6226
Langmuir 1997, 13, 6226-6233
Characterizing the Micropore Size Distribution of Activated Carbon Using Equilibrium Data of Many Adsorbates at Various Temperatures K. Wang and D. D. Do* Department of Chemical Engineering, University of Queensland, Brisbane QLD 4072, Australia Received May 6, 1997. In Final Form: August 18, 1997X In this paper, we present an approach to characterize the micropore size distribution (MPSD) of activated carbon by using multiple-temperature isotherm data of many adsorbates simultaneously. The characteristics of the model are as follows: (1) The MPSD is treated as the intrinsic property of activated carbon (independent of adsorbate) and the sole source of heterogeneity. (2) Different adsorbates have access to different pore size ranges because of their difference in molecular sizes. (3) The adsorbate-adsorbent interaction energy is related to the micropore size through the Lennard-Jones potential theory. This method is examined with the extensive equilibrium data of many adsorbates measured on two different activated carbons under a wide temperature range. The effects of the configuration of the slit-shaped micropores and the local adsorption isotherm are also investigated. It is shown that, with the intrinsic structural parameters, the model proposed can describe very well the adsorption equilibria of different species on each activated carbon and predict reasonably well the multicomponent adsorption equilibria for a number of systems studied in this paper. The physical significance of the results and the applicability of the model are also discussed.
Introduction It is generally accepted that activated carbon is composed of amorphous and graphitic carbons. The former contains void space (macropores and mesopores) for intercrystalline transport, while the latter provides a microporous network in which most adsorption capacity resides. The basic structure of the graphite crystal is graphitic-like aromatic microcrystallites, which form slitshaped micropores between graphite layers (pore walls). The width of the micropores is of molecular dimension, such that the dispersive interactions between the adsorbate molecule and both sides of the pore walls are enhanced, resulting in a strong adsorption potential. The variation of micropore size strongly affects the adsorbatepore interaction, as a small variation in the pore size can result in a very large change in the potential energy of interaction and hence a large change in the adsorption affinity. As a result, the overall adsorption equilibrium is strongly affected by the distribution of the micropore size (MPSD). Characterizing the microporous structure of activated carbon is a very challenging issue. Until now, there was no method which could be accepted as a general tool to characterize the MPSD even though there were many methods proposed in the literature. The popular methods, such as MP,1 TVFM,2 H-K,3 SWQ,4 etc., are very useful in their own ways, but all have their limitations because each model has its own assumptions about the porous structure, the properties of the adsorbed phase, the adsorbate-adsorbent or adsorbate-adsorbate interactions, etc. in the model development. The other properties of activated carbon such as functional groups, defects, pore connectivity or pore shape irregularity, etc. still remain unaccounted for. Furthermore, under most cir-
cumstances the results for the same activated carbon from these methods are incompatible with each other.5 It is difficult to judge the applicability of different theories until we have some advanced surface analysis techniques which can give us an unambiguous picture of the carbon surface and porous structure. An understanding of solid heterogeneity is extremely important for practical applications such as adsorption or catalytic reaction processes, and therefore it is the purpose of this paper to present a simple and easy to use method to characterize the MPSD of activated carbon for the purpose of subsequent use of the MPSD in dynamics studies. Theory Among all the methods characterizing the structure of activated carbon, the vapor/gas adsorption equilibrium is probably the most commonly used.6 In this method, there are two general approaches to address the surface heterogeneity. One is to assume an energy distribution F(E) and a patchwise topology, and then the overall adsorption equilibria can be expressed as the integral of the local isotherm Cµ(E,T,P) over the complete energy distribution range; that is,
〈Cµ〉 )
∫
Emax
Emin
Cµ(E,T,P) F(E) dE
(1a)
where Emin and Emax are the lower and upper limits of the energy distribution, respectively. The other approach is based on the fact that the energy heterogeneity is a result of the variation in the micropore size (represented hereafter in terms of pore half-width r). Thus, by assuming a distribution function of f(r) for the MPSD, where f(r) dr is the volume fraction of the micropore having a pore size in the range r to r + dr, the overall equilibria can be expressed in the same way as in eq 1a:
〈Cµ〉 )
∫
r2
r1
Cµ(r,P,T) f(r) dr
(1b)
* Author to whom all correspondence should be addressed. X Abstract published in Advance ACS Abstracts, October 1, 1997.
where r1 and r2 are the lower and upper limits of the pore halfwidth, respectively. In writing eq 1b, we have assumed that the
(1) Lippens, B.; De Boer, J. H. J. Catal. 1965, 4, 319. (2) Jaroniec, M.; Madey, R.; Choma, J. Carbon 1989, 27, 77. (3) Horvath, G.; Kawazoe, K. J. Chem. Eng. Jpn. 1983, 5, 333. (4) Seaton, A. N.; Walton, J.; Quirke, N. Carbon 1989, 27, 853.
(5) Russell, B. P.; LeVan, M. D. Carbon 1994, 32, 845. (6) Carrott, P. J. M.; Roberts, R. A.; Sing, K. S. W. Characterisation of Porous Solid; Elsevier: Amsterdam, 1994.
