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A New Dubinin-Radushkevich-Modified BET Combined Equation To Correlate with a Single Procedure the Full Relative Pressure Range of 77 K N2 Isotherms of Solids with Different Textural Properties Antonio Marcilla, Amparo Go´mez-Siurana,* and Francisco J. Valde´s Departamento de Ingenierı´a Quı´mica, UniVersidad de Alicante, Apdo. 99, 03080 Alicante, Spain
In this work, a model capable of adequately correlating and reproducing experimental adsorption isotherm data in the overall relative pressure range is suggested. The equation includes a Dubinin-Radushkevich term and a modified BET term that avoids the discontinuity of BET equation at high relative pressures. Thus, the typical independent application of the Dubinin-Radushkevich and BET equations, involving the need of a selection of the respective relative pressure range, is substituted by a single optimization procedure, permitting the calculation of the values of the parameters characterizing the textural properties of solids. The validity of the procedure has been checked through its application to several experimental examples, corresponding to types I, II, and IV isotherms, showing excellent results in all of the cases. 1. Introduction Gas adsorption is a very important unit operation in chemical engineering and a worldwide accepted methodology for determining the textural properties of solids. Such properties are basic parameters to understand the mechanisms involved in catalytic processes and the role of heterogeneous catalysts. Many theories have been developed to explain observed isotherms and to account for the different possible adsorption mechanisms.1-4 The Dubinin-Radushkevich (DR) equation, for micropore filling, and the BET theory, for monolayer coverage, are the base for the software of a lot of commercial adsorption equipment and represent the first choice for the analysis of solids. Dubinin-Astakhov and many other equations5-7 have been developed for analyzing the microporosity (i.e., the volume of micropores and micropore surface area). The t or R methods7,8 are also widely applied to determine the micro- and mesopore surface area. Other methods are based on the assumption of a distribution of pores of different size, each one with its own contribution to the overall porosity of the sample. To describe a bimodal micropore size distribution, Dubinin applied a two-term equation, based on the assumption that there are two separate ranges of pore size,7 and an important advance in the application of the DR equation to heterogeneous micropore structures was made by Stoeckli, assuming a Gaussian distribution to obtain a continuous distribution of pore size.7,9 Recently, the limitations of the computational procedures for the pore size analysis have been pointed out,3,4,10 and the density functional theory (DFT) has become an important tool for the characterization of porous materials.11-13 The approach is based on the established principles of statistical mechanics and assumes a model solid structure and pore topology.3,12 However, even this approach is of limited value unless the adsorbent has a known bulk and surface structure together with rigid pores of known shape (or shape distribution).4 Currently, it seems that two different tendencies coexist for interpreting and reporting textural characteristics of solids. On one side, there are researchers in adsorption processes, heterogeneous catalysis, and materials science applied to porous materials, which apply models and theories very well developed, * To whom correspondence should be addressed. Tel.: +34 96 590 2953. Fax: +34 96 590 3826. E-mail:
[email protected].
