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Langmuir 2002, 18, 9726-9734
Experimental Observation of Critical Depletion: Nitrous Oxide Adsorption on Silica Gel Arvind Rajendran, Thomas Hocker, Orazio Di Giovanni, and Marco Mazzotti* ETH Swiss Federal Institute of Technology Zurich, Institut fu¨ r Verfahrenstechnik, Sonneggstrasse 3, CH-8092 Zu¨ rich, Switzerland Received March 4, 2002. In Final Form: October 1, 2002 The adsorption of nitrous oxide (N2O) on silica gel has been studied using a gravimetric apparatus under near critical conditions. In gravimetric measurements, the determination of the excess amount adsorbed depends on a buoyancy correction factor which accounts for the volumes of the adsorbent and the solid parts in the cell. The accurate measurement of the volume of the sorbent and the metal parts in the measuring cell plays an important role in the precise measurement of the excess adsorbed amount especially at high densities. To this aim, a new protocol has been proposed which accounts for non-negligible helium adsorption and thermal expansion of the sorbent and the solid parts of the balance. Helium isochores under moderate temperature and densities have been experimentally measured. The adsorption behavior of helium has been characterized using a nonlinear mobile adsorption model. The proposed protocol has then been applied for the measurement of the excess adsorbed amount of supercritical N2O on silica gel very close to the critical point. Clear experimental evidence of critical depletion, i.e., the decrease of the excess adsorption along an isochore with decreasing temperature, has been obtained. Unlike the previous observation, where critical depletion was seen only in the direct vicinity of the critical isochore, in the present study this is observed on isochores in the range of 0.74 e Fr e 1.15.
1. Introduction The behavior of fluids in pores has received widespread attention in recent years.1 Data from adsorption experiments have been correlated to models based on statistical mechanics to understand the microscopic picture of fluid behavior in pores. In these experiments the truly measurable quantity is the adsorption excess, which for a flat, homogeneous surface is defined as
Γ(Fb,T) ) A
∫0∞ (F(z) - Fb) dz
(1)
where A is the area of the sorbent surface, F(z) is the mass density profile along the axis perpendicular to the sorbent surface, and Fb is the mass density in the bulk.2 Excess isotherms at temperatures well above the critical temperature exhibit shapes that initially rise with density, reach a maximum, and then fall off with it.3 The shape of the isotherms is determined, among other factors, by the pore size distribution of the sorbent. A description of typical isotherms at different supercritical temperatures on sorbents of different pore sizes is given elsewhere.4 Under near-critical conditions, excess isotherms exhibit peculiar behavior as summarized in the following. For a flat, open surface (i.e., a semi-infinite system) Fischer and de Gennes showed that as the critical temperature is approached from above when moving along the critical isochore (Fr ) Fb/Fc ) 1.0), the correlation length goes to * To whom correspondence should be addressed. Phone: +41-1-632 2456. Fax: +41-1-632 1141. E-mail: mazzotti@ ivuk.mavt.ethz.ch. (1) Gelb, L. D.; Gubbins, K. E.; Radhakrishnan, R.; SliwinskaBartkowiak, M. Rep. Prog. Phys. 1999, 62, 1573-1659. (2) Sircar, S. J. Chem. Soc., Faraday Trans. 1 1985, 81, 1527-1540. (3) Specovious, J.; Findenegg, G. H. Ber. Bunsen-Ges. Phys. Chem. 1980, 84, 690-696. (4) Di Giovanni, O.; Hocker, T.; Rajendran, A.; Do¨rfler, W.; Mazzotti, M.; Morbidelli, M. Measuring and describing adsorption under supercritical conditions. In Fundamentals of Adsorption; Kaneko, K., Kanoh, H., Hanzawa, Y., Eds.; IK International: Chiba, Japan, 2001; Vol. FOA7.
infinity and the adsorption excess diverges according to the following functional dependence
Γ ∝ t-x
(2)
where t ) (T - Tc)/Tc and x is a critical exponent, x ≈ 0.30.5 This phenomenon, termed “critical adsorption”, has been confirmed by experiments.3 Findenegg and co-workers studied the adsorption of SF6 on graphitized carbon black (Vulcan 3-G) and controlled pore glass (CPG-10) with a mean pore size of 31 nm.6 They found that along the critical isochore, the expected increase of Γ with decreasing temperature is shown only up to about T - Tc ) 3 K. Moving closer to the critical temperature produced a sharp unexpected fall of the excess isochore, whereas noncritical isochores (both above and below the critical density) exhibited the expected behavior. This phenomenon was called “critical depletion” and has been the subject of several theoretical studies.7-11 However, theoretical approaches have not yet yielded conclusive evidence or explanation of the critical depletion phenomenon. Since all modeling efforts dealt with single, idealized pores so far, it has been conjectured that critical depletion might arise due to the presence of a network of interconnected pores.1 From the experimental point of view, these first observations were confirmed by further data about the adsorption of SF6 on CPG-350 with a mean pore diameter of 31.3 nm10 and on CPG-100 with a mean (5) Fisher, M. E.; de Gennes, P. C. R. Acad. Sci. Ser. B 1978, 287, 207-209. (6) Thommes, M.; Findenegg, G. H.; Lewandowski, H. Ber. BunsenGes. Phys. Chem. 1994, 98, 477-481. (7) Maciolek, A.; Ciach, A.; Evans, R. J. Chem. Phys. 1998, 108, 97659774. (8) Maciolek, A.; Evans, R.; Wilding, N. B. Phys. Rev. E 1999, 60, 7105-7119. (9) Schoen, M.; Thommes, M. Phys. Rev. E 1995, 52, 6375-6386. (10) Thommes, M.; Findenegg, G. H.; Schoen, M. Langmuir 1995, 11, 2137-2142. (11) Wilding, N. B.; Schoen, M. Phys. Rev. E 1999, 60, 1081-1083.
