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Parameters from a new kinetic equation to evaluateactivated carbons efficiency for water treatment

S. Gasparda,�, S. Altenora, N. Passe-Coutrina, A. Ouensangaa, F. Brouersb

aCOVACHIMM, EA 3592 Universite des Antilles et de la Guyane, BP 250, 97157 Pointe a Pitre Cedex, GuadeloupebInstitut de Physique, B5 Sart-Tilman, 4000, Universite de Liege. 4000 Liege, Belgique

a r t i c l e i n f o

Article history:

Received 19 September 2005

Received in revised form

7 July 2006

Accepted 17 July 2006

Available online 18 September 2006

Keywords:

Fractal kinetic

Activated carbon

Adsorption

Fractal dimension

A B S T R A C T

The fractal dimension of some commercial activated carbon (AC) was determined in the

micro-, meso- and macropore range using mercury porosimetry and N2 adsorption data.

We studied the kinetic of adsorption of phenol, tannic acid and melanoidin on those ACs.

The typical concentration–time profiles obtained here could be very well fitted by a general

fractal kinetics equation qn;aðtÞ ¼ qe½1� ð1þ ðn� 1Þðt=tn;aÞaÞ�1=ðn�1Þ

� deduced from recently

new methods of analysis of reaction kinetics and relaxation. The parameter n is the

reaction order, a is a fractional time index, qe measures the maximal quantity of solute

adsorbed, and a ‘‘half-reaction time’’, t1=2, can be calculated, which is the time necessary to

reach half of the equilibrium. The adsorption process on AC is clearly a heterogeneous

process, taking place at the liquid–solid boundary, and the diffusion process occurs in a

complex matrix with a fractal architecture as demonstrated here. In fact, these systems

belong to what has been called ‘‘complex systems’’ and the fractal kinetic, which has been

extensively applied to biophysics, can be a useful theoretical tool for study adsorption

processes.

& 2006 Elsevier Ltd. All rights reserved.

1. Introduction

The theory of fractals becomes more and more widely used

for describing the structures of porous materials and the

processes occurring in such materials (Mandelbrot, 1982;

Pfeifer and Obert, 1990) Among those materials, activated

carbon (AC) consists, on a geometrical point of view, of solid

(mass) and pore space. Solids may have fractal surfaces, i.e.

boundaries separating mass and pore spaces fractals (So-

ko"owska et al., 2001). However, several physical properties of

the systems may depend not only on the fractal character of

the surface, but also on the scaling behaviour of the entire

mass and of the entire pore spaces (Soko"owska et al., 2001).

Fractality of the ACs arises from the pore network devel-

oped during the activation process and can be described by

fractal geometry (Avnir et al., 1992; Neimark, 1992). It

represents an important factor that has an influence on the

adsorption properties of the adsorbent. There are several

methods for calculating the fractal dimension of a solid on

the basis of different experiments, for example, gas adsorp-

tion, mercury porosimetry, scanning electron microscopy and

from scattering (light, X-ray, neutron) measurements (Avnir

et al., 1983; Pfeifer and Avnir, 1983; Avnir and Jaroniec, 1989;

Neimark, 1992; Ehrburger-Dolle, 1994, 1999; Terzyk et al., 2003;

Gauden et al., 2001). According to the kind of fractal analysis

or technique used, irregular objects can be characterized by

different fractal dimensions: the pore fractal, the surface

fractal and the mass fractal (Diduszko et al., 2000, Soko"owska

et al., 2001). Among those, the methods based on analysis of

adsorption isotherm play an important role (Mahamud et al.,

2004; Avnir et al., 1992), since they require only one complete

adsorption isotherm for a given adsorbent to calculate the

ARTICLE IN PRESS

0043-1354/$ - see front matter & 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.watres.2006.07.018

�Corresponding author. Tel.: +590 590 93 86 64; fax: +590 590 93 87 87.E-mail address: sarra.gaspard@univ-ag.fr (S. Gaspard).

WAT E R R E S E A R C H 4 0 ( 2 0 0 6 ) 3 4 6 7 – 3 4 7 7

surface fractal dimension. The simplicity and the general

applicability of the fractal approach attracted much interest

among surface scientists at the end of the 1980s, but only few

reviews were published (Lee and Lee, 1996, Terzyk et al., 2003)

due to the limited range of applicability of the fractal

behaviour and the complexity of some approaches. However,

some convenient methods allow to assess fractal dimension

of a given porous solid from a single adsorption isotherm and

many equations to fit experimental data have been published

(Terzyk et al., 2003), among them some simple equations are

the well-known fractal analogues of Frenkel Hasley–Hill (FHH)

(Avnir and Jaroniec, 1989; Yin, 1991) or the fractal analogue of

Dubinin–Astakhov (FRDA) equation adapted by Ehrburger-

Dolle (1994), Gauden et al. (2001) and Terzyk et al. (2003).

Furthermore, data collected from mercury porosimetry can

also be used in order to determine the fractal dimension

of ACs.

Due to their high porosity, ACs are the most widely used

materials for adsorption of chemicals (Cooney, 1999; Radovic

et al., 2001). The adsorption process of a solute on an AC takes

place at the liquid–solid boundary; it is thus clearly a

‘‘heterogeneous’’ reaction, and the interface of the two phases

represents a special environment under dimensional or

topological constraints. Furthermore, the adsorption of a

dissolved compound on an AC is a complex phenomenon

involving electrostatic and dispersive interactions, Van der

Waals and hydrogen bounding, ligands exchange; therefore,

these systems can be considered to belong to the class of

complex system. Thus, in a recent article Meilanov et al.

