A thermodynamic analysis of gas adsorption on microporous materials: Evaluation of energy heterogeneity

Journal of Colloid and Interface Science 331 (2009) 302–311

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A thermodynamic analysis of gas adsorption on microporous materials: Evaluation of energy heterogeneity Joan Llorens a,∗ , Marc Pera-Titus b a b

Chemical Engineering Department, Faculty of Chemistry, University of Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain Institut de Recherches sur la Catalyse et l’Environnement de Lyon, UMR 5256 CNRS, Université Claude Bernard Lyon 1, France

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 11 September 2008 Accepted 30 October 2008 Available online 19 December 2008

This paper presents a thermodynamic isotherm derived from solution thermodynamics principles to describe gas adsorption on microporous materials. This isotherm relies on a potential relationship between the integral free energy of adsorption relative to saturation, Ψ/ R T , expressed by the Kiselev equation, and the variable Z = 1/ − Ln(Π ), being Π the relative pressure. A mathematical analysis reveals that the adsorption energy heterogeneity in the micropores is collected in a characteristic parameter of the isotherm, m, that can be related to the α parameter of the Dubinin–Astakhov isotherm in a simple way (m = α + 1). The isotherm also predicts a plateau in Ψ/ R T at extremely low pressures (Π < 10−7 ). Neimark’s thermodynamic equation accounting for gas adsorption on mesoporous solids is found to be a particular case of the isotherm presented in this study. The Langmuir isotherm only shows consistency with the thermodynamic isotherm for a reduced combination of values of the relevant parameters, not usually found in common adsorbents. The suitability of the thermodynamic isotherm is experimentally assessed by testing a collection of microporous materials, including activated carbons, carbon nanotubes, and zeolites. © 2008 Elsevier Inc. All rights reserved.

Keywords: Microporous material Gas adsorption Energy heterogeneity Fractal Dubinin–Astakhov Langmuir

1. Introduction Microporous materials (e.g., activated carbons, zeolites) are used in a great variety of gas separation, purification, and catalytic processes. The adsorption capacity of these materials is strongly determined by their textural properties, such as their “apparent” internal surface, pore geometry, pore size distribution, and surface irregularity. The microstructure of microporous materials can be inferred from physical adsorption of gases and vapors of different sizes and polarities (most often nitrogen at 77 K). A critical review about porous material characterization by gas adsorption methods can be found in Ref. [1]. Considerable attention has been devoted in the past to the development of suitable isotherm models to describe gas adsorption on microporous materials. Early models such as the Langmuir isotherm of idealized monolayer adsorption are not applicable to physical adsorption by microporous adsorbents despite the characteristic form of the isotherm (Type I) [1–3]. In fact, the form of this isotherm is ascribed to a micropore volume-filling process (3D) instead of a layer-by-layer surface coverage (2D) [2,4], and the plateau does not correspond to monolayer completion.

*

Corresponding author. E-mail address: [email protected] (J. Llorens).

0021-9797/$ – see front matter doi:10.1016/j.jcis.2008.10.086

© 2008

Elsevier Inc. All rights reserved.

The vacancy solution theory (VST) [5–9] is one of the first approaches to describe gas adsorption on homogeneous microporous solids as a pore-filling process. In this theory, the adsorbed and gas phases are treated as two vacancy solutions. The vacancy is an imaginary solvent occupying spaces that will be filled by the sorbate. The composition relationship between the two phases is derived from “osmotic” equilibrium criteria. Additional assumptions concerning vacancy properties and activity coefficients of the vacancy solution lead to some fundamental isotherm equations (e.g., Langmuir, Henry, Volmer, and Fowler–Guggenheim) [10]. This theory offers a unified representation of single as well as mixture adsorption within the same framework. Another approach is based on the volume-filling theory of micropores (TVFM) developed by Dubinin and co-workers [11–13]. The well-known Dubinin–Radushkevich (DR) isotherm [14,15] can be deduced from this theory combined with the Polayni’s adsorption potential. Several authors have postulated that the DR isotherm applies only to solids with a uniform structure of micropores [16,17]. Alternative models such as the Dubinin–Astakhov (DA) isotherm [12,18] have been proposed to account for gas adsorption in microporous solids, including bound impurities and functional groups. The Dubinin–Astakhov isotherm has two parameters: the characteristic exponent, α , and the energy of adsorption, E 0 . The characteristic exponent of this isotherm (usually in the range 1–5) can be linked to the degree of heterogeneity of the microporous

J. Llorens, M. Pera-Titus / Journal of Colloid and Interface Science 331 (2009) 302–311

system [19–21]. On the other hand, the energy of adsorption can be linked to the average micropore width [22]. The DR equation can be regarded as a particular case of the DA isotherm for α = 2. Energy heterogeneity can also arise from structural heterogeneity in the form of the micropore size distribution, with adsorption energy varying with pore size. According to the theory of adsorption in micropores, such a distribution can be obtained by solving the Fredholm equation of the first kind defined by [2,23]

θt =

zmax  θtheor (z, p ) f (z)δ z,

(1)

303

spreading pressure, collects the information related to the energy at which the sorbate molecules adsorb. This isotherm provides relevant information about the adsorption energy heterogeneity of the material, involving a separate analysis of the contribution of the physical structure of the adsorbent and the sorbate–adsorbent interaction. Neimark’s equation can be obtained as a particular case of the thermodynamic isotherm presented here when applied to mesoporous materials with surface heterogeneity. This analysis has proven useful to correlate the reaction yields for the Si dissolution process of dealuminated kaolin in sodium–potassium hydroxide [41].

