8 Furmaniak Et Al

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Journal of Colloid and Interface Science 291 (2005) 600–605

www.elsevier.com/locate/jcis

Letter to the Editor

Parameterization of the corrected Dubinin–Serpinsky adsorptionisotherm equation

Sylwester Furmaniak, Artur P. Terzyk ∗, Piotr A. Gauden, Gerhard Rychlicki

Physicochemistry of Carbon Materials Research Group, Department of Chemistry, N. Copernicus University, Gagarin St. 7, 87-100 Toru´ n, Poland

Received 13 June 2005; accepted 29 July 2005

Available online 19 September 2005

Abstract

A recently proposed new modification of the Dubinin–Serpinsky adsorption isotherm equation, the CDS formula, is analyzed. We develop the

equation describing the isosteric enthalpy of adsorption, and we give the meaning of the empirical parameters occurring in the CDS model. Finally

the application of the CDS equation and related enthalpy formula describing experimental water adsorption and enthalpy data measured on two

microporous carbons is shown. The simultaneous fit of the theoretical CDS isotherm and related enthalpy formula to experimental data is very

good.

© 2005 Elsevier Inc. All rights reserved.

Keywords: Adsorption; Activated carbon; Water; Dubinin–Serpinsky equation

1. Introduction

Water adsorption on carbons is still the subject of many ex-

perimental and theoretical studies [1–3]. Recently Gauden [4]

proposed a new adsorption equation that was a continuation of

the studies started by Dubinin and Serpinsky [5], and contin-

ued by Barton and co-workers [6] and others [7–9]. This new

equation, called the CDS isotherm, has the form

a = c(a0 + a)

1 + A1a + A2a3 + A3

ln aaunit

a2

+ A4 exp−

a

aunith

(1)= c(a0 + a)(1 + F 1 + F 2 + F 3 + F 4)h

and contains the empirical term given in large parentheses. In

Eq. (1) a is adsorption, a0 is the so-called concentration of pri-

marily active surface centers, h is the relative water pressure,

and c, A1, . . . , A4, are the equation parameters. The parameter

aunit containing the unit of adsorbed amount (i.e., 1 mol g−1,

* Corresponding author. Fax: +48 56 654 2477. E-mail address: [email protected] (A.P. Terzyk).

1 mmolg−1, 1 g g−1, etc.) is introduced to avoid the findingof the logarithm and exponent from the denominate number.

Thus, there are four correction terms appearing in the CDS

equation: F 1 = A1a, F 2 = A2a3, F 3 = A3[ln(a/aunit)/a2], and

F 4 = A4 exp[−a/aunit].

The empirical term of the CDS equation is related to the

maximum adsorption (as) (assuming in Eq. (1) for h = 1) as

as = c(a0 + as)

1 + A1as + A2a3

s + A3

ln asaunit

a2s

(2)+ A4 exp

−

as

aunit

.

The major aim of this letter is to present results concerning

the meaning of the empirical parameters of the CDS isotherm

(Eq. (1)). To achieve this we analyze the influence of the cor-

rection terms, and also we derive an equation describing the

isosteric enthalpy of adsorption related to the CDS model.

Fig. 1a shows an comparative plot of the CDS isotherm and

the plot of the equation proposed by Dubinin and Serpinsky

(called the DS1 equation [5]). In both equations the same values

of parameters c and a0 are assumed. Fig. 1b explains how the

plot of the empirical terms occurring in Eq. (1) changes from

the start to the beginning of the adsorption isotherm. It should

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S. Furmaniak et al. / Journal of Colloid and Interface Science 291 (2005) 600–605 601

Fig. 1. (a) The typical plots of isotherms generated from DS1 and CDS

adsorption equations (c = 1.7; a0 = 8 mmolg−1; A1 = 0.015 gmmol−1;

A2 = −0.0001 g3 mmol−3; A3 = −5 mmol2 g−2; A4 = 9; aunit = 1

mmolg−1). (b) The effect of the corrections terms appearing in the CDS

equation (F 1 = A1a; F 2 = A2a3; F 3 = A3[ln(a/aunit)/a2]; F 4 = A4 ×

exp[−a/aunit]).

