A Macrothermodynamic Approach to the Limit of Reversible Capillary

Inorganica, Fisica e dei Materiali, Università degli Studi di Torino, Torino, Italy, and Dipartimento di Chimica dei Materiali, Politecnico di To...
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A Macrothermodynamic Approach to the Limit of Reversible Capillary Condensation Philippe Trens,† Nathalie Tanchoux,† Anne Galarneau,† Daniel Brunel,† Bice Fubini,‡ Edoardo Garrone,§ Franc¸ ois Fajula,† and Francesco Di Renzo*,† Laboratoire de Mate´ riaux Catalytiques et Catalyse en Chimie Organique, UMR 5618 CNRS-ENSCM-UM1, Institut C. Gerhardt, FR 1878, ENSCM, 8 rue Ecole Normale, 34296 Montpellier, France, Istituto di Chimica Inorganica, Fisica e dei Materiali, Universita` degli Studi di Torino, Torino, Italy, and Dipartimento di Chimica dei Materiali, Politecnico di Torino, Torino, Italy Received March 24, 2005. In Final Form: June 16, 2005 The threshold of reversible capillary condensation is a well-defined thermodynamic property, as evidenced by corresponding states treatment of literature and experimental data on the lowest closure point of the hysteresis loop in capillary condensation-evaporation cycles for several adsorbates. The nonhysteretical filling of small mesopores presents the properties of a first-order phase transition, confirming that the limit of condensation reversibility does not coincide with the pore critical point. The enthalpy of reversible capillary condensation can be calculated by a Clausius-Clapeyron approach and is consistently larger than the condensation heat in unconfined conditions. Calorimetric data on the capillary condensation of tert-butyl alcohol in MCM-41 silica confirm a 20% increase of condensation heat in small mesopores. This enthalpic advantage makes easier the overcoming of the adhesion forces by the capillary forces and justifies the disappearing of the hysteresis loop.

A hysteresis loop is such a common occurrence in the adsorption-desorption cycles on mesoporous adsorbents (see curve a of Figure 1) that its presence has been included in the IUPAC definition of the type IV isotherm.1 It was early observed that no hysteresis loop can extend below a given relative pressure threshold, p/p0 ) 0.42 in the case of N2 at 77 K.2 Any desorption loop begun at higher pressure is terminated by a sudden evaporation when this threshold of relative pressure is reached (see curve b of Figure 1). This effect, suggestively defined as catastrophic desorption, is at the basis of a frequent artifact in the evaluation of pore size: the sudden desorption is attributed to a narrow distribution of pores around 4 nm in diameter, albeit the actual porosity has smaller size and is more broadly distributed.2,4 For smaller mesopores, in which capillary condensation takes place below the lowest possible closure point of the hysteresis loop, pores are filled and emptied following the same reversible path, as shown by curve c of Figure 1.5,6 For each pore size and shape, the pressure of the hysteresis closure point depends both on the nature of the adsorbate and on the temperature.7 The absence of hysteresis in the adsorption-desorption cycle for small mesopores was attributed to a tensional instability of the meniscus at low curvature radius.8-10 The smaller the mesopores are, * Corresponding author. E-mail: [email protected]. † ENSCM Montpellier. ‡ Universita ` di Torino. § Politecnico di Torino. (1) Sing, K. S. W.; Everett, D. H.; Haul, R. A. W. Pure Appl. Chem. 1985, 57, 603. (2) Harris, M. R. Chem. Ind. (London) 1965, 268. (3) Di Renzo, F.; Galarneau, A.; Trens, P.; Tanchoux, N.; Fajula, F. Stud. Surf. Sci. Catal. 2002, 142, 1057. (4) Broekhoff, J. C. P.; van Beek, W. P. J. Chem. Soc., Faraday Trans. 1 1979, 75, 42. (5) Franke, O.; Schulz-Ekloff, G.; Rathousky, J.; Starek, J.; Zukal, A. J. Chem. Soc., Chem. Commun. 1993, 724. (6) Branton, P. J.; Hall, P. G.; Sing, K. S. W.; Reichert, H.; Schu¨th, F.; Unger, K. K. J. Chem. Soc., Faraday Trans. 1991, 90, 2965. (7) Ravikovitch, P. I.; Neimark, A. V. Langmuir 2002, 18, 9830.

