Mathematical Description of Constrained BET-like Adsorption and Its

Langmuir 2002, 18, 7503-7514

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Mathematical Description of Constrained BET-like Adsorption and Its Approximation with LgBET and LcBET Formulas Jan T. Duda† and Janina Milewska-Duda*,‡ Institute of Automatics and Faculty of Fuels & Energy, University of Mining and Metallurgy, Al. Mickiewicza 30, 30-059 Krako´ w Poland Received February 1, 2002. In Final Form: May 24, 2002 The paper develops a mathematical description of constrained BET-like adsorption of small nearly spherical molecules in microporous materials. Possible formation of adsorbate clusters more compact than BET stacks (implying non-BET configurational effects) and constraints for the cluster size are taken into account. A rigorous mathematical model based on the BET approach and fundamental thermodynamics is written in a recurrent form and analyzed from a formal side. It gives an insight into links between pore geometry and shape of isotherms at high-pressure range. A general modeling scheme of the constrained multilayer adsorption is shown. A new approximation of the rigorous model aimed at examination of microporous systems in wide pressure range is proposed (LcBET formula). An attempt to handle adsorption peculiarities at the highest relative pressure (condensation point) is also made. The new LcBET formula covers an inadequacy area of the models published before (LBET and LgBET). It can provide quantitative information on a shape and size of pores, if high-pressure adsorption data are available. Exemplary results of the model application to analysis of N2 adsorption on synthetic activated carbons are discussed.

1. Introduction The BET equation, applied to adsorption isotherms at a low relative pressure range (up to ca. 0.3), is often recommended as a useful tool for a surface analysis of microporous materials, when small (nearly spherical) molecules are used as the probes.1 For higher pressures the original BET formula becomes often highly inadequate,2 as it admits only rather unrealistic stacklike adsorbate clusters and disregards physical constraints for multilayer adsorption3 (evident in microporous systems). In our earlier papers4,5 we showed that the BET approach, when used with thermodynamic theory of solutions,6,7 is a very effective way to obtain a consistent mathematical description of sorption processes in microporous materials, including swelling phenomena in submicropores,7 sorbate-sorbate interactions,8 and geometrical constraints for adsorbate clusterization in larger pores.4 A general mathematical model for multilayer adsorption, handling such effects, was derived,4 then an essential generalization of the BET theory was made5 to take into account possible formation of more compact * To whom correspondence should be addressed: tel, (48 12) 617 2117; fax, (48 12) 617 2066, e-mail, [email protected]. † Institute of Automatics. ‡ Faculty of Fuels and Energy. (1) Kruk, M.; Jaroniec, M.; Choma, J. Adsorption 1997, 3 209. To`th, J. Adv. Colloid Interface Sci. 1995, 55, 1-229. Sing, K. S. W.; Everett, D. H.; Haul, R. A. W.; Moscou, I.; Pierotti, R. A.; Rouquerol, J.; Siemieniewska, T. Pure Appl. Chem. 1985, 57, 603. Rittner, F.; Boddenberg, B.; Bojan, M. J.; Steele, W. A. Langmuir 1999, 15, 1456. (2) Kruk, M.; Jaroniec, M.; Gadkaree, K. P. Langmuir 1999, 15, 1442. (3) Rudzinski W., Everett D. H, Adsorption of Gases on Heterogeneous Surfaces; Academic Press: London, San Diego, 1992. (4) Milewska-Duda, J.; Duda, J.; Jodłowski, G.; Kwiatkowski, M. Langmuir 2000, 16, 7294. (5) Milewska-Duda, J.; Duda, J. Langmuir 2001, 17, 4548. (6) Prigogine, I. The Molecular Theory of Solutions; North-Holland: Amsterdam, 1957. Flory, J. P. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. (7) Milewska-Duda, J.; Duda, J.; Nodzen˜ski, A.; Lakatos, J. Langmuir 2000, 16, 5458. Milewska-Duda, J.; Duda, J. Langmuir 1993, 9, 3558. Milewska-Duda, J.; Duda, J. Langmuir 1997, 13, 1286. (8) Duda, J.; Milewska-Duda, J. Colloids Surf., in press.

adsorbate clusters than BET stacks (causing non-BET configurational effects). The resultant rigorous model consists of a large number of nonlinear equations. Hence, we proposed simple approximating formulas (LBET and LgBET equations) enabling for reliable evaluation of a material surface area and giving semiquantitative information on the pore structure.4,5,7,8 In this paper the rigorous model is written in a convenient recurrent form and discussed in detail from its formal side. Important links between pore geometry and isotherm shape are outlined. A new useful approximating formula (the LcBET equation) is proposed. It describes very well the constrained multilayer adsorption over a wide relative pressure range, by combining Henry’s and Langmuir’s isotherms on consecutive layers. An attempt to handle adsorption peculiarities at the highest relative pressure (condensation point) is also presented. New possibilities for interpretation of adsorption isotherms are shown by using exemplary literature data on nitrogen adsorption in synthetic microporous carbons.9 2. Constrained Multilayer Adsorption and Its Formal Description The mathematical model of adsorption of small nearly spherical molecules discussed in this paper is based on the generalized BET theory.4,5 One considers an equilibrium state of adsorption at a temperature T and relative fugacity π ) f/fs, where fs, stands for the fugacity of adsorbate in its liquidlike reference state under a pressure Psssee ref. 10 (for vapors Ps is the saturated vapor pressure P0, and π = P/P0). The adsorbate molecules placed in pores may be divided into subsets (adsorption subsystems) containing mpa moles of the molecules of the same energy. At any equilibrium state the following fundamental equation is satisfied6 (9) Choma, J.; Jaroniec, M. Langmuir 1997, 13, 1026. (10) Milewska-Duda, J.; Duda, J.; Jodłowski, G.; Wo´jcik, M. Langmuir 2000, 16, 6601.

10.1021/la020110e CCC: $22.00 © 2002 American Chemical Society Published on Web 08/20/2002

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Langmuir, Vol. 18, No. 20, 2002

RT ln(π) )

∂∆H ∂∆S -T ∂mpa ∂mpa

Duda and Milewska-Duda

(1)

where ∆H and ∆S denote the total energy and entropy changes due to the replacing of adsorbate molecules from their reference state to the adsorption system; R is the gas constant. Adsorbate molecules are distributed over the adsorption space (pores) in the form of clusters of different size. In the BET approach such clusters are reconstructed by a virtual multistep adsorption, starting with molecules placed directly on a pore surface due to adhesive forces (primary adsorption sites) and joining further molecules mainly due to cohesive forces. In step n the adsorbate molecules are arranged only at sites newly created by the molecules placed at the n - 1 step (n ) 0 refers to the primary sites). In this way the nth adsorption layer is created. The number of so-placed molecules is determined by the equilibrium condition (eq 1) related to particular subsystems properly specified. The fundamental assumption of the BET approach is that any adsorbate molecule can be placed at any site available for its subsystem (on the lower layer) with the same probability, and this probability does not depend on occupancy of the sites. It means4,5 that the molecule can occupy only one site, each cluster has to be created only in a presumed space separated from that attributed to other clusters, and energetic interactions of molecules placed at the same layer are negligible. Additionally, the modeling procedure itself needs each cluster to be started with a unique primary site. However, there are no formal obstacles to assume that a molecule placed on the nth layer may offer more than one site for its higher layer. Thus clusters more compact than BET stacks can be considered.5 Acceptability area of such assumptions is discussed in our earlier papers.5,7 The approach was found to be useful for modeling of adsorption of spherical molecules in microporous materials. Otherwise, more complicated modeling techniques may be applied.3,11 In clusters of the multilayer structure an ath adsorption subsystem may be defined as consisting of molecules placed at the same layer of all clusters having identical energy profiles across the layers (at given π, T). The main difference in the adsorption energy is due to different properties of primary sites and geometrical restrictions for a size of clusters attributed to these sites. Let k denote the maximal number of a cluster layers, at which placing of a molecule does not require expansion of the pore; i.e., the adsorption energy at n > k is much larger4,5 than that specific for lower layers n e k. The number k will be referred to as the cluster size or the primary site capacity. Let us split the set of primary sites (containing mhA moles of sites) into subsets, each of them containing mhAk moles of sites of the same capacity k (kth type sites). Let us assume that the adsorption on each primary site contributes to the quantity ∆H (eq 1) with the same molar energy QA. In this case the adsorption subsystems may be pointed to with the pair (k,n) showing an adsorptive capacity (k) of the primary site and a number (n) of the considered layer (n e k). Let Qkn denote the adsorption energy (Qk1 ≡ QA) and mpkn stand for the number of adsorbate molecules in a knth subsystem. Let us take that the number of sites offered by the (n - 1)th layer for the nth one is mpkn-1‚βkn (βk1 ≡ 1). The parameter βkn characterizes an averaged width of pores by possible (11) Steel, W. Langmuir 1999, 15, 6083. Tovbin, Yu., K. Langmuir 1999, 15, 6107.