be pointed out that if in Eq. (1) we assume that A1, . . . , A4 = 0,

i.e., F = 1, the original DS1 formula is obtained from the CDS

one. At the start of the adsorption isotherm the third, F3 (ini-

tially increasing adsorption value and next decreasing it), and

fourth, F4 (rising adsorption value), correction terms are mainly

responsible for the differences between CDS and DS1 equa-

tion plots. Both correction terms are meaningful only for low

adsorption values. In contrast, at larger adsorptions the first,

F1, and second, F2, correction terms are decisive, thus they

are responsible for the appearance of the plateau observed on

the adsorption isotherm. Therefore similarly to the second Du-

binin and Serpinsky isotherm equation, DS2 [5–9], and simi-

larly to Barton’s equations [5–9] those correction terms make

the CDS model useful for description of data measured on car-

bons containing limited pore space and/or possessing the high-

energy-adsorption sites manifested by the Langmuir-type shape

of adsorption isotherm observed at low relative pressures. The

conclusions mentioned above confirm the results of the investi-

gations presented recently by Gauden [4].

2. Enthalpy of adsorption from the CDS equation

It is obvious that the temperature dependence of the kinetic

parameter c is given by [7–9]

(3)c = c0

exp q

RT ,

where q is the enthalpy connected with the DS1 equation, and

c0 is the slightly temperature-dependent entropic factor.

Since the temperature dependence of the empiric terms

A1–A4 is unknown, we only denote here that

(4)Ai = RT 2

dAi

dT .

Application of the Clausius–Clapeyron equation

(5)qst− L = RT 2

∂ ln h

∂T

a

and differentiation of Eq. (1) with respect to the temperature

leads, after simple manipulation, to

(6)

qst − L = q −A

1a + A2a3 + A

3

ln aaunit

a2 + A4 exp

− a

aunit

1 + A1a + A2a3 + A3

ln aaunit

a2 + A4 exp

− aaunit

,

where qst is the isosteric enthalpy of adsorption, and L is the en-

thalpy of water condensation. (The detailed derivation of Eq. (6)

is given in Appendix A.)

3. The physical meaning of the parameters of the CDS

equation

3.1. Parameters A1 and A2

Parameters A1 and A2 determine the shape of the adsorption

isotherm at larger adsorption values (Fig. 1b). Therefore they

are responsible for the finite pore volume of an adsorbent.

3.2. Parameter A3

In the limit a → 0 the isosteric enthalpy of adsorption

(Eq. (6)) should tend to the so-called enthalpy of adsorption

at zero coverage (q0). Therefore, from Eq. (6) we have

(7)qst− L −→

a→0q0 = q −

A3

A3.

Taking into account Eq. (4) and the integral/differential calculus

leads to

(8)A3 = A03 exp

q0 − q

RT

,

where A03 is related to the integral constant. From this equation

it is easy to show that

(9)A3 = (q − q0)A3.



602 S. Furmaniak et al. / Journal of Colloid and Interface Science 291 (2005) 600–605

3.3. Parameter A4

To consider the physical meaning of this parameter one can

neglect the third correction term in Eq. (1) (i.e., the term con-

nected with A3). Moreover, for low adsorption values we can

assume that

(10)a0 + a ≈ a0,

1 + A1a + A2a3 + A4 exp

−

a

aunit

(11)≈ 1 + A4 exp

−

a

aunit

.

The application of the approximation exp[±x] ≈ 1 ± x leads to

further reduction of Eq. (1) to

(12)h ≈a

ca0

1 + A4

1 − a

aunit

.

Here two cases can be considered. First is for the systems where

1 A4. For this case the simplification of Eq. (12) leads to

(13)h ≈

aaunit

ca0A4aunit

1 − a

aunit

.

Equation (13) is the Langmuir-type isotherm [10],

(14)h =Θ

KL(1 − Θ),

where KL is the Langmuir constant, and

(15)a

aunit∼ Θ,

(16)ca0A4

aunit∼ KL.

Therefore, from Eq. (16) it is seen that for the cases 1 A4

the parameter A4 is connected with the averaged Langmuir

constant responsible for interaction of water molecules with ad-

sorption sites,

(17)A4 ∼KLaunit

ca0.