10.1021/la025696j CCC: $22.00 © 2002 American Chemical Society Published on Web 11/05/2002
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Langmuir, Vol. 18, No. 25, 2002 9727
pore diameter of 10.0 nm.12 In this work, we explore highpressure, near-critical adsorption behavior of a new system, namely, nitrous oxide, N2O, on silica gel, thus providing further experimental evidence of the critical depletion phenomenon. The presented data for the adsorption excess is based on a new protocol to accurately determine the volume of all solid parts inside the measuring cell, V0. Since in gravimetric measurements the excess amount adsorbed depends on the “buoyancy correction” term, FbV0, the accuracy with which V0 needs to be obtained generally depends on the densities at which the adsorption experiments are carried out. For example, when operating at low values of Fb, a rather rough estimate of V0 might be appropriate, whereas at very high bulk densities, small changes in V0 will have a large effect on the excess. The most common approachswhich will be adopted here as wellsis to estimate V0 from adsorption measurements using some weakly adsorbing reference material such as helium. Until recently it was assumed that helium adsorption could be neglected at the temperature at which V0 was estimated. However, as recently shown by Sircar, this assumption breaks down at higher densities.13,14 Thus, a new expression for V0 is presented which takes into account nonnegligible adsorption of helium (as the reference material) in the nonlinear range of the adsorption isotherm, as well as thermal expansion of the various solid parts inside the measuring cell. 2. Experimental Section 2.1. Materials. Silica gel (Kieselgel 60, 0.063-0.20 mm, Batch #1.07734.1000) was obtained from Merck KGaA (Darmstadt, Germany). The adsorbent was regenerated at 150 °C before being used in the experiments. The adsorbates, N2O (99.0% purity) and He (99.999% purity), were obtained from PanGas AG (Luzern, Switzerland). The critical properties of the adsorbates are as follows: Tc(He) ) 5.26 K, Pc(He) ) 2.26 × 105 Pa, Fc(He) ) 69.3 kg/m3, Tc(N2O) ) 309.6 K, Pc(N2O) ) 72.2 × 105 Pa, and Fc(N2O) ) 453.2 kg/m3. 2.2. Setup. In the present study a Rubotherm magnetic suspension balance with an absolute accuracy of 0.01 mg was used.15,16 The balance consists of a permanent magnet to which a basket containing the sorbent and a titanium sinker element (whose volume is calibrated) are attached. This permanent magnet is magnetically coupled to an electric magnet (located outside the measuring cell) which is connected to the control system. The distance between the two magnets is a measure of the weight of the system. The balance readings are obtained at two balance positions. In position “1”, the basket alone is lifted while the sinker is at rest (i.e., position 1 measures the weights of the permanent magnet, the metal suspension, the basket, and the sorbent material). In position “2”, both the basket and the sinker are lifted. Consequently, at given bulk conditions (Fb,T), the density of the (pure) bulk fluid inside the measuring cell follows from the difference between the balance readings in position 1, M 1(Fb,T), and position 2, M 2(Fb,T)
Fb )
msinker + M 1(Fb,T) - M 2(Fb,T) Vsinker
(3)
where msinker and Vsinker are the known sinker mass and sinker volume, respectively. Note that the ability to directly measure (12) Machin, W. D. Langmuir 1999, 15, 169-173. (13) Sircar, S. AIChE J. 2001, 47, 1169-1176. (14) Sircar, S. Role of helium void measurement in estimation of Gibbsian surface excess. In Fundamentals of Adsorption; Kaneko, K., Kanoh, H., Hanzawa, Y., Eds.; IK International: Chiba, Japan, 2001; Vol. FOA7. (15) Dreisbach, F.; Staudt, R.; Tomalla, M.; Keller, J. U. In Fundamentals of Adsorption; Meunier, F., Ed.; Elsevier: Paris, 1996; FOA6. (16) Di Giovanni, O.; Do¨rfler, W.; Mazzotti, M.; Morbidelli, M. Langmuir 2001, 17, 4316-4321.
the density of the bulk fluid is especially important when operating the balance under near-critical conditions where small changes in pressure result in large density changes. Consequently, calculating the fluid density from pressure measurements using an equation of state can lead to a serious amplification of the measurement error.