S0743-7463(97)00474-5 CCC: $14.00
© 1997 American Chemical Society
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surface heterogeneity is induced solely by the MPSD. The MPSD approach is more fundamentally sound than the approach of energy distribution, since the energy distribution F(E) may not follow any particular type of distribution function. The pore size distribution, on the other hand, results from the random processes of carbonization and activation, and therefore it can be characterized by some distribution functions, for example the lognormal or Gamma distribution. Hence we assume a function for f(r) is theoretically sound and more reasonable than arbitrarily imposing a function of F(E). If we know the pore size distribution f(r), the energy distribution can be related to the micropore size distribution according to
F(E) dE ) f(r) dr
(2)
Thus when we know the relationship between the energy of interaction E and the pore size r, the function F(E) can be calculated. For given adsorption equilibrium data, eq 1b can be utilized (inverted) to derive the MPSD if the local isotherm is specified. But such an inverse has been known as an ill-posed problem and constitutes some difficulties in application.7 For example, for a given adsorbate and a given local adsorption isotherm, one would find many different distribution functions that can yield the same overall adsorption capacity 〈Cµ〉 (within experimental errors). Furthermore, if the isotherm equation (eq 1b) is applied separately on each adsorbate, we might obtain very different MPSDs for the same activated carbon! This violates the basic premise that the MPSD is the intrinsic parameter of the given carbon; that is, we must demand that the MPSD be a constant parameter, independent of adsorbates used. Realizing this inconsistency problem often seen in the literature, we take an approach in which we apply eq 1b to multiple-temperature data of many adsorbates simultaneously while constraining the MPSD as the intrinsic characteristic of the solid adsorbent. For a given MPSD, the adsorption energies of interaction are different for different adsorbates due to their difference in molecular properties. So before proceeding with the adsorption isotherm calculation, it is imperative to understand the adsorbate-adsorbent interaction, and this will be addressed in the next section. 1. Adsorption Potential and Slit Pore Configuration. In physical adsorption of nonpolar molecules, the main interaction between an adsorbate molecule and a carbon atom on the graphite layer is the dispersive force. This potential energy is adequately described by the Lennard-Jones 12-6 potential theory. The interaction potential energy between an adsorbate molecule and all the carbon atoms on the pore wall is then the summation of all the pairwise interaction energies. This process of summation can be replaced by an integration operation if the distance between the adsorbate molecule and the pore wall is longer than the distance between carbon-carbon centers. Three pore configurations will be dealt with in this paper: (1) two parallel lattice layers; (2) two parallel slabs; and (3) a slit-shaped pore with multiple lattice layers on each wall of the pore. 1.1. Two Parallel Lattice Layers. In case 1, if the lattice layers are assumed to have an infinite length in extent, the result of the integral is the local well-known Lennard-Jones 10-4 potential:8
[ ( ) ( ) ( ) ( )]
5 2 σsk φ(r,z) ) *sk 3 5 r-z
10
+
2 σsk 5 r+z
10
-
σsk r+z
4
-
σsk 4 r-z (3a)
where φ(r,z) is the local potential energy of an adsorbate molecule and
6 *sk ) πnsskσ2sk 5
(3b)
is the minimum interaction energy between the adsorbate molecule k and a single lattice layer containing solid atoms s. The variable z is the distance between the adsorbate and the central plane of the pore, ns is the number density of carbon centers per unit area of graphite surface (38.2 nm-2), and sk and (7) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: San Diego, CA, 1994. (8) Everett, D. H.; Powl, J. C. J. Chem. Soc., Faraday Trans. 1976, 72, 619.