Since the temperature dependence of the Langmuir constant is

known,

(18)KL = K 0L exp

qL

RT ,

where K 0L is slightly dependent on the temperature entropic fac-

tor, and qL is the enthalpy of interaction between water and

Langmuir-type sites. Following Eq. (17), A4 is given by

(19)A4 = (q − qL)A4.

Another case occurs for the systems where 1 A4. From

Eq. (12) we have

(20)h ≈a

ca0≈ Ki Θ0,

where Ki is the Langmuir or Henry constant, depending on the

localized or mobile character of adsorption, and Θ0

≈ a/a0

is

the relative adsorption on primary surface centers.

Summing up, the parameter A4 is strictly related to the shape

of adsorption isotherm in the low-pressure limit. If it is larger

than unity, the adsorption isotherm has Langmuirian shape in

this range. In contrast, if the value of A4 is smaller than unity,

the isotherm is linear in the low-pressure limit. For the first

case A4 is related to the value of adsorption energy. In the sec-

ond case this parameter almost vanishes from the adsorptionisotherm equation.

4. The description of experimental data

Fig. 2 shows the influence of the parameters A1, A

2, A3 =

f (q0), and A4 = f (qL) on the plot of the isosteric enthalpy of

adsorption (Eq. (6)) generated for the CDS isotherm shown in

Fig. 1. At low adsorption values only q0 and qL (i.e., the val-

ues connected with the parameters A3 and A4) influence the

enthalpy plot. In contrast, the parameters A1 and A

2 have influ-

ence at larger adsorptions. Since the temperature dependence of

parameters A1 and A2 is unknown, we assumed arbitrarily (i.e.,A

1 = 0 = A2). We studied the experimental data of water ad-

sorption isotherms and enthalpy published by us previously and

measured for water adsorption on two synthetic and chemically

modified microporous carbons [7,8,11]. Adsorption data has al-

ready been fitted by Gauden [4]. However, it is well known

that the simultaneous fitting of adsorption and enthalpy of ad-

sorption data leads to different set of parameters. Table 1 and

Fig. 3 present the results obtained from such a simultaneous fit-

ting. The adsorption isotherm and related enthalpy relationships

were fitted to experimental data by applying the minimization

procedure using the differential evolution (DE) algorithm pro-

posed by Storn and Price [12,13], and described in detail re-

cently [15]. It should be pointed out that the difference between

values of the two sets of fitted parameters is insignificant. For

both studied carbons the CDS model leads to higher values of

the determination coefficients (DC) than obtained previously

from the DS1, DS2, and/or Barton models [4,8]. Moreover the

empiric CDS model leads to relatively good quality of the fit of

adsorption enthalpy. Obtained results suggest chemisorption of

water on carbon D. For both systems some steps on experimen-

tal enthalpy data are not reproduced (see Fig. 3). We showed

previously that this effect can be achieved if one takes into

account the differences in the energy of water adsorption on

surface-active centers [14,15]. In the case of data measured on

carbon D the Langmuirian part of the isotherm is strongly pro-nounced; thus the value of the parameter A4 is high.

5. Conclusions

The equation describing the isosteric enthalpy of adsorption

related to the CDS model is derived. The physical meaning of

the parameters of this empirical equation is analyzed. Thus,

A1 and A2 are strictly related to the finite volume of adsorp-

tion space. As a consequence, they influence the isotherms and

enthalpy only at larger adsorption values. In contrast, the pa-

rameters A3 and A4 are responsible for the initial part of the

isotherm and enthalpy plot. Parameter A3

is related to the value

of the enthalpy at “zero” coverage. On the other hand, A4 (if




Fig. 2. The influence of the CDS parameters A1 , A

2 , q0 (A3 = f (q0)), and qL ( A

4 = f (qL)) on the plots of enthalpy generated from Eq. (6). q = 6 kJ mol−1;

A1 = 0; A

2 = 0; q0 = 14 kJmol−1; qL = 15 kJmol−1; other parameters as in Fig. 1a.