based on the recent studies of the molecular interactions between the adsorbent and the adsorbate (see refs 3, 4, 10-13). On the other hand, there is another group of researchers who are more interested in the behavior during the applications of such materials than in the molecular mechanisms involved, which report the textural parameters as a way for the bulk characterization of solids. In this way, Marcilla et al.14 report a study where the existence of many works applying conventional models to characterize porous solids as well as the drawbacks of such models is pointed out. The main problem arises from the fact that it is relatively frequent to find cases where the models are applied and the parameters are reported without critique nor analysis of the validity and the coherence of the results obtained. According to Groen et al.,15 the existence of different physical phenomena during adsorption measurements could significantly affect the adsorption isotherm, leading to the incorrect assessment of both micropore and mesopore size calculations. This is particularly so in the case of the novel materials with small mesopores and the modified materials with combined microand mesopores, which are frequently influenced by phenomena such as the tensile strength effect and fluid-to-crystalline-like phase transitions of the adsorbed phase.15-19 As previously stated, numerous works have been published focusing on the study of the difficulties in measuring the textural properties of mesoporous materials by conventional methods. However, currently, there are many works applying the t-plot method to such materials and concluding that the samples have no microporosity based on the procurement of a negative intercept in the t-plot. Sonwane and Bhatic20 state that many classical models are popular because they are embodied in the software supplied along with commercial adsorption equipment, and a similar observation has been carried out by Groen et al.,15 which states that the analysis of the more sophisticated materials should not be used following a “press button” approach in commercial software. As an example, Selvam et al.17 recognize the problems related to the application of the t-plot method to mesoporous materials and the fact that the success of the method depends on the choice of the reference isotherm selected to determine the dependence of t on the relative pressure. In this way, several models have been developed, based on different approaches,17-20 to adequately represent the behavior of the porous solids. Nevertheless, it is not impossible to
10.1021/ie900912x CCC: $40.75 2009 American Chemical Society Published on Web 11/06/2009
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represent the full range isotherm for any solid with a very simple equation. This will be proven in the following paragraphs and may be found very convenient, especially from an engineering point of view, because it would allow having a continuous equation representing the overall isotherm. Additionally, the behavior of the different solids could be more conveniently compared and correlated with the characteristic parameters of such equations. Thus, the objective of the present Article is to suggest and present such equations and to illustrate with some examples their capability to represent the experimental isotherms of very different solids. 2. Suggested Model In a previous paper,14 typical primary methods were discussed to analyze the texture of different adsorbents. We have illustrated some drawbacks of such methods and have suggested a procedure to improve the results on the basis of a combined Dubinin-Radushkevich and BET equation. Nevertheless, such a combined equation is not capable of representing type IV isotherms and cannot be applied in the full range of relative pressures, failing at high relative pressures close to 1. To overcome such drawbacks, the following equation has been suggested: 2
V)
∑V
[
0i · exp
i)1
-ki · ln
P P0
+
(
)
P n-1 P 1P0 P0 P 1 + (C - 1) P0
WmC
( )] ( 2
)
(1)
Such an equation represents a linear combination of three terms, that is, two corresponding to a contribution of two DR (i.e., grouped in the summatory of eq 1), and the second corresponds to a modified BET term. In this equation, V is the amount of N2 adsorbed at a relative pressure P/P0, V0i is the volume of the pores corresponding to the i DR term, ki is the corresponding characteristic parameter, Wm is the volume of N2 corresponding to the monolayer coverage (according to the BET theory for the parameter n ) 0), C is the typical C constant of the BET equation, and n is a parameter that avoids the discontinuity of the BET equation at P/P0 ) 1. The parameters of this equation have certain restrictions and must be all positive and n must be lower than 1. Alternatively, two Dubinin-Astakhov7 terms could be used, if required, instead of the two DR terms. This approach is inspired in the procedure suggested by Remy and Poncelet,21 which analyzed the problems related to the ambiguity of the external surface area obtained by the t-plot method due to the standard material selected, the range of supposed linearity of the t-plot, and different possible mechanisms, and presented a new procedure to obtain the adsorption characteristics of different materials. The method they suggested improves the results for the parameters obtained for the volume of micropores, avoiding spurious values of the C constant of the BET equation. Unfortunately, this procedure has not been widely extended, and the application of the DR and BET methods separately is still very popular. Moreover, papers can frequently be found reporting external surfaces from data yielding negative values for the C constant, and only the micropore volume and the external surface with no mention of the characteristic energy of adsorption of the DR equation or the C constant of the BET equation.14 According to Remy and Poncelet,21 the adsorptions on the external surface and in the micropores are independent processes, and the experimental isotherm can be decomposed into two isotherms corresponding to each type of adsorption.
Figure 1. Experimental isotherm of the 5A sample and prediction from the DR and the BET equation with the parameters corresponding to the conventional analysis.