zmin

where z describes the structural heterogeneity of the adsorbent, f ( z) is the pore size distribution designed in the micropore region (zmin , zmax ), and θtheor is the local adsorption isotherm (i.e., kernel) that describes adsorption on a homogeneous patch of the adsorbent. The function f ( z) can be selected to include the effect of surface impurities and irregularities embedded in model constants [24,25]. As proposed by Pfeifer and Avnir [26–28], fractal geometry can be applied in combination with Eq. (1) to account for gas adsorption on heterogeneous materials. In this way, fractal analogues of the Frenkel–Hasley–Hill (f-FHH), Brunauer–Emmet–Teller (fn-BET), and DA (FRDA) isotherms have been obtained [29–32]. The fractal dimension of a surface accessible to adsorption, D S , is a global measure of structural and surface irregularities of a given solid, remaining invariant over a certain degree of resolution (self-similitude) [33]. It should be emphasized that, for highly porous systems, the fractal dimension does not reflect the structure of the basic objects such as pores or clusters, but their distribution [34]. The fractal dimension can vary from 2 for a perfectly regular smooth surface to 3 for a complex surface. In addition to gas adsorption, the fractal dimension of a surface can be determined from several types of experiments, including porosimetry, small-angle X-ray and neutron scattering (SAXS and SANS), and nuclear magnetic resonance (NMR) [28,29,35–37]. In addition to these fractal analogue adsorption isotherms, Neimark [38] has proposed a thermodynamic equation for the determination of the fractal dimension of mesoporous solids from adsorption data. The theoretical basis for this method is a relationship between the surface area of the adsorbed liquid film, S, and the mean pore radius, r: Ln( S ) = const + ( D s − 2) Ln(r ).

(2)

Equation (2) can be expressed as a function of the relative pressure, P / P 0 , by relating it to the mean pore radius,





Ln( S ) = const + ( D s − 2) Ln − Ln

P P0



.

(3)

The surface area of the adsorbed film can be calculated by the Kiselev equation, S (Π) =

RT

γ

qmax

− Ln



P P0

 δq,

(4)

q

where qmax denotes the amount adsorbed at P / P 0 → 1, and γ is the surface tension of the sorbate. This thermodynamic method is compatible with the classical FHH theory in the capillary condensation regime, since both rely on the Kelvin equation in a fractal context [35,38]. In this paper, we present a thermodynamic model to describe gas adsorption on microporous materials, relying on the DA isotherm and the solution thermodynamics approach proposed by Myers for pure [39] and mixture adsorption [40]. In this analysis, the concept of surface potential, Φ , instead of the classical

2. Theory 2.1. Description of adsorption on microporous materials using solution thermodynamics From the standpoint of solution thermodynamics, an adsorption system is regarded as consisting of three phases: a gas phase (g), a solid phase or “solvent” (s), and an adsorbed phase or “solute” (a). The adsorbed phase has no volume; i.e., V a = 0. This phase, together with the solid phase, constitutes the “condensed phase” (v). The volumes of all phases are assumed not to change during the adsorption process. The volume of the gas phase, V g , including that related to micropores, is determined assuming that helium, as a reference gas, does not adsorb at near-ambient temperature and atmospheric pressure. For such a system, the specific free energy of the adsorbed phase is given by [40] G¯ a = U¯ a − T S¯ a = μq + Φ,

(5)

where U is the internal energy, S the entropy, μ the chemical potential, and Φ the surface potential, which equals the difference between the actual surface potential of the adsorbent and the chemical potential of the adsorbent without loading, i.e., μ − μ S . The symbol ‘–’ on top refers to specific variables related to the mass of adsorbent. The specific free energy of the adsorbed phase includes two contributions: (1) the free energy of q moles per kg of adsorbent adsorbed at equilibrium with the gas phase, namely with the same chemical potential, and (2) the surface potential, Φ , which depends on the sorbate–adsorbent interaction and tends to zero when there is no adsorption. The sorbate–adsorbent interaction alters the surface potential of the adsorbed phase. The differentiation of Eq. (5) at constant temperature allows obtaining δ G¯a = μδq and δΦ = −q · δ μ (Gibbs–Duhem equation), relating the surface potential to the chemical potential of the condensed phase. Assuming δ μ = R T δ Ln( P ) in the case of ideal gases (otherwise the pressure has to be substituted by a fugacity), we can obtain

Φ RT

P =−

P qδ Ln( P ) = −q M

0

θδ Ln( P ),

(6)

0

where θ = q/q M and q M = q( P 0 ). As can be deduced from Eq. (6), the surface potential is always negative and increases in absolute value as this approaches 0. The integral free energy of the adsorbed phase, G¯ a , is defined as the difference between the internal free energy of the adsorbed phase and the free energy of the same amount of sorbate at saturation pressure, μ0 ,

G¯ a = G¯ a − qμ0 .

(7)

Combining Eqs. (6) and (7), we obtain

  G¯ a = q μ − μ0 + Φ.

(8)

¯a

The differential free energy of adsorption, g , can be obtained by differentiating the integral free energy,

g¯ a =

  ∂ G a P 0 . = μ − μ = R T Ln ∂ q T P0

(9)

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2.2. Variation of the integral free energy of adsorption relative to saturation The variation of the integral free energy relative to saturation, Y , can be expressed by Y = G¯ a (q M ) − G¯ a (q)

 



= G¯ a (q M ) − G¯ a (q) − μ0 q M − q( P ) ,

(10)

which turns into

 

 −Y = Φ( P ) − Φ P 0 − q μ0 − μ .

(11)

using Eq. (8). Equation (11) can be rewritten, introducing the variable Ψ = Y /q M , and expressed in dimensionless form by

−

Ψ RT

=−

[Φ( P 0 ) − Φ( P )] RT

−θ

[μ0 − μ] RT

.

(12)

Note that −Ψ/ R T is always positive, reaching a maximum value when P → 0 (or θ → 0) and decreasing to zero as adsorption proceeds. At a given pressure, the first term in the right-hand side of Eq. (12) is always positive and provides the required surface potential to allow the adsorption process. The second term accounts for the compression work to condensate the sorbate. The variation of the integral free energy relative to saturation can be also calculated by integration of the differential free energy of adsorption (Eq. (9)) and further combination with Eq. (10). The following expression is obtained in dimensionless form, using the relative pressure Π = P / P 0 :

−

Ψ RT

=−

1

1

1 g¯ a δθ =

RT θ



− Ln(Π) δθ.

