current study is to present the state of art in the field of analytical description of water sorption on carbons. We discuss the
most important and promising models proposed recently (for example by Mahle; Talu and Meunier; and Malakhov and Volkov) as well as some
older theoretical models inspired by the pioneering ideas proposed in the papers of Dubinin, Serpinsky, Barton, D'Arcy, Watt, Do and Do and
others. The applicability, advantages, and defects of all these analytical formulas are pointed out and some new approaches in this field are
presented. The special attention is paid to the finite adsorption space and the possible involvement of partial chemisorption, i.e. the existence of
various types of the hydrophilic centres. Since the calculation of isosteric enthalpy from an adsorption equation, and the comparison of theoretical
enthalpy plot with the values measured calorimetrically, is the fundamental condition for the verification of the correctness of an adsorption model,
for all considered models we show the corresponding adsorption enthalpy equations. The validity of all mentioned above models is verified for the
data measured for five water-activated carbon systems. Finally, a summary of obtained results and some perspectives and suggestions for the
description of experimental data are presented. From the analysis of experimental data it is seen that developed recently- the heterogeneous Do and
Do model is probably the most successful for the simultaneous description of water adsorption and enthalpy of adsorption results.
© 2007 Elsevier B.V. All rights reserved.
Keywords: Activated carbon; Water adsorption; Chemisorption; Hydrophilic centres; Primary adsorption sites; Modelling; Microporosity; Enthalpy of adsorption;
Calorimetry
Contents
1.
2.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Semi-empirical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.The Dubinin and Serpinsky approach and its improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2. Mahle isotherm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Theoretical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.The improvements of the D'Arcy and Watt isotherm — from original model to generalized D'Arcy and Watt (GDW) and
the multi-site generalized D'Arcy and Watt (MSGDW) equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.The Talu and Meunier isotherm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.3. The CMMS (cooperative multimolecular sorption) approach of Malakhov and Volkov. . . . . . . . . . . . . . . . . . . . 103
3.4. Polymodal versions of the CMMS (PCMMS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.5.The Do and Do (DD) isotherm and its heterogeneous version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.6. Heterogeneous DD with continuous energy distribution (HDDCD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Fitting models to experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.1. Adsorbents — their surface and porosity properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.2. Water adsorption isotherms and the differential enthalpy of adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
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85
85
90
933.
93
4.
Advances in Colloid and Interface Science 137 (2008) 82–143
www.elsevier.com/locate/cis
☆This paper is dedicated to Professor Władek Rudziński on the occasion of his birthday.
⁎Corresponding author. Tel.: +48 56 611 43 71; fax: +48 56 654 24 77.
E-mail address: aterzyk@chem.uni.torun.pl (A.P. Terzyk).
URL: http://www.chem.uni.torun.pl/~aterzyk/ (A.P. Terzyk).
0001-8686/$ - see front matter © 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.cis.2007.08.001

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4.3.
4.4.
4.5.
4.6.
4.7.
4.8.
4.9.
The comparative analysis of results and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Numerical procedure of the fitting of theory to experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dubinin and Serpinsky related models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mahle isotherm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
D'Arcy and Watt related models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Talu and Meunier isotherm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The CMMS model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The DD and HDD isotherms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
125
127
131
131
134
134
134
137
140
140
141
5.
1. Introduction
The properties of water are of importance in many scientific
disciplinessuchaschemistry,biology,geology,nanotechnology,
and materials technology. Moreover, the adsorption of water on
activated carbonsisanimportanttopicinmanydifferentareas of
science and technology because water is the most common
solvent in nature. While the adsorption and phase behaviour of
polar fluids in carbon pores has been studied extensively, our
understanding regarding adsorption of water in carbonaceous
materials is still incomplete [1–10].
In recent years a number of experimental and simulation
studies of adsorption of water in pores have appeared in the lit-
erature.Somestudieshaveassumedthattheadsorptionbehaviour
of water in graphite pores is of hydrophobic nature. Although
principallyhydrophobicadsorbentsmaycontainsignificantnum-
bers of adsorption centres that can interact with water, it is
generally believed that the combination of weak carbon-water
dispersive attractions and strong water-water associative interac-
tions is responsible for the complex behaviour of water confined
in carbonaceous pores. On the other hand, the heterogeneity (for
example somepolargroups) canbeintroducedonthe surface and
those active sites usually lead to enhanced water adsorption. The
lossofwater-waterH-bonds ingraphitic pores iscompensated by
the ability of water molecules to form H-bonds with the surface
sites in activated carbon pores, resulting in the change of the
molecular level adsorption phenomena from capillary condensa-
tion in graphite pores, to continuous pore filling in activated
carbon pores [1,3,10].