intensity of the non-BET clusterization of adsorbate molecules. A value βkn = 1 may be expected in narrow channel or slitlike pores. Larger values of βkn (ranging to βmax ≈ 3) are likely in more compact or dendrite-like pores,5,8 but βkn < 1 is also possible at higher layers. If an average distance between the sites available for the nth layer (n > 1) is greater than the adsorbate molecule diameter, one may assume that Qkn is not affected by mpkn-1. In practice, it may be accepted if the sites are geometrically well separated (narrow clusters, βkn = 1) or sparsely distributed over a set of Mtkn-1 sites preassigned to the k(n - 1)th layer of wide clusters. Otherwise, the value Qkn (constant, specific for low π values) should be corrected by adding a term ∆Qkn = -qknmpkn-1, where qkn denotes a parameter: qkn > 0, if mpkn-1 < θCMtkn-1, θC is a critical coverage ratio (see θ defined in eq 6). In the case of a large flat square lattice θC = 0.25 makes possible significant additional intermolecular contacts; hence for typical clusters θC > 0.25 may be expected. With the above denotation, the energy term ∆H in eq 1 may be written as ∞

∆H )

k

∑ ∑ mpkn(Qkn - qknmpkn-1) k)1 n)1

(2)

Changes in the internal entropy of adsorbate molecules may be included into the quantity Qkn, viewed further as the molar free energy of adsorption. The configurational component ∆S (see eq 1) is of the form5

∆S )

∞

-R

k

∑ ∑ [(βknmpkn-1 - mpkn) ln(βknmpkn-1 - mpkn) + k)1 n)1 mpkn ln mpkn - bnmpkn-1 ln(βkn mpkn-1)] (3)

where bn ) 0 for n ) 1 and bn ) βkn for n > 1. By use of eqs 1-3 the following mathematical description of adsorption isotherms for consecutive layers n placed on kth type primary sites (n ) 1, ..., k) can be derived5

-Πkn*βknmpkn-1 + (Πkn* + 1)mpkn -

1 m )0 βkn+1 pkn+1 (4)

where

Πkn* )

Πkn π ) (1 - θkn+1)1-βkn+1 βkn+1-1 Ckn BknCkn(1 - θkn+1) def

Πkn )

π Bkn

(5)

θkn stands for the layer coverage ratio, Bkn is energetic parameter, and Ckn is correcting factor def

θkn )

def

mpkn βknmpkn-1

( )

Bkn ) exp def

(

Ckn ) exp

Qkn RT

)

-qkn+1(mpkn-1 + mpkn+1) RT

(6)

and mpk0 ≡ mhkA (mhkA is the number of kth type primary sites), mpkn ≡ 0 for n > k.

Mathematical Description of BET-like Adsorption

Langmuir, Vol. 18, No. 20, 2002 7505

The formulas (4-6) describe rigorously an abstract multilayer adsorption in the systems fulfilling strictly the generalized BET theory assumptions discussed above. Having values for Qkn, qkn, βkn, and mhAk eqs 4 and 5 can be solved for specified π values with respect to mpkn, thus producing a theoretical adsorption isotherm. For systems in which qkn ) 0 (i.e., Ckn ) 1), the set of equations (4) can be solved rigorously in the following recurrent way

θkk )

π π + Bkk

and

θkn )

Πkn* π ) (7) βkn+1 1 + Πkn* - θkn+1 π + B (1 - θ kn kn+1) Figure 1. Applicability areas for different versions of the mathematical model describing multilayer adsorption in porous structures of irregular shape (see eq 11).

for

n ) k - 1, ..., 1 then

mpk1 ) mhAkθk1 and mpkn ) mpkn-1βknθkn

(8)

for

n ) 2, ..., k The local and total adsorption isotherms may be calculated as

[

k

mpk ) mhAkθk1 1 +

n

]

∑ ∏(βkj θkj)

n)2 j)2

(9)

N

mp )

∑ mpk

k)1

where N stands for the maximum primary site capacity. Such a model may be recommended for examination of materials with well-recognized porous structures (e.g., structures considered in zeolites12), for which a rather small number N may be assumed, and corresponding subsystems may be distinguished with individual parameters mhAk, βkn, Qkn. Also qkn * 0 can be considered, with eqs 7 and 8 being solved in an iterative way. Nevertheless, in the case of microporous structures of irregular shape a wide spectrum of pore size should be assumed, including micropores of molecular size and primary sites of very large (infinite) capacity. To elaborate any mathematical model of adsorption enabling for effective examination of such structures, it is necessary to emerge at list three types of adsorption subsystems, as shown in Figure 1. The first one (pointed to with the subscript s) contains all small clusters, placed on primary sites of limited capacity k ) 1,.., K, which was discussed above. In the bigger clusters a number L of the lowest layers n ) 1, ..., L e K can be still treated rigorously with eq 7, provided that the value for θkL+1 is properly evaluated. Properties of lower layers are of main importance in eq 8, and they often differ significantly from those of higher layers. In particular, for the first layer we have always βk1 ) 1 and Bk1 ) BA * Bk2 (usually BA < Bk2). Hence, at least L ) 1 must be taken. In rather wide and long channel-like pores, the values for βkn on higher layers are very likely (12) Rudzinski, W.; Narkiewicz-Michałek, J.; Szabelski, P.; Chiang, A. S. T. Langmuir 1997, 13, 1095.

lower than these on lower layers due to limited width of pores. Thus, emergence of a number L > 1 of lower layers makes it possible to apply a more realistic description of the pore structure. The values θkk, θkk-1, ..., θkk-t for a number t of the highest layers may be also calculated with eqs 7, but for remaining upper layers n ) L + 1, ..., k - t (k f ∞) it is necessary to apply the same (properly averaged) value θkn ) θk. Due to evident uncertainty of any description of the highest layers in bigger clusters, we propose to take alternatively t ) 0 or t ) 1, thus avoiding excessive complexity of the model. To derive any effective formula for the total adsorption isotherm, the distribution of the primary site capacity mhAk of bigger clusters (k > K) must be expressed in a simple analytical form. We proposed to accept the exponential distribution4,5

k ) K + 1, ..., ∞ (10)

mhAk ) mhAb(1 - R)Rk-K-1

R stands for an empirical parameter, and mhAb denotes the total number of the bigger clusters. Notice that the distribution of mhAk for k ) 1, ..., K may be shaped in any way, as the small clusters can be treated individually with the rigorous model (7-9). Moreover, if there is a finite number J of bigger cluster subsets of kj > K + 1 (e.g., in mesopores), containing a much greater number of clusters than calculated with eq 10, the excessive clusters may be treated individually with eqs 7-9 like the small ones. Thus the number N in eq 9 is N ) K + J. Assuming the same profile Bkn ) Bn and βkn) βn across the lower layers n ) 1, ..., L < K, and the same values Bkn ) BC, βkn+1 ) β for all subsystems in upper layers n ) L + 1, ..., k - t (with individual treatment of θkk), the adsorption mpb in bigger pores may be calculated as

{∑∏ (∏ ) ( [ L-1 n

mpb ) mhAb

L

βlθl +

n)1 l)1

1+

βlθl ×

l)1

βθkk

1 - Rβθ

R + (1 - R)