(13)

Ψ RT

1 =

− Ln(Π)δθ = θ

At higher pressures, Eq. (14) turns into −Ψ/ R T ∝ Z −m , involving a linear correlation of −Ψ/ R T with Z in double logarithmic axes, with slope m and intercept Ln(G 0 /k). 2.3. Thermodynamic analysis of the Dubinin–Astakhov isotherm The Dubinin–Astakhov isotherm relates the fractional occupancy to the relative pressure by



Ln(θ) = − −

RT

θ

It is interesting to note that Eq. (13) is formally the same as the Kiselev equation used in Neimark’s model [38], except for the physical meaning of variable Ψ . As a matter of fact, in Neimark’s model, the variable Ψ can be expressed as a function of the surface tension and surface area, i.e., Ψ = γ S / R T , referring to a 2D process [42]. Equation (13) constitutes the fundamental equation used in this study to describe gas adsorption on microporous solids. If the dependence of −Ψ/ R T on Π is known, Eq. (13) defines an isotherm, θ = θ(Π). We have verified that many microporous solids obey a “scaling law” when representing −Ψ/ R T vs (1/ − Ln(Π)), −Ψ/ R T being proportional to [1/ − Ln(Π)]−m . The exponent m can vary from 1 to 7. High values of m correspond to a rapid decrease of the available surface potential for adsorption in a narrow range of relative pressures as adsorption proceeds, indicating a narrow distribution of the structural (energy) heterogeneity of the adsorbent. At sufficiently low relative pressures, no adsorption occurs. This means that, under these conditions, −Ψ/ R T should tend to a plateau value corresponding to the maximum surface potential available for adsorption, i.e., −Φ( P 0 )/ R T . Therefore, there should be a critical relative pressure value, Πk , separating the plateau from the scaling law region. A thermodynamic adsorption isotherm, or functional dependence of −Ψ/ R T on Π , as a function of parameters m, Πk , and −Φ( P 0 )/ R T , that fulfills the above stated requirements is given by

−

Fig. 1. Representation of −Ψ/ R T vs Z = 1/ − Ln(Π) in double logarithmic axes.

G0 1 + kZm

(15)

.

The parameters of this isotherm are the characteristic energy, E 0 , and α . The surface potential can be expressed by combining Eqs. (6) and (15),

Φ RT

=− =−

1 E0

α RT 1 E

  

0

α RT



1

α 1

α

 , − Ln(θ)  α  RT , , − 0 Ln(Π)

(16)

E

where ∞ a−1 −t is the Gamma incomplete function, i.e., (a, z) = e dt. z t An expression equivalent to Eq. (16) has been developed considering spreading pressure (instead of surface potential) from the thermodynamics approach to surface coverage (instead of micropore volume filling) [43,44]. The integral free energy of the sorbate, G¯ a , can be obtained from Eq. (8) for a perfect gas,

G¯ a qM R T

=−

1 E0

α RT

    1/ α

1  . , − Ln(θ) + θ α − Ln(θ)

α

(17)

The integral free energy of the sorbate is always negative and reaches its maximum value when θ = 1, equaling the surface potential,

G¯ a q RT M

= θ=1

  1 E0 1 . = −  , 0 q M R T θ=1 α RT α Φ

(18)

Equation (17) can be rewritten in the form of Eq. (14), the relevant parameters of the thermodynamic isotherm having the expressions

(14)

with G 0 = −Φ( P 0 )/ R T , k = [− Ln(Πk )]m , and Z = 1/ − Ln(Π). Equation (14) relates −Ψ/ R T with Π in such a way that −Ψ/ R T → G 0 when Π → 0 and −Ψ/ R T ∝ Z −m at higher Π values (potential trend or scaling law; see representation in Fig. 1).

E0

α

Ln(Π)

G0 = −

Ψ RT

= θ=0

1 E0

α RT

 

1

α

 , 0 > 0,

m = (α + 1) > 1, k=



−1 E 0 −α  0 exp −( RE T )−α + α1 E αα , ( RT )



−1 E 0  0 G 0 − exp −( RE T )−α + α1 E αα , ( R T )−α

> 1,

J. Llorens, M. Pera-Titus / Journal of Colloid and Interface Science 331 (2009) 302–311

Ψ RT

=−

1 E0

α RT

305

    1 1  ,0 −  , − Ln(θ)

α

  1/ α − θ α − Ln(θ) .

α



(19)

The integral free energy relative to saturation reaches its maximum value at θ = 0 (Π = 0), equaling the value of surface potential and the integral free energy of the sorbate at θ = 1 (Π = 1):

Ψ RT

=

θ=0

G¯ a RT

= θ=1

  1 E0 1 . = −  , 0 q M R T θ=1 α RT α Φ

(20)

3. Experimental

(a)

A series of microporous commercial granulated activated carbons, zeolites, and crystalline single-wall and multiwall carbon nanotubes, the latter supplied by the Spanish Carbon Institute (ICB-CSIC), were used to test the suitability of the thermodynamic isotherm presented in this study. Table 1 summarizes the main textural properties of these materials, tested by N2 adsorption at 77 K using a Micromeritics ASAP 2010 and an Autosorb Quantachrome Instruments porosimeter. Prior to any measurement, the samples were subjected to an outgassing protocol to remove any adsorbed species (150–650 ◦ C for 1–6 h). The specific surface was determined from N2 adsorption at 77 K using the BET method in the Π range 0.01–0.15. Microporous volume and surface pore size distributions were obtained by the t-Lippens or V –t [42] and the DFT [45–48] methods from N2 adsorption data at Π < 0.10. In this region, the adsorption measurements were carried out by low-pressure doses of 3–6 N cm3 g−1 with equilibrium times in the range 20–40 s. The fittings to the thermodynamic isotherm were performed for N2 and Ar adsorption in the Π range 10−2 –0.15 in the case of activated carbons and 10−7 –0.10 for zeolites and carbon nanotubes. 4. Results

(b) Fig. 2. Integral free energy relative to saturation, Ψ/ R T , calculated by the thermodynamic isotherm for Π values in the range 10−5 –0.97 and α = 1.5, 2.0, and 2.5. (a) E 0 = 20R T , (b) E 0 = 5R T .

where E [n, z] is the integral exponential function, i.e.,

∞

exp(− zt ) tn

δt .