However, the method described by Remy and Poncelet21 corrects the influence of the micropores contribution to the BET equation, but does not take into account the possible contribution of the adsorption on the external surface at low pressures. Moreover, the validity of their approach depends on the type of solid being analyzed or, mathematically, on the value of the constants of the equations used to describe the two mechanisms, and can only be applied to type I + type II isotherms. Contrarily, eq 1 can be adapted to every adsorbent material, modifying the number of terms of the summatory, and fitting the appropriate n value. Calculation Procedure. In this work, eq 1 is suggested with the objective of representing the adsorption isotherm in the full relative pressure range, that is: 0.005 < P/P0 < 0.995. At very low pressures, diffusional problems have been described, and other adsorbates may be preferred.22,23 At pressures closer to 1 than 0.995, other effects may be present. An equation such as that suggested cannot be linearized, and nonlinear parameter estimation procedures must be used. To obtain the parameters of eq 1 that better fit certain experimental adsorption isotherm data, the solver tool of the excel calculation sheet has been used with the following objective function: nd
OF )
∑ (V
c i
- Vie)2
i)1
e VP/P 0)0.5
(2)
where Vic represents the calculated amount adsorbed at each pressure, Vie is the corresponding experimental value, and e VP/P is the experimental value of the adsorbed volume at 0)0.5 the relative pressure of 0.5. 3. Examples We have selected several examples to illustrate the possibilities of this equation, corresponding to typical isotherms of type I, I + II, and IV of the IUPAC classification.24 Example 1: IUPAC Type I Isotherms. Figure 1 shows a typical quasi-type I isotherm corresponding to a 5A molecular sieve, with a CAS number of 308080-99-1 supplied by PROLABO. The parameters of the DR and BET typical analysis are shown in Table 1. Figure 1 also shows the calculated isotherms in the full range of relative pressure, as obtained from the DR and the BET equations with the parameters corresponding to the conventional analysis. As can be seen, a good
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Table 1. Parameters of the DR and BET Typical Analysis of the Experimental Isotherms Shown in Figures 1, 7, and 11 parameter 3
V0 (cm liquid N2/g) k Wm (g N2/g) C
5A sample
HBeta
MCM-41
0.215 0.00404 0.112 -31.6
0.241 0.00930 0.140 -46.2
0.308 0.0230 0.325 26.9
agreement can be observed for the DR equation up to relatively high relative pressures, but the BET equation fails and only correlates a very narrow relative pressure range. In addition, the BET equation provides a negative value of C and the corresponding discontinuity at low pressures. In this way, the application of an equation such as (1), with one DR term and one BET term, could permit the fitting of the adsorption isotherm in the overall relative pressure range (see Figure 2 and Table 2): the DR term mainly accounts for the adsorption behavior of the microporosity of the sample, and the corresponding parameters, V0 and k, characterize such microporosity, whereas the BET term contributes with the possibility of additional adsorption occurring in the very high relative pressure range. In this case, the Wm parameter represents the contribution of the monolayer to the adsorption outside of the micropores. At this point, it is interesting to observe that, when the BET analysis of the sample shown in Figure 1 is carried out, all of the adsorption accounted for in the 0.05-0.35 range is attributed to the monolayer. However, it is obvious that for this sample, the main adsorption occurs in the micropores, and there is only little contribution of adsorption outside the micropores. Therefore, when eq 1 is applied, the calculated value for Wm would represent the “true” monolayer weight.14 Figure 2 and Table 2 show the results obtained with the application of such an equation to the experimental data with the optimization of the objective function (2). In this case, a value of n ) 0 has been considered, and then the calculated curve tends toward infinite as the relative pressure tends to 1. The optimized values for the parameters of eq 1 are V0 ) 0.224 cm3 liquid N2/g, k ) 0.00656, Wm ) 0.0004 g N2/g, C ) 2.80, and OF ) 2.79. Figure 2 and Table 2 also show the results obtained when the parameter n has been optimized as well. As can be seen, in this case, even the final section of the adsorption isotherm can be perfectly reproduced. The values of the parameters obtained when n * 0 (n ) 633) are V0 ) 0.217 cm3 liquid N2/g, k ) 0.00415, Wm ) 0.0114 g N2/g, C ) 0.193, n ) 0.633, and OF ) 0.0273. Figure 2 reflects that the application of the procedure proposed in this work permits us to perfectly reproduce an experimental type I isotherm as that shown by the 5A sample. As previously discussed, the DR term contribution explains the microporosity
