Adsorption of Supercritical Gases in Porous Media: Determination of

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J. Phys. Chem. B 1999, 103, 6900-6908

Adsorption of Supercritical Gases in Porous Media: Determination of Micropore Size Distribution C. Nguyen and D. D. Do* Department of Chemical Engineering, UniVersity of Queensland, St. Lucia, Queenslandd 4072, Australia ReceiVed: February 23, 1999; In Final Form: May 25, 1999

New concepts are introduced in this paper to account for the enhancement of the potential energy of interaction between adsorbate molecules and surface atoms within the pore interior. With these concepts, a structure based model is developed to describe the adsorption equilibria in heterogeneous activated carbon. The model requires only molecular properties of the adsorbate and adsorbent, and the structural heterogeneity is accounted for with the distribution of micropore size. This model is validated with the adsorption data of nitrogen and methane on carbonaceous materials under supercritical conditions. It is found that the model is capable of describing very well the adsorption of supercritical gases in porous media and could be used as a tool to analyze the micropore size distribution.

I. Introduction A measure of the interaction between a nonpolar adsorbate molecule and a surface atom is the Lennard-Jones potential energy. Since a solid surface is consisted of individual atoms, the interaction between a molecule and a solid surface is commonly calculated as the summation of the Lennard-Jones potential energies exerted by individual surface atoms against the molecule. In a porous medium, the adsorbate molecule is occluded in the pore interior, and it experiences the interaction with all surrounding surfaces, which form the pore walls. When the pore dimension is small enough, the dispersion force fields acted by various parts of the surface overlap each other. Since these force fields are additive (in vector sense), the resulting potential energy of interaction is enhanced. The extent of this enhancement depends on the degree of the force field overlapping, which is determined by the size and shape of the pore and the position of the adsorbate molecule. The binding force of adsorption in porous media and that on a flat surface are of the same nature; the only difference is the enhancement of adsorption characteristic for porous media. In other words, it is possible to consider the adsorption in porous media as an extension to that on a flat surface. This approach has been applied successfully in various studies of subcritical vapor adsorption. For example in the t or R plot methods, the subcritical adsorption in a pore system is investigated by comparing it with that occurring on a flat surface.1 Utilizing the same principle, we have recently developed a method for pore characterization by using adsorption equilibrium data of subcritical vapors.2 We introduced the concept of enhanced multilayering, which in conjunction with the Kelvin equation is capable of determining the pore size distribution of meso and micropores indiscriminately. This approach is novel in the sense that we do not need to distinguish micropores from mesopores. We rather argue that adsorption mechanism is the same in all pores and that the difference is in the degree of enhancement of adsorption in pores of different sizes. Our approach has been applied successfully to numerous adsorption data of nitrogen to determine the PSD of a range of carbonaceous adsorbents. It * Corresponding author.

is however important to note that the approach was limited to subcritical vapors, and data of supercritical fluids were excluded from the analysis. This is because in difference from adsorption of subcritical vapors, there is not a clear surface layering process nor a condensation phase in supercritical adsorption. The lack of a capillary condensation stage and an insignificant adsorption in larger pores make adsorption of supercritical gases less practical for mesopore characterization purposes. This is one of the reasons why the analytical method for adsorption of a supercritical gas is much less advanced compared to that of adsorption of vapors.1,3 The fact that adsorption of supercritical gases occurs mainly in pores of micropore range gives rise to the argument that it could be useful in the determination of micropore size distribution (MPSD). This becomes more advantageous in cases where PSD characterization by adsorption of subcritical fluids, which is commonly carried out at very low temperatures, is too slow to achieve true equilibrium within the time frame of experiments. In the literature, isotherms of supercritical gases are often described by the type I Langmuir equation.1 Other semiempirical equations such as those of Toth, Unilan, etc., are also used to describe the adsorption of supercritical gases when the Langmuir equation fails to describe the data adequately due to some inhomogeneity of the system. Such an approach is useful only as a means to describe and/or to fit the experimental data. It cannot, for example, be used to derive any information about the solid structure. Another approach, which is more “structure” oriented in solving the problem of supercritical adsorption, was introduced by Dubinin in his earlier work.4,5 Starting with the DR equation, which had been developed for subcritical fluids, Dubinin introduced a hypothetical saturated vapor pressure so that the adsorption of gases can be considered as a process of micropore filling. The weak point of this theory rests on the introduction of the hypothetical saturated vapor pressure which is treated as a “fitting” parameter. Recently, Kaneko and Murata6 had pointed out that this hypothetical vapor pressure did not have a sound physical basis. To derive a new method, they introduced instead the quasisaturated vapor pressure and the inherent micropore volume for supercritical gases while retaining the application of the DR equation. The theory was tested using

