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Corrected thermodynamic description of adsorption via formalism of the theory of volume filling of micropores
Journal of Colloid and Interface Science 298 (2006) 66–73 www.elsevier.com/locate/jcis
Corrected thermodynamic description of adsorption via formalism of the theory of volume filling of micropores Artur P. Terzyk, Piotr A. Gauden ∗ , Gerhard Rychlicki Physicochemistry of Carbon Materials Research Group, Department of Chemistry, N. Copernicus University, Gagarina St. 7, 87-100 Toru´n, Poland Received 19 October 2005; accepted 1 December 2005 Available online 19 January 2006
Abstract Based on the series of benzene adsorption and related enthalpy of adsorption data measured on porous carbons that possess various porous structures, we show that the creation of a solidlike structure in pores depends on the average pore diameter of an adsorbent. Taking into account the solidlike adsorbed phase in the thermodynamic description of the adsorption process via the formalism of the theory of volume filling of micropores (TVFM) leads to very good agreement between the data measured experimentally and those calculated from TVFM. Finally we show that the boundary between solidlike and liquidlike structures of benzene molecules in carbon pores is located around the average pore diameter, close to ca. 2.1–2.4 nm. © 2005 Elsevier Inc. All rights reserved. Keywords: Adsorption; Activated carbon; Porosity; Theory of volume filling of micropores; Thermodynamics
1. Introduction It is well known that the enthalpy of adsorption measurements can provide valuable information concerning the mechanism of adsorptions, the energetic and structural heterogeneity of adsorbents, the adsorbate–adsorbent interactions, and the structure of adsorbate confined in pores [1–8]. However, the importance of this research is underestimated, among other things due to the time-consuming nature of experiments. Moreover, it should be pointed out that significant differences between the isosteric (where the bunch of isotherms are taken into account) and differential (i.e., directly measured experimentally) enthalpy of adsorption have been observed for many systems [5–7]. The indispensability of the use of calorimetric measurements (in comparison with those calculated from the Clausius– Clapeyron equation) for the description of adsorption processes has been widely discussed by us [5–7]. The theory of volume filling of micropores of Dubinin et al. [9,10] is still one of the most important tools for the evaluation of the parameters * Corresponding author. Fax: +48 056 654 2477.
E-mail address: [email protected] (P.A. Gauden). 0021-9797/$ – see front matter © 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2005.12.007
of adsorbents. Although it is thermodynamically inconsistent [11–14], this theory has been widely applied, modified, and extended on heterogeneous porous structures [15–17]. The fundamental paper by Chen and Yang [18] showed that the fundamental equation of the theory of volume filling of micropores (TVFM), called the Dubinin–Radushkevich equation (or in the generalized form the Dubinin–Astakhov equation), can be derived from statistical thermodynamics. The application of the most advanced methods of porosity calculation revealed some other interesting features of the DA adsorption isotherm equation [4,15,19]. Namely, we showed that DA, the isotherm equation (depending on the values of the parameters) can describe not only the primary, but also the secondary micropore filling process. While checking the thermodynamic validity of the TVFM, we noticed the similarity in the shapes of experimental and theoretical enthalpy of adsorption plots, and we observed that the theoretical enthalpy plot was shifted (approximately by a constant value) towards lower values [4,20]. Taking into account the creation of quasi-solids in micropores (via the addition of the value of enthalpy of crystallization of benzene to theoretical TVFM enthalpy) led to improvement in theoretical description for some systems. Therefore, we decided to perform some additional measurements and check how this effect
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depends on the structure (i.e., pore size distribution) of the carbonaceous adsorbents. 2. Experimental Eight samples of activated carbons, differing in porosity, are used in this paper: strictly microporous film Cf, obtained from cellophane by Zawadzki and co-workers [20–23]; synthetic microporous carbon A obtained from polyfurfuryl alcohol [4–7, 23]; commercial strictly microporous carbon D55/2 (CarboTech, Essen, Germany) [23]; commercial micromesoporous carbons AHD (Hajnówka, Poland) [23–25], WD (Hajnówka, Poland) [23–25], D43/1 (Carbo-Tech, Essen, Germany) [23– 25], and PICA PCO (PICA Carbon, Australia) [26–29]; and micromesoporous commercial carbon, possessing wider pores than all the materials mentioned previously, PICA HP (PICA Carbon, Australia) [26–29]. Although some of the adsorbents are not accessible commercially, detailed descriptions of their preparation can easily be found in respective references. Nitrogen adsorption isotherms at T = 77 K were measured using an ASAP 2010 volumetric adsorption analyzer from Micromeritics (Norcross, GA, USA) in the relative pressure range from about 1 × 10−6 up to 0.999. Benzene adsorption isotherms were measured at 298 K (1 × 10−6 –0.6 p/ps ) applying a volumetric apparatus with Baratron pressure transducers (MKS Instruments, Germany). The related enthalpy of adsorption was obtained by applying the isothermal Tian–Calvet microcalorimeter described previously [7,30]. The errors of the measurements are as follows: for the adsorption isotherms ±1% and for the enthalpy ±1.5% [7,30]. Before adsorption measurements the samples were outgassed at 473 K. 3. Textural characterization of activated carbons Experimental nitrogen adsorption isotherms measured at 77 K for all studied samples are compared in Fig. 1. Adsorption isotherms for D55/2 and Cf are of type I in the Brunauer, Emmet, and Teller classification; the remaining are of type II. In the other words, it is seen from Fig. 1 that the shape of isotherms progressively changes with the decrease in microporosity (for example, the comparison of Cf with PICA HP). Moreover, looking at these plots, the shapes of isotherms for some materials become upward at higher relative pressure region (for PICA HP, especially). This behavior confirms that meso- and macropores are present in some samples. However, this upward deviation is rather insignificant for most materials. For some nitrogen–adsorbent systems the hysteresis loops are also insignificant (A, WD, and AHD). The visible difference between the adsorption and desorption branches is clearly observed only for PICA HP. Hence, the pore structure in all carbons consist mainly of micropores; however, the number of mesopores cannot be neglected for some samples. It should be pointed out that adsorption properties of studied materials toward benzene (in the range presented in Fig. 1, i.e. up to 0.6 p/ps ) decrease in a sequence similar to that of low-temperature nitrogen adsorption isotherms.