1

Fig. 2 plots the integral free energy relative to saturation,

−Ψ/ R T , against the variable Z in double logarithmic axes using Eq. (15) for α = 1.5, 2.0, and 2.5 and E 0 = 5R T and 20R T in the Π range 10−5 –0.97. It is interesting to note that the energy E 0 is restricted to values higher than 10R T , so that the plateau zone does not appear in the representation of the isotherm. Furthermore, the plateau zone tends to higher Π values with the reduction of energy E 0 , gas adsorption being practically avoided for E 0 → R T (for instance, for E 0 / R T = 1, α = 2, and Π = 0.1, θ < 5 × 10−3 ). The variation of the integral free energy relative to saturation, Ψ , as defined by Eq. (12), can be accounted for by

Figs. 3–5 show the experimental N2 adsorption isotherms (Type I) obtained for the active carbon ABEK, multiwall carbon nanotubes (MWNT), and a collection of zeolites with different frameworks and Si/Al ratios, as well as the representation of their corresponding thermodynamic isotherms. Two different potential trends can be clearly distinguished in the thermodynamic isotherms of zeolites and carbon nanotubes, the former ascribed to low relative pressures (i.e., <10−5 ). Accordingly, both potential trends are characterized by two different slopes (m values). Hereafter, the m values corresponding to the first and second potential trends have been termed m1 and m2 . For all the tested zeolites, parameter m1 shows higher values than parameter m2 , while the opposite trend is observed for carbon nanotubes. Isotherm data in the range of very low relative pressures may be affected by diffusion processes. However, this lack of accuracy does not exert much influence on the exponent, m, of the thermodynamic adsorption isotherm. This is because the integral defined in Eq. (13) has a fixed integral limit, the maximum coverage, θ = 1, and a variable integral limit, the actual coverage, θ . Therefore, the integral calculated at a specific coverage only contains information related to higher coverage and it is not affected for lower coverage. In general, there are only a few points of very low relative pressure in the range of relative pressures where the scaling law holds. The thermodynamic adsorption isotherm, Eq. (14), approaches an asymptotic plateau for extremely low pressures. In all cases, a potential behavior is observed until 90% surface coverage, corresponding approximately to − Ln(Π) ≈ 1.4 or Π ≈ 0.25. Because the interest of these adsorbents relies on their micropores and these are completely filled beyond 90% surface

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Table 1 Textural properties of activated carbons, carbon nanotubes, and zeolites characterized from N2 adsorption/desorption isotherms at 77 K combined with the DFT technique. Sample

Specific surface [m2 g−1 ]

Volume [cm3 g−1 ]

dm [Å]

Supplier

0.165 0.197 0.022 0.020 0.103 0.028 0.113 0.134 0.126 0.218 0.136 0.143 0.200

29.5 37.1 34.4 33.7 35.2 34.2 40.5 37.2 45.6 34.9 36.4 35.6 27.8

Picactif Aguas de Levante Aguas de Levante Warwick Benbassat – Betaquimica Panreac Norit Betaquimica Norit Betaquimica Norit Picactif

0.065 0.004

0.081 0.017

5.7, 14.0 11.2, 29.0

ICB ICB

0.068 0.100 0.217 0.110 0.273 0.069 0.020 0.038 0.038 0.215 –

0.000 0.000 – 0.000 – 0.000 – 0.193 0.061 – –

7 .7 4 .0 8 .6 7 .7 7 .4 7 .4 7 .4 3.9, 5.9 8 .5 6 .0 7 .0

BET

<14.8 Å

>14.8 Å

<14.8 Å

>14.8 Å

ABEKa,b BPL GH12132 GH6112 GMA HROa PANRG RBAA1b RGBa,b RGG08b RGKb RZN1b TAa

1135 1232 953 843 1294 1021 784 862 1331 1222 1179 1087 1451

658 818 814 715 1037 874 559 542 1005 737 857 745 905

158 153 17 16 92 25 51 122 118 176 115 142 205

0.296 0.325 0.303 0.267 0.373 0.320 0.237 0.233 0.372 0.316 0.325 0.305 0.385

SWNTc MWNTd

265 29

57.4 10.7

5A 4A Faujasite NaY NaX ZM-510 AIPO4-39 ZSM-5 ZSM-12 Beta MOR

752 255 611 882 716 470 318 351 337 632 279

0

a b c d

159 8.6 349 – – 529 – 346 100 144 190 – –

– – 0 – 0 – 221 219 – –

Grace–Davidson IQE BAM Grace–Davidson Fluka – – – – – Tosoh

Activated with water vapor. Impregnated. Single-wall carbon nanotubes with the presence of amorphous carbon material and Ni/Y metallic particles. Multiwall carbon nanotubes with variable number of carbon layers.

Fig. 3. Activated carbon ABEK: (left) adsorption isotherm of N2 at 77 K, (right) representation of the thermodynamic isotherm (Eq. (14)).

coverage, the value of m has been measured in a Π range accounting for at least 90% of saturation. More specifically, m has been determined in the Π ranges 0.02–0.25 for activated carbons and 10−7 –0.25 for carbon nanotubes and zeolites. Tables 2–4 list the values found for parameters m and G 0 /k from the fittings of the thermodynamic isotherm to experimental N2 and Ar adsorption data for the microporous solids surveyed in this study. As can be seen in Table 4, within the limits of experimental error, the values of parameters m1 and m2 do not change for a given zeo-

lite irrespective of the adsorbing gas (i.e., N2 or Ar). In contrast, the value of the dimensionless integral free energy of adsorption at Π = 1, the value of parameter G 0 /k changes with the sorbate, showing lower values for Ar than for N2 for all the tested zeolites. Note that, in the case of carbon nanotubes and zeolites, the plateau zone of the thermodynamic isotherm cannot easily be measured in the Π range surveyed in this study (a proper measurement would require Π values lower than 10−7 , something difficult to achieve using conventional porosimeters).