10.1021/jp9906536 CCC: $18.00 © 1999 American Chemical Society Published on Web 08/03/1999

Determination of Micropore Size Distribution

J. Phys. Chem. B, Vol. 103, No. 33, 1999 6901

adsorption data of different gases with a good success. Despite this improvement, it cannot be used to derive any information about the MPSD. Progress in the description of inhomogeneous fluid in confined spaces has led to the development of new theoretical tools for porous solid characterization. These tools are the density functional theory DFT, the Monte Carlo MC, and the molecular dynamics MD simulations. The simulation methods are very computer intensive, but they are useful in that their results can be used as benchmark to evaluate other methods. The DFT and grand canonical Monte Carlo (GCMC) methods have been applied successfully for the adsorption of supercritical gases with a simple molecular structure such as methane, argon, etc. These methods, however, still have some inherent drawbacks, for example, they require a significant computational time, and they may become more demanding when applied to molecules having more complex structures. In this work, we present a new approach to analyze adsorption data of supercritical fluids with a goal to derive useful micropore size determination of the adsorbent. We use the Langmuir equation, which is known to be suitable for homogeneous system as the local isotherm, while the heterogeneity of the adsorbent is accounted for by the distribution of the micropore size. To achieve the goal, we introduce a number of concepts such that the isotherm parameters are related to the solid structure and they incorporate the enhancement of adsorption in smaller pores. The model will be tested against the experimental data available in the literature. II.Theoretical Section The overlapping of the interaction force fields of the pore walls affects the potential energies of the adsorbed molecules as well as the pore gas-phase molecules. In this paper, two new concepts are introduced to account for the enhancement effects on the adsorbed and gas phases within the pore interior. We will hereafter refer to them as the enhanced adsorption affinity (bpore), and the enhanced pore pressure (Ppore). Furthermore, we also use the concept of a threshold thickness of the adsorbed phase (z/pore), which determines the boundary of the two phases. 1. Enhanced Adsorption Affinity. In the classical thermodynamics, the interaction between adsorbate molecules and a solid surface is characterized by the affinity coefficient, which is in general a function of the interaction energy and system temperature. The stronger is the interaction, the larger is the affinity coefficient. The dependence of the affinity coefficient on interaction energy and temperature can be derived using the molecular kinetics reasoning as follows. At very low loading, the rate of adsorption is assumed to be proportional to the rate of striking of molecules on the surface, i.e.,

Rads ) RP/x2πMRT

(1)

where R is the proportional coefficient, M is the gas molecular weight, R is the universal gas constant, and T is the temperature. Similarly, the rate of desorption is assumed to be governed by the rate of molecules leaving a surface, which takes the following form:

Rdes ) kd0 exp(-E/RT)θ

(2)

with θ the fractional loading and E the heat of adsorption. At equilibrium, there is an equality between the rate of adsorption and the rate of desorption i.e., Rads ) Rdes, from which we can

derive the following equation for the fractional loading θ:

θ)

R kd0x2πMRT

exp(E/RT)P

(3)

Assuming that Henry’s law θ ) bP is satisfied at low pressure, the affinity can be derived from eq 3 as

b)

R exp(E/RT) kd0x2πMRT

(4)

Since R and kd0 are generally unknown, they can be lumped together into a parameter β ) R/(kd0x2πR), and eq 4 is simplified to

b)

β

xMT

exp(E/RT)

(5)