The strong effect of primary adsorption centres on water
adsorption was indicated on the basis of the computer simu-
lations [3,9,11–23]. The adsorption process of water mole-
cules on hydrophobic surfaces has been also investigated
[9,12,13,16–19,22] applying the advanced computer calcula-
tions. On the other hand, several types of polar oxygen-
containing sites (e.g. carboxyl [20] carbonyl [12,20,21], and
hydroxyl [15,20] groups or H-sites [18], with different densities
and local distributions) are studied on the surface of the carbon
in order to determine the influence on the adsorption of water.
Computer simulation models are still developed and more and
more complicated systems have been analyzed. Brennan,
Thompson and Gubbins [21,23] in order to provide the realistic
pore representation and the description of water adsorption,
recently suggested the application of the model based on reverse
Monte Carlo. In this model surface sites have been added at
random points on the edges of the graphene microcrystals
possessing random sizes and structures. The major drawback to
advanced numerical calculations is the computing power and
time needed to simulate isotherms. Therefore, simple analytical
approaches are often favoured in order to simplify consideration
and cut down significantly the time of computations (for
instance, the so-called local isotherms (obtained from the
computer simulation) are described by the theoretical model
proposed by Talu and Meunier [24]). Muller and Gubbins [14]
showed, that for non-activated carbon slit pore almost no
adsorption occurs until a sharp vertical rise in the adsorption
curve is seen and the pore fills suddenly (i.e. the capillary
condensation occurs). If the surface is doped by active sites,
adsorption isotherm changes drastically, strong bonds are
formed between these sites and water molecules, and adsorbed
molecules become nucleation sites for other water molecules to
adhere. Thus, these results confirm the role of primary surface
sites in the enhancement of water adsorption. Furthermore,
Jorge et al. [20] showed that the local distribution of primary
sites (carboxyl, hydroxyl, and carbonyl ones) has strong
influence on the low-pressure part of the adsorption isotherm,
while the overall site density affects mainly the vapour-liquid
phase transition. The type of oxygen-containing group was
shown not to be of critical importance, since more complex
groups can effectively be represented by simpler sites. On the
other hand, McCallum et al. [15] proposed two alternative
mechanisms for water adsorption onto activated carbon walls
due to regularly arranged surface sites or a random array (the
same site density in both cases was assumed). The five stages
are observed from the analysis of the both types of simulated
adsorption isotherms. Moreover, their behaviour is similar to
experimental ones.
It should be pointed out that one of the most spectacular
method of the investigation of the adsorption energetics and
the mechanism of this process is the adsorption calorimetry
[2,8,10,25–38]. The careful measurements of the energetic
effects accompanying water adsorption processes on carbon
blacks and carbons were studied by scientists from the so-called
“Russian school of adsorption” in the fifties [31,32] and the
eighties [2,33,34] and by others [25–27,29,30,35–38]. Heat
of adsorption of water indicates that adsorption is a strong
function of surface chemistry. The primary high-energy sites
have significant influence on adsorption and the enthalpy of
adsorption at low relative pressures. Three components can be
delineate according to measured differential adsorption
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enthalpy: (i) chemical adsorption, (ii) physical adsorption, and
(iii) condensation and/or micropore filling. From the point of
view of the thermodynamic verification of the theoretical
models the first case is the most interesting and important.
Chemisorption of water on carbons has been often reported
[2,8,10,25–27,37,38]. Summing up, the analysis of the state of
art shows significant discrepancies between the role of specific
surface groups (and calculated from theory the values of the
energy of interaction with water) and a quantitative description
of the isotherm over the entire relative pressure range that is
universal for all carbons with low, medium, and high densities
of surface groups. It should be pointed out that the correctness
of adsorption equations is easy for verification via the com-
parison with the results obtained (for example) from adsorption
calorimetry. The latter method is particularly important since it
allows elucidating the thermodynamic state of the adsorbed
molecules.