(2002), pointed out the necessity of the development of a

radically new approach for the study of sorption kinetic

taking account of the heterogeneity of such systems and

proposed a generalized sorption kinetics within the context of

fractal conception. In an older article, Seri-Levi and Avnir

(1993) pointed out this anomalous kinetic behaviour for

adsorption on fractal surfaces and extended a fractal

equation initially established for describing adsorption on

flat electrodes to heterogeneous chemical with fractally rough

morphology.

Nevertherless, the complex nature of adsorption and its

effects on the kinetic are rarely taken into account and the

adsorption kinetics are still generally described by classical

kinetic equations or by using diffusion equations (Wu et al.,

2001, Do and Wang, 1998). We proposed here the use of a

fractal kinetic equation, in order to describe the kinetic of

adsorption of some solutes on some ACs. Indeed application

of this model presents many advantages with regard to other

more classical ones. From this equation, two parameters that

we call, ‘‘adsorption power’’ and a half-reaction time can be

derived.

2. Experimental section

2.1. Sorbents

In this research, three commercial ACs were used, two from

Pica Manufactory (Saint-Maurice, France) named, respec-

tively, PAC6 and PAC5 and one named PACV from Prolabo

(VWR International, Fontenay-Sous-Bois, France). The ACs

were washed with water and dried at 110 1C in an oven for

24 h, then cooled and stored in a dessicator prior to use.

Two ACs were also prepared in our laboratory from vetiver

roots (Vetiveria zizanioides) collected in Guadeloupe. The

lignocellulosic raw materials were dried at 105 1C for 48 h

using a drying oven, then crushed. The fraction with a

particle size ranging between 0.4 and 1 mm were chosen for

preparation of the ACs. Two conventional methods of

preparation of AC are used, the physical and chemical

activation. For physical activation, approximately 5 g of pre-

treated vetiver roots are initially pyrolyzed in a furnace

Thermolyne F-21100 at 800 1C for 1 h with a heating rate of

10 1C/min. The chars thus prepared, were then activated with

steam under a nitrogen atmosphere at 800 1C for 21 h and

then cooled at ambient temperature under a pure nitrogen

flow, leading to a sample named VH.

For chemical activation, 3 g of the raw material was

impregnated with pure-grade phosphoric acid (H3PO4) 85%

for 24 h. Impregnation ratios of 1.5:1 (g H3PO4/g precursor)

were used. After impregnation, the sample was dried for 4 h

at 60 1C in an oven. The sample was then pyrolyzed under a

nitrogen flow at 600 1C for 1 h in the same furnace. After

cooling, until ambient temperature, under the same nitrogen

flow, the AC thus obtained was carefully washed with distilled

water until stabilization of the pH, and then dried overnight

using a drying oven at 105 1C and then cooled at ambient

temperature. This sample is named VP15.

2.2. Sorbates

Phenol was selected as a model compound for which

adsorption on AC has been extensively studied (Radovic et

al., 2001; Moreno-Castilla, 2004; Terzyk, 2003). Tannic acid and

melanoidin are recalcitrant phenolic organic compounds

found in anaerobically digested molasses spentwash. Re-

agent-grade tannic acid and phenol from Prolabo (VWR

International) were used. Melanoidin was prepared by mixing

4.5 g of glucose, 1.88 g of glycine and 0.42 g of sodium

bicarbonate with 100 ml of distilled water and then heated

for 7 h at 95 1C. After heating, 100 ml of water was added and

dilute solutions of melanoidin were prepared.

2.3. Kinetic study

All kinetic measurements were established at 27 1C in

deionized water. The concentrations of the adsorbates in

the solutions were determined by spectrophotometric mea-

surements, using a UV–visible spectrophotometer (Model

Anthelie from Secomam) at appropriate maximum absor-

bance wavelengths; i.e. 268 and 275 nm for phenol and tannic

acid, respectively. Previously established, linear Beer–Lambert

relationships were used for concentration measurements. For

melanoidin, the solutions were analyzed for UV–visible

absorbance and dissolved organic carbon (DOC) concentra-

tion. The amount of solute adsorbed at time t, q(t) was

calculated as qðtÞ ¼ ðC02CtÞV=W, where C0 and Ct are, respec-

tively, the solute concentration (mg l�1) at time t ¼ 0 and time

t and V the volume of the solution in L, W the weight of the

dry adsorbent in grams. All the experiments were done in

triplicate.

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WAT E R R E S E A R C H 4 0 ( 2 0 0 6 ) 3 4 6 7 – 3 4 7 73468

2.4. Textural characterization

The BET surface area of the ACs (SBET) was measured with a

ASAP Micrometrics sorptiometer. BET areas were calculated

from adsorption data of N2 at 77 K. Microporosity was

estimated by applying the Dubinin–Radushkevich (DR) equa-

tion. Mesoporosity was determined using the BJH method

(Gregg and Sing, 1982). Mercury porosimetry of the samples

was carried out to determine meso- and macroporosity using

a ThermoFinnigan porosimeter, which provides a maximum

operating pressure of 200 Pa.

2.5. Chemical characterization of the AC

The Boehm titration method was used to determine the

number of acidic and basic surface groups (Boehm et al.,

1964). The determination of the pHpzc of the ACs was

carried out as previously described (Lopez-Ramon et al.,

1999).

3. Results and discussion

3.1. Textural characterization and fractal geometrydescription of the ACs porous system

The commercial ACs PAC6 and PAC5, exhibit similar BET

specific surface areas and micropore volumes (Table 1), with

values of 1226 m2/g and 0.50 cm3/g, respectively, for PAC6 and

1184 m2/g and 0.46 cm3/g, respectively, for PAC5. For both

carbons, mesopore volumes were very low, 0.03 and 0.04cm3/g

for PAC6 and PAC5, respectively. The surface area of carbon

was slightly lower for PACV (909 m2/g) on the other hand

micropore and mesopore volumes were of 0.21 and 0.53 cm3/g,

respectively, showing that this carbon is principally mesopor-

ous. Macropore volumes obtained by mercury porosimetry for

PAC6, PAC5 and PACV were 0.204, 0.162 and 1.7 cm3/g,

respectively (Table 2).