Parameter β is assumed to characterize the solid properties, while the interaction between the adsorbate and the solid is reflected through parameter E. Equation 5 can be used to calculate the heat of adsorption E by measuring the coefficient b at several temperatures. This calculated heat of adsorption is the overall heat contributed by the heat released from various parts of the solid. If the solid is homogeneous, those heats are identical. However, when the solid is heterogeneous, for example solid with micropores of different sizes, the heat released by each pore will depend on the pore size because of the difference in the interaction energy between the adsorbate molecule and the pore walls. Let us consider a slit like pore, which is usually taken as a model for micropores in carbonaceous adsorbents. An adsorbate molecule occluded in a micropore is attracted to both sides of the pore. Such an interaction depends of the distance between the walls as well as the position of the admolecule. At equilibrium, this interaction and hence the heat of adsorption is a function of the pore width. We denote Es and bs as the heat released and the affinity of adsorption on a flat surface, respectively. Similarly, we denote Epore and bpore as those of adsorption in a pore. The affinity coefficient of adsorption in the pore bpore then can be calculated from the affinity coefficient of adsorption on a flat surface bs by the formula

(

bpore ) bs exp

Epore - Es RT

)

(6)

The heat of adsorption at zero loading can be calculated theoretically using common potential energy equations, such as Σ12-6, 9-3, or 10-4 potential.7 In this work, we use the Σ12-6 potential to calculate the zero loading heat of adsorption on a flat surface as well as and in pores having different sizes. Details of the calculation are presented elsewhere.2 Furthermore, the affinity coefficient of a flat surface bs can be calculated using eq 5. Thus, the dependence of the heat of adsorption and the affinity coefficient on pore size can be established. In Figure 1 we plot the calculated heat of adsorption of nitrogen at zero loading and the corresponding affinity coefficient at 303 K versus the half-width of a model carbon pore. 2. Enhanced Pore Pressure. The effects of the closeness of the pore walls onto the confined gas phase is partly addressed by Drago et al.,8 who proposed the definition of the “effective pressure” which could be used to illustrate the concentrating effect of the porosity. The effective pressure is calculated as the pressure, which would have to be applied on an equal amount of moles of an ideal gas to obtain a similar density to that in the pore. They have shown that, for activated carbon,

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Nguyen and Do

Figure 1. Nitrogen heat of adsorption Epore and affinity coefficient bpore at 303 K as functions of the pore half-width. Figure 3. The enhanced pore pressure of nitrogen at 303 K as a function of the pore half-width.

dispersion and the gravitational forces are of different natures. If we take the potential of molecules in bulk at pressure pbulk as a standard, then the pressure at a position where the potential energy of the gas molecule is Eg can be calculated by

( ) -Eg RT

p ) pbulk exp

Figure 2. Potential of the gas molecules at different locations relative to a surface (left) and (right) a pore.

this effective pressure can be hundred times larger than the bulk pressure. In this paper, we accept that the pressure of the gas confined in the pore interior is different from the pressure of the bulk phase. Let us consider a gas phase, which is in contact with a solid surface as shown in Figure 2. The potential energy of the gas molecules, which can be calculated using any of the potential equations is presented qualitatively in the figure. As seen, the closer to the surface is a molecule, the lower is its potential; that is, molecules are attracted toward the surface and their density is higher at locations close to the solid surface. The variation of gas-phase density with distance from the solid surface has been mentioned by Fischer.9 He argued that since the probability of finding a molecule at a given point in the pore is essentially determined by the pore potential Ψ. If the bulk gas density is nb(r), the local density at location x is given by

[ ]

n(x) ) nb(r) exp

Ψ(x) kT

(7)

From the molecular kinetic point of view, a higher number of molecules per unit volume means a higher pressure. That is, the gas-phase pressure varies with distance from the solid surface. If a molecule is moved closer toward a surface, its potential is lowered, and at the same time it is exposed to a higher pressure local environment. A similar scenario is observed in the Earth atmosphere, where the air pressure decreases with the height (distance) from the Earth surface. This can be used to support our argument despite the fact that the

(8)

If there are pores in the solid, molecules will be dragged further into the pore interior by a stronger dispersion force due to the enhancement of interaction as discussed earlier (Figure 2). Provided these pores are not very small, some of the molecules will be adsorbed very close to the solid surface while the others, which are referred to as confined gas molecules will remain in the gas phase within the pore. Applying the above argument, we can see that the pressure of the gas-phase confined in a pore is higher than that of the bulk. As before, it can be calculated as follows:

( )

ppore ) pbulk exp

-Egpore RT

(9)

where Egpore is the potential energy of the gas-phase molecules confined in the pore interior. It is important to note that even, within a pore, the potential changes in the radial direction, and so does the gas-phase pressure. However, for the sake of simplicity we will take the potential and the enhanced pore pressure at the center line as the first estimate for Egpore and Ppore. A typical plot of the enhanced pore pressure versus the pore size is shown in Figure 3. As seen in the figure, the pore pressure is significantly higher than the bulk gas-phase pressure. This prompts the need to use fugacity instead of the general pore pressure. Nevertheless, in this paper again for simplicity, we will use the pore pressure to account for the effects of the pore walls on the gas phase. The observed adsorption equilibrium then can be broken into two processes: equilibrium between the bulk gas phase and the confined pore gas phase and the equilibrium between the pore gas phase and the adsorbed phase. 3. Adsorbed Phase Threshold Thickness. As seen in Figure 2, adsorbate molecules are attracted to the solid surface. If the attraction force is strong enough, the molecules are said to be adsorbed. The number of molecules adsorbed depends on the system temperature and the bulk gas-phase pressure. Molecules



Figure 4. Concept of the threshold potential energy E* and the threshold thickness of the adsorbed phase z*.

having potentials close to the depth of the potential well will be adsorbed first. This means that if the molecule-molecule interaction is small, there must be a threshold potential energy E* such that all molecules having potentials lower than E* can be considered adsorbed while molecules with potentials higher than E* are not. Since the potential of a molecule decreases with the distance z from a surface, the above observation also means that there is a threshold distance z* such that all molecules with z < z* can be assumed adsorbed while molecules with z > z* are not (cf. Figure 4a). This concept is further applied for the case of adsorption in porous media. In Figure 4, adsorption on a flat surface (a) and in pores belonging to different ranges: macropores (b), mesopores and supermicropores (c), and ultramicropores (d) are schematically presented together with the corresponding gas molecule potential energy diagrams. Using the threshold potential E* concept, the threshold distance z/pore can be determined as graphically shown in the figure. Here we can see that when there is practically no enhancement as it is the case of macropores (Figure 4b), z/pore ) z*. In narrower pores, the enhancement takes effect, and z/pore > z* (Figure 4c), while in even narrower pores (ultramicropores) the potentials of molecules are so low (the pore pressure is so high) that practically all of them are adsorbed by the solid (Figure 4d). Using the concept of energy demarcation, a plot of the adsorbed phase threshold thickness as a function of the pore size can be established. Figure 5 shows an example of the system nitrogen-slit carbon pore. In this example, the threshold energy was chosen to be equal to the heat of adsorption at zero loading. In principle, however, the threshold energy can be calculated from the equilibrium data of adsorption on a flat surface. III. The Model Equation If f(r) is the pore size distribution function and H(p,r) is the single pore isotherm equation, the amount adsorbed at pressure p can be calculated as

Cµ(p) )

∫0∞ H(p,r)f(r) dr

(10)

In our model, the single pore isotherm is assumed to follow the Langmuir theory, and it takes the following form when

Figure 5. Threshold thickness of the adsorbed phase versus the pore half width.

written for adsorption in pores of radius r:

Cµ(r) ) Cµs(r)

bpore(r)ppore(r) 1 + bpore(r)ppore(r)

(11a)

or

θ(r) )

Cµ(r) Cµs(r)

)

bpore(r)ppore(r) 1 + bpore(r)ppore(r)

(11b)

where Cµs(r) is the maximum capacity of all pores of size r and θ(r) is the fractional loading of pore having size r. As an example, eq 11b is applied for methane adsorption at 303 K with ppore and bpore calculated using eq 9 and 6, respectively. Results are the single pore isotherms, and they are presented in Figure 6. Substituting eq 11a into eq 10, we obtain

Cµ(p) )

b

(r)ppore(r)

∫0∞ Cµs(r) 1 +pore b

pore(r)ppore(r)

f(r)dr

(12)

Equations 6, 9, and 12 are the three model equations, which will be used to fit the experimental data. It is worth noting that the result of the optimization process is a pore size distribution in terms of the adsorbed capacity but not in terms of the pore volume. To convert one to another, we can use the concept of


Nguyen and Do TABLE 1.