3. Procedure for Gravimetric Adsorption Measurements 3.1. Excess Adsorption from Gravimetric Measurements. The mass of the sorbent material plus all metal parts inside the cell (permanent magnet, metal suspension, basket), m0, is measured at high temperatures and very low densities (vacuum), i.e.
m0 ) M 1(Fbf0,Tf∞) ≡ M 1(0,∞)
(4)
With eq 4 the adsorbed mass at bulk conditions (Fb,T) follows as
ms ) M 1(Fb,T) - M 1(0,∞) + FbVbuo
(5)
Here, Vbuo represents the “buoyancy volume” which is the sum of the volumes of the sorbent material and the metal parts
V0 ≡ Vsorb + Vmet
(6)
plus the volume of the adsorbed fluid “attached” to the sorbent, Vs, i.e.
Vbuo ) V0 + Vs
(7)
Using eq 7, eq 5 becomes
ms ) M 1(Fb,T) - M 1(0,∞) + Fb(V0 + Vs)
(8)
Note, however, that Vs on the right-hand side of eq 8 neither can be measured nor can it be estimated with sufficient accuracy from theory (i.e., from a microscopic model of the adsorbed phase) at conditions where multilayer adsorption occurs (i.e., at rather high bulk densities and rather low temperatures).17 Instead, the excess amount adsorbed (in units of mass of adsorbate), Γ, is considered
Γ(Fb,T) ≡ ms - FbVs ) M 1(Fb,T) - M 1(0,∞) + FbV0 (9) The only further unknown in eq 9 is the volume of the solid parts inside the measuring cell, V0, as given by eq 6. Since in eq 9 it is multiplied by the bulk density, the accuracy with which V0 needs to be obtained generally depends on the densities at which the adsorption experiments are carried out. 3.2. Accurate Description of the Volume of the Solid Parts, V0. The volume of the sorbent material plus all metal parts, V0, is a function of temperature when thermal expansion is significant. In this case eq 6 can be written as
V0(T) ) Vsorb(T*) exp[3Rsorb(T - T*)] + mmet exp[3Rmet(T - T*)] (10) met F (T*) where T* represents some (arbitrary) reference temperature, whereas Rsorb and Rmet are the linear expansion coefficients of the sorbent material and the metal parts, (17) Sircar, S. Ind. Eng. Chem. Res. 1999, 38, 3670-3682.
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respectively. Note also that the volume of the metal parts at T ) T* has been written as Vmet(T*) ) mmet/Fmet(T*). Here the mass of the metal parts, mmet, follows from gravimetric measurements (with empty basket) in the limit where (Fb f 0, T f ∞), whereas the metal density, Fmet(T*), can be obtained from standard tables. Note also that eq 10 can be readily adapted in case different types of metals with different expansion coefficients are present in the measuring cell. To proceed further, the sorbent volume at the reference temperature, Vsorb(T*), needs to be determined. This can be done, for example, by using a weakly adsorbing reference substance such as helium and by performing adsorption measurements in some reference state (FHeb*,T*). According to eq 6, Vsorb(T*) then follows as
Vsorb(T*) ) V0(T*) - Vmet(T*) ) ΓHe + M 1(0,∞) - M 1(FHeb*,T*) FHeb*
-
mmet (11) F (T*)
V (T) )
ΓHe(FHeb*,T*) + M 1(0,∞) - M 1(FHeb*,T*) FHeb*
exp[3Rsorb(T - T*)] +
×
mmet {exp[3Rmet(T - T*)] F (T*) exp[3Rsorb(T - T*)]} (12) met
Finally, by substituting eq 12 into eq 9, the excess adsorption can be recast as
Γ ) M 1(Fb,T) - M 1(0,∞) + Fbmmet {exp[3Rmet(T - T*)] Fmet(T*) Fb exp[3Rsorb(T - T*)]} + [ΓHe(FHeb*,T*) + b FHe * M 1(0,∞) - M 1(FHeb*,T*)] exp[3Rsorb(T - T*)] (13) Besides material properties such as the thermal expansion coefficients of the various solid parts inside the measuring cell, the only further unknown in eq 13 is the excess amount adsorbed of the reference material (i.e., helium) under reference conditions, ΓHe(FHeb*,T*). Usually, the following two assumptions are made to simplify eqs 12 and 13. First, one assumes that the excess amount of adsorbed helium at (FHeb*,T*) is so small that ΓHe(FHeb*,T*) ≈ 0. Second, one neglects the change in V0 caused by thermal expansion of the solid parts due to the difference between the reference temperature T* and the temperature at which the actual adsorption measurements are carried out. Under these two assumptions, eq 12 simplifies to
V0 ≈
M 1(0,∞) - M 1(FHeb*,∞) FHeb*
(14)
Γ ≈ M 1(Fb,T) - M 1(0,∞) + [M1(0,∞) - M
FHeb*
To estimate V0, consider a series of isochoric helium adsorption measurements, i.e., a series of values of M 1(FHeb,T). According to eq 9
V0(T) -
ΓHe(FHeb,T) FHeb
)
M 1(0,∞) - M 1(FHeb,T) FHeb
(16)