σsk are the cross L-J parameters for adsorbate-adsorbent interaction, which are calculated using a geometric mean and an arithmetic mean (Lorentz-Berthelot rule), respectively:
sk ) xss × kk σsk )
σss + σkk 2
(3c)
1.2. Two Parallel Slabs. In case 2, if the pore walls are assumed to be two parallel semi-infinite slabs, the potential energy between an adsorbate molecule and two semi-infinite slabs is the Lennard-Jones 9-3 potential:7
φ(r,z) )
[ ( )
3 2 σsk *sk 15 r + z x10
9
+
( ) ( ) ( )]
2 σsk 15 r - z
9
-
σsk r+z
3
-
σsk r-z
3
(4a)
where
*sk )
2x10 πn′sskσ3sk 9
(4b)
is the interactive energy minimum between the adsorbate molecule and a single slab (pore wall) and n′s is the number of carbon molecules per unit volume of the slab. 1.3. Pore Made Up of Multiple Lattice Layers. In case 3, if the pore walls are assumed as the combination of many parallel lattice layers separated by a distance ∆, the potential energy will take the form of the 10-4-3 potential given below:
φ(r,z) )
{ [( ) ( ) ] [( ) ( ) ] ) ( )]} [( σsk r-z
10
3∆(0.61∆ + r + z)3
-
2 σsk 5 * 3 sk 5 r + z
10
+
-
σsk r+z
σ4sk
4
+
σsk r-z
σ4sk
3∆(0.61∆ + r - z)3
4
-
(5a)
where
6 *sk ) πFsskσ2sk∆ 5
(5b)
is the interaction energy minimum between an adsorbate molecule and a single lattice layer and Fs is the number density of carbon molecules per unit volume. The values of ∆ and Fs are taken as 0.335 nm and 114 nm-3, respectively.9 The adsorbate-adsorbent interaction energy can be taken as the negative of the potential energy minimum inside that pore.10-12 Solving eqs 3, 4, or 5 for the minimum potential energy, the relationship between the interaction energy and the pore half-width can be obtained for each micropore configuration, and for a given MPSD, the corresponding energy distribution can be derived from eq 2. 2. Adsorption Equilibria and MPSD. Having obtained the relationship between the adsorbate-adsorbent interaction energy and the pore size, we turn to the relationship between the adsorption equilibrium and the MPSD. Activated carbon is assumed to be composed of micropores of various sizes. For a particular pore of size r, the adsorbate-adsorbent interaction energy is E(r) obtained from the previous section. The local adsorption isotherm is assumed to take the form of the Langmuir equation:
Pb(r) Cµ(r,T,P) ) Cµs(T) 1 + Pb(r)
(6a)
(9) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon: Oxford, 1974. (10) Jagiello, J.; Schwarz, J. A. J. Colloid Interface Sci. 1992, 154, 225. (11) Jagiello, J.; Schwarz, J. A. Langmuir 1993, 9, 2153. (12) Hu, X.; Do, D. D. Langmuir 1994, 10, 3296.
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The affinity b(r) is the local adsorption affinity in the pore with a half-width of r. It has the following temperature dependence:
( )
b(r) ) b∞ exp
E(r) RT
(6b)
where b∞ is the adsorption affinity at infinite temperature and E(r) is the adsorbate-pore interaction energy, which is taken as the negative of the adsorption potential minimum in that pore and is a function of the pore size. This relationship depends on the assumption about the pore configuration, as discussed in section 1. If the adsorbate-adsorbate interaction is taken into account, the adsorption affinity in the isotherm equation can take the form
(
b ) b∞ exp
)
E(r) + nuθ RT
(6c)
where θ is the surface fractional coverage of the adsorbate, u is the adsorbate-adsorbate interaction energy and n is the number of neighboring adsorbed molecules. With eq 6c, the local adsorption isotherm is known as the Fowler-Guggenheim isotherm equation. The adsorption affinity b as given in eq 6b has the energy of interaction E(r), which is a function of the specific adsorbateadsorbent pair. Its value at infinity b∞ takes the following form to account for its dependence on different adsorbates
b∞ )
β xMT
(7)
Equation 7 is derived assuming the collision of gaseous molecules at the pore mouths of micropores, and it is known from the kinetic theory of gases that such a collision rate is inversely proportional to the square root of the molecular weight and to the square root of the temperature. The affinity parameter β in eq 7 is specific only to the solid adsorbent, and it has the unit of (K × g/mol)1/2/ kPa. The theoretical value of β was reported as 4.26 × 10-4 (K × g/mol)1/2/kPa.13 The maximum adsorption capacity of each species, Cµs, is assumed to take the following temperature dependence form:
Cµs(T) ) C°µs exp[δ(T0 - T)]
(8)
where C°µs is the adsorption capacity at some reference temperature T0 and δ is the thermal expansion coefficient. The MPSD in this study is assumed to follow the gamma distribution:
f(r) )
qγ+1rγe-qr Γ(γ + 1)
(9)
where q and γ are the two distribution parameters. There are no reasons for this particular choice of this distribution function. Other distributions can also be used to describe the micropore size distribution. With all the parameters defined, the overall adsorption isotherm of eq 1b can be expressed explicitly in terms of the MPSD for the adsorbate k as
Cµ,k )
∫
rmax
rmin
Cµs,k(T)
Pbk(r) qγ+1rγe-qr dr 1 + Pbk(r) Γ(γ + 1)
(10a)
where
bk(r) )
β
xMkT
( )
exp
Ek(r) RT
(10b)
The parameters rmin and rmax are the lower and upper limits for the pore size range, respectively. The lower limit rmin is the half-width of pores in which the adsorbate molecule has the same adsorption potential energy as that when it interacts with a single wall of the pore. Because for pores having a size smaller than (13) Hobson, J. P. J. Chem. Phys. 1965, 43, 1934.