Figure 6. Methane single pore isotherms at 303 K in pores of differing half width (reading from left to right: 0.37, 0.4, 0.43, 0.47, 0.52, and 1 nm).

the adsorbed phase boundary. It is easy to prove that, for a pore of size r, the ratio of the pore volume V(r), and the volume of the adsorbed phase W(r) is given by

V(r) r ) W(r) z/pore(r)

(13)

W(r) ) Cµs(r)δ

(14)

Further,

where δ is the molar volume of the adsorbed phase. Thus, the pore volume can be calculated from the adsorbed capacity by the following equation:

V(r) )

r C (r)δ az/pore(r) µs

(15)

The volumetric PSD thus can be calculated from the distribution obtained by fitting eq 15 against experimental data. This model will be tested on supercritical nitrogen and methane adsorption data, which are obtained from the literature.6-12 In the optimization process, no distinction is made in regard to the range of the pore size. That is, if the whole pore spectrum is divided into many subranges, then the optimization procedure is invoked to find the volume of each sub-range such that the calculated isotherm matches the experimental data. This method does not assume any particular form for the distribution function, and as a result the PSD reflects the “real” distribution better than methods assuming an a priori form for the distribution. IV. Adsorption of Supercritical Gases in Carbon Pores 1. Carbon Pore Model and the Heat of Adsorption. The heat of adsorption includes the changes in the intramolecular energy upon adsorption. The ordinary molecular potential does not take into account the intramolecular motions such as vibration and rotation. Hence, the heat of adsorption at zero loading can be calculated as the potential energy well depth of the adsorbate molecule, which in this paper is evaluated as the summation of the pair wise Lennard-Jones (12-6) interaction between the adsorbate molecule and individual atoms of the solid surface. That is, the heats of adsorption on a flat surface

parameter

value

ref

σcarbon σnitrogen σmethane carbon/k nitrogen/k methane/k

0.34 nm 0.3798 nm 0.3758 nm 28 K 71.4 K 148.6 K

7 13 13 7 13 13

and in a pore (Es and Epore in eqs 5 and 6, respectively) can be estimated if the molecule - molecule interaction parameters are known and a configuration can be assumed for the solid structure. The calculation method is detailed as follows. The collision diameters and the interaction energy between the adsorbate molecules and the carbon surface are taken from the literature.7,13 These values are listed in Table 1. The adsorbentadsorbate collision diameter σ12 and interaction energy 12 are calculated by using the Lorentz-Berthelot mixing rule.7 It has been widely accepted that micropores of members of the carbonaceous family are slitlike in shape, with two parallel pore walls consisted of a few graphite layers. Very often in the literature7,14-16 the pore wall is modeled as a structure-less semiinfinite slab, which allows the application of the derived potential equations such as the 10-4-3 and 9-3 potentials. In our view, due to the proximity of the adsorbate molecule to the pore walls, any assumption of an homogeneous distribution of the mass centers over the graphite plane or in the space of the graphite domain may lead to the underestimation of the effect of the repulsive forces exerted by individual carbon atoms nearest to the adsorbate molecule. A more realistic model of the pore wall structure is used in this paper, here the micropore is visualized as the gap between the graphite sheets, which are stacked on top of each other with an interlayer spacing of 0.3354 nm. The stacking of the graphite layers is in a hexagonal arrangement, which is the most common form of staking in graphite structure.17 The number of graphite layers of a pore wall is limited to 2 or 3,18 corresponding to a realistic pore wall thickness of about 1.1-1.5 nm. This model structure of the pore wall makes possible the use of the Lennard-Jones pair potential between the adsorbate molecule and individual carbon atoms of the pore walls. 2. Methane Adsorption. Methane is probably the most studied supercritical adsorbate. The reason is 3-fold: (i) Methane molecules are small and they can penetrate into smaller pores. (ii) Methane molecules are spherical in shape, which is a favorable factor for molecular simulation processes. (iii) Methane is the most potential fuel gas for gas storage studies. Methane adsorption data are available in the literature in a large number. In this paper, we use methane adsorption data from the work of Kaneko,6 Keller,19 and Gusev.11 We also use data of methane adsorption on Ajax AC measured in our laboratory. a. Parameter β. To proceed further, we need to know the value for β in eq 5. As mentioned above, this parameter is characteristic for the solid. Hobson20 reported a value of 4.26 × 10-1 for β to yield the affinity having unit of inverse MPa. In other work, Do and Do21 have obtained the optimized value for β of carbon surface as much as 10-15 times smaller than the value reported by Hobson. A possible reason is that Hobson calculated this value for adsorption on a flat surface, where there is no restriction for the gas-phase molecules movement. On the other hand, in confined spaces such as inside the pore interior, the freedom of movement of gas molecules is reduced significantly. Our model can fit the selected experimental data with β ranging from 1 × 10-4 to 1 × 10-1, with smaller value of β



Figure 7. Model fitting of subcritical nitrogen isotherm and the PSD of sample AX21.