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307

Fig. 4. Multiwall carbon nanotubes (CNT2): (left) adsorption isotherm of N2 at 77 K, (right) representation of the thermodynamic isotherm (Eq. (14)).

5. Discussion 5.1. Physical interpretation of parameter m As stated in Section 2, parameter m can be related to parameter α in the DA isotherm through the relation m = α + 1. This implies that parameter m might be related to a certain extent to the energy heterogeneity of the solid. This idea is supported by the fact that the experimental values of parameters m1 and m2 obtained in this study remain invariant for a given solid when the sorbate changed (see Table 4). Moreover, the fact that, for a given sorbate, parameters m1 and m2 change with the solid reflects that, although the sorbate–adsorbent interaction remains unchanged, its distribution might change. It is also interesting to note that, when the thermodynamic isotherm presented in this study (potential zone in Eq. (14)) is compared with Neimark’s model for mesoporous solids with selfsimilar fractal geometry [38], both equations are formally the same, since the starting point of both models, the Kiselev equation (Eq. (4)), is also the same. Therefore, Neimark’s model could be regarded as a particular case of the thermodynamic isotherm, with the parameter m being related to the fractal dimension of the solid (m = D S − 2). As the fractal dimension of a solid always lies in the range 2–3, m values in the range 0–1 are expected for these materials. As a matter of fact, these values are much lower than those obtained for parameters m1 and m2 in the microporous materials tested in this study (see Tables 2–4). 5.2. Energy heterogeneity in microporous materials: two potential trends The two differentiated potential zones in the representations of the thermodynamic isotherm for carbon nanotubes and zeolites (see Figs. 4 and 5, respectively) suggest a different pore-filling or free-energy-dissipation pattern in the early stage of adsorption and at higher loadings. Parameter m1 , obtained at lower relative pressures (relative pressures <10−5 ), when available, appears to be more adequate to characterize the structure of microporous materials, since its values are obtained from the interaction between an extremely reduced number of sorbate molecules and the adsorbent. Indeed, parameter m1 provides a picture of the energy heterogeneity of the raw material. In contrast, m2 values provide

Table 2 Activated carbons characterized by the thermodynamic isotherm (Eq. (14)) using N2 adsorption data at 77 K. Sample

G a /k × 102 [—]

m2 [—]

r 2 [—]

ABEK BPL GH12132 GH6112 GMA HRO PANRG RBAA1 RGB RGG08 RGK RZN1 TA

3.1 ± 0.4 3.0 ± 0.1 1.72 ± 0.09 2.12 ± 0.05 2.2 ± 0.2 1.8 ± 0.1 7.83 ± 0.2 2.5 ± 0.2 2.3 ± 0.2 2.8 ± 0.2 2.6 ± 0.2 2.4 ± 0.2 1.75 ± 0.04

2.2 ± 0.1 2.23 ± 0.05 2.10 ± 0.06 1.88 ± 0.02 2.2 ± 0.1 1.98 ± 0.08 1.21 ± 0.02 2.2 ± 0.1 2.1 ± 0.2 2.36 ± 0.07 2.3 ± 0.1 2.4 ± 0.09 2.87 ± 0.03

0.9943 0.9988 0.9985 0.9994 0.9965 0.9974 0.9995 0.9974 0.9915 0.9984 0.9956 0.9978 0.9999

Note. Fittings in the Π range 0.0300–0.2500.

information about the energy heterogeneity of an adsorbent partially filled with the sorbate, since they are obtained from fittings at higher loadings, lying in the narrow range 0.86–3.5 for all the tested materials. According to Table 4, parameter m1 shows higher values than parameter m2 in the case of zeolites, where values as high as 10.6 have been obtained for zeolite 5A. This result could be interpreted in terms of surface smoothing at higher loadings. In contrast, in the case of single-wall and multiwall carbon nanotubes, m1 values are much lower. These differences between zeolites and carbon nanotubes in terms of m1 values reflect a different gas adsorption pattern of the two materials. Furthermore, although no m1 values could be measured for activated carbons, a difference in their adsorption pattern compared to that of zeolites can be foreseen (i.e., m1 < m2 ). In general, no correlations can be expected to exist between the specific surface and volume, for each class of material, and their corresponding parameters m1 and m2 . This is because energy heterogeneity concerns characteristics of points in space, while specific surface and volume involve the entire characteristics of the material. However, a slight correlation can be observed between specific surface and the m2 parameter. On the whole, m2 increases with specific surface for each class of material (see Tables 1–4).

308

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Fig. 5. Zeolites ZM-510, faujasite, 5A, and ZSM-12: (left) adsorption isotherm of N2 at 77 K, (right) representation of the thermodynamic isotherm (Eq. (14)).

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309

Table 3 Carbon nanotubes characterized by the thermodynamic isotherm (Eq. (14)) using N2 adsorption data at 77 K. First potential trenda

Sample

G 01 /k

a

× 10 [—]

12 ± 1 54 ± 3

SWNT MWNT b

Second potential trendb

2

m1 [—]

r12

1.52 ± 0.05 1.00 ± 0.04

0.9968 0.9947

[—]

G 02 /k × 102 [—]

m2 [—]

r22 [—]

5.88 ± 0.02 0.19 ± 0.03

1.86 ± 0.05 3.5 ± 0.1

0.9984 0.9965

Fittings in the Π range 5 × 10−7 –7 × 10−5 . Fittings in the Π range 7 × 10−5 –0.2500.