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Further information about the pore structure of the investigated carbons can be obtained from the analysis of the differential (Jdiff (Heff )) and/or cumulative (Jcum (Heff )) pore size distributions (Fig. 2) calculated via the Nguyen and Do model [31–34]. It should be pointed out that only four representative PSDs are shown, for Cf, A, PICA PCO, and PICA HP. The analysis of Jdiff (Heff ) and/or Jcum (Heff ) calculated for all materials indicates that the most microporous is the carbonaceous film, Cf. Samples A and D55/2 are adsorbents having a more heterogeneous pore structure than the former one. However, PSDs show only the contribution of micropores. The analysis provided by the ND method confirms a more gradual development of dispersion of microporosity in the whole range with an important extension into the mesopores range for the remaining adsorbents: WD, AHD, D43/1, and PICA PCO. From Fig. 2 it is seen that PICA HP is activated carbon with a significant number of mesopores. 4. Quantitative description of the derivation of benzene from the liquid state in carbon pores As was shown recently [4,20,35] for adsorbate–adsorbent systems where adsorbate molecules approach the state of quasisolid, the fundamental equations of Dubinin’s theory of the volume filling of micropores (TVFM) do not satisfactorily describe the calorimetric data. Theoretical differential enthalpy values are usually too low compared to the experimental ones. These differences are probably caused by the incorrect postulation of the liquid state of an adsorbate (treated as the reference)—the fundamental assumption of the potential theory. This problem was discussed by Rudzi´nski and Everett [3] and by Rudzi´nski et al. [36]. They stated that it has been a common practice to put p0 = ps (p0 and ps are the standard and saturation pressures, respectively), in both the DR and the DA isotherm equations. However, it is to be noted that p0 cannot, in general, be identified with the value of the saturation pressure at the investigated temperature. The assumption that p0 = ps comes from the classical viewpoint of Polanyi that adsorption is a compression of adsorbate molecules by gas–solid forces to a liquidlike state in pores. On the other hand, it is easy to show that p0 cannot be treated as the best-fitting DA parameter due to “the compensation effect.” In other words, it is impossible to find a stable solution (i.e., for various values of p0 the various sets of other parameters can be evaluated, and similar values of goodness of fit are observed). Therefore we assume that p0 = ps , similarly to Rudzi´nski et al. [36]. In the case of heterogeneous surfaces, the adsorption takes place in pores having limited dimensions. The results of computer simulation and of adsorption in pores, carried out so frequently, show that the state of the adsorbate in pores may be distinctly different from that in a bulk liquid [37,38]. To investigate the influence of porosity on the fit of the DA theory to the experimental data, the fundamental equations of this theory are presented below. Assuming the fulfillment of the main condition of the original potential theory (first of all the temperature invariance condition (∂Apot /∂T )ΘDA = 0), we ob-
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Fig. 1. Nitrogen (T = 77 K) and benzene (T = 298 K) adsorption isotherms for the studied carbons.
differential enthalpy of adsorption is given by [9,10]
tain [9,10] H ads = −Apot −
αT ΘDA , F (Apot )
(1)
where F (Apot ) = −dΘDA /dApot is the differential adsorption potential distribution, ΘDA = W/W0DA = exp (Apot /(βE0DA ))n is the Dubinin–Astakhov (DA) equation [9,10], α is the coefficient of the adsorbate thermal expansion, Apot = RT ln(ps /p), E0DA is the characteristic energy of adsorption, and n is the exponent of the DA equation. Therefore, for the DA equation, the
Apot 1−n αT βE0DA qdiff = Apot + + L − RT , n βE0DA
(2)
where L is enthalpy of condensation. First, applying the C6 H6 adsorption data presented in Fig. 1, we calculated the parameters of the DA isotherm equation for β equal to unity in the range of relative pressure up to 0.1 p/ps . The nonlinear fitting procedure was used. The goodness of the fit of the theoretical model to experimental data is satisfactory (the values of the determination coefficient are close to 0.98 or
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Fig. 2. Differential (Jdiff (Heff )) and cumulative (Jcum (Heff )) pore size distributions generated on the basis of the Nguyen–Do method and nitrogen adsorption data (T = 77 K).
larger regardless of the kind of porosity of investigated materials). Next, we computed the enthalpy of adsorption (Eq. (2)) on the basis of the best-fitting DA parameters for α = 0.00116 K−1 and L = 33.54 kJ/mol. We fitted the data applying the genetic algorithm constructed by Storn and Price [39,40] and previously applied by us with great success [41,42]. Figs. 3 and 4 show the comparison of qdiff (Eq. (2); short dashed lines) generated for four representative carbons (the same as in Fig. 2) with experimental data (closed circles). It should be pointed out that we extent of the generation of qdiff data with the modified parameters to wider range of relative pressures, comparing with the suggestion of Dubinin. It was done to show the behavior of the differential enthalpy of adsorption at higher loadings (i.e., the abrupt rise of qdiff ). The analysis of the experimental data leads to the conclusion that the best fit between experimental and theoretical curves of the enthalpy is obtained for adsorbents possessing the greatest number of mesopores (i.e., PICA HP). However, for all adsorbate–adsorbent systems too low qdiff values are observed. Summing up, the largest differences between theoretical end experimental enthalpies of adsorption observed for Cf and A carbons are caused by the incorrect assumption of the state of adsorbed phase. It can be noticed that the liquid state of adsorbate is usually assumed in models and theories describing the mechanism of adsorption on the carbonaceous adsorbents. Therefore, the above-mentioned condition leads to the general inequality [4, 43–45]: |T Sdiff | |qdiff |.
(3)
Thus qdiff can be easily calculated based on the simplified relation [4,43–45]: qdiff = Gads − H vap ,
(4)
where Gads = Apot and H vap is the enthalpy of vaporization (equal with minus sign to L). Horvath and Kawazoe [43,44] noticed the similarity of the data calculated based on Eq. (4) to the “experimental” isosteric heat of adsorption obtained from low-temperature nitrogen isotherms measured at different temperatures. Using the same procedure, the differential enthalpy of C6 H6 and CCl4 adsorption was calculated by Gauden et al. [4] applying the formalism presented above (i.e., Eq. (4)). Final results showed that for the studied microporous A and B carbon samples Eq. (4) describes the experimental qdiff data (C6 H6 , T = 313 K) inadequately, i.e., this relation leads to lower, enthalpy than measured experimentally, by about 20 kJ/mol. Significantly different situation was observed for CCl4 data (T = 308 K), where the similarity is larger, especially for lower values of adsorption. These results suggest that one of the most important and above mentioned postulation of some theoretical models is not always correct for strictly microporous adsorbents. It is obvious that the same conclusion is valid for carbonaceous films possessing only micropores and homogeneous PSD [20]. Final results of the computation of the differential enthalpy of C6 H6 adsorption (T = 298 K) for Cf, A, PICA PCO, and PICA HP carbons applying Eq. (4) are shown in Figs. 3 and 4, crosses. It should be emphasized that for strictly microporous Cf sample this equation again describes inadequately the experimental C6 H6 enthalpy data. Similar discrepancy is observed for carbon A. Significantly different results are obtained for PICA HP, where the similarity is larger. Summing up, the differences between theoretical and experimental curves are constant for investigated range of adsorption for the given adsorbent and they are probably the function of porosity. On the other hand, as it is shown in this figure the curves generated on basis of Eq. (2) (with the DA equation) and Eq. (4) have the same shape for lower values of adsorption due to the pos-
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Fig. 3. Comparison of the differential molar enthalpy data measured calorimetrically (closed circles) measured for C6 H6 on Cf and A carbons at T = 298 K and calculated based on Eqs. (2) and (4)–(6). L is enthalpy of condensation equal to 33.54 kJ/mol. For other details, see text.