this value the adsorption potential energy is higher than that corresponding to a single wall, adsorbate molecules would preferentially adsorb on a flat surface rather than in such pores. For 10-4 and 9-3 potentials, rmin is of 0.8885σsk and 0.7450σsk, respectively. For the 10-4-3 potential, however, rmin must be calculated numerically from eqs 5. The value of rmax takes the upper limit of the micropore range, but in this study it is taken as 3.5σsk for convenience, since the adsorption potential in pores with half width larger than this value is small and the contribution of those pores toward adsorption capacity is negligible. It should be pointed out that the value of the collision diameter σsk for each species governs the accessibility of that species in the MPSD and strongly affects the model performance. The model equations have two distinct groups of parameters to be optimized: (1) three structural parameters which characterize the carbon particle (they are q and γ (MPSD structural parameters) and β (solid affinity parameter)) and (2) the adsorbate-adsorbent parameters, C°µs(k), °sf(k), and δ. By simultaneously optimising the multiple temperature data of many adsorbates measured on the same carbon, it is possible to derive the results which are consistent with the structural information and equilibrium information contained in all experimental equilibria data. 3. Equilibrium Data and Solution Methodology. The experimental isotherm data of 11 species on two activated carbons, Ajax14 and Nuxit,15 are employed to examine the model equation. The isotherm data were measured using a volumetric technique. The molecular properties of adsorbates and the experimental temperatures are listed in parts a and b of Table 1 for Ajax-activated carbon and Nuxit-activated carbon respectively. As we stated before, the Lennard-Jones diameter of the adsorbate is a key parameter in the model, which dictates the accessibility of that species in the micropore network of the adsorbent. Different values for the same adsorbate do appear in different literature. In this study, we use the values reported by Breck.16 We take the sizes of propane and butane to be the same, and the size of benzene is 3.7 Å instead of the 5.34 Å reported in the thermodynamics because adsorption molecules tend to orient in such a way that their smallest configuration is oriented along the diffusion path into the micropore. Also due to this reason the diameter of toluene is taken as 3.8 Å, the same as that of methane. The nonlinear optimization technique in MATLAB is employed to fit the model to the experimental data. The integrals with respect to the micropore size over the required micropore size range are evaluated by using the Raudau quadrature. Since the interaction energies are very sensitive to the change in micropore size, 50 node points in the MPSD range are used to ensure the integration accuracy. All the simulations are performed on a Pentium 133 MHz. With the Langmuir equation as the local isotherm, the optimization takes about 3 h for all the equilibria data of eight adsorbates and at least three temperatures for each adsorbate on Ajax carbon with the optimization tolerance of 1 × 10-4.
Results and Discussions 1. Ajax-Activated Carbon. The optimization is first performed for the isotherm data of eight species measured on Ajax-activated carbon (Table 1a) with at least three different temperatures for each species. To investigate the effect of the slit pore configuration, the 10-4, 9-3, or 10-4-3 potentials are employed individually in the model equation with the local Langmuir isotherm equation. The reference temperature of the adsorption capacity for all species, T0, is taken as 273 K. The model fitting (continuous lines for the 10-4 potential, dotted lines for the 9-3 potential, and dashed lines for the 10-4-3 potential) and the experimental data (symbols) are plotted in Figure 1 for all eight adsorbates. The fitting is regarded as excellent in light of the fitting done on all isotherm data simultaneously (a total of 27 isotherm curves). The (14) Do, D. D.; Do, Ha D. Chem. Eng. Sci. 1997, 52, 297. (15) Valenzuela, D. P.; Myers, A. L. Adsorption Equilibrium Data Handbook; Prentice Hall: New Jersey, 1989. (16) Breck, D. W. Zeolite Molecular Sieves: Structure, Chemistry and Use; John Wiley & Sons: New York, 1974.
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Langmuir, Vol. 13, No. 23, 1997 6229
Table 1. Adsorbates and Experimental Temperatures
σk (Å) /κ (K) T (K)
(a) Ajax-Activated Carbon propane butane benzene
methane
ethane
3.8 148.6 258, 273 283, 303
3.9 300.0 283, 303 333
4.3 237.1 283, 303 333
methane
acetylene
ethylene
ethane
3.8 148.6 293, 313 333, 363
3.3 231.8 293, 313 333, 363
3.9 224.7 293, 313 333, 363
3.9 300.0 293, 313 333, 363
4.3 331.4 283, 303 333, 473
CO2
SO2
303, 363 473
3.3 195.2 258, 273 303
3.6 335.4 298, 323 373
propylene
propane
butane
CO2
4.5 248.9 293, 313 333, 363
4.3 237.1 293, 313 333, 363
4.3 331.4 293, 313 333, 363
3.3 195.2 293, 313 333, 363
3.7 412.3 303, 333 363
toluene 3.8
Nuxit-activated Carbon σk (Å) /κ (K) T (K)
Figure 1. Model fittings and experimental data on Ajaxactivated carbon: (s) 10-4 potential; (‚‚‚) 9-3 potential; (- -+) 10-4-3 potential; (a) methane; (b) ethane; (c) propane; (d) butane; (e) benzene; (f) toluene; (g) CO2; (h) SO2.