Figure 1. Adsorption-desorption isotherms of N2 at 77 K on SBA-15 (a), trimethylbenzene-swelled MCM-41 (b), and MCM41 (c) silicas: filled symbols, adsorption; empty symbols, desorption. Lowest closure point of the hysteresis loop at p/p0 ) 0.42.3

the larger the capillary tension experienced by the liquid. For very small mesopores, the tension can exceed the tensile strength of the liquid, which cannot help evaporation. Conditions in which a liquid-gas interface is unstable are strongly reminiscent of the definition of the critical point of a fluid. Since the beginning of the 1980s, a large research effort has been devoted to the evaluation of the (8) Schofield, R. K. Discuss. Faraday Soc. 1948, 3, 105. (9) Kadlec, O.; Dubinin, M. M. J. Colloid Interface Sci. 1969, 31, 479. (10) Burgess, C. G. V.; Everett, D. H. J. Colloid Interface Sci. 1970, 33, 611.

10.1021/la0507838 CCC: $30.25 © 2005 American Chemical Society Published on Web 07/28/2005

Reversible Capillary Condensation

critical properties of confined fluids. Nonlocal density functional theory (NLDFT) calculations allowed simulation of the adsorption and desorption branches of the isotherm in porous systems.11-15 It was calculated that a mesopore is filled and emptied along the same path above a threshold of temperature and pressure defined as a capillary critical point. Comparisons between calculations and measurements became much easier when ordered adsorbents in the size range of the small mesopores were made available by micelle-templated synthesis.16 Adsorption experiments at several temperature levels allowed establishment of the hysteresis closure point for several adsorbates in the new MCM-41 adsorbents with monodispersed pore size.17,18 Contrary to the expectations, no quantitative consistency was found between the capillary critical point as calculated by NLDFT and the hysteresis closure point. For instance, in the case of the adsorption of N2 at 77 K, the experimental closure point at p/p0 ) 0.42 corresponds to the filling of pores with diameter about 4 nm (see Figure 1), whereas the capillary critical point was calculated at p/p0 < 0.05, corresponding to the filling of pores with diameter 1.8 nm.19 Attempts to correlate the capillary critical point with other features of the isotherms20,21 confirmed that the instability of the meniscus is by no way a necessary condition for the absence of hysteresis.22 Further confirmation of the difference in nature between the capillary critical point and the hysteresis closure point came from the measurement of the enthalpy of condensation in small mesopores of constant diameter. The latent heat of condensation decreases when pressure and temperature approach the critical point of a fluid, where the difference between the properties of liquid and gas phases disappears. On the contrary, the heat of condensation in small mesopores was found to be significantly higher than the heat of condensation on a flat liquid surface.23-25 The heat of condensation is actually higher when the mesopores are smaller, an effect attributed to the interaction of the condensate with the pore walls26 or to the surface energy of the adsorbed film.27 NLDFT calculations indicate that the desorption branch corresponds to the equilibrium recession of a meniscus for mesopores larger than 5 nm.19,28,29 Can this be told with the same consistency in the case of the smaller (11) Nakanishi, H.; Fisher, M. E. J. Chem. Phys. 1983, 78, 3279. (12) Evans, R.; Marini Bettolo Marconi, U.; Tarazona, P. J. Chem. Phys. 1986, 84, 2376; 1987, 86, 7138. (13) Ball, P. C.; Evans, R. Langmuir 1989, 5, 714. (14) de Keizer, A.; Michalski, T.; Findenegg, G. H. Pure Appl. Chem. 1991, 63, 1495. (15) Peterson, B. K.; Walton, J. R. B.; Gubbins, K. E. J. Chem. Soc., Faraday Trans. 2 1986, 82, 1789. (16) Beck, J. S.; Vartuli, J. C.; Roth, W. J.; Leonowicz, M.E.; Kresge, C. T.; Schmitt, K. D.; Chu, C. T. W.; Olson, D. H.; Sheppard, E. W.; McCullen, S. B.; Higgins, J. B.; Schlenker, J. L. J. Am. Chem. Soc. 1992, 114, 10834. (17) Morishige, K.; Fujii, H.; Uga, M.; Kinukawa, D. Langmuir 1997, 13, 3494. (18) Rathousky, J.; Zukal, A.; Franke, O.; Schulz-Ekloff, G. J. Chem. Soc., Faraday Trans. 1994, 90, 2821. (19) Ravikovitch, P. I.; O’Domhnaill, S. C.; Neimark, A. V.; Schu¨th, F.; Unger, K. K. Langmuir 1995, 11, 4765. (20) Gross, S.; Findenegg, G. H. Ber. Bunsen-Ges. Phys. Chem. 1997, 101, 1726. (21) Morishige, K.; Shikimi, M. J. Chem. Phys. 1998, 108, 7821. (22) Machin, W. D. Langmuir 1999, 15, 169. (23) Rathousky, J.; Zukal, A.; Franke, O.; Schulz-Ekloff, G. J. Chem. Soc., Faraday Trans. 1995, 91, 937. (24) Ja¨nchen, J.; Stach, H.; Busio, M.; van Wolput, J. H. M. C. Thermochim. Acta 1998, 312, 33. (25) Qiao, S. Z.; Bhatia, S. K.; Nicholson, D. Langmuir 2004, 20, 389. (26) Neimark, A. V.; Ravikovitch, P. I.; Gru¨n, M.; Schu¨th, F.; Unger, K. K. J. Colloid Interface Sci. 1998, 207, 159. (27) Trens, P.; Tanchoux, N.; Maldonado, D.; Galarneau, A.; Di Renzo, F.; Fajula, F. New J. Chem. 2004, 28, 874.