tulate of liquid state of the adsorbed phase. For higher values of adsorption the influence of the coefficient of the adsorbate thermal expansion is predominating. Consequently, observed differences between these plots are larger because qdiff generated on the basis of Eq. (2) tends to infinity for higher values of adsorption. 5. The procedure of calculation of the correction term The analysis of the results generated on the basis of Eqs. (2) and (4) (Figs. 3 and 4) shows that the inequality |T Sdiff |
Fig. 4. The comparison of the differential molar enthalpy data measured calorimetrically (closed circles) measured for C6 H6 on PICA PCO and PICA HP carbons at T = 298 K. The symbols the same as in Fig. 3.
|qdiff | cannot be accepted for some benzene–microporous adsorbent systems. In the other words, the correction term related to the partially ordered structure of adsorbed phase should be taken into account. Thus, significant improvement in the description of the benzene experimental data for microporous adsorbents should be observed, when Eq. (4) is rewritten as qdiff = Gads − H vap + H corr = Apot + L + H corr ,
(5)
where H corr is the above mentioned correction term. To find the value of the correction term in Eq. (5) the following cases were analyzed in the current studies:
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(i) First, H corr can be assumed by us to be a constant value and equal to the enthalpy of crystallization, H cryst = −9.84 kJ/mol for C6 H6 [46]. This behavior is justified by the approach in which the adsorbed molecules can be treated as in the quasi-solid state in the micropores and narrow mesopores, as was described above in detail. The comparison of calculated qdiff (Eq. (5), open small circles) and experimental data (closed symbols) is presented in Figs. 3 and 4. Analyzing both kinds of curves it is seen that the theory underestimates the experimental data for Cf and A. Opposite results are observed for PICA PCO and PICA HP. Thus, it can be concluded that the “correction term” assumed as H cryst in Eq. (5) is not absolutely correct. (ii) From the discussion mentioned above, it becomes obvious that H corr (the term of Eq. (5)) is generally a function of the pore structure (e.g., PSD, average pore widths), and it slightly depends on adsorption (and, of course, p/ps ). Therefore, we decided to optimize this term and it is treated as the global parameter. Final results are presented in Figs. 3 and 4 (bold solid lines). The analysis of plots leads to the conclusion that better similarity for all investigated adsorbents is observed for fitted H corr in comparison with the assumption of H corr = H cryst . The values of the correction term change from 16.04 kJ/mol (Cf, DC = 0.82) to 4.20 kJ/mol (PICA HP, DC = 0.94). For A and PICA PCO carbons the following values are calculated: H corr = 11.90 and 7.72 kJ/mol, and DC = 0.85 and 0.91, respectively. General tendencies occur, i.e., the values of the determination coefficients increase and/or the correction terms decrease with the increase of the contribution of mesopores to the total porosity. Ricca et al. [47,48] claimed that for some adsorption systems at temperatures below the triple point of the adsorptive, better agreement with the DA equation is obtained if ps is identified with the vapor pressure of the solid adsorptive rather than with the value extrapolated from the liquid. Consequently, these authors proposed taking into consideration the enthalpy of sublimation or vaporization (depending on whether the adsorbed phase was assumed to be solid or liquid) in the relation describing the isosteric enthalpies of adsorption (TVFM). Their studies showed that this assumption leads to improvement of the description of experimental data by the theoretical model. Thus, in the case where the state of adsorbed molecules is intermediate between liquidlike and solidlike, Eq. (2) can be rewritten as qdiff = Apot + L Apot 1−n αT βE0DA sol + H − RT , + n βE0DA
(6)
where H sol is the enthalpy associated with the creation of the quasi-solid phase in pores. Comparing Eqs. (5) and (6) leads to Apot 1−n αT βE0DA + H sol − RT . H corr = (7) n βE0DA Since (RT ) = const and α → 0 the correction term has the form H corr = H sol − RT .
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Fig. 5. The dependence between H corr and Heff,av,ND,tot (closed points—the values calculated for three studied cases, (ii) circles, (iii) squares, and (iv) triangles; for other details, see text) and their linear correlation (solid lines); the horizontal dashed line—H cryst = −9.84 kJ/mol; open symbols are corresponding to H corr = H cryst and H corr = 0 kJ/mol, respectively.