K-1

)]}

∑ (βθ)k

k)0

(11)

β1 ≡ 1 The adsorption in small pores mps is expressed by eqs 7-9 and the isotherm is mp ) mps + mpb. To find an appropriate averaging formula for the coverage ratios of the upper layers (θ ≈ θkn: n ) L + 1, ...,

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k - t), let us define an auxiliary sequence si for i ) k + 1 - n (i ) 0, 1, ..., k): def

si ) Πi(1 + si-1)β i-1

- Π) and sg ) ∞. By virtue of eqs 13 and 18, for β g 1 and Π ∈ (0, Πm] the sequence si may be rewritten as

si ) ss[1 + Π∞*(si-1/ss - 1)] β

(12)

Let us consider the sequence

def

Πi ) π/Bkn ) π/Bi

def

∆si ) (si/ss - 1)

s0 ≡ 0

and the function |∆sI| ) f (∆sI-1) (see Figure 2)

β0 is any real number

∆si ) [1 + Π∞*(∆si-1)] β - 1

The following equalities hold:

θi )

(19)

(20)

The convergence

si 1 + si

(13)

Πi* ) Πi(1 + si-1)β i-1-1

80 ∆si 9 if∞ implies

8 Π∞* Πi* 9 if∞

si ) Πi*(1 + si-1) and

i.e.

Πi* )

8 Π∞* θi 9 if∞

si 1 + si-1

Let us consider the sequence {si: i ) I, I + 1, ..., k - 1} in a pore of large capacity k ≈ ∞, assuming that Bkn ) BC and βkn ) βkn+1 ) β g 1 (β, BC are constants) for n ) 2, ..., k - I + 1, i.e., βi ) βi-1 and Bi ) BC for i ) I, ..., k - 1. Let Π∞* denotes a value specific for the layers i g I at

as by virtue of eqs 13, 19, and 20, both sequences, (Π∞* - Πi*) and (Π∞* - θi), may be bounded as follows:

Π∞*|∆si| g |Π∞* - Πi*| and Π∞*|∆si| g |Π∞* - θi| (21) In turn, a divergence

def

8∞ ∆si 9 if∞

Π ) Πi such that

means

Π∞*(1 - Π∞*)β-1 ) Π

and

8∞ si 9 if∞

Π∞* g Π (14)

Apart from the trivial cases with β ) 1 or Π ) 0 (when Π∞* ≡ Π), the value for Π∞* exists and Π∞* > Π (see Figure 4), if Π e Πm, where Πm is a value, for which def

so by virtue of eq 13

81 θi 9 if∞ and

Π∞*(Πm) ) Π∞* ) 1/β

8∞ Π∞* 9 if∞

i.e.

Πm ) β-β(β - 1)β-1

(15)

According to eq 12 the considered sequence {si: i ) I, ..., k - 1} is expressed as

si ) Π(1 + si-1)

β

(16)

with an initial value sI-1 > 0 resulting from properties of the higher layers (i < I). It can be proved that for β g 1 and Π ∈ (0, Πm) there are two real roots {ss, sg} of the equation

s(1 + s)-β - Π ) 0

(17)

such that sg g ss > 0, and the smaller one ss is

ss ) Π∞*/(1 - Π∞*)

(18)

For Π ) Πm there is one real root sg ) ss ) 1/(β - 1), and for Π > Πm eq 17 has no real solutions. For β ) 1 we have Π∞* ≡ Π and Πm ) 1 (see eqs 14 and 15); hence ss ) Π/(1

The sequence ∆si converges to 0 if and only if |∆si|/ |∆si-1| < 1 for i ) I, ..., ∞. For β ) 1 we have |∆si| ) Π|∆si-1|, hence

80 ∆si 9 if∞ provided that Π < 1. If β > 1 and ∆sI-1 < 0, the convergence is evident, as the following relation holds:

∆si > βΠ∞*∆si-1, i.e., |∆si| < βΠ∞*|∆si-1| (22) For ∆sI-1 > 0 and β > 1 the slope of the function |∆sI| ) f(∆sI-1) is greater than βΠ∞* and increases monotonically; thus ∆sI > ∆sI-1, provided that the relation ∆sI-1 < ∆smax ) (sg/ss - 1) is satisfied; see Figure 2. This condition implies (by mathematical induction applied to eq 20) the inequality ∆si < ∆si-1 for i > I, as shown in Figure 2. In turn, ∆sI-1 > ∆smax implies ∆si > ∆si-1, so that

8∞ ∆si 9 if∞


Langmuir, Vol. 18, No. 20, 2002 7507

Πi* 9 8∞ if∞

Figure 2. Convergence conditions for the sequence ∆si ) (si - ss)/ss (i ) k - n + 1) with exemplary βi and Π∞*. The bold lines show the function |∆sI| ) f(∆sI-1), dotted lines show ∆sI-1, the steps show changes in ∆si at consecutive layers; ∆smax is the convergence limit.

For β ) 1 the value of ∆smax is infinite. For β > 1 it tends to 0 with Π∞* f 1/β, and decreases quickly with growing β (for β ) 2 we have ∆smax ) [1 - βΠ∞*]/[(β - 1)(Π∞*)2]). Thus generally

∆si 9 80 if∞ if and only if sI-1 ∈ [0, sg). The convergence is slower for sI-1 > ss. The ratio ∆si/∆si-1 approaches to βΠ∞* < 1 when ∆si-1 ≈ 0. Hence, at low pressures ∆si tends to 0 quickly, so that θi ≈ Π∞* can be reached even in small clusters. At Π ) Πm, where Π∞*β ) 1, the convergence becomes the slowest. For Π > Πm the formula (20) may be written as follows:

∆smi ) Π/Πm(1 + Π∞*∆smi-1)β - 1 ) Π/Πm(1 + ∆smi-1/β)β - 1 (23) def

∆smi ) [si(β - 1) - 1] is the relative distance of si from the double root of eq 17 at Π ) Πm. For ∆smi-1 > -1, the following inequalities are valid:

∆smi > (Π/Πm - 1) + (Π/Πm)∆smi-1 > (Π/Πm)∆smi-1 (24) If ∆smI-1 g 0 we have ∆smI > ∆smI-1, so that

8∞ ∆smi 9 if∞ The slope of ∆smi ) f(∆smi-1) at ∆smi-1 ) 0 is Π/Πm > 1, and f(0) ) (Π/Πm - 1). Hence, due to (24), if ∆smI-1 < 0, there is always a negative ∆smi g (1 - Π/Πm) producing ∆smi+1 > 0, so

8∞ ∆smi 9 if∞ (the divergence ratio is greater than Π/Πm). Thus, for Π > Πm the values si tend to ∞, which implies (see eq 13)

81 θi 9 if∞ and

In porous structures subsets of layers having β ∈ (0, 1) may also occur (if the average cluster width decreases with growing n due to pore size constraints). In this case eq 17 has a unique positive solution ss ) Π∞*/(1 - Π∞*), where Π∞* fulfills the equality Π∞*(1 - Π∞*)β-1 ) Π, and Π∞* < min(Π, 1); see eqs 14 and 18. The function (20) with β ∈ (0, 1) is convex and |∆si|/|∆si-1| < 1. Hence, the convergence condition is always met, so that si tends to ss on consecutive lower layers, thus producing Πi* and θi approaching to the stable value Π∞* < Π. The above considerations give an insight into possible behavior of multilayer adsorption in constrained porous structures. In real porous structures the parameters Bkn and βkn change from layer to layer, which implies perturbations in consecutive skn. Nevertheless, in larger pores there are sequences of layers, where Bkn and βkn do not differ significantly from their averaged values BC and β. If the value skk-I coming from higher layers to such a sequence does not exceed the convergence area for ∆skn, the coverage ratio θkn tends to its steady value θ∞ ) Π∞*. Any exceeding of the convergence area (in particular at any Π > Πm) causes θ∞ ) 1. Thus, for pores of β > 1, there is a discontinuity in θ∞ at Π ) Πm (for Π e Πm, we have