The BET specific surface of VH and VP15 was 558 and

959 m2/g, respectively. Like PACV, the porous volume of VH

and VP15, is distributed between a microporous and a

mesoporous network of 0.24 cm3/g and 0.16 cm3/g for VH

and 0.36 cm3/g and 0.48 cm3/g for VP15, respectively. Macro-

pore volumes obtained by mercury porosimetry for VH and

VP15 were 1.58 and 0.79 cm3/g, respectively (Table 2).

Recently, several researchers reported calculation of the

fractal dimension of various ACs (Gomez-Serrano et al., 2005;

Laszlo et al., 1998; Mahamud et al., 2004). The fractal

approach used in order to analyze the porous structures

relies upon the concept of self-similaritiy, in which the

appearance of self-similar objects does not depend on the

scale at which they are observed. The fractal dimension, D, of

the AC used was determined using nitrogen adsorption, and

mercury porosimetry data. The surface fractal dimension D is

2pDp3, for a perfectly smooth surface D ¼ 2, the dimension

of a Euclidean surface, while a higher rough, very irregular

surface has a D ¼ 3.

In the meso- and macropore range, the surface fractal

dimension of the ACs can be calculated from mercury

porosimetry data. Among the several equations proposed,

from the one of Friesen and Mikula (1988) the fractal

dimension is given by the slope of the quasi-linear curve

Ln(dV/dP) versus Ln P. In a method described by Mahamud et

al. (2004), the real cumulated volume V is obtained by taking

account of the compression of the AC samples, from the

experimental (Fig. 1) Ve volume values, then plotting Ve

ARTICLE IN PRESS

Table 1 – Textural characteristics and fractal dimensions of PAC5, PAC6, PACV, VH, VP15, from N2 adsorption isotherms at77 K

Activated carbon SBET (m2/g) Vmi (cm3/g) Vme (cm3/g) DFHHmic DFHHme DE DT

PAC6 1226 0.50 0.03 2.65 2.75 2.36 2.61

PAC5 1184 0.46 0.04 2.55 2.75 2.49 2.14

PACV 909 0.21 0.53 2.25 2.25 — —

VH 558 0.24 0.16 2.40 2.28 — —

VP1-5 959 0.36 0.48 2.58 2.58 — —

Table 2 – Textural characteristics and fractal dimensions of PAC5, PAC6, PACV, VH, VP15, from mercury porosimetry data

Activated carbon Mercury porosimetry results

VT Vme (cm3/g) Vma (cm3/g) Compressibility Factor c 10�12 m3 kg�1 Pa�1 DHg

PAC6 0.178 0.204 1.03 2.97

PAC5 0.147 0.162 0.96 2.69

PACV 2.023 1.7 1.63 2.23

VH 1.94 1.576 1.72 2.72

VP15 1.261 0.789 3.74 2.93

WAT E R R E S E A R C H 40 (2006) 3467– 3477 3469

against the pressure and, using the equation V ¼ Ve�cP. The

slope of the linear curve obtained is c, the compressibility

factor, calculated at high pressures (100–200 MPa). The

corrected V values are also shown in Fig. 1. The calculated

values of the compressibility factor are given in Table 1, all the

r2 coefficients obtained were around 0.999. The slope of the

curve giving ln(dV/dP) vs ln(P) is (DHg�4), DHg being the mean

fractal dimension over the meso- and macropores size range.

The highest fractal dimension DHg, close to 3 are obtained for

PAC6 (2.97), VP15 (2.93), VH (2.72) and PAC5 (2.69) whereas a

DHg value of 2.23 is calculated for PACV. These results indicate

that PACV has a smoother surface than all the other carbons,

for which a more complex physical surface measured by

mercury porosimetry could be assumed.

The surface fractal dimension of the ACs can be calculated

from nitrogen adsorption data using several methods (Terzyk

et al., 2003). Since the FHH method is very simple, it is widely

used for determining the fractal dimensions of ACs. The

method is based on an expression for the surface fractal

dimension from an analysis of multilayer adsorption to a

fractal surface such as ln(V/Vm) ¼ C+S. [ln ln(P0/P)] (Ismail and

Pfeifer, 1994). C is a constant, V, the gas volume adsorbed at

equilibrium pressure P, Vm the volume of gas in the

monolayer and P0, the saturation pressure of nitrogen. The

C constant is a pre-exponential factor and S a power-law

exponent dependant on DFHH, the surface fractal dimension.

There are two limiting cases: at lower end of the isotherm,

representing the early stages of the multilayer build-up, the

ARTICLE IN PRESS

Pressure (Pa)

Vol

ume

(m3 /k

g)V

olum

e (m

3 /kg)

Pressure (Pa)

V PAC5V PAC6Ve PAC5Ve PAC6

V PACV

Ve PACV

(A)

Ve VP15

Ve VHV VP15V VH

(B)

0.0025

0.002

0.0015

0.001

0.0005

0

0 5 × 107 1 × 108 1.5 × 108 2 × 108 2.5 × 108

0 2 × 107 4 × 107 6 × 107 8 × 107 1 × 108 1.2 × 108

0.003

0.0025

0.002

0.0015

0.001

0.0005

0

Fig. 1 – Cumulated mercury volume versus pressure for (A) PAC5 (J), PAC6 (B), and PACV (&), and (B) VH (B) and VP15 (J). Ve:

experimental volume (closed symbols), V: corrected volume value after compressibility determination (open symbols).