It should be pointed out that in this case H corr depends on the pore structure and the adsorption amount. Therefore, the analysis of Eq. (7) leads to the conclusion that two cases of Eq. (6) can be discussed: (iii) Applying the C6 H6 adsorption and related differential enthalpy of adsorption data simultaneously, we calculated the parameters of the DA isotherm equation and H sol (using Eq. (6)) for β equal to unity, α = 0.00116 K−1 , and L = 33.54 kJ/mol in the range of relative pressure up to 0.1 p/ps . (iv) Finally, we assumed that the value of the coefficient of adsorbate thermal expansion, α, remains unknown, and it can be treated as the best-fitting parameter. Thus, an analysis of the results of the simultaneous fitting of the adsorption and related enthalpy data similar to that for (iii) was studied. The final results of the calculations for the cases (iii) and (iv) are summarized in Figs. 3 and 4. It can be seen that the assumption of constant and/or optimized α leads almost to the same results in the low range of adsorbed amount. For higher values of adsorption the influence of the coefficient of adsorbate thermal expansion on qdiff (Eq. (6)) is considerable. In both cases a remarkably better fit to the experimental data is observed in comparison with the original enthalpy of adsorption (Eq. (2)). On the other hand, it should be pointed out that for (iv) the optimized values of α are close to zero. Thus, in this case the following relationship is true: H corr = H sol − RT . Finally, in Fig. 5 (points) the values of H corr calculated from the studied cases (i.e., (ii) circles, (iii) squares, and (iv) triangles) in the function of the average pore diameters, Heff,av,ND,tot are presented. Heff,av,ND,tot was calculated on the basis of the numerical integration of PSDs (Jcum (Heff )) for all pores taken into consideration in the ND method (Fig. 2, N2 , T = 77 K). We assume a slitlike shape of pores. In order to determine H corr on the basis of Eq. (7) for the α = 0.00116 K−1 (third case) we simplified considerations and assumed that
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H corr = H sol − RT . It should be emphasized that the contribution of the first term of Eq. (7) (related to the coefficient of the adsorbate thermal expansion) to the correction term is negligible. Additionally, a trend line plotted through these data is marked as a solid one (Fig. 5). All adsorbents are taken into account. It can be noticed that the increase in the average pore width leads to a decrease in the value of the correction term. Moreover, in this figure the curve of a enthalpy of crystallization equal to −9.84 kJ/mol is plotted (horizontal dashed line). The detailed analysis of the behavior for the data shown suggests that only for D43/1, AHD, PICA PCO, and PICA HP, are the values of H corr equal to or lower than the enthalpy of crystallization. Summing up, the border value of the effective pore diameter corresponding to H corr = H cryst = −9.84 kJ/mol is equal to 1.14 nm for (ii), 0.90 nm for (iii), and 1.22 nm for (iv) (Fig. 5, the intersection of respective solid and dashed lines). On the other hand, taking into consideration the linear correlation between Heff,av,ND,tot and H corr , one can calculate the critical value of the effective pore width for H corr = 0, Heff,c = 2.20 nm for (ii), 2.11 nm for (iii), and 2.40 nm for (iv). Thus, the main conclusion drawn from the results of this part of the study is that Eqs. (2) and (4) are proper for mesoporous adsorbents (i.e., the “liquid” state of confined benzene is reliable only for them). The above mentioned values of Heff,c are comparable to those evaluated by Chakrabarti and Kerkhof [49]. Those authors systematically investigated by Monte Carlo simulations the role of the wall structure on a fluid of flat hexagonal molecules confined between two graphite walls at T = 300 K. Their results show that the centers of mass of the molecules in different layers undergo an order–disorder transition as the wall separation increases, irrespective of the details of the wall structure. They investigated adsorbate confined between two graphite walls with widths Heff from 1.943 down to 0.971 nm. Chakrabarti and Kerkhof [49] concluded that the system undergoes an orderdisorder transition as Heff increases and the coverage at the substrate layer decreases as the critical value Heff,c increases. The calculations suggest that this critical value should lie within the bounds 1.619 < Heff,c < 1.943 nm. Additionally, the order– disorder transition point appears to be insensitive to temperature in the range we consider. 6. Conclusions Detailed knowledge of the behavior of adsorbed benzene or its derivatives will improve the understanding of the mechanisms of adsorption and the formulation of the proper theories and theoretical models. We extend our previous studies from adsorbents possessing only micropores [4,20] up to the materials possessing significant amount of wider pores. Presented results show that, generally, there is a significant influence of the pore structure of adsorbents on the behavior of adsorbed benzene in pores. Adsorbate molecules formed a partially ordered structure in the micropore even near room temperature. It should be pointed out that the solid and cooled liquid benzene behavior should be taking into theoretical considerations. This hypothesis is verified for the well-known and still very
popular Dubinin theory of the volume filling of micropores. However, the differences between theoretical and experimental data are still significant. For the materials possessing significant amounts of mesopores the satisfactory correlation is obtained. Taking into account solidlike adsorbed phase in the thermodynamic description of adsorption process via the formalism of the TVFM leads to very good agreement between the data measured experimentally and those calculated from the TVFM. Moreover, the main conclusion drawn from the comparison of theoretical and experimental data is the statement that the inequality |T Sdiff | |qdiff | is proper for adsorbents possessing average effective pore width larger than ca. 2 nm. So the results of this paper and those presented previously show that calorimetry techniques are helpful for the real thermodynamical description of the adsorption process. It should be underlined that similar behavior of adsorbed benzene molecules was observed by Pendleton [50]. He investigated the temperature dependence of benzene adsorption on microporous silica. The interpretation of the differential molar adsorption entropy at 298 K suggests that strongly localized adsorption occurs in the primary micropores and two-dimensional translational motion with rotation in the plane of the ring occurs in the secondary micropores. Summing up, results presented by us can be easily extended to other adsorbate–adsorbent systems, however, on the basis of adsorption isotherm and related calorimetrically measured enthalpy of adsorption. Acknowledgments The authors thank Dr. Gayle Newcombe (Australian Water Quality Centre, Salisbury, Australia) for the PICA PCO and PICA HP samples. It should be pointed out that those adsorbents were extensively investigated at the Carbon Round Robin International Inter-Laboratory Trial for the Determination of Activated Carbon Pore Volume Distributions (http://www. waterquality.crc.org.au/carbon_rr/round_robin_index.htm). References [1] F. Rouquerol, J. Rouquerol, K.S.W. Sing, Adsorption by Powders and Porous Solids. Principles, Methodology and Applications, Academic Press, London, 1999. [2] J.N. Israelachvili, Intermolecular and Surface Forces, Academic Press, London, 1991. [3] W. Rudzi´nski, D.H. Everett, Adsorption of Gases on Heterogeneous Surfaces, Academic Press, London, 1992. ´ [4] P.A. Gauden, A.P. Terzyk, G. Rychlicki, P. Kowalczyk, M.S. Cwiertnia, J.K. Garbacz, J. Colloid Interface Sci. 273 (2004) 39. [5] A.P. Terzyk, G. Rychlicki, Adsorpt. Sci. Technol. 17 (1999) 323. [6] G. Rychlicki, A.P. Terzyk, J. Thermal Anal. 45 (1995) 961. [7] J.K. Garbacz, G. Rychlicki, A.P. Terzyk, Adsorpt. Sci. Technol. 11 (1994) 15. [8] P.I. Babaev, M.M. Dubinin, A.A. Isirikyan, Izv. Akad. Nauk SSSR 9 (1976) 1929 [in Russian]. [9] M.M. Dubinin, Adsorption and Porosity, WAT, Warsaw, 1975 [in Polish]. [10] M.M. Dubinin, Chem. Rev. 60 (1961) 235. [11] J. Toth (Ed.), Adsorption: Theory, Modeling, and Analysis, Surfactant Science Series, vol. 1, Dekker, New York, 2002, p. 1. [12] J. Toth, J. Colloid Interface Sci. 163 (1994) 299.