Figure 9. PSD of sample AX21 by methane adsorption at 303 K.

Figure 8. Fitting of methane isotherm at 303 K in (a) normal and (b) logarithm scale.

giving larger median of the resulted PSD. Thus, if the median of the MPSD is available, the value of β can be estimated. We chose the sample AX21 AC for the purpose of estimating the value for β. This carbon sample has been characterized by nitrogen adsorption at 77 K in the work of Lastoskie et al.16 Further, using their data we successfully generated the PSD of this activated carbon sample (see ref 2). Results of our work is shown in Figure 7, and it is worth noting that this PSD resembles the one obtained by Lastoskie et al.16 Methane adsorption data on sample AX21 is taken from work of Kaneko and Murata6 to determine the parameter β. The exact value of β is not critical as we have mentioned earlier that it affects mainly the median, but not much the shape of the distribution. For activated carbon, the value of β is estimated to be 0.021 (1/20 of the value reported by Hobson). The fitting of methane adsorption data at 303 K on AX21 is shown in Figure 8 in both normal and log scale to highlight the goodness of fit at higher and lower pressure ranges. As is seen, the fitting is quite satisfactory at both pressure ranges. The resulted PSD is shown in Figure 9a. To make the comparison easier, the molar PSD of the sample is converted

to volumetric PSD (Figure 9b) using the liquid molar volume of methane at 111.6 K, which is estimated to be 37.87 cm3/ mol.19 The mean pore diameter of the sample is estimated to be ∼12 Å, about the same as the mean pore diameter obtained from analysis of subcritical nitrogen adsorption (cf. Figure 7b). The value of 0.021 for β thus will be used for further analysis of methane as well as nitrogen adsorption on various carbon materials. b. Multitemperature Isotherm Fitting. The multitemperature fitting is carried out on methane adsorption data onto BPL AC at 212.7, 260.2, and 301.2 K, which are taken from ref 12. The thermal expansion coefficient was found to be rather small (in the order of 1 × 10-5 mol/deg),22 which means the change of the molar capacity within the given range of temperature can be ignored. The isotherm fitting is carried out for all temperatures simultaneously. Results are presented in Figure 10a, and the calculated PSD of the BPL AC sample is shown in Figure 10b. It is noted that the pore width used in this paper is the center to center distance of the first carbon layers of the opposite pore walls. The molar PSD in Figure 10b indicates that most of the adsorption occurs in pores of the micro range, which is characteristic for supercritical adsorption. The procedure is further tested against data of methane adsorption onto Ajax AC measured in our laboratory. Results presented in Figure 11 confirm the suitability of the model. The calculated PSDs of the BPL and Ajax AC are quite similar with slightly different mean pore sizes. 3. Supercritical Nitrogen Adsorption. Nitrogen adsorption at its liquid temperature (77 K) is the most frequent method used in solid characterization by adsorption.23 However, this method has suffered from some drawbacks, such as the problem of very slow activated diffusion at low temperature. This particular problem is encountered mainly when pores are in micropore range. In such instances, supercritical adsorption provides an alternative way to characterize the microporosity. This is because being carried out at near ambient temperatures,


Nguyen and Do

Figure 12. PSDs of samples (up) P5 and (down) P25 calculated from nitrogen adsorption at 303 K.

Figure 10. Multitemperature fitting of methane adsorption at 212, 260, and 301 K, and the PSD of BPL AC.

Figure 11. Multitemperature fitting of methane adsorption at 273, 283, and 303 K, and the PSD of Ajax AC.

the adsorption dynamics is not retarded that much by the quadruple effects of the pore mouth. As mentioned above, supercritical adsorption is significant only in narrow pores, and only very narrow pores can be assumed to be filled with the adsorbate. Thus, in difference from subcritical adsorption, supercritical adsorption provides only information on PSD at the lower end of the pore spectrum.