8 1/β θ∞ ) Π∞* 9 ΠfΠ m

while for Π > Πm, θ∞ ) 1 > 1/β). In hypothetical pores of infinite capacity (k ) ∞) with constant BC and β > 1, the same θkn is always reached at all layers (effect of the highest layers is of no importance). Thus the isotherm becomes vertical at Π ) Πm (see Figure 4) that results in total filling of pores. It may be viewed as a rapid condensation of adsorbate in pores caused by configurational effects; hence πC ) ΠmBC will be referred to as the condensation pressure (specific for the adsorption system). In pores of limited capacity assuming of constant BC and βkn ) βkn+1 ) β for all the layers n ) 2, ..., k - 1 seems to be acceptable for pores of regularly but slowly growing width (β ) 1-2). However, in larger slitlike and holelike pores the parameter Bkk at the highest layer may be significantly lower than Bkn for n < k. It is due to possible additional interactions of adsorbate molecules with opposite wall of a slitlike pore, or with a cluster growing from the opposite wall surface in a compact holelike pore (see ref. 5). It produces a rather large initial value s1 for the sequence si at lower layers, where constant Bkn ) BC and βkn ) β are more realistic. In the particular case, if θkk ) Πkk/(Πkk + 1) ) Π∞* for n < k, i.e., Bkk fulfills the equation BC ) Bkk(1 + π/Bkk)βk1, the coverage ratio reaches the same value θkk ) Π∞* at all layers up to the second one. Thus, at a pressure Π e Πm the adsorption on the layers n >1 fulfills the generalized Henry’s law:

mpn ) mpn-1βθn θn ) θ ) Π∞*

(25)

and θkn approaches to 1 if Π > Πm (i.e., π > πC). As we noticed before, in flat and large clusters one may expect significant decrease in Bkn at higher coverage ratio θkn > θC (θC > 0.25; see comments above eq 2), due to more intensive interlayer interactions. It reduces Πm and forces very rapid growth in adsorption in highest pressure range. Such a phenomenon, if observed, suggests rather big β (β ) 1.1-2). In small or channel-like pores, where purely geometrical constraints for the cluster size are of primary

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Assuming that θk2 ) θ, the value for Π∞* at the first layer (n ) 1) may be calculated as follows:

Π∞* ) Πk1(1 - θk2)1-β ) π(1 - θ)1-β(BC /BA) ) Π*/BA* (28) Thus, the value for θk1 can be obtained with the rigorous adsorption formula given in eq 7:

θk1 ) Π*/[Π* + BA*(1 - θ)]

(29)

On the basis of eqs 11 and 26-29 with K ) {1, 2} and t ) {0, 1} the adsorption isotherm may be expressed in the form4,5

Figure 3. Coverage ratio θk1 of the first layer in clusters limited to k layers with the same BC and β at different pressures π. The consecutive curves from bottom to top correspond to Π ) π/BC ) {0.1, 0.2, 0.35, 0.5, 0.65, 0.8, 0.99}. Dotted lines are used for Π > Πm. Horizontal lines show θ∞ ) Π∞*, dotted lines show the maximum value for Π∞*, i.e., Πm* ) 1/β.

importance, the adsorption energy at the highest layer is rather close to that at lower layers, i.e., Bkk ) BC. In such cases the values of θkn approach to Π∞* only at low and moderate pressures, where the convergence is fast (see Figure 2). When Π tends to Πm the convergence becomes slow, so that θkn = Π∞* can be reached on hardly low layers of very large pores. On the other hand, in this pressure range exceeding the convergence area is likely; hence in large pores the coverage ratio θkn may approach to 1 at low layers. Thus, if pores of rather small volume are dominant, at Π ≈ Πm no specific (or a bit faster but smooth) growth of adsorption may be expected. In turn, a rapid change in the isotherm slope around Π ) Πm suggests that there is a noticeable amount of pores of very large volume with β > 1. It is illustrated in Figure 3, where the coverage ratio of layers with constant BC and β is studied. The values for θkn were calculated by eqs 12 and 13 from I ) 1. In this case, the sequence si starts with s0 ) 0, thus θkn tends to Π∞* at any Π e Πm. At Π > Πm the coverage ratio goes slowly to 1. Notice that in pores of low and moderate capacity the volume filling ratio remains at low level, even for Π > Πm. Hence, at higher relative pressures π, in the model (11) a value θ < θ∞ depending on the parameters BC, β, and R should be applied. In our earlier papers4,5,8 we showed that effects of geometrical and energetic constraints on multilayer adsorption in natural microporous adsorbents may be handled adequately by applying the exponential distribution (eq 10) to all primary sites (k ) 1, ..., ∞), and using the model of the form (11) with the number K (defining the small clusters) limited to 2 (K ) {1, 2}) and with the set of lower layers in the bigger clusters reduced to the first layer only (L ) 1). Let BA denote the energetic parameter Bk1 for the first layer, BA* its value related to BC: def

def

BA ) exp(QA/RT); BA* ) BA /BC

(26)

[

def

(27)

)

]

where d ) (K - 1) ) {0, 1}, and θt ) θ if t ) 0, or θt ) θkk ) Π/(1 + Π) if t ) 1 (see eqs 7 and 11). For many systems it is enough to take d ) 0 and t ) 0 (see refs. 5, 6) that leads to the formula

mp (1 - R)‚Π Π/ ‚R β‚θ + / ) 1+ mhA BA* + Π 1 - R‚β‚θ Π + BA*(1 - θ) (31)

[

]

In our earlier papers4,5 we proposed to accept the generalized Henry’s isotherms (eq 25) for the layers n ) 2, ..., k - t, i.e., to take θ ) Π∞* and Π* ) Π∞*. The main advantage of such a model (referred to as the LgBET equation) is that there are two special cases, for which application of θ ) Π∞* produces rigorous solutions of the model (7, 8), i.e.: (1) if there are only primary sites of infinite capacity (R ) 1); (2) if the top layer coverage ratio is θt ) θkk ) Π∞*. Thus, it provides a convenient (and formally justified) reference point for qualitative evaluation of real porous structures5,8 and makes possible detection of the non-BET effect (β > 1) on adsorption in microporous materials.5 It is also applicable to any system at Π < 0.7. Nevertheless, apart from these special cases, the resultant formulas (30, 31) yield significantly greater mp at higher pressures, than mp produced by eqs 7-9 with a properly large K and Bkn ≈ BC for n ) 2, ..., k - t. Moreover, the model applicability is limited to the range Π < Πm, in which a value for Π∞* can be calculated (see eq 14), while empirical data for larger Π values may be available. We showed before (see Figure 3) that adsorption isotherms in pores of finite volume remain continuous around the condensation pressure πC (where Π ≈ Πm). In particular, for pores of strongly limited capacity (rather small R), the isotherms do not exhibit any special properties at Π ≈ Πm, and the only way to handle with LgBET the adsorption at π > πC is by proper extrapolation5,8 of Π∞*. But in fact, it does not give essentially new information on the system. Hence, to make possible more precise examination of microporous systems at high π values (especially if Bkk ≈ BC may be expected), we propose to apply the following new formula for θ

and Π* stand for a quantity corresponding to Π∞* in eq 14, but defined for the average θ:

Π* ) Π(1 - θ)1-β ) π/BC(1 - θ)1-β

(

mp R (1 - R)Π (1 - R)Π βΠ +d ) 1+ + mhA BA* + Π 1+Π BA*(1 + Π)-β + Π βθt Π*Rd+1 1+ (1 + dβθ(1 - R)) (30) 1 Rβθ Π* + BA*(1 - θ)

θ ) Π* R.