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Corrected thermodynamic description of adsorption via formalism of the theory of volume filling of micropores Artur P. Terzyk, Piotr A. Gauden ∗ , Gerhard Rychlicki Physicochemistry of Carbon Materials Research Group, Department of Chemistry, N. Copernicus University, Gagarina St. 7, 87-100 Toru´n, Poland Received 19 October 2005; accepted 1 December 2005 Available online 19 January 2006
Abstract Based on the series of benzene adsorption and related enthalpy of adsorption data measured on porous carbons that possess various porous structures, we show that the creation of a solidlike structure in pores depends on the average pore diameter of an adsorbent. Taking into account the solidlike adsorbed phase in the thermodynamic description of the adsorption process via the formalism of the theory of volume filling of micropores (TVFM) leads to very good agreement between the data measured experimentally and those calculated from TVFM. Finally we show that the boundary between solidlike and liquidlike structures of benzene molecules in carbon pores is located around the average pore diameter, close to ca. 2.1–2.4 nm. © 2005 Elsevier Inc. All rights reserved. Keywords: Adsorption; Activated carbon; Porosity; Theory of volume filling of micropores; Thermodynamics
1. Introduction It is well known that the enthalpy of adsorption measurements can provide valuable information concerning the mechanism of adsorptions, the energetic and structural heterogeneity of adsorbents, the adsorbate–adsorbent interactions, and the structure of adsorbate confined in pores [1–8]. However, the importance of this research is underestimated, among other things due to the time-consuming nature of experiments. Moreover, it should be pointed out that significant differences between the isosteric (where the bunch of isotherms are taken into account) and differential (i.e., directly measured experimentally) enthalpy of adsorption have been observed for many systems [5–7]. The indispensability of the use of calorimetric measurements (in comparison with those calculated from the Clausius– Clapeyron equation) for the description of adsorption processes has been widely discussed by us [5–7]. The theory of volume filling of micropores of Dubinin et al. [9,10] is still one of the most important tools for the evaluation of the parameters * Corresponding author. Fax: +48 056 654 2477.
E-mail address: [email protected] (P.A. Gauden). 0021-9797/$ – see front matter © 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2005.12.007
of adsorbents. Although it is thermodynamically inconsistent [11–14], this theory has been widely applied, modified, and extended on heterogeneous porous structures [15–17]. The fundamental paper by Chen and Yang [18] showed that the fundamental equation of the theory of volume filling of micropores (TVFM), called the Dubinin–Radushkevich equation (or in the generalized form the Dubinin–Astakhov equation), can be derived from statistical thermodynamics. The application of the most advanced methods of porosity calculation revealed some other interesting features of the DA adsorption isotherm equation [4,15,19]. Namely, we showed that DA, the isotherm equation (depending on the values of the parameters) can describe not only the primary, but also the secondary micropore filling process. While checking the thermodynamic validity of the TVFM, we noticed the similarity in the shapes of experimental and theoretical enthalpy of adsorption plots, and we observed that the theoretical enthalpy plot was shifted (approximately by a constant value) towards lower values [4,20]. Taking into account the creation of quasi-solids in micropores (via the addition of the value of enthalpy of crystallization of benzene to theoretical TVFM enthalpy) led to improvement in theoretical description for some systems. Therefore, we decided to perform some additional measurements and check how this effect
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depends on the structure (i.e., pore size distribution) of the carbonaceous adsorbents. 2. Experimental Eight samples of activated carbons, differing in porosity, are used in this paper: strictly microporous film Cf, obtained from cellophane by Zawadzki and co-workers [20–23]; synthetic microporous carbon A obtained from polyfurfuryl alcohol [4–7, 23]; commercial strictly microporous carbon D55/2 (CarboTech, Essen, Germany) [23]; commercial micromesoporous carbons AHD (Hajnówka, Poland) [23–25], WD (Hajnówka, Poland) [23–25], D43/1 (Carbo-Tech, Essen, Germany) [23– 25], and PICA PCO (PICA Carbon, Australia) [26–29]; and micromesoporous commercial carbon, possessing wider pores than all the materials mentioned previously, PICA HP (PICA Carbon, Australia) [26–29]. Although some of the adsorbents are not accessible commercially, detailed descriptions of their preparation can easily be found in respective references. Nitrogen adsorption isotherms at T = 77 K were measured using an ASAP 2010 volumetric adsorption analyzer from Micromeritics (Norcross, GA, USA) in the relative pressure range from about 1 × 10−6 up to 0.999. Benzene adsorption isotherms were measured at 298 K (1 × 10−6 –0.6 p/ps ) applying a volumetric apparatus with Baratron pressure transducers (MKS Instruments, Germany). The related enthalpy of adsorption was obtained by applying the isothermal Tian–Calvet microcalorimeter described previously [7,30]. The errors of the measurements are as follows: for the adsorption isotherms ±1% and for the enthalpy ±1.5% [7,30]. Before adsorption measurements the samples were outgassed at 473 K. 3. Textural characterization of activated carbons Experimental nitrogen adsorption isotherms measured at 77 K for all studied samples are compared in Fig. 1. Adsorption isotherms for D55/2 and Cf are of type I in the Brunauer, Emmet, and Teller classification; the remaining are of type II. In the other words, it is seen from Fig. 1 that the shape of isotherms progressively changes with the decrease in microporosity (for example, the comparison of Cf with PICA HP). Moreover, looking at these plots, the shapes of isotherms for some materials become upward at higher relative pressure region (for PICA HP, especially). This behavior confirms that meso- and macropores are present in some samples. However, this upward deviation is rather insignificant for most materials. For some nitrogen–adsorbent systems the hysteresis loops are also insignificant (A, WD, and AHD). The visible difference between the adsorption and desorption branches is clearly observed only for PICA HP. Hence, the pore structure in all carbons consist mainly of micropores; however, the number of mesopores cannot be neglected for some samples. It should be pointed out that adsorption properties of studied materials toward benzene (in the range presented in Fig. 1, i.e. up to 0.6 p/ps ) decrease in a sequence similar to that of low-temperature nitrogen adsorption isotherms.