i.e., the right-hand side of eq 16 contains only quantities that can be directly measured in a gravimetric adsorption device. The idea is now to use a physically sound description of the left-hand side of eq 16 to correlate V0(T) with the He adsorption data. Note that the functional form of V0(T) is already given by eq 10, whereas the ratio of the adsorption excess of helium to its bulk density, ΓHe(FHeb,T)/FHeb, needs to be determined from an adsorption model. Fortunately, since helium is an “ideal” substance from a modeling viewpoint and since we are measuring helium adsorption at rather small densities and moderate temperatures, a rather simple monolayer adsorption model will suffice. The model is based on an adsorbed phase which has the following characteristics: Suppose we have a flat, homogeneous surface of area A available for adsorption. In addition, we assume that each adsorbate can have a maximum of z2D ) 4 adsorbates as nearest neighbors. This implies that each adsorbate occupies an area of rj2, where rj is the intermolecular distance between two neighboring adsorbed atoms. At rather low bulk densities, rj can be identified with the distance for which the adsorbate-adsorbate pairwise interaction energy has its minimum. Adopting, for example, the Lennard-Jones potential yields, rj ) σ21/6, with σ being the Lennard-Jones diameter.18 Since adsorption shall be restricted to the first fluid layer adjacent to the surface, the adsorbed phase has a thickness of rj and a volume of Vs ) Arj. It follows directly from the assumptions made above that the maximum possible number of adsorbed atoms is Nmaxs ) A/rj2, which allows us to define the maximum adsorbedphase density as
Fmaxs )
Nmaxsm V
s
)
m rj3
(17)
with m being the mass of an adsorbate atom and the fractional surface coverage as
θ)
so that substituting eq 14 into eq 9 yields
Fb
4. Estimation of V0 from Helium Adsorption Measurements
met
so that insertion of eq 11 into eq 10 results in 0
As already mentioned in section 3.1, it is usually safe to make the above assumptions when operating at rather low values of Fb. However, since at high bulk densities small changes in V0 will have a large effect on Γ, there it becomes crucial to carefully check the assumptions behind the estimated value for V0. In the following, a new method is presented to estimate V0 from eq 12, i.e., to account for non-negligible helium adsorption as well as for thermal expansion effects. This method correlates a physically sound adsorption model with a small number of isochoric helium adsorption measurements carried out under rather moderate temperatures and densities.
Fs
) s
Fmax
Ns A/rj2
(18)
From eqs 17 and 18, the mass density of the adsorbedb 1(FHe *,∞)]
(15)
(18) Hill, T. L. An Introduction to Statistical Thermodynamics; Dover: New York, 1986.
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Langmuir, Vol. 18, No. 25, 2002 9729
layer, Fs, is given by
Fs ) θFmaxs ) θm/rj3
(19)
Note that in the literature, models based on monolayer adsorption assumption are written in terms of “absolute adsorption”. Therefore, before applying the above model to experimental data, first it has to be recast in terms of excess adsorption, i.e., Γ ) Γ(Fb,T). To this aim, let us consider again eq 1, which simplifies for monolayer adsorption to b
s
b
Γ(F ,T) ) Arj(F - F )
(20)
Substituting eq 19 into eq 20 then leads to
Γ(Fb,T) b
F
(
) Arj
m θ -1 rj3 Fb
)
(21)
which is applicable to any monolayer adsorption model where θ ) θ(Fb,T). Since helium has an extremely low critical temperature of Tc(He) ) 5.26 K, it is reasonable to choose a model based on the microscopic picture of “mobile adsorption”.19 Thus, adsorbed He atoms are assumed to vibrate with a certain bonding frequency in the direction perpendicular to the surface but to move freely in directions parallel to it.20 Therefore, the adsorption behavior of helium at temperatures well above Tc(He) ) 5.26 K can be described using an equation of state for a two-dimensional fluid with additional terms accounting for adsorbate-surface interactions, as well as for the “frustration” of the translational degree of freedom perpendicular to the sorbent surface. The Hill-de Boer model20-22 offers a simple equation that accounts for mobile adsorption in a monomolecular layer
bm(T)Fb )
θ θ exp 1-θ 1-θ
(
)
(22)
Taking into account adsorbate-surface, Ufs, as well as pairwise adsorbate-adsorbate, ff, interactions, from statistical mechanics the coefficient bm(T) in units of volume per mass adsorbate can be written as
bm(T) )
jr2kT/(hνz) m(2πmkT/h2)
(
exp 1/2
)
Ufs + z2Dffθ kT
(23)
Here, νz is the bonding frequency with which an atom vibrates in the direction perpendicular to the surface and h is the Planck constant. For given bulk conditions, eq 22 can now be used to calculate θ which, when inserted into eq 21, specifies the term ΓHe(FHeb,T)/FHeb to be used in eq 16. Taking also into account eq 10, then, the functional form of the left-hand side of eq 16 is fully determined and can be correlated with experimental data. As a first check of the above derived adsorption model, let us compare it with the Langmuir model that is based on the concept of “localized adsorption”. In localized adsorption molecules are assumed to be “trapped” in local minima of the potential energy exerted by the sorbent walls; i.e., upon adsorption all three translational degrees of freedom of atoms in the gas phase are transformed into (19) Hill, T. L. J. Chem. Phys. 1946, 14, 441-453. (20) Hill, T. L. Adv. Catal. 1952, 4, 211-258. (21) de Boer, J. H. The Dynamical Character of Adsorption, 2 ed.; Clarendon Press: Oxford, 1968. (22) Al-Muhtaseb, S. A.; Ritter, J. A. J. Phys. Chem. B 1999, 103, 8104-8115.