A more complex problem considered by quantum methods is
theinteractionofwaterwithgraphite-likeandcarbonaceous-like
structures containing oxygenated hydrophilic groups [39–42].It
is well-known that adsorption is strongly dependent on the
presenceandarrangementofthehydrophilicgroups.Hamadand
co-workers [39] reported a quantum chemical characterization
of a hydrophilic site modelled by a carboxyl group attached to
one of the carbon atoms in the graphite layer. The interaction
between only one water molecule and this surface group was
investigated.Theysupplyvaluesofpointchargesatthe–COOH
sites that have been used to derive a pair potential for the water-
active site interaction. Picaud at el. [11] presented molecular
dynamics simulations based on classical potential issued from
quantum mechanics calculations to provide a view of the
dynamics and the structure of a water adlayer on the model soot
surface consisting of a planar graphitic layer plus different
numbers of active carboxylic sites. The results showed that the
–COOH groups act as strong trapping sites for a few water
molecules that then become nucleation centres for other water
molecules that form larger aggregates tied to the –COOH.
Additionally, they also show that water adsorption on a model
soot surface depends on the type, number, and the relative
positions of active carboxyl surface sites on the surface. On the
other hand, Tarasevich and Aksenenko [40] used the semi-
empirical PM3 method to study the interaction of very few
water molecules with a partially oxidized graphite surface. The
main conclusion of this work is that water molecules are
adsorbed at the hydrophilic centre (i.e. the carboxyl) with
the formation of microclusters consisting (at initial coverages)
of 2 up to 3 molecules. In the light of these results, Tarasevich
and Akseneko [40] suggested that water adsorption measure-
ments cannot be used to determine the number of active centres
even at hydrophobic, and, far less, at hydrophilic surfaces
(instead methanol molecule was recommended for this
purpose). However, the studies of Tarasevich and Akseneko
[40] and Hamad and co-workers [39] are limited only to one
type of surface groups (i.e. –COOH). The confirmation of very
complicated mechanism of water adsorption on –COOH, –O–,
–OH, and –H surface groups and formation of water
microclusters was also given [42].
Inouropinion,itisdifficulttodrawunambiguousconclusions
about the applicability of water adsorption measurements to
determine the number of surface active centres. In other words,
the relationship between the amount of primary centres and the
numberof adsorbed water molecules isstill open question[2]. To
solve this problem, further detailed experimental and simulation
studies of water adsorption are necessary. The next important
problem is the structure of water adsorbed in pores. Some
experimental, i.e. X-Ray Diffraction (XRD) [43], Differential
Scanning Calorimetry (DSC) [44], Dielectric Relaxation Spec-
troscopy (DRS) [9,44,45] and theoretical studies [9,44,45]
showedthepresenceofawell-orderedstructureofwateradsorbed
in the graphite micropores at ambient temperature. Adsorbed
waterhasmoreorderedstructurethanbulkliquid,butlessthanice
andthestructureofwatermoleculesdependsofthewidthofpores
[9,43]. Summing up the progress in computer simulations and
quantum methods led to very interesting results making possible
the better understanding of the experimental observations and
water adsorption mechanisms.
Another important problem in the theoretical description of
water sorption on carbons is still unknown model of bulk water.
This was recently discussed by Gun'ko et al. [7]. They pointed
out the facts that the properties of bulk and interfacial water are
very complex and, in general, unusual that have been shown in
detail using experimental and theoretical methods. They also
presented the recent progress in theory of description of bulk
water, showing that it can be described in terms of a two-state
mixture model including high and low density water being in
equilibrium. Moreover, they concluded that solutes, surface
functionalities of macromolecules and solid surfaces can be
separated into two groups with chaotropic (water structure
breakers) and cosmotropic (i.e. structure makers).
On the other hand, as it was mentioned by Cerofolini and
Rudziński [46] despite the progress in computer simulations, it is
still necessary to look for the elegant analytical formulas de-
scribing adsorption isotherms and being suitable for a wide range
of scientist and engineers. Analytical approaches will always be
competitive as far as the computational time is considered (this is
especially true in the case of simulations of water sorption where
the computations of the energy of long range electrostatic
interactions are very time consuming). That is the reason why
in this review we try to present different approaches based on
analytical formulas. Some older as well as the most sophisticated
ideas are shown and their weak points are stressed. Simulta-
neously the new formulas describing the isosteric adsorption
enthalpyrelatedtoallmodelsarederivedinthisstudy(ifwerenot
presentedintheoriginalpapers).Insomecasestheroughanalysis
of generated adsorption enthalpy plots is sufficient to show how
some models are far from reality — generating completely
unrealistic(ifcomparedwithexperimental)shapesoftheenthalpy
of water sorption curves.