optimized results for each potential form are listed below in Table 2. For the 10-4 potential, the optimized structural parameters are q ) 21.57 Å-1, γ ) 97.87 (-), and β ) 1.03 × 10-4 (K × g/mol)1/2/kPa, and the isotherm parameters are listed in Table 2a. For the 9-3 potential,
the optimized structural parameters are q ) 14.75 Å-1, γ ) 60.25 (-), and β ) 1.01 × 10-4 (K × g/mol)1/2/kPa, and the isotherm parameters are listed in Table 2b. For the 10-4-3 potential, the optimized structure parameters are q ) 19.59 Å-1, γ ) 88.69 (-), β ) 1.04 × 10-4 (K × g/mol)1/2/kPa, and the isotherm parameters are listed in Table 2c. It is seen in Figure 1 that, using any of the three potentials, the adsorption equilibrium model using the MPSD can accurately describe the overall adsorption equilibria of each species over a wide temperature range. The micropore size distributions (MPSDs) derived from the 10-4, 9-3, and 10-4-3 potentials are shown in Figure 2. It can be seen that the derived MPSDs are similar for the cases of the 10-4 and 10-4-3 potentials. The MPSD derived from the 9-3 potential is shifted to the lower pore size range compared to those derived from the 10-4 and 10-4-3 potentials. This is because, in the case of the 9-3 potential, the maximum adsorption potential energy occurs at smaller pores, r ) 0.8584σsk, compared to r ) σsk in the case of the 10-4 potential. It is interesting to note that the affinity parameters β derived from the three potentials are almost the same, 1 × 10-4 (K × g/mol)1/2/ kPa, indicating that this affinity parameter characterizes the surface properties of carbon and that it is independent of adsorbate. The resulted average pore width of Ajax carbon is about 9.2 Å for the 10-4 and 10-4-3 potentials and 8.3 Å for the 9-3 potential. Although they all fall in the range of micropore size determined by experimental methods for activated carbon,17 the 10-4 potential (and the 10-4-3 potential) has been proven to be superior to the 9-3 potential in describing the microporous structure of activated carbon.18 Let us now turn to the isotherm parameters optimally obtained for each potential. It can be seen that, for the same species, the adsorption maximum capacities at 273 K, C°µs, obtained for the three potentials are very comparable, for example, for propane the values are 6.04, 5.96, and 6.00 mmol/g for the 10-4, 9-3, and 10-4-3 potentials, respectively. The thermal expansion coefficient δ is very small, suggesting that the saturation capacity is not affected by the change of temperature and the choice of pore configuration. The adsorption energy distributions for each species are obtained according to eq 2, and the average adsorption energies, Emean, for each species are then calculated from the resulting energy distributions and tabulated in Table 2 for the three potentials. It is noted that, for each potential, the minimum interaction energy of the adsorbate molecule with a single lattice layer (or slab), *sk, and the average adsorption energy, Emean, increase with the carbon number for a given class of adsorbate family (ethane, propane, butane). Aromatics have a higher energy of interaction, *sk, and Emean than those of paraffins, with toluene being stronger adsorbing (17) Innes, R. W.; Fryer, J. R.; Stoeckli, H. F. Carbon 1989, 27, 71. (18) McEnaney, B. Carbon 1988, 26, 267.
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Table 2. Optimized Isotherm Parameters for Each Speciesa methane C°µs (mmol/g) *sk (kJ/mol) δ (1/K) Emean (kJ/mol) C°µs (mmol/g) *sk (kJ/mol) δ (1/K) Emean (kJ/mol) C°µs (mmol/g) *sk (kJ/mol) δ (1/K) Emean (kJ/mol)
7.310 12.22 1.81 × 10-5 16.13 7.582 12.04 1.48 × 10-5 16.12 7.491 10.47 1.711×10-5 16.11
ethane
propane
benzene
toluene
CO2
SO2
8.040 16.85
(a) 10-4 Potential 6.040 4.881 19.73 23.84
4.851 34.36
4.412 35.80
11.34 15.64
18.72 18.94
22.84
28.76
45.12
48.07
17.91
26.93
7.961 16.73
(b) 9-3 Potential 5.961 4.842 20.03 24.17
4.850 33.63
4.414 35.21
12.29 14.81
17.80 18.60
22.86
29.11
45.15
48.56
17.83
24.90
8.018 14.39
(c) 10-4-3 Potential 6.004 4.863 16.70 20.18
4.845 29.49
4.393 30.68
12.11 13.40
14.91 16.92
22.21
27.96
44.56
47.16
18.75
24.02
Figure 2. MPSDs for Ajax-activated carbon derived from different potentials.
Figure 3. Optimized * sk from different potentials against their theoretical values.
species than benzene. Table 2 also shows that the values of Emean for the same species are not sensitive to the choice of the potential form; for example, for butane the mean interaction energies are 35, 35, and 34 kJ/mol for the 104, 9-3, and 10-4-3 potentials, respectively. Figure 3 shows plots of the optimized interaction energy *sk obtained from the fitting versus the theoretical interaction energy calculated from eqs 3b, 4b, and 5b for the 10-4, 9-3, and 10-4-3 potentials, respectively. The theoretical parameters are calculated using the molecular properties given by Reid et al.19 and Steele.9 It is noted that the optimal and theoretical values for *sk are very (19) Reid, R. C.; Prausnitz, J. M.; Polling, B. E. The Properties of Gases and Liquids; McGraw-Hill: New York, 1987.