vibrational modes. Of course this picture is justified only at rather low temperatures and/or for highly attractive fluid-wall interactions. The Langmuir model, when derived from statistical mechanics, can be written as
bl(T)Fb )
θ 1-θ
(24)
with bl(T) given by18
bl(T) )
(kT)3/(hνz)3 2 3/2
m(2πmkT/h )
( )
exp
Ufs kT
(25)
Note that the density dependence of adsorption isotherms predicted from models based on localized adsorption often is similar to those based on mobile adsorption. With respect to the temperature dependence of Γ(Fb,T), however, for very weakly adsorbing substances such as helium a model such as the Langmuir isotherm might even give physically meaningless results at ambient conditions. This is illustrated in Figure 1, where the T dependence of the ratio of the excess adsorption to the bulk density, Γ/Fb, is plotted for four different bulk densities. Shown in Figure 1a are the predictions of Langmuirian adsorption based on eq 24 in combination with eq 25. These results are compared with those based on the mobile adsorption model, eqs 22 and 23, shown in Figure 1b. To simulate the expected behavior for a weakly adsorbing substance such as helium a very small fluid-wall interaction energy of Ufs/k ) -325 K has been assumed (where the minus sign denotes attraction). All other parameters are given in Table 1. In both diagrams the solid lines represent Henry’s law limits where Γ/Fb|Fbf0 ) H(T). While the isochores based on mobile adsorption show the expected behavior and decrease monotonically with increasing temperature, those based on localized adsorption show the exact opposite trend which is obviously physically meaningless. For the given situation, the localized adsorption model predicts the right trend (i.e., decreasing Γ/Fb|Fb with increasing T) only when |Ufs/k| > 700 K. In Figure 2, isochoric data for the excess adsorption of helium on Kieselgel 60 (shown as symbols) are compared with modeling results based on substituting eqs 10, 22, and 21 into eq 16. The parameters and the material properties used to correlate the model with the experimental data are given in Table 1. Besides V0(T*), the fluidwall interaction energy, Ufs, as well as the bonding frequency of adsorbed helium atoms, νz, have been treated as adjustable parameters to fit the data. However, the assumed values for Ufs and νz lie within a physically meaningful range (see Appendix A for details). Note also that the obtained volume of the solid parts, V0(300 K) ≈ 2.935 mL, is rather insensitive with respect to the value used for the specific surface area of the sorbent, asorb ) A/msorb. To investigate this point, we varied asorb between 400 and 600 m2/g (the specific surface area of Kieselgel 60 provided by the supplier is 500 m2/g). To maintain a best fit to the experimental data, variations in asorb by (20% only required an adjustment of Ufs by (7% but left the fitted values for V0(T*) and νz unchanged. On the other hand, the predicted value for V0(T*) would be overestimated by about 10% if a model based on absolute adsorption rather than on excess adsorption were used to fit the data. This observation stresses the importance of distinguishing between absolute and excess adsorption for weakly adsorbing substances such as helium. As expected from a rather simple adsorption model such as eq 22, the agreement with the experiments is good at the rather low density of FHeb ) 1.435 mol/L but becomes
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Figure 1. Temperature dependence of the ratio of the adsorption excess to the bulk density, Γ/Fb, for four different bulk densities. Shown in part a are the predictions of Langmuirian adsorption based on eq 24 in combination with eq 25. These results are compared with those based on the mobile adsorption model, eqs 22 and 23, shown in part b. In both models a very small fluid-wall interaction energy of Ufs/k ) -325 K has been assumed (where the minus sign denotes attraction).
Figure 2. Comparison of isochoric data for the excess adsorption of helium on silica gel with modeling results that account for mobile adsorption of helium as well as for thermal expansion of the solid parts inside the measuring cell: (0), FHeb ) 3.745 mol/L; (4), FHeb ) 2.785 mol/L; (O), FHeb ) 1.435 mol/L; (- ‚ -), FHeb f 0; (- - -), V0(T). The last curve shows that V0(310 K) ≈ 2.935 mL. Table 1. Model Parameters and Material Properties Used To Correlate the Model Based on Inserting Equations 10, 22, and 21 into Equation 16 with Isochoric Data for the Excess Adsorption of Helium on Kieselgel 60 model parameters
adsorbate properties
z2D ) 4 Ufs/k ) -325 K νz ) 1.90 × 1012 Hz V0(310 K) ≈ 2.935 mL
jr(He) ) σ(He)21/6 ) 2.869 Å (ref 18) ff/k ) -10.22 K (ref 18) Mw(He) ) 4 g/mol
sorbent properties ) 2.18459 g Rsorb ) 11 × 10-6 K-1 A ) msorbasorb ) 1092 m2 (msorb ) 2.18459 g, asorb ) 500 m2/g)
msorb
a
Figure 3. Supercritical isotherms for the excess adsorption of N2O on Kieselgel 60 as a function of the reduced bulk density, Fb/Fc. The symbols represent experimentally measured values and are connected by lines to guide the eye. Note that for a density range 0.74 e Fb/Fc e 1.15, nex is larger at T/Tc ) 1.0145, compared to T/Tc ) 1.0077, giving evidence of the “critical depletion” phenomenon.