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Figure 2. Corresponding state graph for the limit of reversible condensation for Ar (9), Xe (2), N2 (0), O2 (]), CO2 (4), benzene (×), cyclopentane (O), and 2,2-dimethylbutane (+) on several adsorbents.

mesopores in which adsorption-desorption cycles follow a reversible path or the absence of hysteresis loop for small mesopores has to be attributed to tensional instabilities of the liquid phase?30,31 The purpose of this communication is to investigate if macrothermodynamic methods and calorimetric measurements can shed some light on the nature and properties of the threshold of reversible condensation in mesoporous systems. Experimental Section tert-Butyl alcohol adsorption was carried out on a MCM-41 sample (surface area 700 m2 g-1, mesopore volume 0.56 cm3 g-1, pore diameter 3.0 nm, hexagonal cell size 5.2 nm) synthesized in the presence of cetyltrimethylammonium and calcined at 823 K in air flow.32 Powder X-ray diffraction (XRD) patterns were collected using a CGR Theˆta-60 diffractometer with monochromated Cu KR radiation. Textural properties have been determined by N2 sorption at 77 K in a Micromeritics ASAP 2000 apparatus. The lowest closure point of the hysteresis loop has been verified in the same apparatus by adsorbing nitrogen, oxygen, or argon at 77 and 87 K. Data for the adsorption of water on the same sample have already been published.32 Calorimetric and volumetric data were obtained by means of a Tian-Calvet microcalorimeter (Setaram) connected to a volumetric apparatus allowing simultaneous measurement of adsorbed amount (molar uptake, Na), heat released (Q), and equilibrium pressure (p) for small increments of tert-butyl alcohol dose to the solid sample previously outgassed for 2 h at 423 K.32-34 A second adsorption run was carried out after intermediate outgassing. The temperature of the calorimeter was maintained at 303 K throughout the adsorption experiment.