of the sample, whereas the BET term explains the possibility of adsorption outside the pores, especially at high relative pressures. The modification of the BET equation when multiplying by the term (1 - P/P0)n avoids the discontinuity shown by the BET equation, which tends to infinite when P/P0 tends to 1. This modification provides very high flexibility, and the curvature degree as well as the point where such a curvature appears can be perfectly correlated. To illustrate the abovementioned behavior, Figure 3 shows the predicted isotherms considering the previous values of the parameters (V0 ) 0.217 cm3 liquid N2/g, k ) 0.00415, Wm ) 0.0114 g N2/g, C ) 0.193) and different values of n. As can be seen, eq 1 with a DR term and a modified BET term can be adapted for very different situations, and, obviously, a pure type I isotherm would be fitted adequately with a value of n ) 0, whereas a pure type II isotherm will require n * 0 values. Example 2: Series of Activated Carbons with Different Burn Off Degree. Figure 4 shows the experimental N2 adsorption isotherms corresponding to a series of activated carbons25 with different burn off degrees (BO). The corresponding DR and BET parameters, as well as the corresponding burn off degrees, are shown in Table 3. Figure 5 shows, as an example, the calculated isotherms as obtained from the DR and the BET equation with the parameters corresponding to the conventional analysis for sample 1. Similar results are obtained for the other samples. As in the previous cases, a good agreement can be observed in the range where the linearization of the equations has been applied, but fails out this range. As expected, a linear relation exists between the micropore volume and the burn off degree as well as between the monolayer weight and the burn off degree, leading to the following relationships: V0 (cm3 liquid N2/g) ) 0.134 + 0.385 · BO R2 ) 0.991 (3) Wm (g N2/g) ) 0.0782 + 0.234 · BO R2 ) 0.993
In this case, the application of the proposed model has been carried out by simultaneously fitting the four experimental adsorption isotherms, and applying eq 1 in the overall pressure range with the following considerations: setting n * 0; a DR type term and a BET type term; the same values of the parameters k and C for the four curves; and the values of the parameters V0 and Wm as linear functions of the burn off degree, and fitting the corresponding slope and intercept. Therefore, the resulting equation is as follows:
[ ( )]
V ) (iV + sV·BO) · exp -k · ln
P P0
2
+
(
P P 1P0 P0 P 1 + (C - 1) P0
(iW + sW · BO)C
(
Figure 2. Comparison between the experimental isotherm shown in Figure 1 and that calculated by fitting eq 1.
(4)
)
)
n-1
(5)
where iV and iW are the intercepts and sV and sW are the slopes, and the subindexes V and W refer to the linear relationships corresponding to the micropore volume and the monolayer weight, respectively. It must be pointed out that the shape of the experimental isotherms suggests that the contribution of the BET terms must be low, and, therefore, the use of eq 5 with n ) 0 could be enough for adequately fitting the set of isotherms. However, the agreement between the experimental and the calculated isotherms in the high relative pressure range is relatively poor, and therefore it is preferable to use an n *
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Table 2. Comparison among the Experimental and Calculated Values for the Different Adsorption Isotherms of the Samples Studied in This Work through the Root Mean Square of Residuals (rms)a rms of residuals 3
sample 5A (example 1)
activated carbon-sample 1 (example 2) activated carbon-sample 2 (example 2) activated carbon-sample 3 (example 2) activated carbon-sample 4 (example 2)
HBeta zeolite (example 3)
MCM-41 (example 4)
DR prediction
BET prediction
Vcalculated (cm STP/g) (eq 1, n ) 0)
Vcalculated (cm3STP/g) (eq 1, n ) 0.633)
109.7
5 685 028
8.3
0.1
DR prediction 13.1
BET prediction 547.6
Vcalculated (cm3 STP/g) (eq 1) 1.8
15.9
638.5
4.6
65.8
523.1
4.5
24.1
813.9
2.8
DR prediction
BET prediction
115.07
5705.36
DR prediction 255.2
BET prediction 8602.6
Vcalculated (cm3 STP/g) (fit of Figure 8) 15 238.10
Vcalculated (cm3 STP/g) (fit of Figure 9) 52.09
Vcalculated (cm3 STP/g) (fit of Figure 10) 4.24
Vcalculated (cm3 STP/g) (eq 1) 9.6
a The calculated curves are shown in Figures 1, 2, and 4-12. The values of rms have been calculated as follows: rms ) [∑(calculated values measured values)2/number of points)]1/2.