The models of water adsorption discussed in this paper can be
dividedintotwo groups(Scheme 1).First (calledsemi-empirical)
originates from the pioneering idea of Dubinin and Serpinsky.
We discuss here the both historical DS models (DS1 and DS2),
some related approaches (CDS) and the models originating from
the DS idea and formed by Barton et al. (B1, B2 and GB1). All
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those modelsare placedtothe semiempirical groupsince herethe
derivations often use empirical corrections to fit the shapes of
isotherms to experimental data. This kind of “derivation” of the
isotherm equation was also applied recently by Mahle and this is
the reason why his isotherm is also discussed in this group.
Second group form the models having more or less reliable
theoretical derivations. Here they are discussed starting from the
oldest to the newest ones, i.e. we start from the D'Arcy–Watt
(DW) model and its generalization (GDW) and modifications
(MSGDW, MSDWGB). Next Talu–Muenier approach is dis-
cussed. Finally, wepresent the newestapproachesi.e.the CMMS
approach formed by Malakhov and Volkov (with modification
called PCMMS), and the Do–Do model (DD) (with its hetero-
geneous versions HDD and HDDCD).
Theapplicability,advantagesanddefectsofalltheseanalytical
formulas are pointed out and some new approaches in this field
are proposed. The special attention is paid to the finite adsorption
space and the possible involvement of partial chemisorption and
heterogeneity,i.e.theexistenceofvarioustypesofthehydrophilic
centres. All mentioned above models are verified (the simulta-
neous description of adsorption and enthalpy of adsorption data)
for five water-activated carbon systems in Section 4. Finally,
Section5givesasummaryofobtainedresults,someperspectives,
and suggestions for the description of experimental data.
2. Semi-empirical models
2.1. The Dubinin and Serpinsky approach and its improvements
The Dubinin and Serpinsky (DS) approach [32,47–49],
leading to the explanation of the mechanism of water adsorption
on carbonaceous materials, is still very attractive and often im-
proved[26,27,50–54].Althoughthe fundamentalassumptions of
the resulting model were introduced almost fifty years ago it is
still widely applied due to simplicity and giving reliable results
[29]. In the mid-1950s Dubinin et al. [32,47,48] introduced a
phenomenological model of adsorption of water molecules on
energy privileged sites acting as primary adsorption centres for
wateronpredominantlyhydrophobiccarbonsurface.Ontheother
hand, water molecules adsorbed on these sites create some new
centres (called elsewhere the secondary ones) for adsorption of
the subsequent molecules via the cluster formation. The original
DS1 equation can be written as [32,47,48]:
a ¼ c a0þ a
where a0is the surface concentration of the energy privileged
hydrophilic adsorption centres, h is the relative pressure (=p/ps,
where p and ps is the equilibrium and saturation pressure,
respectively),andc isthe ratiooftherateconstants describingthe
kinetics of adsorption and desorption (kads/kdes). On the other
hand, k represents the loss of the secondary sites in the course of
adsorption and thus the value of this constant affects the maximal
adsorption capacity. In the original DS1 equation it is equal to
unity, and it is treated as a part of the constant c (in other words,
thisparameterisalwaysomittedinEq.(1)).Itshouldbenotedthat
k is introduced in order to compare the decrease in the number of
adsorption centres postulated by this simple model and by other
models inspired by the Dubinin and Serpinsky concept (see Eqs.
(2)–(4) and the legend in Fig. 1). It is well known that Eq. (1)
describes well only the data measured on non-porous adsorbents
and on strongly hydrophobic carbons [1,2,26,27,32,50,54,55].
Therefore, the main disadvantage of this simple model is
the assumption of unlimited adsorption space (or the lack of
saturation of secondary adsorption centres). The applicability
of the DS1 model is limited to the hyperbolic behaviour of
the adsorption isotherms (i.e. the water adsorption generated on
the basis of Eq. (1) leads to infinity if the reduced pressure tends
to (1/c)).
So the original DS concept was improved by Dubinin et al.
[48], and by Barton and co-workers [52,53].The initial stages of
water adsorption mechanism on carbonaceous solids are similar
ðÞkh
ð1Þ
Scheme 1. The models discussed in this study divided into two basic groups.
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asassumedintheoriginalDS1equation(i.e.watermoleculesare
initially strongly adsorbed via hydrogen bonding on surface
sites considered as specific or active groups). However, further
adsorption causes the creation of water clusters at larger relative
pressures and this process decreases the number of secondary
adsorption sites. Thus, Dubinin et al. [48,56,57] proposed the
following equation:
a ¼ c a0þ a
which is frequently called the DS2 adsorption equation. More-
over, Barton and co-workers [52] published an alternative modi-
fication of the DS1 isotherm. Resulting Barton's isotherm (B1)
can be written as:
?