Results and Discussion Corresponding state graphs of the limit of reversibility for adsorption-desorption cycles are drawn in Figure 2 as ln(prpf/pc) vs (Tc/Trpf) curves for several fluids, the index rpf indicating the reversible pore filling. The experimental data from which the curves of Figure 2 are drawn are representative of adsorbate/adsorbent systems as different as argon on silica,17,35-38 xenon on silica,10,37,39 nitrogen on (28) Neimark, A. V.; Ravikovitch, P. I. Microporous Mesoporous Mater. 2001, 44, 697. (29) Kornev, K. G.; Shingareva, I. K.; Neimark, A. V. Adv. Colloid Interface Sci. 2002, 96, 143. (30) Coasne, B.; Grosman, A.; Ortega, C.; Simon, M. Phys. Rev. Lett. 2002, 88, 256102-1. (31) Sonwane, C. G.; Bhatia, S. K. Langmuir 1999, 15, 5347. (32) Cauvel, A.; Brunel, D.; Di Renzo, F.; Garrone, E.; Fubini, B. Langmuir 1997, 13, 2773. (33) Fubini, B. Thermochim. Acta 1988, 135, 19. (34) Fubini, B.; Bolis, V.; Cavenago, A.; Ugliengo, P. J. Chem. Soc., Faraday Trans. 1992, 88, 277. (35) Emmett, P.; Cines, M. J. Phys. Chem. I 1946, 56, 735. (36) Llewellyn, P. L.; Grillet, Y.; Schu¨th, F.; Reichert, H.; Unger, K. K. Microporous Mater. 1994, 3, 345.

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Table 1. Linear Correlation Coefficients for the ln(p0/pc) ) A + B[1/(T/Tc)] Plots of the Limit of Reversible Condensation for Several Adsorbates adsorbate

A

B

Ar Xe N2 O2 CO2 dimethylC4 C6H6 cycloC5

7.01 6.77 7.66 7.67 8.78 8.57 8.86 10.06

-6.86 -6.69 -7.39 -7.30 -8.55 -8.57 -8.61 -9.19

silica,2,17,18,38,40 zirconia,41 and magnesia,42 oxygen on silica,17,38,43,44 carbon dioxide on silica,10 benzene on carbon,10 cyclopentane on silica,23 and 2,2-dimethylbutane on carbon.10 In the case of a first-order phase transition, a ln(p°/pc) vs 1/(Tb/Tc) graph is usually linear over a very large field of temperature. Guggenheim has shown that, in the case of unconfined gas-liquid condensation, this kind of boiling point graph is linear also extremely close to the critical point, owing to a compensation between the deviation due to the nonideality of the vapor and the decrease of the condensation enthalpy near the critical point.45 He has also shown that the linear correlations can be extrapolated toward a point not far from the origin. However, the extrapolated straight lines are not expected to pass exactly through the origin because the assumptions of the Clausius-Clapeyron equation on the negligible molar volume of the liquid and the ideal gas state of the vapor become less and less valid when the critical point is approached.46 In the case of the limit of reversible condensation, the corresponding state graphs of Figure 2 are linear in a field of temperature going from nine tenths to less than half the critical temperature and can be extrapolated in the vicinity of the critical point. The coefficients of the least-squares linear regressions ln(p°/pc) ) A + B[1/(T/Tc)] are reported in Table 1. Slope B is steeper for fluids with stronger deviations from the corresponding state behavior. This is in good qualitative agreement with the behavior of the same fluids in nonconfined conditions. In the case of nonconfined fluids, the deviation from the corresponding state relationship can be quantified by the Pitzer acentric factor ω. This factor is defined by the logarithm of the reduced saturation pressure measured at a given reduced temperature, namely, ω ) -1 - log(pcond/pc)T)0.7Tc.47,48 In Figure 3, the slopes, B, for all adsorbates at the reversibility limit are compared to the Pitzer acentric factors for the corresponding nonconfined fluids. The linear correlations for argon and xenon, whose virtually null (37) Machin, W. D. Langmuir 1994, 10, 1235. (38) Authors’ data. (39) Machin, W. D.; Golding, P. D. J. Chem. Soc., Faraday Trans. 1990, 86, 171. (40) Joyner, L. G.; Emmett, P. J. Am. Chem. Soc. 1948, 70, 2359. (41) Gregg, S. J.; Langford, J. F. J. Chem. Soc., Faraday Trans. 1 1977, 73, 747. (42) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity, 2nd ed.; Academic Press: London 1982; 304 pp. (43) Barrer, R. M.; MacLeod, D. M. Trans. Faraday Soc. 1954, 50, 980. (44) Barrer, R. M.; McKenzie, N.; Reay, J. S. S. J. Colloid Sci. 1956, 11, 479. (45) Guggenheim, E. A. Thermodynamics, 5th ed.; North-Holland: Amsterdam, 1967; 390 pp. (46) Vidal, J. Thermodynamics; Technip: Paris, 2003; 492 pp. (47) Pitzer, K. S.; Lipmann, D. Z.; Curl, R. F.; Huggins, C. M.; Petersen, D. E. J. Am. Chem. Soc. 1955, 77, 3433. (48) Tester, J. W.; Modell, M. Thermodynamics and its applications, 3rd ed.; Prentice Hall: Upper Saddle River, NJ, 1997; 936 pp.