Table 4 Zeolites characterized by the thermodynamic isotherm (Eq. (14)) using N2 and Ar adsorption data at 77 K. First potential trenda

Sample

G /k × 10 [—] 0

ZM-510 AIPO4 Mordenite a

c d

11 ± 1 7±3 6.2 ± 0.4 7±3 4.3 ± 0.1 4.0 ± 0.2 6.4 ± 0.3 6.3 ± 0.7d 6.5 ± 0.4 5.4 ± 0.8d 4.8 ± 0.2 9±1 9 ± 2d –

0. 3 ± 0 . 3 3±2 2±1 (8 ± 2) × 102 (1.5 ± 0.3) × 103 3±1 12.4 ± 0.9d 2.1 ± 0.9 (1.0 ± 0.9) × 10d (1.7 ± 0.4) × 102 (5 ± 4) × 10−4 0.1 ± 0.3b –

ZSM-5

Second potential trendb m1 [—]

(3 ± 3) × 10−5

Zeolite 5A Zeolite 4A Faujasitec Zeolite X Zeolite Y Zeolite β ZSM-12

b

7

r12

[—]

0.9994 0.9376 0.9869 0.9005 0.9907 0.9931 0.9843 0.9986d 0.9939 0.9702d 0.9961 0.9959 0.9590d –

G 0 /k × 102 [—]

m2 [—]

r22 [—]

(3.9 ± 0.3) × 10−2

3 .3 ± 0.1 1 .5 ± 0.5 2 .4 ± 0.1 3 .3 ± 0.1 2.56 ± 0.08 1 .4 ± 0.1 1.88 ± 0.03 2.09 ± 0.04d 2 .0 ± 0 .3 2.4 ± 0.1d 2.74 ± 0.03 0.86 ± 0.02 1.16 ± 0.03d 2 .4 ± 0.1

0.9989 0.9733 0.9948 0.9976 0.9990 0.9750 0.9989 0.9980d 0.9721 0.9854d 0.9993 0.9995 0.9978d 0.9971

1 .6 ± 0.5 0.33 ± 0.07 (6.8 ± 0.6) × 10−2 0.29 ± 0.01 4 .6 ± 0.5 2.32 ± 0.01 2.13 ± 0.01d 1 .6 ± 0.6 1.0 ± 0.1d 0.30 ± 0.01 27.2 ± 0.5 6.44 ± 0.1d 0.48

Fittings in the Π range 5 × 10−7 –7 × 10−5 . Fittings in the Π range 7 × 10−5 –0.25. Data supplied by the Federal Institute for Materials Research and Testing (BAM, Unter den Eichen 87, 12205-Berlin, Germany). Data obtained from Ar adsorption at 77 K.

Finally, in the case of zeolites, it is interesting to point out that although no definite correlation can be established between the values of parameters m1 and m2 listed in Table 4 and the zeolitic structure, zeolites with cage-like structures, namely LTA (4A and 5A) and faujasite (NaX and NaY), apparently show higher m1 values (range 4.3–10.6) than zeolites based on cylindrical pore frameworks (i.e., beta, ZSM-5, ZSM-12, ZM-510), lying in the range 3.99–6.4.

Table 5 Isotherms derived from the generalized Langmuir isotherm. Adsorption isotherm

β [—]

δ [—]

Equation

Single-site Langmuir

1

1

Generalized Freundlicha Langmuir–Freundlich Tóthb

0β 1 β =1 β >1

δ=1 0δ1 0δ1

q = q M 1+K KP P q = q M ( K P )1/n

a b

5.3. Thermodynamic analysis of the Langmuir isotherm Although the surface thermodynamics approach (2D) does not seem to provide a suitable framework to describe gas adsorption in microporous materials, the Langmuir isotherm will be here used to prove its inconsistency with the thermodynamic isotherm. In the most general situation, the generalized Langmuir (GL) isotherm can be used to account for gas adsorption on an energetically heterogeneous surface, where the fractional loading can be obtained by solving Eq. (1) using the Langmuir equation for local fractional loading. The generalized Langmuir (GL) isotherm can be obtained [49],

θ=

q qM

 =

y 1+ y

β (21)

, 

y = ( K P )δ = ( K ∞ P )δ exp −δ

H 0 RT

 ,

(22)

where K is the adsorption constant, K ∞ is a pre-exponential factor that contains the entropy of adsorption, and H 0 is the adsorption enthalpy. The resolution of the GL isotherm provides a number of adsorption isotherms depending on the relative values of parameters β and δ . As shown in Table 5, the GL isotherm tends to the single-site Langmuir (L) isotherm for β = δ = 1, the generalized Freundlich (GF) isotherm for 0  β  1 and δ = 1, the Langmuir– Freundlich (LF) isotherm for β = 1 and δ < 1, and the Tóth (T)

q KP

q = [1+( KM P )δ ]1/δ

β · δ = 1/n  1. β · δ = 1.

isotherm for β > 1 and 0 < δ  1. The surface potential, Φ , can be obtained by

Φ qM R T

=−

yβ

t

δ

t β−1 (1 + t y )−β δt .

(23)

0

The integral free energy of the sorbate can be obtained from Eq. (8) for a perfect gas,



G¯ a qM R T

=

y 1+ y

β Ln(Π) +

Φ qM R T

(24)

.

The integral free energy of the sorbate is negative and increases with Π . Note that a maximum pressure value has to be placed to restrict the pressure at which saturation is achieved. Since the Langmuir isotherm assumes an asymptotic tendency to saturation, we can use, as a practical saturation, θ = 0.99999, in order to attain a finite value for the saturation pressure. In this case, using for instance a Π value involving θ = 0.99999, the saturation pressure can be expressed by P0 =

1 K



(0.99999)1/β 1 − (0.99999)1/β

1/δ

 exp

E0 RT

 .

(25)

310

J. Llorens, M. Pera-Titus / Journal of Colloid and Interface Science 331 (2009) 302–311

Fig. 6. Integral free energy relative to saturation, Ψ/ R T , calculated by the Langmuir isotherm, as a function of the parameter Z = 1/ − Ln(Π) for Π values in the range 10−4 –0.90. The straight line corresponds to the generalized Langmuir isotherm with β = 0.00025 and δ = 1.5.

In addition, the integral free energy relative to saturation can be calculated by

Ψ RT

β

=

− yo δ

1

t β−1 (1 + t y o )−β dt −

0



y 1+ y

β Ln(Π) −

Φ RT

with yo =

1

(0.99999)−1/β − 1

.