Table 1

The values of the parameters obtained during the fitting of CDS equation (Eqs. (1) and (6)) to the experimental data

Carbon c a0

[mmol g−1]

A1

[g mmol−1]

A2

[g3 mmol−3]

A3

[mmol2 g−2]

A4 DCiz q q0 qL DCqst

[kJ mol−1]

D 0.86 7.87 0.123 −3.40 × 10−4 −0.042 9.25 0.98 3.90 46.0 19.2 0.96

E 1.11 5.77 0.084 −2.49 × 10−4 −0.68 1.15× 10−16 0.98 6.65 42.1 – 0.96

it is larger than unity) is related to the Langmuir constant. Theapplication of the CDS equation (and related enthalpy formula)

to description of experimental data lead to very good quality of

the fit.

Finally, the empirical nature of the DS-type equations makes

them applicable to the description of experimental data. The

results presented here show also that different correlations ob-

tained basing on the fitting of the DS-type models to experi-

mental water adsorption data are doubtful, as far as the thermo-

dynamic verification of the experimental data is not provided.

On the other hand, the stepwise character of enthalpy plots is

not recovered by the CDS equation. Thus, in our opinion the

attention should be also paid to new theoretical models such

as, for example, the Do and Do one [16], or its heterogeneousversion developed by us recently [14,15]. On the other hand,

they are more complicated and contain more parameters. Since

the parameters can compensate for each other it is impossi-

ble to obtain the real values if one describes only adsorption

isotherm data. However, if simultaneous description of adsorp-

tion isotherm and related enthalpy data is applied (by advanced

numerical algorithms) this problem disappears.

Acknowledgments

A.P.T. gratefully acknowledges financial support from KBN

Grant 3 T09A 065 26. P.A.G. gratefully acknowledges financial

support from KBN Grant 4T09A 077 24.



604 S. Furmaniak et al. / Journal of Colloid and Interface Science 291 (2005) 600–605

Fig. 3. The results of the fitting of the CDS model (Eqs. (1) and (6): lines) to the experimental data (points).

Appendix A. Step-by-step derivation of the isosteric

enthalpy of adsorption (Eq. (6)) from the CDS model

(Eq. (1))

From Eqs. (1) and (3) it can easily be shown that

(A.1)

dc

dT = c0 exp

q

RT

C

d

dT

q

RT

= c

−

q

RT 2

= −

q

RT 2c.

After differentiation of Eq. (1) with respect to temperature (as-

suming constant adsorption) we have

a = c(a0 + a)

1 + A1a + A2a3 + A3

ln aaunit

a2

(A.2)+ A4 exp

−

a

aunit

h d

dT , a = const.,

0 = −q

RT 2c(a0 + a)1 + A1a + A2a3 + A3

ln aaunit

a2

+ A4 exp

−

a

aunit

h

+ c(a0 + a)

dA1

dT a +

dA2

dT a3

+dA3

dT

ln aaunit

a2

+dA4

dT exp

−

a

aunit

h

+ c(a0 + a)

1 + A1a + A2a3 + A3

ln aaunit

a2

(A.3)+ A4 exp

−

a

aunit

dh

dT .

Double-sided multiplication of Eq. (A.3) by

(A.4)

RT 2

c(a0 + a)1 + A1a + A2a3 + A3

ln aaunit

a2 + A4 exp− a

aunit h




leads to

0 = −q +

RT 2

dA1

dT a + RT 2 dA2

dT a3 + RT 2

dA3

dT

ln aaunit

a2

+ RT 2dA4

dT

exp−a

aunit1 + A1a + A2a3 + A3

ln aaunit

a2 + A4 exp

−

a

aunit

(A.5)+RT 2

h

dh

dT .

Applying the formula defining the isosteric enthalpy of adsorp-

tion (Eq. (5)),

qst− L = RT 2

∂ ln h

∂T

a

a=const.= RT 2

d ln h

dT =

d ln h =

dh

h

(A.6)=

RT 2

h

dh

dT ,

together with Eqs. (4) and (A.5), leads to

0 = −q +A

1a + A2a3 + A

3

ln aaunit

a2 + A4 exp

− a

aunit

1 + A1a + A2a3 + A3

ln aaunit

a2 + A4 exp

− aaunit

(A.7)+ qst − L.

Equation (A.7), after simple manipulation, leads to Eq. (6). If

one of the parameters Ai is temperature-independent dAi /dT

= 0 and Ai = 0.

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8 Furmaniak Et Al