WAT E R R E S E A R C H 4 0 ( 2 0 0 6 ) 3 4 6 7 – 3 4 7 73470

film-gas interface is controlled by attractive Van der Waals

forces between the gas and the solid, and the value of the

constant S is given by S ¼ (DFHH�3)/3. At higher coverage,

however, the interface is controlled by the liquid–gas surface

tension. Fig. 2 shows a plot of ln(amount adsorbed) against

ln(ln(1/relative pressure)). For �1=3oSo� 1, the fractal di-

mension can be obtained from S ¼ DFHH�3 and for higher S

values from S ¼ ðDFHH � 3Þ=3 and that, in this case, the

mechanism of adsorption is dominated by Van der Waals

forces.

Fig. 3 shows the linear fit of ln V versus ln ln(P0/P) plot of the

activated carbon samples. DFHH can be calculated for two

ranges of relative pressures (Garnier et al., 2005): the first

range ð0oP=P0o0:1Þ, where micropore filling assumed to be

complete, and the second range ð0:1oP=P0o1:0Þ, were meso-

pore filling was supposed to be predominant.

In the micropore range, DFHH,mic values of 2.65, 2.55, 2.58

and 2.4 were found for PAC6, PAC5 VP15 and VH, respectively.

For the mesoporous ACs, PACV a lower DFHH,mic value of 2.25,

was calculated, indicating that the mesoporous AC PACV has

a smooth surface in the micropore range.

In the mesopore range, a fractal dimension DFHH,mes value

of 2.75 was found for both PAC6 and PAC5, respectively. For

the mesoporous ACs, PACV, VP15 and VH, lower values of 2.25,

2.28 and 2.58, respectively, were calculated for their fractal

dimensions.

An attempt of calculating the fractal dimension in the

micropore range was made, using two equation one

proposed by Ehrburger-Dolle (1994) and the other one

propoposed by Gauden et al. (2001) and Terzyk et al. (2003).

The fractal dimension from the N2 adsorption isotherms

could only be be calculated for the microporous AC, PAC6 and

ARTICLE IN PRESS

ln (P)

ln (P)

ln (

dV/d

P)

ln (

dV/d

P)

ln (P)

ln (

dV/d

P)

ln (P)

ln (

dV/d

P)

ln (P)

ln (

dV/d

P)

(a)

12 13 14 15 16 17 188 10 12 14 16 18

(b)

12 13 14 15 16 17(c)

12 13 14 15 16 17 18 19 20(d)

13 14 15 16 17 18 19 20(e)

-18

-20

-22

-24

-26

-28

-30

-32

-18

-20

-22

-24

-26

-28

-30

-32

-18

-20

-22

-24

-26

-28

-30

-22

-24

-26

-28

-30

-32

-23

-24

-25

-26

-27

-28

-29

Fig. 2 – Plot of ln(dV/dP) (10�6 m3 kg�1 Pa�1) versus ln(P) (Pa) for determination of the fractal dimension of PAC5, PAC6 and PACV

from mercury porosimetry data of (a) PAC6, (b) PAC5, (c) PACV, (d) VH and (e)VP15.

WAT E R R E S E A R C H 40 (2006) 3467– 3477 3471

PAC5 samples, using an equation proposed by Ehrburger–

Dolle:

DE ¼ 32½24=E0FRð1� E0FR=E0Þ, and for E0 and E0FR are char-

acteristic adsorption energies obtained from the DR equation

W ¼W0 exp½�ðA=bE0Þ2�, where W is the volume of micropores

filled at temperature T and relative pressure P=P0, W0, the total

volume of micropore and A ¼ RT lnðP=P0Þ is the differential

molar work, b a similarity coefficient, and E0 the character-

istic adsorption energy and the Freundlich equation, respec-

tively, W ¼ kPa, with a ¼ RT/E0FR. Using DE ¼ 3� ð24=E0FRÞðE0FR=

E0Þ or DE ¼ 328=E0FR, for E0 o18 kJ/mol and/or E0FRo 12 kJ/mol.

The fractal dimension DE could only be calculated for the

microporous AC PAC5 and PAC6 for which the Freundlich

equation could be fitted (Table 1). The DE values obtained for

PAC5 and PAC6 (2.36 and 2.49, respectively) were lower than

DFHHmic values (2.57 and 2.55 for PAC6 and PAC5, respectively).

Fractal dimensions of the ACs, DT, could be calculated using a

more sophisticated equation derived from the Dubinin–

Astakov equation W ¼W0 exp½�ðA=bE0ÞnDA� (Gauden et al., 2001;

Terzyk et al., 2003), but which can also only be applied only for

strictly microporous AC, within a strict range of nDA and E0

values: 21.2436oE0o23.1579 kJ/mol and 1.9999onDAo2.7542.

As expected, the following equation could be used only for

calculating DT of PAC5 and PAC6 (Table 1):

DT ¼E0ð15:3897� 3:6083� 10�4nDAÞ � 283:3356þ 6:3019� 10�3nDA

E0ð0:9396þ 0:1054nDAÞ þ 1:0557þ 1:8407nDA.

Overall, a fractal dimension could be calculated for all the

carbons studied using mercury porosimetry and N2 adsorp-

tion data. For the ACs used in this work, among the most

convenient equations reported in the literature and tested

here, the FHH method was more appropriate, for the

determination of the fractal dimension as those carbons

have various porous textures. The difference between

DFHH mes, and DHg is higher for the microporous than for the

mesoporous AC. Nevertheless, it is generally difficult to find

significant correlations between DHg and DFHH mes fractal

dimensions (Terzyk et al., 2003). One reason for lack of these

correlations between DHg and DFHH mes values is that mercury

intrusion and adsorption methods provide information on

geometry of adsorbant surface at different scales. Thus

fractal dimension resulting from mercury intrusion data are

complementary to those obtained by nitrogen adsorption

(Sokolowska et al., 1999). Overall, our results indicate that the

geometry of the ACs studied is fractal in both micro- meso-

and macropore range.