metal parts in cell ) 14.7597 g Rmet ) 10-6 K-1 (footnote a)
mmet
Fmet(300 K) ) 7.65 kg/m3
Fabian, J.; Allen, P. B. Phys. Rev. Lett. 1997, 79, 1885-1888.
less satisfying at the higher bulk densities. Also shown is the isochore in the limit FHeb f 0, which represents the Henry’s law region, i.e., Γ(Fb,T)/Fb|Fbf0 ) H(T). Assuming that the model predictions are accurate at small densities, one sees that the experimental data clearly lie outside the region where Henry’s law is valid, i.e., where a helium adsorption protocol such as the one proposed by Sircar could be applied.13,14 As mentioned above, when using the model to extrapolate the adsorption data to temperatures where ΓHe(FHeb,T)/FHeb ≈ 0, a value of V0(310 K) ≈ 2.935 mL is predicted. This value is about 3.5% higher than what one would obtain from neglecting He adsorption at 410 K (neglecting He adsorption at 310 K would result in
Figure 4. Supercritical isotherms for the excess adsorption of N2O on Kieselgel 60 as a function of the reduced bulk density, Fb/Fc. The symbols represent experimentally measured values and are connected by lines to guide the eye. However, since the isotherm at T/Tc ) 1.0042 does not have any experimental points around the critical density, only the symbols are shown. For a density range 0.74 e Fb/Fc e 1.15, isotherms at higher temperatures lie above those at lower temperatures suggesting the occurrence of “critical depletion”.
an underestimation of V0 as large as 4.7%). Note also that the thermal expansion of the solid parts inside the measuring cell causes only a weak temperature effect on V0 due to the small values for Rsorb and Rmet. For example, a temperature increase of 100 K gives V0(410 K) ≈ 2.942
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Langmuir, Vol. 18, No. 25, 2002 9731
Figure 5. Supercritical isochores, extracted from isotherms shown in Figures 3 and 4, for the excess adsorption of N2O on Kieselgel 60 as a function of, t ) (T - Tc)/Tc. For a density range 0.74 e Fb/Fc e 1.15, the “critical depletion” phenomenon occurs. Table 2. Absolute Values of the Interaction Parameters, E/k (in K) Used for the Calculation of the Fluid-Solid Interaction Parameter through the Application of the L-B Rule parameter/k
value (K)
source
Ufs*
-1207.0
ff* ff Ufs
-190.0 -10.2 -280.0
estimated from isosteric heat data for system reported16 a a calculated using the L-B rule
a Bird, B.; Stewart, W.; Lightfoot, E. Transport Phenomena; John Wiley & Sons: New York, 1964.
mL, which corresponds to an increase in V0 of only about 0.2%. 5. Supercritical Adsorption of Nitrous Oxide on Kieselgel 60 In Figures 3 and 4, the excess adsorbed amount of N2O on Kieselgel 60, nex (in millimoles of adsorbate per gram of sorbent), is plotted against the reduced density, Fr ) Fb/Fc, accounting for helium adsorption and thermal expansion. The symbols represent experimentally measured values and are connected by lines to guide the eye. However, since the isotherm at T/Tc ) 1.0042 does not have any experimental points around the critical density, only the symbols are shown. In Figure 3, the isotherms correspond to reduced temperatures, T/Tc, from 1.503 to 1.0077. The isotherms, which are well above the critical temperature, exhibit a behavior that is typical of supercritical isotherms: they rise with the density, reach a maximum, and then fall off with it, thus showing no crossover with other isotherms. The isotherms at higher temperatures show a near-linear fall on their descending branch. On the contrary, the isotherm corresponding to a temperature of T/Tc ) 1.0145 slightly deviates from this pattern of behavior, especially closer to the critical density (Fr ) 1.0). This effect is further pronounced in the isotherm at T/Tc ) 1.0077, which crosses the isotherm at T/Tc ) 1.0145 at a reduced density of about 0.74. For 0.74 e Fr e 1.15, the isotherm at T/Tc ) 1.0145 yields a smaller excess adsorbed amount than the one at T/Tc ) 1.0077. As shown in Figure 4, this effect becomes even more pronounced when moving closer to the critical temperature. In this figure, isotherms at lower temperatures are plotted along with three isotherms (T/Tc ) 1.087, 1.0145,
Table 3. Corrected Experimental Excess Adsorption Data of N2O on Silica Gel F nex F nex F nex T (K) (g/L) (mmol/g) T (K) (g/L) (mmol/g) T (K) (g/L) (mmol/g) 310.9 140 228 248 310 329 343 357 364 370 532 558 611 630 752 311.4 43 109 141 189 225 253 278 310 319 335 360 367 391 395 430 472 474 518 524 539 592 665 673 750 837 926 940
6.38 8.26 8.41 7.77 7.21 6.52 5.24 4.65 4.16 4.12 4.01 3.65 3.50 2.70 3.04 5.34 6.33 7.50 8.09 8.26 8.16 7.70 7.45 7.03 5.83 5.45 4.43 4.20 4.00 4.02 4.07 4.13 4.11 4.13 3.75 3.22 3.15 2.69 2.34 2.16 2.14
312
43 84 132 165 223 241 259 281 300 321 339 367 376 387 402 421 441 461 557 618 786 896 314.1 12 40 53 70 86 104 138 153 204 271 307 317 338 430 524 605 690 776 886
3.01 4.52 5.98 6.86 7.94 8.10 8.16 8.07 7.87 7.53 7.09 6.18 5.83 5.47 5.09 4.80 4.59 4.49 4.11 3.60 2.52 2.21 1.37 2.85 3.37 3.98 4.47 5.04 5.97 6.32 7.37 7.76 7.47 7.39 7.04 5.20 4.30 3.65 3.01 2.53 2.19