Figure 3. Relation between the slopes -B of the corresponding state graphs of Figure 2 and the Pitzer acentric factors for Ar (9), Xe (2), N2 (0), O2 (]), CO2 (4), benzene (×), cyclopentane (O), and 2,2-dimethylbutane (+).

acentric factors indicate an excellent agreement with the corresponding state principle, present the less steep slopes. Nitrogen and oxygen, still in fairly good agreement with the corresponding state law, present slightly steeper slopes. The other fluids, which feature larger deviations from the corresponding state law, present the steepest slopes. The corresponding state plots and the deviation from ideality of the reversible condensation thresholds correspond to the expected behavior for a first-order gasliquid phase transition. However, the threshold of reversible condensation differs from the boiling point of the unconfined fluid by a different slope of the reduced pressure-reduced temperature relationship. For instance, in the case of argon and xenon, the slope of the ln(p°/pc) vs 1/(T/Tc) graphs in nonconfined conditions is 5.31,45 to be compared with slopes 24-28% steeper for the ln(p°/prpf) vs (Tc/Trpf) plots during reversible condensation in a porous system. This difference is a manifestation of the peculiar enthalpy of reversible condensation, which can be better examined through a different representation of the reversibility threshold data of Figure 2. The energetics of a phase transition can be evaluated from the variation of temperature with pressure through the ClausiusClapeyron relation.45 The Clapeyron relation

dp/dT ) ∆Hrpf/T∆Vrpf

(1)

can only be used for isothermal transitions at constant pressure between equilibrated phases. The threshold of reversible condensation is far enough from the capillary critical point19,20 to allow for the use of the Clapeyron relation. If the molar volume of the liquid is neglected by comparison with the molar volume of the gas and the adsorptive is taken as a perfect one, the ClausiusClapeyron equation

d(ln prpf)/d(1/T) ) -∆Hrpf/R

(2)

can be written. The Clausius-Clapeyron plots for several adsorbates are reported in Figure 4. The slope of the linear correlation for each adsorbate allows determination of the enthalpy of pore filling at the threshold of reversibility through eq

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Figure 4. Clausius-Clapeyron graphs for the limit of reversible pore filling of Ar (9), Xe (2), N2 (0), O2 (]), CO2 (4), benzene (×), cyclopentane (O), and 2,2-dimethylbutane (+) on several adsorbents. Table 2. Pore-Filling Enthalpies at the Limit of Reversibility from the Clausius-Clapeyron Graphs for Several Adsorbates and Their Ratios with the Condensation Enthalpies

Ar Xe N2 O2 CO2 benzene cyclopentane 2,2-dimethylbutane

∆Hrpf (kJ mol-1)

∆Hcond (kJ mol-1)

∆Hrpf/ ∆Hcond

-8.6 -16.1 -7.8 -9.4 -21.6 -40.2 -39.1 -34.5

-6.7 -13.0 -5.6 -6.8 -17.2 -34.1 -29.0 -30.4

1.28 1.24 1.39 1.38 1.26 1.18 1.35 1.13

Figure 5. Adsorption isotherms of tert-butyl alcohol on MCM41 at 303 K: empty symbols, first run; filled symbols, second run. The adsorbed amount in the second run includes 0.5 molecules nm-2 not desorbed after the first run.