butane
34.75
35.14
33.54
comparable to each other with the 10-4-3 potential giving the closest agreement. This may suggest that the pore configuration of the 10-4-3 potential is the closest to describe the structure of the activated carbon micropores. It is also found that the model works better for hydrocarbons than for SO2 and CO2. Although the model fitting for SO2 and CO2 is good by using all three potentials, the derived adsorption capacities are a bit high and the values of Emean calculated from different potentials are not the same. Two reasons may account for this. First, we do not have enough experimental data in the highpressure range which can exert some constraints on the adsorption capacity during optimization. Second, the dispersive force may not be enough to describe the interaction between polar adsorbates and the carbon surface. The effect of the local isotherm equation in this model is also investigated by applying the Folwer-Guggenheim equation to allow for the adsorbate-adsorbate interaction. After intensive computations, it is found that the adsorbate-adsorbate interaction energy for each species is very small compared with the adsorbate-adsorbent interaction and that this interaction for different species shows no regularity. In addition, the fitting goodness is not much improved by the introduction of the extra parameter. So it is concluded that the adsorbate-adsorbate interaction does not play an important role in this model. 2. Nuxit-Activated carbon. Next, the model is employed to fit the equilibrium data of the other activated carbon, Nuxit-activated carbon, of which the equilibria data for eight species are available, each of which has isotherm data at four temperatures (Table 1b). To show the potential of the model as well as its predictability, we choose the isotherm data of six species collected at three temperatures (283, 303, and 333 K) for optimization, and then we use the optimized parameters to predict the remaining equilibria data of all eight species. The model fittings (continuous lines for the 10-4 potential, dotted lines for the 9-3 potential, and dashed lines for the 104-3 potential) and the isotherm data (symbols) of the six species are shown in Figure 4. Three potential forms are employed in the simulation, and we presented below only the results from the optimization of the 10-4-3 potential. The optimized structural parameters are q ) 17.78 Å-1, γ ) 82.10 (-), β ) 3.80 × 10-5 (K × g/mol)1/2/Pa. The optimized isotherm parameters from the 10-4-3 potential are tabulated in Table 3. It can be seen in Figure 4 that the fitting is excellent for all six adsorbates studied over the range 283-333 K. Like the case for the Ajax-activated carbon studied in the last section, the thermal expansion coefficient for the saturation capacity (δ) is very small. The affinity parameters β resulting from different potentials (10-4, 9-3, 10-4-3) are about the same, 3.80 × 10-5 (K × g/mol)1/2/kPa, which is a bit smaller than that
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Langmuir, Vol. 13, No. 23, 1997 6231
(2) C°µs: This value is calculated by assuming that, for adsorbates of similar structures, the adsorption capacities on the same carbon are related to the liquid molar volumes, which can be evaluated by the formula proposed by Kapoor and Yang.20 Thus, by taking ethylene and propylene as the reference adsorbates, the adsorption capacities of ethane and propane on Nuxit carbon at 273 K are calculated as 6.279 and 5.191 mmol/g, respectively, using the related thermodynamic parameters of Perry and Green.21 Figure 6 shows the model predictions (dotted lines) and experimental data (symbols) for ethane and propane. Considering the way we choose the theoretical parameters, the predictions are satisfactory. This simple demonstration does not mean that we can rely on this method for equilibria information. But it does show the consistency of the MPSD toward each species, lending good support for this model as the tool to study the adsorption equilibria. 3. Multicomponent Equilibria. Finally, to test the MPSD derived in this paper and the adsorption equilibria model, the model is further explored by studying the multicomponent adsorption equilibria. By employing the extended Langmuir equation as the local isotherm, eq 10a can be expressed as
Cµ(k) )
∫r(k)
rmax min
P(k) b(r,k) Cµs(k,T) 1+
NC
P(i) b(r,i) ∑ i)1
qγ+1rγe-qr
dr
Γ(γ + 1)
(k ) 1, 2, 3, ..., NC) (11)
Figure 4. Model fittings and experimental data on Nuxitactivated carbon: (s) 10-4 potential; (‚‚‚) 9-3 potential; (- -)10-4-3 potential; (- ‚ ‚ -) model prediction with 10-4-3 potential; (a) methane; (b) acetylene; (c) ethylene; (d) propylene; (e) butane; (f) CO2.
of Ajax carbon (β ) 1.04 × 10-4 (K × g/mol)1/2/Pa), suggesting that the adsorption affinity parameter β at infinite temperature on Nuxit carbon is slightly weaker than that on Ajax carbon. The derived MPSD for this Nuxit carbon is plotted in Figure 5. Also plotted in the same figure is the MPSD of Ajax-activated carbon, and we see that the MPSDs of these two activated carbons are extremely comparable. The extracted optimized parameters obtained above are now employed to predict the adsorption equilibria of these six species at the other temperature (363 K). The model predictions using the 10-4-3 potential (dashed-dotted line) and the experimental data (inverted triangular symbols) are shown in Figure 4. The good prediction of the model at 363 K clearly demonstrates that the temperature dependence of the model is very satisfactory. To further test the model for its predictability, we use the model to predict the adsorption equilibria of the other two adsorbates, ethane and propane, on Nuxit-activated carbon at all four temperatures (293, 313, 333, 363 K). To perform this prediction, we need to calculate two parameters (*sk and C°µs) for ethane and propane. They are estimated as follows. (1) *sk: The optimized values of *sk for the six species used in the fitting are, on average, about 1.15 times the theoretical values calculated from eq 5b. So for ethane and propane, their *sk values are taken as their theoretical values multiplied by this factor (1.15). The resulting estimated values for *sk for ethane and propane are 15.75 and 19.30 kJ/mol, respectively.