In this paper we use nitrogen adsorption data onto AC samples from ref 6. The model fitting of nitrogen isotherms at 303 K on samples P5 and P25 are shown in Figure 12. The PSD histograms in the figures show that the mean pore size of the sample P5 is significantly smaller than that of the sample P25, which is consistent with the fact that the sample P25 is more thoroughly activated. 4. Comparison between the PSDs Obtained from Nitrogen and Methane Adsorption. As discussed above, the PSD can be characterized by either methane or nitrogen isotherm data. A question then arises: are these PSDs comparable with each other? To address that, we test the model against the nitrogen and methane adsorption data on Norit A1 Extra AC at 298 K, which are available from ref 19. The fitting results are shown in Figure 13 accompanied by the corresponding PSDs. We reemphasize that the same value for β in the model eq 6 is used in each case. It is interesting to observe that, like methane, supercritical nitrogen adsorption occurs exclusively in the narrowest pores and that the calculated PSDs are in a fairly good agreement with each other. The difference between them can be attributed to the fact that the probe molecules have different sizes and shapes, which may bring about differing sieving effects. 5. Prediction of Adsorption at Other Temperatures. The reverse problem of predicting an isotherm is addressed in this section. Here we demonstrate that the molar PSD calculated by using our model can be used to predict the adsorption at other temperatures. First, the PSD of a BPL AC sample (ref 11) is calculated by using the methane adsorption data at 333 K (Figure 14). The isotherm of methane at 308 K is then simulated, and the result is presented in Figure 15. The figure shows the fitting in both normal and logarithm pore size scale to emphasis the goodness of fit of the model. As is seen, the model predicts methane adsorption at 308 K accurately, at both low- and high-pressure regions. It is interesting to compare the PSD of the BPL AC in Figure 14 with that in Figure 10. They are quite similar in the micropore region, however no contribution of supermicropores or narrow mesopores is seen in the former. An explanation for this is that adsorption data onto the


J. Phys. Chem. B, Vol. 103, No. 33, 1999 6907 for example. It is rather a PSD in terms of molar volume adsorbed. The volumetric PSD however can be estimated from the molar one with some idealizations; for example, the adsorbed phase behaves like a liquid, and its density is not a function of neither the pore size nor distance from the pore walls. In some papers,6,19 the liquid like status of the adsorbed phase is assumed, and the density is normalized to that at a subcritical temperature or the boiling point. In this paper, we use the liquid density of methane at its boiling point (111.6 K).19 The concept of a threshold thickness of the adsorbed phase is then applied to calculate the volumetric PSD of sample AX21. Results are shown in Figure 9b. As is seen, the pore volume of individual pore fraction of this PSD histogram is lower than that of the PSD histogram in Figure 7b. An obvious explanation is that if adsorbed methane were in liquid form at the experiment temperature, then its density would have been much higher than that at its normal boiling point. It would be necessary to note that the shapes of the two types of PSD are essentially the same in the micropore region since these pores can be “filled” with adsorbed phase. Despite its simplifications, this provides a connection between the PSD obtained from supercritical adsorption and the “effective” PSD of the solid adsorbents measured by liquid nitrogen adsorption.

Figure 13. Comparison of the PSD by (up) nitrogen and (down) methane supercritical adsorption.

V. Conclusions In contrast to molecules in the bulk, those occluded in the pore interior, both in the adsorbed state or in the confined fluid state are under the influence of the overlapped interaction force field exerted by the pore walls. The concepts of enhanced adsorption affinity and enhanced pore pressure are found to be suitable to account for this fact. The model developed based on those concepts is capable of describing adsorption of supercritical gases in porous media, and it can be used to derive the MPSD. Application of this model has been demonstrated to supercritical adsorption of nitrogen and methane on a number of carbonaceous adsorbents.

Figure 14. Model fitting of methane adsorption at 333 K and the PSD of BPL AC.

Acknowledgment. Support from the Australian Research Council is gracefully acknowledged. References and Notes

Figure 15. Prediction of methane adsorption onto BPL AC at 308 K (data from ref 11).

sample in Figure 10 were collected up to a higher pressure than the other, meaning adsorption may have started in larger pores. Furthermore, we also cannot exclude uncertainty arising from the fact that data were measured by different groups using carbon from different batches. 6. Converting the Adsorption PSD into the Volumetric PSD. As presented above, the PSD calculated using the above fitting procedure is not the volumetric PSD of the material, which we normally obtain by using liquid nitrogen adsorption

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