3.2. Kinetic studies

The analysis of the kinetics of sorption (adsorption, chimio-

sorption, biosorption) in liquid phase relies also traditionally

on the simple classical theory of chemical kinetics. The data

are fitted to first- or second-order equations. However, they

may be many types of chemicals adsorbent–solute interac-

tions due to the large number of different functional groups

that may be present on carbon surfaces (carboxylic, carbonyl,

hydroxyl, ether, lactone, anhydride, quinine, etc.y). Conse-

quently, the classical kinetic or mass transfert representation

is likely to be global. Many models such as homogeneous

surface diffusion model, pore diffusion model and hetero-

geneous diffusion model have also been used to describe the

transport of solutes inside the AC (Cooney, 1999). The

mathemathical complexity of those models make them

inconvenient for practical use (Ho and McKay, 1998).

The adsorption process on AC is clearly a heterogeneous

process taking place at the liquid–solid boundary, and the

diffusion process occurs in a complex matrix with a fractal

architecture as demonstrated here. The reaction is thus

‘‘heterogeneous’’, and the interface of the two phases

represents a special environment under dimensional or

topological constraints. The complex adsorption phenomen-

on may involve chemical interaction between the solute and

the chemical groups on the AC surface, that may involve

electrostatic interactions, Van der Waals, hydrogen bounding,

ligand exchange, hydrophobic interactions. In fact, these

systems belong to what has been called ‘‘complex systems’’

and recently new methods of analysis of reaction kinetics and

relaxation have been derived to account for their non-usual

ARTICLE IN PRESS

4

5

6

7

8

9

10

11

0.8 1 1.2 1.4 1.6 1.8

ln ln (P0/P)(A)

4

5

6

7

8

9

10

11

-5 -4 -3 -2 -1 0 1 2

ln ln (P0/P)(B)

In V

In (

V)

Fig. 3 – Plot of log V (cm3/g) versus log log (P0/P) (Pa) for

determination of the fractal dimension from nitrogen

adsorption data on PAC5 (&), PAC6 (� ), PACV (B), VH (+),

VP15(J), A, in the range of relative pressure (0.0–0.1) and B

(0.1–1).

WAT E R R E S E A R C H 4 0 ( 2 0 0 6 ) 3 4 6 7 – 3 4 7 73472

behaviour. Futhermore, Kopelman (1988) proposed a phenom-

enological fractal-like kinetics to account for reaction in

materials prepared as fractals and recently following the

works of Frauenfelder et al. (1999), and using their work on

relaxation in complex systems, Brouers et al. (2004) and

Brouers and Sotolongo-Costa (2006), have introduced a more

general fractal kinetic. This kinetic equation generalizes the

usual reaction equation introducing a fractal time index, a,

and a fractional reaction order n, where c is the concentration

of the compound involved.

�dcdta¼ Ka;ncn, (1)

whose solution is given as a BurrXII (generalized Pareto law)

distribution by

ca;nðtÞ ¼ cð0Þ½ð1þ ðn� 1ÞðKa;ntÞa��ð1=n�1Þ. (2)

Some typical situations are recovered:

1. If a ¼ 1; n ¼ 1; we have

�dcðtÞdt¼ K1cðtÞ ! cðtÞ ¼ cð0Þ expð�K1tÞ, (3)

which is a first-order kinetics

2. If aa1; n ¼ 1; we have

�dcaðtÞdta

¼ KacaðtÞ ! caðtÞ ¼ cað0Þexpð�KataÞ, (4)

which is a pseudo-first-order (Weibull) kinetics. If

0oao1, it is a ‘‘stretched exponential’’ kinetics.

3. If a ¼ 1; n ¼ 2; we have

�dcðtÞdt¼ K2c2ðtÞ !

1cðtÞ�

1cð0Þ¼ K2t. (5)

This is the second-order kinetics.

4. If aa1; n ¼ 2; we have

�dcaðtÞdta

¼ K2acaðtÞ2! caðtÞ ¼ cað0Þ½1þ expð�Kat

a��1. (6)

This is a pseudo-second-order kinetics.

Cases (2) and (4) have been discussed previously (Brouers

and Sotolongo-Costa, 2003; Jurlewicz and Weron, 1999;

Frauenfelder et al., 1999). The solutions can be obtained from

the simple exponential kinetic (case (1)) by assuming a

distribution of the rate constant K due to fluctuations of the

exponent of the Arrhenius law:

Ka ¼ n expð�E=RTÞ.

This was suggested in Brouers et al. (2004) and discussed in

details mathematically in Brouers and Sotolongo-Costa

(2006).

If one analyses, the kinetics of the difference between the

dissolved molecule absorbed at time t, q(t) the mass of solute

adsorbed per gram of AC and the maximum adsorbed

quantity, qeðtÞ, the sorption time evolution can be fitted to

the general equation which we will call in the future of the

paper BSWðn; aÞ kinetic equation:

qn;aðtÞ ¼ qe½1� ð1þ ðn� 1Þðt=tn;aÞaÞ�1=ðn�1Þ

�, (7)

when n and a are different from 1, one can no longer define a

time-independent rate constant and the relevant quantity

characterizing the time evolution of the process is the

characteristic time tn;a.

The quantity qe measures the adsorption power, and one

can also define a ‘‘half-reaction time’’ t1=2 which is the time

necessary to adsorb half of the equilibrium quantity by

solving the equation:

ð1þ ðn� 1Þðt=tn;aÞaÞ�1=ðn�1Þ

¼12

.

In the case, n! 1; aa1; BSWð1; aÞ can be written as a

Weibull distribution. Such a stretched exponential kinetic

ðao1Þ was already discussed in Jurlewicz and Weron, (1999)

for biological systems:

qaðtÞ ¼ qe½1� expð�t=taÞa�, (8)

With t1=2 ¼ ðln ð2ÞÞ1=ata.