Here NC is the number of species in the mixture. It should be pointed out that eq 11 suggests an ‘adsorbate-pore interaction’ mechanism for the adsorbate-adsorbate interaction in the adsorbed phase; that is, the adsorbates compete with each other only in the pores accessible to them. The smaller pores may be inaccessible to larger adsorbates due to the effect of size exclusion. For those small pores, the local extended Langmuir equation only accounts for the species accessible to these pores. The effect of size exclusion is reflected in the lower integration limit, r(k)min. The adsorbate-adsorbent interaction energy and the effect of size exclusion are shown graphically in Figure 7 for a methane (1)-propane (2) system in the case of the 10-4-3 potential. From this figure, in pores having halfwidths greater than rmin(2), both methane and propane are accessible, while in pores with half widths less than rmin(2), the single-component adsorption mechanism of methane will prevail due to the effect of the size exclusion of propane. The multicomponent equilibria of methane-ethylene and ethylene-propylene systems on Nuxit-activated carbon are first studied by the model using the single component equilibria parameters in Table 3. The model fittings (solid lines) and experimental data (symbols) are presented in Figure 8a and b for ethylene-methane and propylene-ethylene mixtures, respectively. As a reference to judge the model performance, the related simulation results from IAST22 theory are also illustrated in this figure (and figures hereafter) as dotted lines. The parameters used in IAST are taken from Valenzuela and Myers.15 We see from Figure 8 that the predictions are very satisfactory in describing the amount adsorbed of each component as a function of the gas mole fraction. (20) Kapoor, A.; Yang, R. T. Gas Sep. Purif. 1989, 3, 187. (21) Perry, R. H.; Green, H. Perry’s Chemical Engineer’s Handbook; McGraw-Hill: Singapore, 1985. (22) Myers, A. L.; Prausnitz, J. M. AIChE J. 1965, 11, 121.
6232 Langmuir, Vol. 13, No. 23, 1997
Wang and Do
Table 3. Optimized Isotherm Parameters of Six Species (Nuxit-Activated Carbon) C°µs (mmol/g) *sf (kJ/mol) δ (1/K) Emean (kJ/mol)
methane
acetylene
ethylene
propylene
butane
CO2
6.023 11.98 1.369 × 10-5 16.27
6.693 17.76
6.305 15.68
5.485 17.27
4.123 21.64
8.954 15.84
22.21
22.06
27.55
34.20
19.96
Figure 5. MPSDs of Nuxit-activated carbon and Ajax-activated carbon derived from the 10-4-3 potential.
Figure 8. Binary equilibria on Nuxit-activated carbon: (a) ethylene-methane, 293 K; (b) propylene-ethylene, 293 K; (c) propane-ethane, 293 K; (d) propane-ethane, 333 K.
Figure 6. Model prediction of the adsorption equilibria of ethane and propane on Nuxit carbon by using the optimized structural parameters: (a) ethane; (b) propane; (s) this model; (‚‚‚) IAST.
Figure 7. Energy matching and effect of size exclusion for the methane-propane system.
Next we study the binary equilibria of the ethanepropane system on Nuxit-activated carbon. The singlecomponent equilibria parameters for ethane and propane are obtained by optimizing the experimental data of eight species on Nuxit carbon simultaneously. In doing so, we obtain the optimized isotherm parameters for ethane and propane. The optimized structural parameters obtained from this fitting of eight species are almost identical to those obtained earlier when fitting was done with only six
species. The model predictions of multicomponent equilibria of ethane-propane on Nuxit-activated carbon are shown in Figure 8c and d at two different temperatures, 293 and 333 K, respectively. Like the cases of ethylenemethane and propylene-ethylene, the prediction for this binary mixture is also good. Finally, the multicomponent adsorption equilibria on Ajax-activated carbon collected in our laboratories are studied.23 The optimized parameters for the single component from the 10-4-3 potential (Table 2c) are utilized in the model prediction. The parameters derived from the Langmuir isotherm equation are used in the IAST simulation. The following mixtures are studied: methane-ethane, methane-propane, and methane-CO2 (273 and 303 K). The model predictions (continuous lines) and experimental data (symbols) are presented in Figure 9. It is seen from this figure that, compared with the binary equilibria predicted by the IAST, the model predictions are regarded as good. 4. Discussion of the Optimized Parameters. 4.1. MPSDs. The derived MPSDs from the 10-4-3 potential for the two activated carbons studied in this paper are compared in Figure 5. It is seen that Nuxit-activated carbon has a slightly larger average pore size and variance than Ajax-activated carbon does. However, the difference is not very significant. Let us now discuss the physical meaning and the reliability of the derived MPSD. The microporous structure of activated carbon is very complex; the assumption of any form of f(r) will likely impose some constraint upon the inverse process of eq 1b. Furthermore, the assumption of the slit-shaped micropore with infinite length is also an oversimplified picture of a real micropore, in which the other sources of surface heterogeneity such (23) Ahmadpour, A.; Wang, K.; Do, D. D. Carbon Conference, 1997, State College, PA.