(

)

1 + wHΠ* 1 + Π*

(32)

where wH stands for the weighting factor depending on


Langmuir, Vol. 18, No. 20, 2002 7509

The factor wH represents a fraction of the upper layers (n > 1), for which the generalized Henry’s law (eq 25) is accepted, assuming that adsorption on remaining ones fulfill the Langmuir’s equation with Π* used instead of Π. We stated that the individual treatment of the top layer (t ) 1) gives no improvement of the model, so that t ) 0 is taken in the sequel. Hence, to reach a good approximation of the model (7-9) with constant βkn ) βkn+1 ) β and Bkn ) BC for n ) 2, ..., k, the factor wH should be a bit lower than a number fraction vK_1 of layers n ) 2, ..., k - 1 placed in clusters of k > K (the kth layer fulfills strictly the Langmuir equation), related to the total number of layers n ) 2, ..., k in such clusters. It is also reasonable to take wH a bit larger than the number fraction vK_2 of layers n ) 2, ..., k - 2 in the upper layers n ) 2, ..., k. In systems fulfilling eq 10 the values for vK_2 are v1_1 ) R, v1_2 ) R2, and v2_1 ) 1/(2 - R). We found by numerous calculations that the following formulas for wH are the most suitable:

{

R(1 + R - R2) for d ) 0 (K ) 1) 2-R wH ) 2+R for d ) 1 (K ) 2) 3(2 - R)

(33)

The value for Π* has to be calculated by solving eqs 27, 32, and 33, which may be easy done by direct iterations (starting with Π* ) Π), provided that the following relation is satisfied:

F(Π*) ) wHβ(Π*)3 + wH(2β - 1)(Π*)2 + (β - 2)Π* - 1 e 0 (34) It produces Π* e Πmax*, and Π* > Π for Π ∈ (0, Πmax], where Πmax* and Πmax fulfill the equalities

F(Πmax*) ) 0

(35)

Πmax ) Πmax*(1 - θmax)β-1 θmax ) θ(Πmax*) Notice that

81 wH 9 Rf1 i.e.

Π* 9 8 Π∞* Rf1 8 Π∞* θ9 Rf1 and generally, wH ) 1 gives the LgBET formula. Properties of the transformation Π*(Π) and of the averaging formula (32) are illustrated in Figure 4. It may be seen that the curves (Π*/Πmax*) vs (Π/Πmax) with different R and β are very close to (Π∞*/Πm*) vs (Π/Πm) for the same β. Thus practically, the new formulas (27, 32, and 33) only rescale the function (14) with Πmax* > Πm* and Πmax > Πm. Also θmax values are merely a bit larger than Πm* ) 1/β (see Figure 4D). Hence, the model LgBET applied to π ) πmax may be forced to give the adsorption close to that produced with the new formulas (27, 32, and 33) by taking properly enlarged BC (in eqs 14 and 17 BC is only the scaling factor). However, essential differences are in the shape of θ(Π) and Π∞*(Π) (see Figures 4A and 4B) that is of consequences in eq 36.

Figure 4. Transformation Π*(Π) (eq 14) and averaged coverage ratio θ (eq 32) for different β. The numbers on curves point to the value for β: (1,1θ) β ) 1.1; (2,2θ) β ) 1.3; (3,3θ) β ) 2.0. The letters (a-e) point to R and wH; (b,c) R ) 0.5; (d,e) R ) 0.1; (a) wH ) 1; (b,d) wH calculated with eq 33 for d ) 1; (c,e) wH calculated with eq 33 for d ) 0. Horizontal and vertical lines in subfigure (B) show Πm* ) 1/β and Πm. In the subfigure (D) the curves 1,2,3 show Πmax*; 1θ, 2θ, 3θ show θmax, and curves 1a, 2a, 3a show the values of Πm* ) 1/β. In subfigures (A) and (B) the curves (1a, 2a, 3a) are identical.

Isotherms calculated by the new adsorption formulas (eqs 31-33) were compared to those produced by the rigorous model (7-9) with the same parameters mhAk, BA, BC, R, β, and K satisfying the relation RK < 10-10. The study was carried out within an area of the model parameters covering all cases met in our practice. To outline the approximation accuracy bounds, exemplary worse case results are presented in Figure 5. They were obtained with rather high BA (QA ) -0.5) that makes eqs 30 and 31 more sensitive to θ1 evaluation errors. For typical adsorption systems QA < -2, thus discrepancies are mostly a bit smaller. Isotherms calculated with the model LgBET (eqs 31 and 25) are shown too. It may be seen that the new approximation is very accurate for relative pressures π e πC and remains acceptable up to Π < (Πm + Πmax)/2, provided that Rβ < 1. At larger π values, discrepancies are hardly predictable. For example, the rapid adsorption growth shown in Figure 5 (β ) 1.9, R ) 0.75) just over πC (filling of very big pores) is not handled, although the adsorption is well evaluated up to πC. It is noteworthy that the applicability area of the new model covers practically the full pressure range, in which any reliable formal description, with constant parameters, may be proposed. In fact, at higher pressures changes in adsorption energy and cluster reconstruction seem to be very likely, which implies changes in the system parameters. Thus, up to π a bit larger than πC, the eqs 30-33 are practically equivalent to eqs 7-9 with constant BC, β, and the exponential distribution of mhAk (eq 10). Hence they will be referred to as the LcBET model (combining Langmuir’s and constrained BET-like isotherms). It is recommended for microporous systems as an adequate description of adsorption in channel-like and dendritelike pores, where intercluster interactions are of low intensity, thus Bkk ≈ BC ≈ Bkn may be expected. In turn, in slitlike and larger holelike pores, energetic interactions of clusters placed at the same pore can produce θkk ≈ Π∞*; hence the LgBET model may be advantageous. The LcBET model may be fitted to empirical isotherms by iterative adjusting of its parameters with a “three point” identification procedure5 (elaborated for LgBET). In this

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The procedure applied to the data produced by the fitted model evaluates the presumed parameters with negligible errors. To get insight into reliability of the quantitative information gained in this way on real porous systems, we fitted the both, LgBET and LcBET, to adsorption isotherms calculated by eqs 7-9 for π e πC. The results obtained for the worse case discussed above are depicted in Figure 5. The corresponding parameters are gathered in Table 1. It is seen in Figure 5 that in the both cases very good fitting is attainable. Within the fitting area (π e πC) the model data mismatches are little and they often remain acceptable for π > πC as well. In fact for LcBET discrepancies are mostly a bit smaller, but the fitting quality itself does not give evidence for the advantage of LcBET over LgBET in the studied cases. Benefits from application of LcBET are clearly seen in Table 1. The parameters determined with LcBET are much closer to their actual values than those obtained with LgBET. For typical systems (QA/RT < -2) the surface parameters, mhA and QA, may be very well evaluated with both models (errors are about 1% or less), but LcBET gives mostly much better evaluation of R (typical errors do not exceed 20%) and a bit better evaluation of β and BC (the results are also less sensitive to changes in πR, πB, πβ). The determined parameters β and BC are noticeably different from their actual values. Nevertheless, the evaluation error for the ratios Rβ/BC and R/BC are moderately low (comparable to that for R, see Table 1). We stated that deviations of such a magnitude as shown in Table 1 are unavoidable, if π e πC, as the isotherms produced by eqs 7-9 in this pressure range with different sets of parameters (properly linked) are practically indistinguishable. The same effect may be expected in examination of real systems. As the matter of fact, at high pressures, when the primary sites are totally covered (θ1 ) 1) eq 31 brings to the following form:

Figure 5. Theoretical constrained multilayer adsorption (eqs 7-9) at high-pressure range (bold lines), its approximation by the LgBET and LcBET formulas with the same parameters (see Table 1), and fitting results with parameters shown in Table 1. Curves: thin-solid lines, LcBET equations (31-33) with parameters of eqs 7-9; dotted lines, fitting of LcBET (if not shown, covered by bold line); dotted lines, LgBET equations (31, 25) with parameters of eqs 7-9; double-point-dotted lines, fitting of LgBET (if not shown, the curves for LcBET and LgBET are identical). System parameters: the numbers on curves show the value of a: (1) R ) 0.25; (2) R ) 0.50; (3) R ) 0.75; β is shown in subfigures; BA ) exp(-0.5), BC ) max(1,1/Πmax) with Πmax calculated with eq 35 for the model (30-33) (i.e., in each case πmax ) 1); mvti, pore volume related to mhA. Horizontal lines show the condensation pressure πC ) BCΠm for the curves 1, 2, and 3 (see eq. 15) where the LgBET model (dotted lines) meets its applicability bound.