ðÞ 1 ? ka
ðÞh
ð2Þ
a ¼ c a0þ a
ðÞ 1 ? ka2
?h
ð3Þ
In the both equations the finitely of the adsorption space is taken
into account by the terms: (1−ka) and (1−ka2). It should be
pointedout thatthese bothterms takeintoaccount the decreasein
thenumberofadsorptionsiteswithincreasingadsorption.Eqs.(2)
and (3) do not describe satisfactorily the final parts of water
adsorption isotherms measured for strictly microporous carbons
[52]. Therefore, Barton et al. [53] postulated similar relationship
introducing the term (1−exp[−k2(a−ac)2]) and proposing the
following adsorption equation (B2):
h
Thevaluesoftheconstantkarecalculatedfromtheconditiona=as
(where asis the maximum adsorption) if h=1 [47,52,54,56,57].
In the opinion of Barton and co-workers [53], the parameters
k and acserve to trigger the start of the decline in adsorptive
power, and k, as before, governs the rate of this decline with
increasing adsorption. Obviously, both these parameters are
associated with the Gaussian-like distribution of the adsorption
sites or pore sizes of the carbon substrate. Barton et al. [53]
observed that thevalues ofacareall largerthanthose ofas.They
also stated that it is difficult to assign a physical meaning to ac.
Therefore,inordertoexplainsomeofthedoubtssurrounding
the interpretation of this parameter, water adsorption isotherms
weregeneratedonthebasisoftheoriginalDSisotherm(Eq.(1)),
and on the basis of the (presented above) three improvements
of this equation (Eqs. (2)–(4)). The results are compared in
Fig. 1(a). The values of the parameters are similar to those
calculated from the fitting of Eqs. (1)–(4) to experimental data
[26,50,52,53]. From Fig. 1(a) it is seen that the DS1 equation
(unlike the others) predicts infinite adsorption when h tends to
(1/c). This result can be easily explained in view of the fact that
the concentration of the secondary sites is constant during
adsorption (i.e. k=1; Fig. 1(b)). On the other hand, considering
the improvements of the original DS1 model reported so far,one
can generate the finite values of the adsorbed quantity (equal or
very close to as) for relative pressures tending to unity, as a result
of the gradual saturation of the secondary adsorption centres.
Summingup,Eqs.(2)–(4)introducetheempiricalfactorF(a,h)dec
equal to unity, (1–ka), (1−ka2), and (1−exp[−k2(a−ac)2]), re-
spectively(moreover,seethelegendofFig.1(b)),whichdescribes
thedecreaseintheconcentrationofadsorptionsites.Additionally,
Fig. 1(b) shows hypothetical adsorbed quantities greater than the
maximum adsorption (i.e. as=20 mmol/g). Analyzing the results
shown in this figure, one can see that acis strictly linked to the
decrease in the concentration of the secondary adsorption centres
(the Gaussian-like shape). For adsorption equal to ac, F(a,h)decis
equal to zero. In other words, for a=acall active sites are
saturated. In the original Gaussian function this parameter is the
average value of a. For acNaswe consider only the decreasing
part of F(a,h)dec(the half of Gaussian–like function is con-
sidered). In the opposite case F(a,h)decdecreases to acand next
increases tending to as(for h=1 the values of adsorption are
smaller than as). In other words, from the analysis of the ex-
perimental data [53,58] it can also be noticed that the pore filling
oftheavailableadsorptionspaceforwatermoleculesoccursprior
to the measurement of ac(i.e. it is greater than as). Therefore, this
a ¼ c a0þ a
ðÞ 1 ? exp ?k2a ? ac
ðÞ2
i??
h
ð4Þ
Fig. 1. Generated numerically water adsorption isotherms (a) and the functions
describing the decrease in the number of adsorption sites (F(a,h)dec) (b). They
were generated basing on DS-like equations (Eqs. (1)–(5), respectively). The
parameters used in calculations: c=3, a0=0.5 mmol/g, as=20 mmol/g, and
ac=45 mmol/g. Additionally, in (b) the hypothetical values of adsorbed amount
greater than the maximum adsorption (as) are presented (they are limited by the
bold solid vertical line).
86S. Furmaniak et al. / Advances in Colloid and Interface Science 137 (2008) 82–143