336.6
23 38 52 64 80 100 112 127 176 213 266 299 353 438 523 611 699 804 350.7 48 68 86 98 111 147 224 307 391 476 566 655 750 388.7 24 47 60 80 92 136 215 300 385 474 568 456.6 31 74 100 168 248 337
1.56 2.13 2.56 2.89 3.26 3.69 3.91 4.16 4.78 5.07 5.24 5.21 4.98 4.39 3.69 3.08 2.59 2.23 2.09 2.56 2.91 3.12 3.34 3.79 4.32 4.38 4.09 3.58 3.04 2.59 2.27 0.87 1.39 1.65 1.95 2.12 2.58 3.02 3.12 2.97 2.70 2.39 0.55 1.05 1.27 1.68 1.91 1.98
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Figure 6. Supercritical isotherms for the excess adsorption of N2O on Kieselgel 60 as a function of the reduced bulk density, Fb/Fc. The symbols represent the results corrected for helium adsorption (i.e., the same data as in Figures 3 and 4), whereas the dashed curves represent the uncorrected data.
Figure 7. Error in the excess adsorption when neglecting helium adsorption and thermal expansion for estimating the volume of the solid parts inside the measuring cell. Connecting lines are drawn to guide the eye only.
1.0077) that were already shown in Figure 3. Considering the isotherms in the temperature range 1.0042 e T/Tc e 1.0145, it can be seen that in the density range 0.74 e Fr e 1.15 isotherms at lower temperatures exhibit smaller excess adsorbed amounts compared to the isotherms at a higher temperature. This is in contrast to the isotherms at higher temperatures, where over the entire experimental range of densities, isotherms at lower temperature always yielded higher excess amounts. To get a clearer picture of this phenomenon, consider the isochoric results shown in Figure 5 which have been extracted from the isothermal data reported in Figures 3 and 4. Figure 5 shows plots of nex against t ) (T - Tc)/Tc for six different bulk densities. At Fr ) 0.6 the excess increases as the critical temperature is approached. The isochores in the density range 0.74 e Fr e 1.15 exhibit a behavior in which the excess increases as the temperature is reduced to t ) 0.015. When the temperature is further reduced, it is clearly seen that the excess drops. Unlike the previous results, where critical depletion has been observed only along isochores with Fr ≈ 1.0,6,10 in the present system it can be seen that critical depletion occurs between Fr ≈ 0.74 and Fr ≈ 1.15. Also for isochores at higher densities, the excess seems to flatten out at temperatures where t < 0.0145. For the sake of complete-
ness, the experimental data presented in Figures 3 and 4 are reported also in Table 3. The comparison between the isotherms corrected for both helium adsorption and thermal expansion and the uncorrected ones is illustrated in Figure 6; the error made if using the uncorrected isotherms is plotted in Figure 7. As seen from Figure 6, at low bulk densities, the helium adsorption and thermal expansion corrections are negligible, whereas at higher densities they lead to significant deviations between the corrected and the uncorrected results. This behavior is expected since, in eq 9, V0 is multiplied by the bulk density. Even though the error in the adsorption excess becomes as large as 40%, all the basic features of the isotherms remain unchanged. Namely, the phenomenon of critical depletion is present in the corrected as well as in the uncorrected data. 6. Concluding Remarks This work contributes to the field of adsorption under near-critical conditions in two ways. On one hand it provides clear experimental evidence of critical depletion, i.e., a highly debated phenomenon that has been reported before only in very few cases. We have observed critical depletion in the case of adsorption of nitrous oxide on
Experimental Observation of Critical Depletion
silica gel; this phenomenon occurs for a broader range of temperature and density values than previously reported for other systems. On the other hand we propose a new measurement protocol aimed at estimating as accurately as possible the volume of the adsorbent and of the solid parts in the measuring cell, i.e., V0. This piece of information is crucial for an accurate determination of the excess adsorbed amount, as recognized already in the literature. Our estimate is based on adsorption measurements carried out using helium, i.e., a weakly adsorbing species. In the past, two different approaches were followed. The simpler approach is to assume that helium does not adsorb at all. The second approach is to account for helium adsorption, by measuring it under proper conditions and by assuming a constant isosteric heat of adsorption and linear adsorption behavior over the whole range of temperature and pressure of interest.14 This provides a first-order correction to the estimate of the V0 value. In the present case, we have shown that, under the chosen experimental conditions, helium indeed adsorbs, and that it does in a nonlinear way, i.e., the Henry’s law assumption is not valid. On the basis of these observations and using a rather simple, “mobile adsorption” model it was possible to provide a second-order correction to the estimate of the V0 value, which improves the accuracy and the reliability of the experimental measurements. Glossary A a bm bl h H k m Mw M1 M2 N nex NAv rj T t T* Ufs V x z z2D