We studied here the kinetic of adsorption of phenol, tannic

acid and melanoidin on the ACs. PAC5, PAC6 and VH

contained much more basic groups and exhibit pHpzc values

of 10, 10 and 11.5, respectively. The pHpzc of PACV is 7.5, and it

contains a high quantity of acidic groups (Table 3). VP15,

which was prepared by chemical activation with H3PO4, as an

activating agent is very acidic, its pHpzc is 2.45 (Table 3). The

working pH of the solution was, respectively, in the range of

4.5–7 during phenol adsorption experiments, from 3 to 6.7 for

tannic adsorption experiments and 7 for melanoidin adsorp-

tion on PACV experiment. This indicates that only VP15 was

negatively charged and that PACV, PAC5, PAC6 and VH were

positively charged during the liquid adsorption experiments.

Typical concentration–time profiles where obtained for

adsorption of phenol and PAC5 and PAC6 (Fig. 4). In order to

fit our experimental data, Eq. (7) was applied using Matlab

software. For the fitting procedure, the starting values used

were, n ¼ 0:99; a ¼ 0:01, qe ¼ qmax þ 0:01, and ta ¼ 0:01. The

curves obtained could be very well fitted by Eq. (8) (Fig. 4). The

values calculated, qe, a, ta and t1/2, found are presented in

Table 4.

The experimental results reported for adsorption of phenol

on the five ACs (Fig. 4, Table 4) can be interpreted using

BSWð1; aÞ as value of n ¼ 1 was obtained for the best fit for

adsorption of phenol on all carbons, which shows that a

pseudo-first-order kinetics was obtained and the adsorption

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Table 3 – Acido-basic groups and pHpzc of PAC6, PAC5,PACV, VH and VP1-5

Chemical analyses

Activatedcarbon

pHpzc Total acidicgroups(meq/g)

Total basicgroups(meq/g)

PAC6 10 0.34 0.49

PAC5 10 0.35 0.53

PACV 7.6 0.69 0.09

VH 11.5 0.43 6.5

VP15 2.45 2.75 0.15

WAT E R R E S E A R C H 40 (2006) 3467– 3477 3473

power values qe could be calculated. As expected, the

microporous commercial AC, PAC5 and PAC6 had a

much higher adsorption power qe, than the mesoporous

ACs (Table 4). Although, VP15 has a much higher specific

surface and a higher micropore volume than VH a very

low qe values was obtained for adsorption of phenol on VP15.

The acidic character of VP15 may not favour adsorption

of phenol on VP15 surface (Radovic et al., 2001; Moreno-

Castilla, 2004).

The adsorption data of tannic acid on the mesoporous

ACs, PACV, VH and VP15 could also well be fitted

with a pseudo-first-order 1 kinetic (Fig. 5). On the other hand,

for the adsorption data of melanoidin on PACV which is a

complex mixture containing brown polymers and deshydra-

tion products of sugar, it was difficult to distinguish between

a first- or a second-order kinetic, and the best fit was obtained

with a pseudo-(n, a) order using Eq. (7), with 1ono2. Fig. 6

shows the best fit obtained for adsorption data of melanoidin

on PACV, for which a pseudo-1,5 order with a ¼ 0:56, is

obtained.

We attempted to determine whether the a parameters

calculated by fitting the liquid adsorption data on the

different ACs with the fractal kinetic proposed here could be

correlated to the fractal dimension D of the ACs. Fig. 7A shows

the plot of the a parameters calculated by fitting the

adsorption data of phenol and tannic acid on the ACs against

DFHHmic (Fig. 7A) and DFHHmes (Fig. 7B) and Fig. 8 shows the plot

of the a parameters calculated by fitting the adsorption data

of phenol and tannic acid on the ACs against DHg. A

correlation could be observed between atannic acid and DFHHmes

and also between aphenol and DFHHmes, but only for the

mesoporous ACs (Fig. 7B). The reason of this correlation is

not clear and it still have to be confirmed collecting additional

adsorption data from much more ACs with various textures

and various chemical groups at their surface. Our first

hypothesis concerning such a correlation, is that the aparameter which is contained in the distribution of the rate

constant K (due to fluctuations of the exponent of the

Arrhenius law) may be related with the surface geometry.

This observation has to be related to the results from the

work of Rigby (2002, 2003, 2005). The kinetic of adsorption

processes are generally, diffusion controlled (Terzyk and

Gauden, 2002, Do and Wang, 1998) and recently, Rigby (2003,

2005) developed a theory relating the degree of heterogeneity

of a surface to the rate of diffusion of the adsorbed phase on

that surface. This theory predicted that both of the Arrhenius

parameters for the correlation time are particular function of

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Table 4 – Kinetic data obtained for adsorption of variouscompounds on PAC5, PAC6, PACV, VH and VP1-5 byfitting the experimental data with the BWS (n, a) equation

Activated carbon Kinetic parameters

qe (mg/mg) a ta (h) t1/2 (h)

Phenol

PAC6 0.24 0.60 24 13.1

PAC5 0.25 0.58 26.7 17.6

PACV 0.075 0.22 23.4 4.43

VH 0.087 0.77 22.87 14.2

VP15 0.009 0.35 6.62 2.32

Tannic acid

PACV 0.44 0.33 1.09 0.36

VH 0.163 0.58 16.57 8.82

VP15 0.15 0.44 13.32 5.81

Melanoidin

PACV 0.095 0.56 0.93 0.40

qe (

mg/

mg)

time (h)

0 50 100 150 200 250 300 350 400 450

0.25

0.20

0.15

0.10

0.05

0.00

Fig. 4 – Experimental data and curve fits (line) of phenol adsorption kinetic on, PAC6 (J), PAC5 (n), VH(&), PACV (� ), VP15 (+).