procedure the Langmuir’s component of eqs 30 and 31 is first calculated by subtracting effects of R, BC, and β (using their values found in preceding iterations) from lowpressure adsorption data (π < 0.4). It makes possible to obtain good evaluation of mhA and BA. Then, the values for R, BC, and β are adjusted, so as to force a good fitting of the model to the adsorption isotherm at three fixed pressures πR, πB, and πβ (πR > 0.7, πR < πB < πβ e πmax).

mp|θ1)1 )

mhA ) 1 - Rβθ

mhA Rβ θ 1π BC Π

()

(36)

Table 1. Parameters Obtained by the “Three Point Procedure” (with πr/πC ) 0.75, πB/πC ) 0.93, πβ/πC ) 1) Applied to the Adsorption Isotherms Calculated with Equations 7-9 for mhA ) 1, QA/RT ) -0.5 and Different r, β, and BC (see also Figure 4) (constraints: βe g 1, βCe g 1) parameters of eqs 7-9 β

R

BC

model

mhAe

QAe/RT

βe

Re

BCe

RβBCe/BCReβe

RBCe/BCRe

mVt/mVte

1.00

0.75

1.000

0.50

1.000

0.25

1.000

0.75

1.190

0.50

1.000

0.25

1.000

0.75

1.609

0.50

1.274

0.25

1.000

0.75

3.321

0.50

2.803

0.25

2.177

LgBET LcBET LgBET LcBET LgBET LcBET LgBET LcBET LgBET LcBET LgBET LcBET LgBET LcBET LgBET LcBET LgBET LcBET LgBET LcBET LgBET LcBET LgBet LcBET

1.36 1.02 1.85 1.02 1.36 1.00 1.09 0.98 1.40 0.98 1.31 1.03 1.12 1.04 1.09 0.93 1.27 1.01 1.09 1.05 1.09 1.07 1.03 1.04

-0.20 -0.48 0.12 -0.48 -0.19 -0.50 -0.41 -0.52 -0.16 -0.52 -0.23 -0.47 -0.39 -0.46 -0.42 -0.57 -0.26 -0.49 -0.41 -0.45 -0.41 -0.43 -0.47 -0.46

1.000 1.000 1.000 1.000 1.000 1.000 1.063 1.042 1.000 1.000 1.000 1.000 1.261 1.274 1.000 1.093 1.000 1.158 1.581 2.090 1.292 2.024 1.000 1.673

0.998 0.730 0.157 0.484 0.034 0.251 0.999 0.800 0.480 0.593 0.133 0.300 0.776 0.859 0.756 0.569 0.226 0.291 0.607 0.685 0.434 0.425 0.396 0.262

1.866 1.000 1.292 1.000 1.025 1.000 1.691 1.168 1.520 1.039 1.616 1.190 1.984 1.975 1.746 1.133 1.616 1.100 2.644 3.656 2.064 3.025 2.042 2.359

1.40 1.03 4.11 1.03 7.54 1.00 1.10 0.97 1.74 0.96 3.34 1.09 1.18 1.05 1.13 0.89 2.23 1.02 1.18 1.10 1.25 1.19 1.13 1.17

1.40 1.03 4.11 1.03 7.54 1.00 1.07 0.92 1.58 0.88 3.04 0.99 1.19 1.07 0.91 0.78 1.79 0.95 0.98 1.21 0.85 1.27 0.59 1.03

0.01 1.08 1.69 1.03 1.29 1.00 -0.3 0.95 1.16 0.90 1.20 0.97 0.34 -1.5 0.65 1.01 1.13 0.96 -0.1 1.02 8.79 2.80 1.15 1.07

1.10

1.25

1.90


Langmuir, Vol. 18, No. 20, 2002 7511

Thus, if θ/Π is close to a 1-intercept straight line (at low π, for β ≈ 1 or R ≈ 0, see Figure 4) there is only one parameter (Rβ/BC) affecting effectively the shape of the isotherm in the high-pressure range. It reveals essential barriers for identification of geometry of pores based on typical adsorption data. The more involved the shape of θ/Π, the better capability of an adsorption formula to distinguish between individual effects of R, β, and BC on the isotherm. It is why the LcBET model yields more reliable evaluation of the pore shape parameters than LgBET (see Figure 4). It means also that porous structures for which LcBET is more suitable may be better identified. In each case the higher πβ and the larger (πβ - πR), the more reliable evaluation (the isotherm at πR has to be sensitive enough to R, thus πR ≈ 0.65 is recommended4). More precise information on geometry of pores can be gained, if adsorption data provide a basis to presume the value for πC or πmax. In such a case the parameters R, β, and BC may be forced to satisfy eq 15 or (eq 34) at given πC or πmax and minimize a data-model mismatch at lower pressures. This concept will be exploited in the next section of this paper. The ratio R/BC may be viewed as the parameter showing an effective exponential distribution of a reachable cluster size, with respect to cumulated effects of energetic and geometrical constraints. In fact, it can be seen in eq 9 that the number of layers practically covered at π e πC is very likely subjected to an exponential function. In larger pores the highest layers remain empty, and so they are of no effect on the isotherm shape. In turn, if eq 10 overestimates the number of larger clusters, the model can meet a real adsorption by taking an overevaluated value for BC. That is why eq 10 is acceptable for many microporous systems.4 The influence of β on adsorption is of strictly geometrical nature. If the structure of clusters meets the exponential distribution (eq 10), and Rβ < 1, the total volume of pores mVt (related to the adsorbate volume) is formally expressed as

mVt )

mhA 1 - Rβ

(37)

The cluster volume mvk, its formal distribution ft(mvk), and the effective one fe(mvk) are

mvk )

(1 - βk) 1-β

(38)

(1 - R)Rk-1 1 - Rβ

f(mvk) ) mvk‚

( )

(1 - R/BC) R fe(mvk) ) mvk 1 - β(R/BC) BC

k-1

Notice, that fe(mvk) is almost directly measurable, while the evaluation of ft(mvk) is based only on the assumed (purely geometrical) effects of R and β on the curve (θ/Π) (eqs 27, 32, and 33). In fact, the evaluated R and β should be viewed as describing only averaged properties of this fraction of pore volume, in which a coverage ratio of layers is of significance under pressures close to the maximal π used in the parameter identification procedure.5 In particular, one may obtain Rβ g 1. It may be due to an overevaluation of contents of large pores (eq 10) or by excessively large β. If the pore volume and f(mvk) are of no interest, the above inconsistency may be disregarded,

and infinite pore volume may be assumed. Otherwise, the application of the model (11) should be considered. In the case of Rβ < 1 the total volume of pores can be formally calculated with eq 37. However, the result may be uncertain, as the averaged β are usually far too inadequate to handle a geometry of largest pores. In eq 38 a width of clusters of β >1 is growing exponentially from layer to layer; thus the description of large clusters is always unrealistic. An evaluation of mVt based on direct measurements of adsorption at the highest pressure is usually uncertain as well. The maximal averaged coverage ratio reachable in the LgBET model is θmax ) 1/β (see eq. 14), and in the LcBET formulas θmax is slightly larger (see Figure 4D). Thus, according to eq 36, the adsorption at Πmax is lower than mhA/(1 - R). It may be concluded from eqs 7-15 that the total filling of pores of limited size cannot be attained by adsorption with constant BC at any finite pressure (see Figure 3). As the effect, many microporous adsorption systems remain far below the total filling level up to π ) 1 (compare mVt(.) with mp/mhA at π ) 1 shown in Figure 5). There are two cases in which a high filling degree of pores is possible. First, an averaged θ may be close to 1 in very large pores at π > πC. Hence, if volume of such pores is considerable, sharp growing of adsorption near πC should be observed. Neither, LgBET nor LcBET are able to predict such effects, but it may be handled with the rigorous model (7-9) (see Figure 5). The second case is due to changes in adsorption energy on highly covered layers. It may be expected in systems containing slitlike or larger holelike pores, where flat clusters can be created. In such systems a rapid growth of adsorption can be initiated at a pressure π, for which the coverage ratio of layers θkn exceeds θC (see comments above eq 2). It implies a decrease in Bkn (see eqs 6) making π > πC. If Bkn reaches its bulk phase value (i.e., Bkn ) 1) and corresponding Πkn* is much larger than Π∞*, a high filling degree of the pore can be attained (see divergence condition in eq 24). It should be exhibited by rapid reduction in the isotherm slope, as further growing of adsorption is only due to filling of small pores. If such a shape of isotherm is observed, the maximal adsorption may be used to evaluate more reliably the total pore volume. This effect can be roughly taken into account in the LcBET model by assuming that the parameter BC > 1 is applicable up to the pressure πθC, were θ ) θC, then Bkn declines and reaches 1 at π ) πmax ) Πmax (πmax corresponds to the isotherm slope break-down point). It makes it possible to reach more realistic evaluations for R, β, and BC by fitting of the model at πβ ) πmax and πR ) πB < πθC, with eq 35 being satisfied. The parameters obtained in this way and mVt may be then checked and adjusted by using the rigorous model (eqs 7-9). The above analysis shows that examination of real adsorption systems needs manifold fitting of the both discussed models with different parameters of the “three point” procedure (πR, πB, πβ), and possibly with different πC or πmax being presumed. Additional information on properties of the system components (e.g., pore volume, cohesion energies for adsorbent and adsorbate) should be used to check acceptability and consistency of the energetic parameters BA and BC obtained in each repetition (see refs 4, 5, and 8). 3. Analysis of Empirical Adsorption Data with LgBET and LcBET Formulas Examination of high-pressure adsorption data with the both alternative models, LgBET and LcBET, may be especially useful, when adsorption isotherms at highest