(26)

Fig. 6 shows the evolution of −Ψ/ R T with Z = 1/ − Ln(Π) for the GL isotherm, the saturation pressure being estimated at different fractional loadings. As predicted by the thermodynamic isotherm, this trend only becomes potential for a reduced combination of values of parameters β and δ . It is noteworthy that the representation of −Ψ/ R T vs Z in the case of the single-site Langmuir isotherm (β = δ = 1) does not show the potential zone at any pressure. Fig. 7 plots the evolution of −Ψ/ R T with Z in the interval 10−3 < Π < 0.50 for P 0 calculated at θ = 0.99999 using Eq. (26) and for the specific β and δ values that allow displaying a potential behavior. As can be inferred from this figure, the GL isotherm only provides a potential trend at very specific pressure intervals. The functional dependence between parameter m and parameters β and δ is not straightforward, contrary to what is observed in the DA isotherm, where m = α + 1. As expected, the Langmuir isotherm does not provide a good framework to describe gas adsorption in microporous adsorbents on the basis of its implicit 2D character. 5.4. Predicting gas adsorption on microporous materials The thermodynamic isotherm appears to be a useful tool for inferring the adsorption pattern of microporous materials in relevant industrial applications (e.g., pressure-swing adsorption, energy storage). The parameter m1 that captures the information related to the energy distribution of raw materials, as well as their structure and composition, might address research strategies to synthesize materials with optimized structures suitable for the

Fig. 7. Integral free energy relative to saturation, Ψ/ R T , calculated by the Langmuir isotherm, as a function of parameter Z = 1/ − Ln(Π) for Π values in the range 10−3 –0.50 and P 0 calculated at θ = 0.99999. The points correspond to the values calculated by the generalized Langmuir isotherm: ! (β = 1.5 × 10−3 , δ = 2.0), 1 (β = 2.5 × 10−4 , δ = 1.5), P (β = 4.5 × 10−5 , δ = 1.0), and E (β = 8 × 10−6 , δ = 0.5). The dashed lines refer to fittings to the thermodynamic isotherm, while the straight line refers to the single-site Langmuir isotherm (β = δ = 1).

adsorption of target gases. This is the case, for instance, of hydrogen adsorption on activated carbon. In this case, it appears more convenient to infer the adsorption pattern at these conditions using the parameter m2 , characterizing the second potential trend, since high pressures are usually required to attain high loadings. 6. Conclusions In this study, we have presented a new general thermodynamic isotherm to describe gas adsorption on microporous materials. This isotherm relies on a potential relationship between the integral free energy relative to saturation, −Ψ/ R T , and the parameter Z = 1/ − Ln(Π), where m is a structural parameter that characterizes the energy heterogeneity of the adsorbent. The isotherm has been deduced from solution thermodynamic principles and the DA isotherm. Neimark’s thermodynamic model has been found to be a particular case of the thermodynamic isotherm presented here. The application of the thermodynamic isotherm to experimental N2 and Ar adsorption on a collection of microporous materials (activated carbons, carbon nanotubes, and zeolites) reveals the presence of two differentiated potential trends, the former involving m parameters (m1 ) not dependent on the sorbate, thus providing a picture of the raw material. However, parameter m2 , accounting for the second potential trend, appears to be more appropriate to characterize gas adsorption at higher loadings. The thermodynamic analysis presented in this study provides complimentary information about the intimate structure of microporous materials, which can be further used to infer the adsorption behavior of target gases (e.g., hydrogen). This aspect will be analyzed in detail in a future paper. Acknowledgments The authors express their gratitude to the Spanish Ministry of Education and Science for funding support (Project CTQ200508346-C02-01). The authors are indebted to N. Cristin from IRCELYON for providing a great part of the N2 and Ar adsorption

J. Llorens, M. Pera-Titus / Journal of Colloid and Interface Science 331 (2009) 302–311

isotherm data on zeolites. Professors M.T. Martinez and Dr. A. Anson, both from the Spanish Instituto de Carboquímica (ICB-CSIC, Zaragoza), kindly provided the N2 adsorption data on carbon nanotubes.

Appendix A. Nomenclature Ds E [n, z] E0 f G G0 K k m P0 P q R S T U V y y0 Z z

Surface fractal dimension [—] Integral ∞ −n exponential function: E [n, z] = t exp(− zt ) dt 1 Characteristic energy of the DA isotherm [J mol−1 ] Distribution function, Eq. (1) [—] Gibbs free energy per unit mass of adsorbent [J kg−1 ] Integral free energy of the sorbate at P 0 , Eq. (30) [—] Pre-exponential factor in the GLI, Eq. (35) [Pa−1 ] Parameter in Eq. (35) [—] Exponent in Eq. (35) [—] Saturation pressure [Pa] Pressure [Pa] Sorbate loading [J kg−1 ] Gas constant [8.314 J mol−1 K−1 ] Sorbate entropy per unit mass of adsorbent [J kg−1 K−1 ] Temperature [K] Sorbate internal energy per unit mass of sorbent [J kg−1 ] Sorbate volume per unit mass of adsorbent [m3 kg−1 ] Parameter defined in the GLI, Eq. (34) [—] Parameter defined in Eq. (40) [—] 1/ − Ln(Π)] [—] Variable describing the structural heterogeneity, Eq. (1)

Greek symbols

α β ga G a δ Φ (a, z)

μ μ μ s Π Πk θ θl θt Ψ

Exponent in the DA isotherm, Eq. (14) [—] Exponent in the GLI [—] Differential Gibbs free energy [J mol−1 ] Integral free energy of the sorbate [J kg−1 ] Exponent in the GLI [—] Surface potential [J kg−1 ] ∞ Incomplete Gamma function: (a, z) = z t a−1 e −t dt Chemical potential of the sorbate [J mol−1 ] Chemical potential of the adsorbent [J mol−1 ] Chemical potential of the pure adsorbent [J mol−1 ] P / P 0 [—] Parameter related to k according to Eq. (31) [—] Fractional loading, q( P )/q( P 0 ) [—] Local adsorption loading isotherm in Eq. (1) [—] Fractional loading in Eq. (1) [—] Integral Gibbs free energy relative to saturation [J kg−1 ]