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Further information about the pore structure of the investigated carbons can be obtained from the analysis of the differential (Jdiff (Heff )) and/or cumulative (Jcum (Heff )) pore size distributions (Fig. 2) calculated via the Nguyen and Do model [31–34]. It should be pointed out that only four representative PSDs are shown, for Cf, A, PICA PCO, and PICA HP. The analysis of Jdiff (Heff ) and/or Jcum (Heff ) calculated for all materials indicates that the most microporous is the carbonaceous film, Cf. Samples A and D55/2 are adsorbents having a more heterogeneous pore structure than the former one. However, PSDs show only the contribution of micropores. The analysis provided by the ND method confirms a more gradual development of dispersion of microporosity in the whole range with an important extension into the mesopores range for the remaining adsorbents: WD, AHD, D43/1, and PICA PCO. From Fig. 2 it is seen that PICA HP is activated carbon with a significant number of mesopores. 4. Quantitative description of the derivation of benzene from the liquid state in carbon pores As was shown recently [4,20,35] for adsorbate–adsorbent systems where adsorbate molecules approach the state of quasisolid, the fundamental equations of Dubinin’s theory of the volume filling of micropores (TVFM) do not satisfactorily describe the calorimetric data. Theoretical differential enthalpy values are usually too low compared to the experimental ones. These differences are probably caused by the incorrect postulation of the liquid state of an adsorbate (treated as the reference)—the fundamental assumption of the potential theory. This problem was discussed by Rudzi´nski and Everett [3] and by Rudzi´nski et al. [36]. They stated that it has been a common practice to put p0 = ps (p0 and ps are the standard and saturation pressures, respectively), in both the DR and the DA isotherm equations. However, it is to be noted that p0 cannot, in general, be identified with the value of the saturation pressure at the investigated temperature. The assumption that p0 = ps comes from the classical viewpoint of Polanyi that adsorption is a compression of adsorbate molecules by gas–solid forces to a liquidlike state in pores. On the other hand, it is easy to show that p0 cannot be treated as the best-fitting DA parameter due to “the compensation effect.” In other words, it is impossible to find a stable solution (i.e., for various values of p0 the various sets of other parameters can be evaluated, and similar values of goodness of fit are observed). Therefore we assume that p0 = ps , similarly to Rudzi´nski et al. [36]. In the case of heterogeneous surfaces, the adsorption takes place in pores having limited dimensions. The results of computer simulation and of adsorption in pores, carried out so frequently, show that the state of the adsorbate in pores may be distinctly different from that in a bulk liquid [37,38]. To investigate the influence of porosity on the fit of the DA theory to the experimental data, the fundamental equations of this theory are presented below. Assuming the fulfillment of the main condition of the original potential theory (first of all the temperature invariance condition (∂Apot /∂T )ΘDA = 0), we ob-
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Fig. 1. Nitrogen (T = 77 K) and benzene (T = 298 K) adsorption isotherms for the studied carbons.
differential enthalpy of adsorption is given by [9,10]
tain [9,10] H ads = −Apot −
αT ΘDA , F (Apot )
(1)
where F (Apot ) = −dΘDA /dApot is the differential adsorption potential distribution, ΘDA = W/W0DA = exp (Apot /(βE0DA ))n is the Dubinin–Astakhov (DA) equation [9,10], α is the coefficient of the adsorbate thermal expansion, Apot = RT ln(ps /p), E0DA is the characteristic energy of adsorption, and n is the exponent of the DA equation. Therefore, for the DA equation, the
Apot 1−n αT βE0DA qdiff = Apot + + L − RT , n βE0DA
(2)
where L is enthalpy of condensation. First, applying the C6 H6 adsorption data presented in Fig. 1, we calculated the parameters of the DA isotherm equation for β equal to unity in the range of relative pressure up to 0.1 p/ps . The nonlinear fitting procedure was used. The goodness of the fit of the theoretical model to experimental data is satisfactory (the values of the determination coefficient are close to 0.98 or
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69
Fig. 2. Differential (Jdiff (Heff )) and cumulative (Jcum (Heff )) pore size distributions generated on the basis of the Nguyen–Do method and nitrogen adsorption data (T = 77 K).
larger regardless of the kind of porosity of investigated materials). Next, we computed the enthalpy of adsorption (Eq. (2)) on the basis of the best-fitting DA parameters for α = 0.00116 K−1 and L = 33.54 kJ/mol. We fitted the data applying the genetic algorithm constructed by Storn and Price [39,40] and previously applied by us with great success [41,42]. Figs. 3 and 4 show the comparison of qdiff (Eq. (2); short dashed lines) generated for four representative carbons (the same as in Fig. 2) with experimental data (closed circles). It should be pointed out that we extent of the generation of qdiff data with the modified parameters to wider range of relative pressures, comparing with the suggestion of Dubinin. It was done to show the behavior of the differential enthalpy of adsorption at higher loadings (i.e., the abrupt rise of qdiff ). The analysis of the experimental data leads to the conclusion that the best fit between experimental and theoretical curves of the enthalpy is obtained for adsorbents possessing the greatest number of mesopores (i.e., PICA HP). However, for all adsorbate–adsorbent systems too low qdiff values are observed. Summing up, the largest differences between theoretical end experimental enthalpies of adsorption observed for Cf and A carbons are caused by the incorrect assumption of the state of adsorbed phase. It can be noticed that the liquid state of adsorbate is usually assumed in models and theories describing the mechanism of adsorption on the carbonaceous adsorbents. Therefore, the above-mentioned condition leads to the general inequality [4, 43–45]: |T Sdiff | |qdiff |.
(3)
Thus qdiff can be easily calculated based on the simplified relation [4,43–45]: qdiff = Gads − H vap ,
(4)
where Gads = Apot and H vap is the enthalpy of vaporization (equal with minus sign to L). Horvath and Kawazoe [43,44] noticed the similarity of the data calculated based on Eq. (4) to the “experimental” isosteric heat of adsorption obtained from low-temperature nitrogen isotherms measured at different temperatures. Using the same procedure, the differential enthalpy of C6 H6 and CCl4 adsorption was calculated by Gauden et al. [4] applying the formalism presented above (i.e., Eq. (4)). Final results showed that for the studied microporous A and B carbon samples Eq. (4) describes the experimental qdiff data (C6 H6 , T = 313 K) inadequately, i.e., this relation leads to lower, enthalpy than measured experimentally, by about 20 kJ/mol. Significantly different situation was observed for CCl4 data (T = 308 K), where the similarity is larger, especially for lower values of adsorption. These results suggest that one of the most important and above mentioned postulation of some theoretical models is not always correct for strictly microporous adsorbents. It is obvious that the same conclusion is valid for carbonaceous films possessing only micropores and homogeneous PSD [20]. Final results of the computation of the differential enthalpy of C6 H6 adsorption (T = 298 K) for Cf, A, PICA PCO, and PICA HP carbons applying Eq. (4) are shown in Figs. 3 and 4, crosses. It should be emphasized that for strictly microporous Cf sample this equation again describes inadequately the experimental C6 H6 enthalpy data. Similar discrepancy is observed for carbon A. Significantly different results are obtained for PICA HP, where the similarity is larger. Summing up, the differences between theoretical and experimental curves are constant for investigated range of adsorption for the given adsorbent and they are probably the function of porosity. On the other hand, as it is shown in this figure the curves generated on basis of Eq. (2) (with the DA equation) and Eq. (4) have the same shape for lower values of adsorption due to the pos-
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Fig. 3. Comparison of the differential molar enthalpy data measured calorimetrically (closed circles) measured for C6 H6 on Cf and A carbons at T = 298 K and calculated based on Eqs. (2) and (4)–(6). L is enthalpy of condensation equal to 33.54 kJ/mol. For other details, see text.