Micropore Size Distribution of Activated Carbon
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the demonstration in this study, it is suggested that the species which present the same adsorption mechanism on activated carbon are a good choice, and these species should also have as wide a range of interaction energies with the carbon surface as possible. 4.2. Number of Parameters. The model has 2N + 4 fitting parameters. But as shown before, the thermal coefficient is very small for all species and can be eliminated without seriously affecting the goodness of fit. The parameters to be optimized is then reduced to 2N + 3. The fitting parameter can be further reduced by using the expression
Cµs(T) )
Figure 9. Binary equilibria on Ajax-activated carbon: (a) methane-ethane, 303 K; (b) methane-propane, 303 K; (c) methane-CO2, 273 K; (d) methane-CO2, 303 K.
as the geometric irregularity of the micropores, edge effect, functional groups, etc. are not accounted for (other methods like SWQ, H-K, and TVFM all suffer from the same problem). Other studies24,25 also reveal that, due to the sensitivity of the L-J potential theory to the pore size, the MPSD thus obtained will locate mainly in the region of one to two L-J diameters. So the MPSD derived this way represents the lumped effect of micropores of all sizes on the overall adsorption equilibria. Considering the complexity of the carbon structure and the two-parameter gamma distribution function, the MPSD obtained this way should be treated as qualitatively rather than quantitatively. Nevertheless, it does provide very useful information about the way micropore sizes are distributed, and this information has been utilized very effectively by Do and co-workers in the dynamics calculation.26 Our emphasis in this paper is to present a model which is easy to use and able to account for the role of surface heterogeneity. From a practical point of view, this model offers several advantages: (1) The MPSD derived is based on the extensive equilibria information of many species, which is necessary to account for surface heterogeneity. (2) The model gives perfect fitting for the equilibria of many species over a broad range of temperature. (3) The model has a predictive capability in describing the multicomponent equilibria. It is difficult to specify the optimal number of species for which equilibria information should be included in the data pool to derive the MPSD parameters, since the ‘real’ MPSD of an activated carbon is still unavailable at this stage. It depends on the system of interest. But from (24) Jagiello, J.; Bandosz, T. J.; Putyera, K.; Schwarz, J. A. J. Chem. Soc., Faraday Trans. 1995, 91, 2929. (25) Jagiello, J.; Bandosz, T. J.; Schwarz, J. A. Langmuir 1996, 12, 2837. (26) Do, D. D.; Wang, K. Carbon, accepted.
W0 v(T)
(12)
where W0 is the micropore volume of the activated carbon and v(T) is the molar volume of the adsorbate in the adsorbed phase at temperature T. If W0 is taken as the fitting parameter and v(T) is calculated from the liquid molar volume of adsorbate at that temperature, the fitting parameters are significantly reduced to N + 4! However, some care must be exercised when eq 12 is used, since the density of the adsorbed phase in the micropore can be very much different from that of the pure liquid.4 The mean field theory proposed by Evans and Tarazona27 could be a good choice to calculate the v(T) in micropores, which is theoretically sound and consistent with the assumption in this model. However, the model will become very complicated and time-consuming in the computation, losing its simplicity. Conclusion We have proposed and examined a new approach to characterize the MPSD of activated carbon by using the multiple-temperature isotherm data of many adsorbates simultaneously. The MPSD is assumed to be the intrinsic property of activated carbon and the sole source of adsorption energy heterogeneity. The adsorbate-adsorbent interaction is described by Lennard-Jones potential theory and the pore configuration, of which the 10-4-3 potential is found to be the best selection. The model catches the basic feature of surface heterogeneity on activated carbon, and the MPSD derived is consistent with the equilibria properties of many adsorbates studied. Extensive equilibria data on two activated carbons of many adsorbates over a broad temperature range are employed to validate the model, and it is found that the model can characterize the MPSDs of activated carbons on a qualitative basis. The model shows the correct temperature dependence, and by using the equilibria information of some species, it could approximately simulate the adsorption equilibria of similar species on that carbon. Using the MPSD parameters and the single-component equilibira data, the model is found to be able to predict binary equilibria of systems studied. The simplicity and flexibility of the model makes it an appealing choice in describing the adsorption energy heterogeneity in dynamic studies. Acknowledgment. This project is supported by the Australia Research Council (ARC). LA970474K (27) Evans, R.; Tarazona, P. J. Chem. Phys. 1986, 4, 2376.