2. The pore-filling enthalpies calculated in this way are reported in Table 2, in which they are compared to the condensation enthalpies on a flat liquid surface. When the pore-filling enthalpies are evaluated through the Clausius-Clapeyron plots, their ratios with the condensation enthalpies ∆Hrpf/∆Hcond cover the range 1.13-1.39. The high condensation enthalpies observed in small mesopores have been usually attributed to the effects of the interaction with the pore walls.26 To verify at which point the formation of a monolayer can influence the evaluation of the condensation heat, the evolution of the adsorption enthalpy with coverage has been studied by calorimetric methods for the adsorpion of tert-butyl alcohol on MCM-41 silica. The isotherms of adsorption of tert-butyl alcohol vapor are reported in Figure 5. The isotherms present two wellseparated steps at low relative pressure and at about p/p0 0.2. The low-pressure step corresponds to the formation of a monolayer of tert-butyl alcohol. BET linearization of

Figure 6. Molar differential enthalpy for the adsorption of tert-butyl alcohol on MCM-41 at 303 K: empty symbols, first run; filled symbols, second run. Adsorbed amount as per Figure 5.

the isotherm indicates a monolayer volume of 2.03 mmol g-1, corresponding to 1.73 adsorbed molecules per nm2. The value of the CBET parameter is 465, a very high value which depends on the sharp step at the top of the monolayer adsorption and indicates a strong interaction between the tert-butyl alcohol molecules and the mesopore walls.42 A second step at p/p0 0.19-0.22 corresponds to the filling of mesopores and allows classification of the isotherm as type IV. After the first adsorption run, outgassing of the sample at 303 K for the allotted time did result in a partial evaporation of the adsorbed tert-butyl alcohol. Nearly 0.5 molecules nm-2 from the first run were still adsorbed at the beginning of the second run. The second adsorption run perfectly reproduces the first one, provided that this initial adsorbed amount is taken into account. The differential adsorption heat as a function of the adsorbed amount of tert-butyl alcohol is reported in Figure 6. The high initial heat of adsorption for the first dose (more than 100 kJ mol-1) corresponds to a limited number of highly reactive sites, probably corresponding to surface defects. The formation of the monolayer (coverage up to 1.7 molecule nm-2) develops a differential adsorption heat gradually decreasing from 83 to 73 kJ mol-1. Once the monolayer is completed, the adsorption heat suddenly decreases to a value of 52 kJ mol-1 and remains constant at this level throughout the step of the type IV isotherm (from 2 to 3.4 molecules nm-2). Once the mesopores are filled, adsorption takes place only at the outer surface and the adsorption heat rapidly decreases to the value of the condensation heat on a flat liquid surface, 43.6 kJ mol-1. The results of the second adsorption run are in excellent agreement with the results of the first run. The pore-filling step, which begins at p/p0 0.19, is well separated from the monolayer formation, which is completed about p/p0 0.04. It seems unlikely that the measured values of the pore-filling enthalpy can be affected by any direct interaction with the pore walls. However, the observed pore-filling enthalpy ∆Hpf is significantly higher than the condensation enthalpy on a flat surface ∆Hcond. The ratio ∆Hpf/∆Hcond is 1.20. Similar results have already been observed for the adsorption heat of other molecules in adsorbents with small mesopores. In Table 3, literature data are collected for the adsorption enthalpies of argon,26 cyclopentane,23 acetonitrile,24 and n-hexane24,27 in micelle-templated mesoporous silica. In all cases, a plateau of adsorption heat was observed in correspondence with the filling of mesopores. The values of this plateau for all adsorbates, reported as ∆Hpf in Table 3, correspond to ∆Hpf/∆Hcond ratios between 1.16 and 1.24.