tulate of liquid state of the adsorbed phase. For higher values of adsorption the influence of the coefficient of the adsorbate thermal expansion is predominating. Consequently, observed differences between these plots are larger because qdiff generated on the basis of Eq. (2) tends to infinity for higher values of adsorption. 5. The procedure of calculation of the correction term The analysis of the results generated on the basis of Eqs. (2) and (4) (Figs. 3 and 4) shows that the inequality |T Sdiff |
Fig. 4. The comparison of the differential molar enthalpy data measured calorimetrically (closed circles) measured for C6 H6 on PICA PCO and PICA HP carbons at T = 298 K. The symbols the same as in Fig. 3.
|qdiff | cannot be accepted for some benzene–microporous adsorbent systems. In the other words, the correction term related to the partially ordered structure of adsorbed phase should be taken into account. Thus, significant improvement in the description of the benzene experimental data for microporous adsorbents should be observed, when Eq. (4) is rewritten as qdiff = Gads − H vap + H corr = Apot + L + H corr ,
(5)
where H corr is the above mentioned correction term. To find the value of the correction term in Eq. (5) the following cases were analyzed in the current studies:
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(i) First, H corr can be assumed by us to be a constant value and equal to the enthalpy of crystallization, H cryst = −9.84 kJ/mol for C6 H6 [46]. This behavior is justified by the approach in which the adsorbed molecules can be treated as in the quasi-solid state in the micropores and narrow mesopores, as was described above in detail. The comparison of calculated qdiff (Eq. (5), open small circles) and experimental data (closed symbols) is presented in Figs. 3 and 4. Analyzing both kinds of curves it is seen that the theory underestimates the experimental data for Cf and A. Opposite results are observed for PICA PCO and PICA HP. Thus, it can be concluded that the “correction term” assumed as H cryst in Eq. (5) is not absolutely correct. (ii) From the discussion mentioned above, it becomes obvious that H corr (the term of Eq. (5)) is generally a function of the pore structure (e.g., PSD, average pore widths), and it slightly depends on adsorption (and, of course, p/ps ). Therefore, we decided to optimize this term and it is treated as the global parameter. Final results are presented in Figs. 3 and 4 (bold solid lines). The analysis of plots leads to the conclusion that better similarity for all investigated adsorbents is observed for fitted H corr in comparison with the assumption of H corr = H cryst . The values of the correction term change from 16.04 kJ/mol (Cf, DC = 0.82) to 4.20 kJ/mol (PICA HP, DC = 0.94). For A and PICA PCO carbons the following values are calculated: H corr = 11.90 and 7.72 kJ/mol, and DC = 0.85 and 0.91, respectively. General tendencies occur, i.e., the values of the determination coefficients increase and/or the correction terms decrease with the increase of the contribution of mesopores to the total porosity. Ricca et al. [47,48] claimed that for some adsorption systems at temperatures below the triple point of the adsorptive, better agreement with the DA equation is obtained if ps is identified with the vapor pressure of the solid adsorptive rather than with the value extrapolated from the liquid. Consequently, these authors proposed taking into consideration the enthalpy of sublimation or vaporization (depending on whether the adsorbed phase was assumed to be solid or liquid) in the relation describing the isosteric enthalpies of adsorption (TVFM). Their studies showed that this assumption leads to improvement of the description of experimental data by the theoretical model. Thus, in the case where the state of adsorbed molecules is intermediate between liquidlike and solidlike, Eq. (2) can be rewritten as qdiff = Apot + L Apot 1−n αT βE0DA sol + H − RT , + n βE0DA
(6)
where H sol is the enthalpy associated with the creation of the quasi-solid phase in pores. Comparing Eqs. (5) and (6) leads to Apot 1−n αT βE0DA + H sol − RT . H corr = (7) n βE0DA Since (RT ) = const and α → 0 the correction term has the form H corr = H sol − RT .
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Fig. 5. The dependence between H corr and Heff,av,ND,tot (closed points—the values calculated for three studied cases, (ii) circles, (iii) squares, and (iv) triangles; for other details, see text) and their linear correlation (solid lines); the horizontal dashed line—H cryst = −9.84 kJ/mol; open symbols are corresponding to H corr = H cryst and H corr = 0 kJ/mol, respectively.