Superscripts a g mono s

Sorbate Gas phase Monolayer Solid phase

311

References [1] K.S.W. Sing, Colloids Surf. A Physicochem. Eng. Aspects 241 (2004) 3. [2] S.J. Gregg, K.S.W. Sing, Adsorption, Surface Area and Porosity, Academic Press, London, 1982. [3] D.H. Everett, R.H. Ottewill (Eds.), Surface Area Determination, Butterworths, London, 1970. [4] K.S.W. Sing, D.H. Everett, R.A.W. Haul, I. Moscou, R.A. Pierotti, J. Rouquerol, T. Siemieniewska, Pure Appl. Chem. 57 (1985) 603. [5] V.P. Bering, V.V. Serpinski, Izv. Akad. SSSR Ser. Khim. (1974) 2427 [in Russian]. [6] V.P. Bering, V.V. Serpinski, T.S. Jakubov, Izv. Akad. SSSR Ser. Khim. (1977) 121 [in Russian]. [7] S. Suwanayuen, S.P. Danner, AIChE J. 26 (1980) 68. [8] S. Suwanayuen, S.P. Danner, AIChE J. 26 (1980) 76. [9] L.P. Ding, S.K. Bhatia, Carbon 39 (2001) 2215. [10] S. Koter, A.P. Terzyk, J. Colloid Interface Sci. 282 (2005) 335. [11] B.P. Bering, M.M. Dubinin, V.V. Serpinsky, J. Colloid Interface Sci. 21 (1966) 378. [12] M.M. Dubinin, V.A. Astakhov, L.V. Radushkevich, in: Physical Adsorption of Gases and Vapors in Micropores, in: Progress and Membrane Science, vol. 9, Academic Press, New York, 1975. [13] F. Stoeckli, Adsorpt. Sci. Technol. 10 (1993) 13. [14] M.M. Dubinin, E.D. Zaverina, L.V. Radushkevich, Zh. Fiz. Khim. 21 (1947) 1351. [15] M.M. Dubinin, E.D. Zaverina, Zh. Fiz. Khim. 21 (1947) 1373. [16] H.F. Stoeckli, J. Colloid Interface Sci. 59 (1977) 184. [17] N.D. Hutson, R.T. Yang, Adsorption 3 (1997) 189. [18] M.M. Dubinin, V.A. Astakhov, Izv. Akad. Nauk SSSR Ser. Khim. 11 (1971) 5 [in Russian]. [19] H.F. Stoeckli, T. Jakubov, A. Lavanchy, J. Chem. Soc. Faraday Trans. 90 (1994) 783. [20] F. Carrasco-Marin, M.V. Lopez-Ramon, C. Moreno-Castilla, Langmuir 9 (1993) 2758. [21] P.A. Terzyk, A.P. Gauden, P. Kowalczyk, Carbon 40 (2002) 2879. [22] P.A. Gauden, A.P. Terzyk, G. Rychlicki, P. Kowalczyk, M.S. Cwiertnia, J.K. Garbacz, J. Colloid Interface Sci. 273 (2004) 39. [23] W. Rudzinski, D.H. Everett, in: Adsorption of Gases on Heterogeneous Surfaces, Academic Press, London, 1992. [24] J. Sun, S. Chen, M.J. Rood, M. Rostam-Abadi, Energy Fuels 12 (1998) 1071. [25] J. Jagiello, J.A. Schwarz, Langmuir 9 (1993) 2513. [26] P. Pfeifer, D. Avnir, J. Phys. Chem. 79 (1983) 3558. [27] P. Pfeifer, D. Avnir, J. Phys. Chem. 80 (1984) 4573. [28] D. Avnir (Ed.), The Fractal Approach to Heterogeneous Chemistry, Chichester, Wiley, 2002, pp. 48, 145. [29] A.P. Terzyk, P.A. Gauden, Arab. J. Sci. Eng. 28 (2003) 133. [30] P.A. Gauden, A.P. Terzyk, G. Rychlicki, Carbon 39 (2001) 267. [31] A.P. Terzyk, P.A. Gauden, G. Rychlicki, R. Wojsz, Colloids Surf. A Physicochem. Eng. Aspects 152 (1999) 293. [32] A.P. Terzyk, R. Wojsz, G. Rychlicki, P.A. Gauden, Colloids Surf. A Physicochem. Eng. Aspects 96 (1997) 67. [33] B.B. Mandelbrot, The Fractal Geometry of Nature, Freeman, San Francisco, 1983. [34] O. Malcai, D.A. Lidar, O. Binham, D. Avnir, Phys. Rev. E 56 (1997) 2817. [35] D. Avnir, D. Farin, New J. Chem. 16 (1992) 439. [36] M. Jaroniec, Langmuir 11 (1995) 2316. [37] M. Jaroniec, M. Kruk, J. Olivier, Langmuir 13 (1997) 1031. [38] A.V. Neimark, Physica A 191 (1992) 258. [39] F.R. Siperstein, A.L. Myers, AIChE J. 47 (2001) 1141. [40] A.L. Myers, AIChE J. 48 (2002) 145. [41] F.G. Colina, J. Llorens, Microporous Mesoporous Mater. 100 (2007) 302. [42] B. Sahouli, S. Blacher, F. Brouers, Langmuir 12 (1996) 2872. [43] A. Lavanchy, M. Stockli, C. Wirz, H.F. Stoeckli, Adsorpt. Sci. Technol. 13 (1995) 537. [44] P.A. Gauden, A.P. Terzyk, S. Furmaniak, R.P. Wesołowski, P. Kowalczyk, J.K. Garbacz, Carbon 42 (2004) 573. [45] B.C. Lippens, J.H. Deboer, J. Catal. 4 (1965) 319. [46] P. Tarazona, Mol. Phys. 52 (1984) 81. [47] P. Tarazona, Phys. Rev. A 31 (1985) 2672. [48] J.P.R.B. Walton, N. Quirke, Chem. Phys. Lett. 129 (1986) 382. [49] D.M. Ruthven, in: Principles of Adsorption and Adsorption Processes, Wiley, New York, 1984.