WAT E R R E S E A R C H 4 0 ( 2 0 0 6 ) 3 4 6 7 – 3 4 7 73474

the surface fractal dimension which can be determined from

gas adsorption isotherm. The author also suggested that the

variation in the degree of heterogeneity between different

domains of the surface give rise to resultant variations across

the surface in the heat of adsorption, and the Arrhenius

parameters for surface diffusivity.

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time (h)

q (m

g/m

g)

0.5

0.4

0.3

0.2

0.1

00 50 100 150 200 250 300 350

Fig. 5 – Experimental data and curve fits (line) of tannic acid adsorption kinetic on PACV,(J), VPH (D), VP1-5 (B).

Time (h)

0 10 20 30 40 50 60 70 80

0.1

0.08

0.06

0.04

0.02

0

q (m

gDC

O/g

)

Fig. 6 – Experimental data and curve fits (line) of melanoidin adsorption kinetic on PACV.

WAT E R R E S E A R C H 40 (2006) 3467– 3477 3475

Overall, using this new equation adsorption kinetic data

can be easily fitted, and allows to take into account the

complexity of the adsorption process and of the surface

geometry.

4. Conclusion

Our work describes a new fractal kinetic equationfor describ-

ing adsorption kinetic: qn;aðtÞ ¼ qe½1� ð1þ ðn� 1Þðt=tn;aÞaÞ

�1=ðn�1Þ�, depending on the order of the reaction, n, and the

fractal time exponent, a.

Using this equation, key parameters of practical impor-

tance for adsorption kinetic can be determined: the para-

meter qe which measures the maximal quantity of solute

adsorbed, and the ‘‘half-reaction time’’, t1=2, the time neces-

sary to reach half of the equilibrium.

A value of n ¼ 1 is obtained for the best fit for adsorption of

phenol and tannic acid respectively, on all carbons tested. For

adsorption data of melanoidin on PACV, n ¼ 1:5.

For the mesoporous Acs, a correlation is observed between

atannic acid and DFHHmes, the fractal dimension determined

using the FHH method in the mesopore range, and also

between aphenol and DFHHmes. The reason of this correlation is

not clear and still has to be confirmed.

R E F E R E N C E S

Avnir, D., Jaroniec, M., 1989. An isotherm equation for adsorptionon fractal surfaces of heterogeneous porous material. Lang-muir 5, 1431–1433.

Avnir, D., Farin, D., Pfeifer, P., 1983. Chemistry in non integerdimension between two and three. II. Fractal surfaces ofadsorbents. J. Chem. Phys. 79, 3566–3571.

Avnir, D., Farin, D., Pfeifer, P., 1992. A discussion of some aspectsof surface fractality and of Its determination. New J. Chem. 16,439–449.

Boehm, H.P., Diehl, E., Heck, W., Sappock, R., 1964. Surface oxidesof carbons. Angew. Chem. 76, 742–751.

Brouers, F., Sotolongo-Costa, O., 2003. Universal relaxation innonextensive systems. Europhys. Lett. 62 (6), 808–814.

Brouers, F., Sotolongo-Costa, O., 2006. Generalized fractal kineticsin complex systems. (applications to biophysics and biotech-nology). Physica A 368 (2006) 165–175.

Brouers, F., Sotolongo-Costa, O., Weron, K., 2004. Burr, Levy,Tsallis. Physica A 344, 409–416.

Cooney, D.O., 1999. In Adsorption Design for Wastewater Treat-ment. Lewis Publisher, Boca Raton.

Diduszko, R., Swiatkowski, A., Trznadel, B.J., 2000. On surface ofmicropores and fractal dimension of activated carbon deter-mined on the basis of adsorption and SAXS investigations.Carbon 38, 1153–1162.

Do, D.D., Wang, K., 1998. Dual diffusion and finite mass exchangemodel for adsorption kinetics in activated carbon. AIChE 44,68–82.

Ehrburger-Dolle, F., 1994. Some new correlations between theDubinin–Radushkevich and Freundlich equations and fractaldimension in microporous solids. Langmuir 10, 2052–2055.

Ehrburger-Dolle, F., 1999. A new way to analyse adsorptionisotherms. Langmuir 15, 6004–6015.

Frauenfelder, H., Wolynes, P.G., Austin, R.H., 1999. Biologicalphysics. Rev. Mod. Phys. 71, S419.

Friesen, W.I., Mikula, R.J., 1988. Mercury porosimetry of coals: porevolume distribution and compressibility. Fuel 67 (11),1516–1520.

Garnier, C., Gorner, T., Razafitianamahavaro, A., Villieras, F., 2005.Investigation of activated carbon surface heterogeneity byargon and nitrogen low-pressure quasi equilibrium volumetry.Langmuir 21, 2838–2846.

ARTICLE IN PRESS

(A)

(B)

PAC6

PAC5

DFHH mic

α0.8

0.7

0.6

0.5

0.4

0.3

0.2

α

0.8

0.7

0.6

0.5

0.4

0.3

0.2

2.2 2.3 2.4 2.5 2.6 2.7

DFHH mes

2.2 2.3 2.4 2.5 2.6 2.7 2.8

Fig. 7 – Plot of the kinetic parameter a obtained from

adsorption data of phenol (J) and tannic acid (+) on the

various activated carbons against (A) DFHHmic and (B)

DFHHmic.

DHg

α

0.8

0.7

0.6

0.5

0.4

0.3

0.22.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3

Fig. 8 – Plot of the kinetic parameter a obtained from

adsorption data of phenol (J) and tannic acid (+) on the

various activated carbons against DHg.