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Table 3. Pore-Filling Enthalpies in MCM-41 for Several Adsorbents and Their Ratios with the Condensation Enthalpies adsorbate

ref

method

∆Hpf (kJ/mol)

∆Hcond (kJ/mol)

∆Hpf/ ∆Hcond

argon tert-butanol acetonitrile cyclopentane n-hexane n-hexane

26 a 24 23 24 27

isosteric calorimetric calorimetric isosteric calorimetric isosteric

-8.3 -52 -37 -35 -38 -38

-6.7 -43.6 -32 -29 -31.8 -31.8

1.24 1.20 1.16 1.21 1.20 1.20

a

This work.

A direct comparison between experimental pore-filling enthalpies (Table 3) and pore-filling enthalpies evaluated by the Clausius-Clapeyron equation (Table 2) is available in the cases of argon and cyclopentane. In the case of argon, the two values of the pore-filling enthalpy only differ by 0.3 kJ mol-1, a shift well inside the error bar of the isosteric method. In the case of cyclopentane, the two values of the pore-filling enthalpy differ by 4 kJ mol-1, a more significant figure. However, the experimental enthalpic data are in reasonable agreement with the ∆Hrpf values as determined by the Clausius-Clapeyon equation. If the corresponding states treatment indicates that the reversible pore filling behaves as a first-order phase transition, the enthalpic data indicate that the heat of this phase transition is nearly 20% higher than the condensation heat of the corresponding unconfined fluid. This value of the ∆Hpf/∆Hcond ratio appears to be an inherent property of the limit of reversible pore filling (rpf limit). It has been shown that the condensation enthalpy increases with the confinement in smaller pores.26 This effect has been attributed to an increase of the energy content of the adsorbed layer as the pore curvature increases.27 When capillary condensation takes place, this excess energy is released and the condensation heat is correspondingly raised. The condensation heat at the rpf limit (∆Hpf ≈ 1.20∆Hcond) corresponds to a critical value of curvature in this correlation between pore size and condensation heat.27 The nucleation of the condensed phase in a mesopore has been shown to proceed via a symmetry transition in which bumps of the adsorbed layer transform into lenses through the pore.29,49,50 In fact, the high energy of the adsorbed (49) Saam, W. F.; Cole, M. W. Phys. Rev. B 1975, 11, 1086.

layer lowers the pressure at which capillary forces overcome the adhesion forces. If the energy barrier for nucleation is sufficiently lowered, the capillary condensation can take place at the same equilibrium pressure at which desorption will occur, justifying the absence of any hysteresis loop.51 Conclusions The limit of reversible pore filling (rpf limit) has been empirically defined as the relative pressure level below which adsorption and desorption in a mesoporous system follow the same path. The attempts to identify the rpf limit with the critical point of the confined fluid have failed. Nevertheless, the rpf limit can be defined on the basis of macrothermodynamic properties. Its dependence on temperature follows corresponding state relations and strongly suggests that it has to be considered as a first-order gasliquid transition in the confined environment of the mesopore. The data on which this conclusion is drawn seems quite general, applying to adsorbates as different as argon and dimethylbenzene. The nature of the adsorbent seems not to be a determining factor, experimental data on several adsorbents (silica, zirconia, magnesia, and carbon) having been used. An important property of the rpf limit is a value of transition enthalpy ca. 20% higher than the condensation enthalpy on a flat liquid surface. It seems to correspond to a given threshold in the continuous evolution of adsorption enthalpy with confinement.26,27 The rpf limit can be defined as the threshold of confinement beyond which the properties of the adsorbed layer create an infinite probability of formation of bridges between the layers adsorbed on the opposite walls of the pore.52 In this way, adsorption behaves no more as an activated process and is superposed to the equilibrium desorption curve.51 It is also significant that a macrothermodynamic treatment can provide useful information on the condensation in pores not larger than 10 molecular sizes, a domain usually reserved for statistical thermodynamics.22 LA0507838 (50) Crassous, J.; Charlaix, E.; Loubet, J. L. Europhys. Lett. 1994, 28, 37. (51) Thommes, M.; Ko¨hn, R.; Fro¨ba, M. Appl. Surf. Sci. 2002, 196, 239. (52) Gelb, L. D.; Gubbins, K. E.; Radhakrishnan, R.; SliwinskaBartkowiak, M. Rep. Prog. Phys. 1999, 62, 1573.