7512



Table 2. Fitting of LBET, LgBET, and LcBET Formulas to the Empirical Isotherms of N2 on Active Carbons AMBERSORB at 77.4 K (Choma and Jaroniec, ref 9) for Relative Pressures π g πmin ) 6 × 10-3 a model

mhA

QA/RT

R

β

BC

Rβ/BC

R/BC

mVt

0.254 0.084 0.221 0.244

8.34 22.63 30.92 44.24

0.211 0.067 0.187 0.204

15.13 35.29 31.10 34.78

BET LBET LgBET (a) LgBET (b) LcBET

5.51 6.22 6.09 6.09 6.09

AMB563: mpmax ) 18.7 [mmol/g]; mVcBET ) 44.24 [mmol/g]; θC ) 0.256 -8.65 -7.22 0.254 1 1.000 0.254 -7.23 0.341 2.141 4.040 0.181 -7.23 0.742 1.083 3.353 0.240 -7.23 0.816 1.057 3.346 0.258

BET LBET LgBET (a) LgBET (b) LcBET

10.93 11.94 11.77 11.77 11.77

AMB572: mpmax ) 25.3 [mmol/g]; mVcBET ) 44.78 [mmol/g]; θC ) 0.267 -8.10 -6.78 0.211 1 1.000 0.211 -6.79 0.293 2.274 4.377 0.152 -6.79 0.569 1.092 3.038 0.205 -6.79 0.618 1.071 3.030 0.218

a Adsorbate, nitrogen; cohesion energy, 3577 J/mol; molar volume, 34.7 cm3/mol; molecule section surface area, 83.72 m2/mmol. LgBET (a), LgBET fitted with πβ shown in Figures 7 and 8, and extrapolated to π ) πmax. LgBET (b), LgBET fitted with πβ ) πmax. mpmax, maximal adsorption (at π ) πmax) used in identification of LgBET(b) and LcBET. mVcBET, total volume of pores used to fit eqs 7-9, see curves eqs (7-9)m in Figure 8.

Figure 6. Adsorption of N2 at T ) 77.4 K in the initial relative pressure range on synthetic carbons AMBERSORB563 (AMB563) and AMBERSORB572 (AMB572), measured by Choma and Jaroniec (ref 9). Subfigures (a) and (b), empirical adsorption (circles) for π < 0.05 taken from ref 9 and the theoretical isotherms examined in the paper (see Figures 7 and 8) plotted against log10(π). Subfigures (c) and (d), the adsorption data mp at relative pressures π ∈ 〈πmin ) 6 × 10-3, 0.08〉 depicted in the coordinates (1/π, 1/mp) linearizing the Langmuir equation, and the resultant linear Langmuir plots used in further examination of the models LBET, LgBET, and LcBET.

pressures (π ≈ 1) are available and they suggest significant non-BET effects. To demonstrate advantages of such an extended analysis, we used the empirical isotherms of nitrogen adsorption at its normal boiling point (77.4 K) on two synthetic active carbons AMBERSORB, measured by Choma and Jaroniec9 (see Figures 6-8). The study was focused on explanation of the rapid growth of adsorption observed for π ) P/P0 > 0.85. We employed the model of the simpler type expressed by eq 31. Results are presented in Table 2 and in Figures 6, 7, and 8. Three versions of the formula (31), i.e., LBET (eqs 31 and 25), BC ≡ 1, β ≡ 1), LgBET (eqs 31 and 255, BC g 1, β g 1), and LcBET (eqs 31, 32, and 33, d ≡ 0, BC g 1, β g 1), were fitted to the empirical data for relative pressures π ranging from πmin ) 6.0 × 10-3 to πmax ) 0.985. The lower bound πmin was chosen by analysis of the lowest pressure adsorption shown in Figure 6. Surface heterogeneity effects, clearly seen in parts a and b of Figure 6 and discussed in ref 9, are beyond the scope of this paper (our

approach to this topic is outlined in ref 13). Hence, we found the pressure range π ∈ 〈6 × 10-3, 8 × 10-2), in which an acceptable fitting of the Langmuir linear equation 1/mp ) f(1/π) was reached (see parts c and d of Figure 6), and the resultant surface parameters mhA, QA were practically unaffected by the parameters R, BC, and β of eq 31, when the identification procedures based low and high pressure data were applied (see section 2 and Table 2). In further treatment of higher pressure data we used first the LBET model (eqs 31 and 25, BC ≡ 1 and β ≡ 1) with R minimizing the model error at πR, to demonstrate its usefulness in a lower pressure range.4 In fact, it meets very closely empirical data up to π ≈ 0.8 (see the curves LBET in Figures 7 and 8). Then, all the model parameters mhA, BA, R, BC, and β were determined by fitting the LgBET formula with the “three point” procedure. It was done in two ways: (a) and (b). In (a) we carried out the identification with πβ < πmax, where πβ close to a supposed model applicability limit πCa was taken (πβ ) πCa = πC). Starting from πm ≈ 0.99πCa a supposed Π* profile was calculated by linear extrapolation up to Π*(πmax) ) 1 (see ref 5). The resultant isotherms (LgBETa) are close to empirical data up to πmax, but fitting is rather poor. Moreover the parameters BC and β are unrealistically large (see Table 2). In the second way (b) we attempted to apply the LgBET formula (eqs 31 and 25) with πC > πmax. The parameters were found by the “three point” procedure at πR < πB< πβ ) πmax with a larger model error at πβ ) πmax being admitted, see Figures 7 and 8. In the both systems, at π ≈ 0.8 the coverage ratio θ produced by eq 25 exceeds the value 0.25 enabling for intensification of interlayer contacts in more compact or flat clusters. It may cause the rapid growth of the LgBETb model-data mismatch observed from π ) πθC ) 0.84, at which θ ) θC > 0.25. Hence, the alternative isotherms (LgBETb*) were calculated for π > πθC with Bkn decreasing linearly from Bkn ) BC at πθC to Bkn ) 1 at πmax. Finally, we fitted the LcBET model at πR < πB ) πβ ) πmax by the modified identification procedure, assuming that constant Bkn ) BC is applicable for π < πR, and eq 35 with Bkn ) 1 is satisfied at πmax. Also in this case at πθC ) 0.84 the formulas 32 and 33 yield θ ) θC > 0.25 (see values for θC in Table 2). Hence, the isotherms for π > πθC (i.e., for θ > θC) were calculated with Bkn decreasing linearly from Bkn ) BC at πθC to Bkn ) 1 at πmax. The results are shown in Figures 7 and 8 as the curves LcBET*, while the (13) Milewska-Duda, J.; Duda, J. Ann. Univ. Mariae Curie-Sklodowska, Sect. AA: Chem. 2001, 56, 29. Milewska-Duda, J.; Duda, J. New BET-Like Models for Heterogeneous Adsorption in Microporous Adsorbents, Appl. Surf. Sci., in press.