surface area specific surface area of sorbent coefficient in the isotherm based on mobile adsorption coefficient in the isotherm based on Langmuir adsorption Planck’s constant Henry’s constant Boltzmann’s constant mass of a single atom/molecule molecular weight mass at measuring point 1 mass at measuring point 2 number of atoms excess adsorbed amount Avogadro number intermolecular distance between two neighboring adsorbed atoms temperature (T - Tc)/Tc reference temperature total energy of interaction volume critical exponent length coordinate perpendicular to sorbent surface two-dimensional coordination number
Subscripts and Superscripts * 0 b buo c He l m
reference state solid parts in the measuring cell bulk buoyancy critical helium Langmuir type adsorption mobile adsorption
Langmuir, Vol. 18, No. 25, 2002 9733 max met r s sinker sorb z
maximum metal reduced solid (adsorbent) sinker sorbent z coordinate
Greek Symbols R β Γ ff fs ss θ ν F σ
coefficient of linear thermal expansion geometric factor relating overall fluid-solid potential to individual fluid-solid pair interaction potential adsorption excess in mass units fluid-fluid pairwise interaction potential fluid-solid pairwise interaction potential solid-solid pairwise interaction potential fractional surface coverage bonding frequency density Lennard-Jones diameter
Acknowledgment. Partial support of the Swiss National Science Foundation through grant SNF-2155674.98 is gratefully acknowledged. Appendix. Estimation of Model Parameters The bonding frequency, νz, generally depends on the fluid-surface potential, Ufs. Assuming a Lennard-Jones potential for adsorbed molecule-surface molecule interactions (with a minimum energy of fs for the pairwise potential) and treating the solid as a continuum with a flat surface, one obtains
(
fsrjFsorbNAv νz ) 1.351 mMw
)
1/2
(26)
where Fsorb is the mass density of the sorbent;18 all other parameters in eq 26 are as defined before. Considering a one-dimensional slit-pore as a very rough geometric model for the sorbent material, fs can be written in terms of Ufs as follows:20
fs ) 0.318
-UfsMw jr3FsorbNAv
(27)
With eq 27, eq 26 simplifies to
νz ) 0.762
( ) -Ufs 2
rj m
1/2
,
Ufs < 0
(28)
Finally, for rj(He) ) 2.869 Å and the fitted value of Ufs/k ) -325 K, eq 28 predicts νz ) 2.18 × 1012 Hz, which is rather close to the fitted value of νz ) 1.90 × 1012 Hz reported in Table 1. For helium adsorbed on silica gel, a check on the value of the total energy of interaction between an adsorbate atom and the solid, Ufs, can be made as follows. First, note that Ufs can be related to the individual fluid-solid pair interaction parameter by the expression
Ufs ) βfs
(29)
where β is a geometric factor. It is Ufs which is used as a parameter in fitting the helium isochores to the mobile
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adsorption model. For most common adsorbent-adsorbate systems, this parameter could be estimated from the isosteric heat data at zero coverage. In the absence of such data, the Lorentz-Berthelot (L-B) rule offers a way to approximately estimate the fluid-solid interaction parameter, fs
ij ) (iijj)
1/2
fs ) (ffss)
Ufs* ) β*fs*
(31)
where fs is the helium-silica gel fluid-solid interation parameter. To evaluate fs, we need to obtain the values for ff and ss. While ff can be obtained from standard tables by assuming that helium is a Lennard-Jones atom, an estimate of ss could be obtained if we can evaluate this from another reference sorbate-silica gel system where the fluid-solid interaction parameter is known. For the reference system, the L-B rule takes the form
(33)
Substituting eq 33 in eq 32 and rearranging yields
(30)
where i and j denote the two species whose interaction is being considered.23 Hence, in our case, the L-B rule takes the form 1/2
interaction parameters of the reference system.) For the reference system we can write
(Ufs*/β*)2 ff*
ss )
(34)
This expression can now be substituted into eq 31 to obtain an expression for the total energy of interaction for the He-silica gel system
( )
ff β Ufs ) Ufs* β* ff*
1/2
(35)
(Note that the asterisk has been used to denote the relevant
In the present analysis, we choose the CO2-silica gel system whose interaction parameter, Ufs*, can be independently estimated from reported isosteric heat data at zero coverage. By assuming that β/β* ) 1, Ufs can be evaluated by substituting the values given in Table 2 into eq 35. This simple analysis yields a value of Ufs/k ) -280 K, which compares well with the value obtained by fitting the experimental helium isochores, i.e., Ufs/k ) -325 K.
(23) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids; McGraw-Hill: New York, 1987.
LA025696J
fs* ) (ff*ss)1/2
(32)