It should be pointed out that in this case H corr depends on the pore structure and the adsorption amount. Therefore, the analysis of Eq. (7) leads to the conclusion that two cases of Eq. (6) can be discussed: (iii) Applying the C6 H6 adsorption and related differential enthalpy of adsorption data simultaneously, we calculated the parameters of the DA isotherm equation and H sol (using Eq. (6)) for β equal to unity, α = 0.00116 K−1 , and L = 33.54 kJ/mol in the range of relative pressure up to 0.1 p/ps . (iv) Finally, we assumed that the value of the coefficient of adsorbate thermal expansion, α, remains unknown, and it can be treated as the best-fitting parameter. Thus, an analysis of the results of the simultaneous fitting of the adsorption and related enthalpy data similar to that for (iii) was studied. The final results of the calculations for the cases (iii) and (iv) are summarized in Figs. 3 and 4. It can be seen that the assumption of constant and/or optimized α leads almost to the same results in the low range of adsorbed amount. For higher values of adsorption the influence of the coefficient of adsorbate thermal expansion on qdiff (Eq. (6)) is considerable. In both cases a remarkably better fit to the experimental data is observed in comparison with the original enthalpy of adsorption (Eq. (2)). On the other hand, it should be pointed out that for (iv) the optimized values of α are close to zero. Thus, in this case the following relationship is true: H corr = H sol − RT . Finally, in Fig. 5 (points) the values of H corr calculated from the studied cases (i.e., (ii) circles, (iii) squares, and (iv) triangles) in the function of the average pore diameters, Heff,av,ND,tot are presented. Heff,av,ND,tot was calculated on the basis of the numerical integration of PSDs (Jcum (Heff )) for all pores taken into consideration in the ND method (Fig. 2, N2 , T = 77 K). We assume a slitlike shape of pores. In order to determine H corr on the basis of Eq. (7) for the α = 0.00116 K−1 (third case) we simplified considerations and assumed that
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H corr = H sol − RT . It should be emphasized that the contribution of the first term of Eq. (7) (related to the coefficient of the adsorbate thermal expansion) to the correction term is negligible. Additionally, a trend line plotted through these data is marked as a solid one (Fig. 5). All adsorbents are taken into account. It can be noticed that the increase in the average pore width leads to a decrease in the value of the correction term. Moreover, in this figure the curve of a enthalpy of crystallization equal to −9.84 kJ/mol is plotted (horizontal dashed line). The detailed analysis of the behavior for the data shown suggests that only for D43/1, AHD, PICA PCO, and PICA HP, are the values of H corr equal to or lower than the enthalpy of crystallization. Summing up, the border value of the effective pore diameter corresponding to H corr = H cryst = −9.84 kJ/mol is equal to 1.14 nm for (ii), 0.90 nm for (iii), and 1.22 nm for (iv) (Fig. 5, the intersection of respective solid and dashed lines). On the other hand, taking into consideration the linear correlation between Heff,av,ND,tot and H corr , one can calculate the critical value of the effective pore width for H corr = 0, Heff,c = 2.20 nm for (ii), 2.11 nm for (iii), and 2.40 nm for (iv). Thus, the main conclusion drawn from the results of this part of the study is that Eqs. (2) and (4) are proper for mesoporous adsorbents (i.e., the “liquid” state of confined benzene is reliable only for them). The above mentioned values of Heff,c are comparable to those evaluated by Chakrabarti and Kerkhof [49]. Those authors systematically investigated by Monte Carlo simulations the role of the wall structure on a fluid of flat hexagonal molecules confined between two graphite walls at T = 300 K. Their results show that the centers of mass of the molecules in different layers undergo an order–disorder transition as the wall separation increases, irrespective of the details of the wall structure. They investigated adsorbate confined between two graphite walls with widths Heff from 1.943 down to 0.971 nm. Chakrabarti and Kerkhof [49] concluded that the system undergoes an orderdisorder transition as Heff increases and the coverage at the substrate layer decreases as the critical value Heff,c increases. The calculations suggest that this critical value should lie within the bounds 1.619 < Heff,c < 1.943 nm. Additionally, the order– disorder transition point appears to be insensitive to temperature in the range we consider. 6. Conclusions Detailed knowledge of the behavior of adsorbed benzene or its derivatives will improve the understanding of the mechanisms of adsorption and the formulation of the proper theories and theoretical models. We extend our previous studies from adsorbents possessing only micropores [4,20] up to the materials possessing significant amount of wider pores. Presented results show that, generally, there is a significant influence of the pore structure of adsorbents on the behavior of adsorbed benzene in pores. Adsorbate molecules formed a partially ordered structure in the micropore even near room temperature. It should be pointed out that the solid and cooled liquid benzene behavior should be taking into theoretical considerations. This hypothesis is verified for the well-known and still very
popular Dubinin theory of the volume filling of micropores. However, the differences between theoretical and experimental data are still significant. For the materials possessing significant amounts of mesopores the satisfactory correlation is obtained. Taking into account solidlike adsorbed phase in the thermodynamic description of adsorption process via the formalism of the TVFM leads to very good agreement between the data measured experimentally and those calculated from the TVFM. Moreover, the main conclusion drawn from the comparison of theoretical and experimental data is the statement that the inequality |T Sdiff | |qdiff | is proper for adsorbents possessing average effective pore width larger than ca. 2 nm. So the results of this paper and those presented previously show that calorimetry techniques are helpful for the real thermodynamical description of the adsorption process. It should be underlined that similar behavior of adsorbed benzene molecules was observed by Pendleton [50]. He investigated the temperature dependence of benzene adsorption on microporous silica. The interpretation of the differential molar adsorption entropy at 298 K suggests that strongly localized adsorption occurs in the primary micropores and two-dimensional translational motion with rotation in the plane of the ring occurs in the secondary micropores. Summing up, results presented by us can be easily extended to other adsorbate–adsorbent systems, however, on the basis of adsorption isotherm and related calorimetrically measured enthalpy of adsorption. Acknowledgments The authors thank Dr. Gayle Newcombe (Australian Water Quality Centre, Salisbury, Australia) for the PICA PCO and PICA HP samples. It should be pointed out that those adsorbents were extensively investigated at the Carbon Round Robin International Inter-Laboratory Trial for the Determination of Activated Carbon Pore Volume Distributions (http://www. waterquality.crc.org.au/carbon_rr/round_robin_index.htm). References [1] F. Rouquerol, J. Rouquerol, K.S.W. Sing, Adsorption by Powders and Porous Solids. Principles, Methodology and Applications, Academic Press, London, 1999. [2] J.N. Israelachvili, Intermolecular and Surface Forces, Academic Press, London, 1991. [3] W. Rudzi´nski, D.H. Everett, Adsorption of Gases on Heterogeneous Surfaces, Academic Press, London, 1992. ´ [4] P.A. Gauden, A.P. Terzyk, G. Rychlicki, P. Kowalczyk, M.S. Cwiertnia, J.K. Garbacz, J. Colloid Interface Sci. 273 (2004) 39. [5] A.P. Terzyk, G. Rychlicki, Adsorpt. Sci. Technol. 17 (1999) 323. [6] G. Rychlicki, A.P. Terzyk, J. Thermal Anal. 45 (1995) 961. [7] J.K. Garbacz, G. Rychlicki, A.P. Terzyk, Adsorpt. Sci. Technol. 11 (1994) 15. [8] P.I. Babaev, M.M. Dubinin, A.A. Isirikyan, Izv. Akad. Nauk SSSR 9 (1976) 1929 [in Russian]. [9] M.M. Dubinin, Adsorption and Porosity, WAT, Warsaw, 1975 [in Polish]. [10] M.M. Dubinin, Chem. Rev. 60 (1961) 235. [11] J. Toth (Ed.), Adsorption: Theory, Modeling, and Analysis, Surfactant Science Series, vol. 1, Dekker, New York, 2002, p. 1. [12] J. Toth, J. Colloid Interface Sci. 163 (1994) 299.
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