Figure 7. Theoretical constrained multilayer adsorption (eqs 7-9) and the LBET, LgBET, and LcBET isotherms fitted to the empirical adsorption of N2 on AMBERSORB5639 at T ) 77.4 K in the relative pressure range π ∈ (πmin ) 0.006, πmax ) 0.980), depicted for π > 0.02 (see Figure 6 for lower pressure data). Equations 7-9 are applied with the parameters determined by fitting of the model LcBET; see Table 2. Empirical data9 are shown as the circles. The curves LBET, LgBETa, LgBETb, LcBET, and eqs 7-9 were calculated with constant BC (see Table 2). The corrected adsorption obtained for π > πθC ≈ 0.85 with Bkn decreasing from BC to Bkn(πmax) ) 1 is depicted as the curves LBET*, LgBET*, LcBET*, eqs (7-9)*. The dotted-double point curves show the BET and Langmuir (L) adsorption. Horizontal lines: πR, the relative pressure used in the identification procedures; πCa, condensation pressure presumed for the LgBETa curves. The subfigure (b) shows the cluster volume distributions f(mVk) calculated by eqs 38 with parameters of LcBET: ft, the theoretical distribution; fe, the effective distribution.

Figure 8. Application of eqs 7-9, and the LBET, LgBET, LcBET formulas to the adsorption isotherm of N2 on AMBERSORB5729 at T ) 77.4 K depicted for π > 0.02 (see Figure 6 for lower pressure data). Denotation of curves and πmin the same as in Figure 7, πmax ) 0.983. The curves eqs (7-9)m and eqs (7-9)*m show the theoretical isotherms calculated by eqs 7-9 with the additional pore volume 10 mmol/g included into the pores of k ) 33. The curve ftm at the subfigure (b) shows the cluster volume distribution, based on ft, but modified correspondingly with the additional volume as above. The distribution ftm produces the upper curves eqs (7-9)m and eqs (7-9)*m while ft yields the lower ones, i.e., eqs 7-9 and eqs (7-9)*.

isotherms produced by the model with Bkn ) BC are depicted as LcBET curves. The parameters found in this way were also applied to the rigorous model (eqs 7-9 and

Langmuir, Vol. 18, No. 20, 2002 7513

eqs 7-9*), with the number of layers limited to N, such that RN < 10-6. It is clearly seen in Figures 6-8 that no model with constant Bkn ) BC is able to explain the rapid growth of adsorption in the high-pressure range, although each of them fits very well the empirical data for lower pressures. While the isotherms LgBETa meet the high-pressure data satisfactorily, it is of poor significance in merits, as the extrapolation of Π* is rather arbitrary, and Π*(πmax) ) 1 is certainly too big. Moreover, the resultant parameters β and BC are unrealistically large. The parameters obtained with LgBETb and LcBET seem to be more reliable, and the models are acceptable at a lower pressure range. Nevertheless, to reduce considerably the highest pressure model-data mismatch, it was necessary to apply the LcBET formula with varying Bkn. The critical coverage ratio θC is almost the same for both systems. Its rather low value (θC ≈ 0.26) together with low β suggests that the observed rapid growth of adsorption is mainly in large pores due to creation of flat clusters creeping over an irregular pore surface, which growing at lower pressure is constrained by large BC. It was confirmed by results obtained with the rigorous model, showing the important role of large clusters (k > 30) in the high-pressure adsorption. It is noteworthy that the LcBET formula gives the cumulated factors (R/BC) and (Rβ/BC) very close to those obtained with the LBET model (based only on the lower pressure data). It means that the LcBET* formula describes adequately geometrical properties of both the small and the large pores; hence it provides a reliable evaluation of the pore volume. In the case of AMB563 the volume found in this way makes possible to reach almost perfect fitting of the empirical adsorption at πmax with the both, LcBET* and rigorous formulas (7-9)*. Thus, the structure of pores in this material is adequately described by the exponential distribution with significant contribution of larger pores (R > 0.8, β ≈ 1.06). However, for AMB572 the fitting of LcBET* exhibits an inadequacy of such a simple description. It is also confirmed by the model (7-9)*; see Figure 8. To fit the highest pressure adsorption, it was necessary to apply eqs (7-9)* with the volume of largest pores (k ) 33) enlarged by 10 mmol/g. Further increasing of the volume gave no effect, and enlarging of smaller pores produced overevaluated isotherms for lower pressures. For the both materials, AMB563 and AMB572, the maximal measured adsorption mpmax is much lower than the total pore volume found with LcBET and next adjusted with eqs (7-9)m*scompare values of mpmax with mVt and mVcBET in Table 2. The examination shows that the studied materials contain pores of wide size spectrum, but the adsorption is strongly constrained by adsorption energy at higher layers (BC is very high, but still acceptable5,8). Hence, very small clusters of adsorbate molecules are dominant (see values for R and R/BC). It is why the BET model is insufficient for examination of adsorption mechanisms in wide pressure range (see Figures 7 and 8). 4. Conclusions Non-BET effects due to creation of non-stack-like clusters in constrained multilayer adsorption, can be handled with a model based on fundamental thermodynamic and derived with a BET approach. The resultant rigorous equations may be written in a convenient recurrent form, enabling an effective analysis of systems with a finite number of layers. A formal study of these formulas reveals basic links between a pore geometry and shape of adsorption iso-

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therms. In particular, it shows that there is a “condensation pressure” πC, specific for an individual pore geometry, making possible total filling of very large pores due to configurational effects. It produces stepwise changes of adsorption at πC. In large or flat clusters, rapid growth of adsorption may be caused by a decrease in adsorption energy with growing coverage ratio of layers. However, typical microporous structures remain far below the total filling up to the relative pressure π ) 1, and typical isotherms have no singularities at πC. Hence, the total pore volume is hardly determinable directly by adsorption measurements. Any examination of natural porous structures needs a simple but adequate model approximating the above equations. The formulas LgBET and LcBET, discussed in this paper, may be recommended as the alternative tools for analysis of such structures. Apart from the typical surface characterization provided with the parameters mha and QA, good fitting of the suitable formula LcBET or LgBET (over an appropriately wide pressure range) gives a quantitative characterization of a geometry of pores, showing geometrical (the parameters R, β) and energetic (BC) constraints for adsorbate clusterization. By fitting of the both models to empirical high-pressure adsorption data, one may draw conclusions about what is the dominant shape of pores in the studied material. If the LcBET model may be well fitted with acceptable values for BC and β fitting also the rigorous model equations (7-9), channel-like and dendrite-like pores are probably


dominant (intercluster ineractions are of low probability). In turn, if fitting of the LgBET formula is more advantageous, a dominant role of slitlike and larger holelike pores may be concluded (mutual interactions of clusters placed in the same pore is possible). A shape of constrained adsorption isotherms at higher pressures is affected mainly by the product (Rβ/BC) that implies an unavoidable uncertainty of the pore geometry identification. Detection of individual effects of these parameters needs adsorption measurements at the highest pressures. It makes possible to obtain an adequate evaluation of the total pore volume. Identification of geometrical properties of channel-like and dendrite-like pores is likely more reliable, as the LcBET formula (suitable for such structures) involves in more extent (than LgBET) purely geometrical effects of R and β on high pressure adsorption. In any case, an effective cluster size distribution is adequately characterized by the ratio (R/ BC), showing a combined effect of the geometrical and energetic constraints. More precise examination of porous systems may be performed with the rigorous model completed with the LgBET or LcBET formulas. Acknowledgment. The paper was prepared within the Statutory Research grant UMM-KBN (Krako´wWarsaw, Poland), ref. No. 11.11.210.62 LA020110E

Mathematical Description of Constrained BET-like Adsorption and Its

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