WAT E R R E S E A R C H 4 0 ( 2 0 0 6 ) 3 4 6 7 – 3 4 7 73476

Gauden, P.A., Terzyk, A.P., Rychlicki, G., 2001. New correlationbetween microporosity of strictly microporous activatedcarbons and fractal dimension on the basis of the polanyi–Dubinin theory of adsorption. Carbon 39 (2), 267–278.

Gomez-Serrano, E.M., Cuerda-Correa, M.C., Fernandez-Gonzalez,M.F., Alexandre-Franco, Macıas-Garcıa, A., 2005. Preparation ofactivated carbons from chestnut wood by phosphoric acid-chemical activation. Study of microporosity and fractaldimension. Mater. Lett. 59, 846–853.

Gregg, S.J., Sing, K.S.W., 1982. Adsorption Surface Area andPorosity. Academic Press, London, 90pp.

Ho, Y.S., McKay, G., 1998. Comparison of chemisorption kineticmodels applied to pollutant removal on various sorbents.Process Saf. Environ. Prot. B 76 (4), 332–340.

Ismail, I.M.K., Pfeifer, P., 1994. Fractal analysis and surfaceroughness of nonporous carbon fiber. Langmuir 10, 1532–1538.

Jurlewicz, A., Weron, K., 1999. A general probabilistic approach tothe universal relaxation response of complex systems. Cell.Mol Biol. Lett. 4, 55–86.

Kopelman, R., 1988. Fractal reaction kinetics. Science 241,1620–1626.

Laszlo, K., Bota, A., Nagy, L.G., Subklew, G., Schwuger, M.J., 1998.Fractal approach of activated carbons from solid wastematerials. Colloids Surf. A: Physicochem. Eng. Aspects 138,29–37.

Lee, C.K., Lee, S.L., 1996. Heterogeneity of surfaces and materials,as reflected in multifractal analysisHeterogen. Heterogen.Chem. Rev. 3, 269–302.

Lopez-Ramon, M.V., Stoeckli, F., Moreno-Castilla, C., Carrasco-Marin, F., 1999. On the characterization of acidic and basicsurface sites on carbons by various techniques. Carbon 37,1215–1221.

Mahamud, M., Lopez, O., Pis, J.J., Pajares, J.A., 2004. Texturalcharacterization of chars using fractal analysis. Fuel Process.Technol. 86 (2), 135–149.

Mandelbrot, B.B., 1982. The Fractal Geometry of Nature. Freeman,San Francisco.

Meilanov, R.P., Sveshnikova, D.A., Shabanov, O.M., 2002. Fractalnature of sorption kinetics. J. Phys. Chem. A. 106, 11771–11774.

Moreno-Castilla, C., 2004. Adsorption of organic molecules fromaqueous solutions on carbon materials. Carbon 42, 83–94.

Neimark, L.A., 1992. A new approach to the determination of thesurface fractal dimension of porous solids. Physica A: Stat.Theor. Phys. 191, 258–262.

Pfeifer, P., Avnir, D., 1983. Chemisry in non-integer dimensionbetween two and three. I. Fractal theory of heterogeneoussurfaces. J. Chem. Phys. 79, 3558–3565.

Pfeifer, P., Obert, M., 1990. Fractals: basic concepts and terminol-ogy. In: Avnir, D. (Ed.), The Fractal Approach to HeterogeneousChemistry. Wiley, New York.

Radovic, L.R., Moreno-Castilla, C., Rivera-Utrilla, J., 2001. Carbonmaterial as adsorbent in aqueous solution. In: Radovic, L.R.(Ed.), Chemistry and Physics of Carbon, vol. 27. Marcel Dekker,New York, pp. 1–66.

Rigby, S.P., 2002. Fractal theory for the compensation effectobserved in a surface diffusion process studied using deuter-on NMR. Langmuir 18, 1613–1618.

Rigby, S.P., 2003. A model for the surface diffusion of molecules ona heterogeneous surface. Langmuir 19, 364–376.

Rigby, S.P., 2005. Predicting surface diffusivities of molecules fromequilibrium adsorption isotherms. J. Colloid Interface 262,139–149.

Seri-Levi, A., Avnir, D., 1993. Kinetics of diffusion-limited adsorp-tion. J. Phys. Chem., 10380–10384.

Soko"owska, Z., Hajnos, M., Hoffmann, C., Manfred, M.S.,Soko"owski, S., 1999. Adsorption of nitrogen on thermallytreated peat soils: the role of energetic and geometricheterogeneity. J. Colloid Interface Sci. 219, 1–10.

Soko"owska, Z., Hajnos, M., Hoffmann, C., Manfred, M.S.,Soko"owski, S., 2001. Comparison of fractal dimensions ofsoils estimated from adsorption isotherms, mercury intrusion,and particle size distribution. J. Plant. Nutr. Soil Sci, 591–599.

Terzyk, A.P., 2003. Further insights into the role of carbon surfacefunctionalities in the mechanism of phenol adsorption.J. Colloid Interface Sci. 268, 301–329.

Terzyk, A.P., Gauden, P.A., 2002. The simple procedure of thecalculation of diffusion coefficient for adsorption on sphericaland cylindrical adsorbent. Particles—experimental verifica-tion. J. Colloid Interface Sci. 249, 256–261.

Terzyk, A.P., Gauden, P.A., Kowalczyk, 2003. Fractal geometryconcept in physical adsorption on solids. Arab. J. Sci. Eng. 28(1C), 133–167.

Wu, F.-C., Tseng, R.-L., Juang, R.-S., 2001. Kinetic modeling ofliquid-phase adsorption of reactive dyes and metal ions onchitosan. Water Res. 35, 613–618.

Yin, Y., 1991. Adsorption isotherm on fractally porous materials.Langmuir 7, 216–217.

ARTICLE IN PRESS

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