0 value. The four adsorption isotherms corresponding to the four carbon samples with different burn off degrees have been fitted simultaneously, with only seven parameters: iV, iW, sV, sW, k, C, and n. The results obtained from the fit carried out in the described conditions are shown in Figure 6 and Tables 2 and 4, where the V0 and Wm values corresponding to each
Table 3. Parameters of the DR and BET Typical Analysis of the Experimental Isotherms Shown in Figure 4 sample
burn off degree
V0 (cm3 liquid N2/g)
k
Wm (g N2/g)
C
1 2 3 4
0.237 0.344 0.468 0.546
0.221 0.273 0.311 0.343
0.0071 0.0091 0.0124 0.0124
0.133 0.161 0.184 0.208
-54.4 -58.0 -62.5 -65.4
sample have been obtained with the following equations, and considering the respective burn off degrees:
Figure 3. Comparison between the different adsorption isotherms predicted varying the value of the parameter n.
Figure 4. Experimental adsorption isotherms corresponding to a series of activated carbons with different burn off degrees.
V0 (cm3 liquid N2/g) ) 0.147 + 0.294 · BO
(6)
Wm (g N2/g) ) 0.125 · BO
(7)
As can be seen, the agreement between the experimental and the calculated data is very satisfactory. Example 3: IUPAC Type II Isotherm. Figure 7 shows a typical type II isotherm corresponding to an HBeta zeolite supplied by Su¨d-Chemie, which has been described elsewhere.26 The parameters of the DR and BET typical analysis are shown in Table 1. Figure 7 also shows the calculated isotherms in the
Figure 5. Prediction of the adsorption isotherm of sample 1 of Figure 4, obtained from the DR and the BET equation with the parameters corresponding to the conventional analysis.
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Figure 6. Comparison between the experimental isotherms shown in Figure 4 and those calculated by fitting eq 1 in the overall relative pressure range. Table 4. Results Obtained from the Simultaneous Fit of the Experimental Adsorption Isotherms Shown in Figure 4 with the Model Proposed in the Present Work (n ) 0.885) sample
V0 (cm3 liquid N2/g)
k
Wm (g N2/g)
C
1 2 3 4
0.216 0.248 0.284 0.307
0.00588 0.00588 0.00588 0.00588
0.0295 0.0428 0.0582 0.0680
6.45 6.45 6.45 6.45
Figure 9. Comparison between the experimental isotherm shown in Figure 7 and that calculated by fitting eq 1 with one DR term and n ) 0 in the overall relative pressure range. The respective contribution of each term of eq 1 is also shown.
full range of relative pressure as obtained from the DR and the BET equations and the parameters corresponding to the conventional analysis. The corresponding root mean squares of
Figure 10. Comparison between the experimental isotherm shown in Figure 7 and that calculated by fitting eq 1 with one DR term and n * 0 in the overall relative pressure range. The respective contribution of each term of eq 1 is also shown.
Figure 7. Experimental HBeta isotherm and prediction from the DR and the BET equation and the parameters corresponding to the conventional analysis.
Figure 8. Comparison between the experimental isotherm shown in Figure 7 and that calculated by fitting eq 1 with one DR term and n ) 0 in the 0-0.9 relative pressure range. The respective contribution of each term of eq 1 is also shown.
the residuals are shown in Table 2. In this case, the DR equation shows a good agreement up to relative pressures of around 0.25, and the BET equation only shows a certain agreement in the range where it has been applied, failing out of this range, and providing negative values of C and the two corresponding discontinuities. Figures 8 and 9 show the fitting to the suggested equation by setting n ) 0 and using one DR type term. The respective contributions of the DR and BET terms are also shown in Figures 8 and 9. In the first of these two figures, the fitting has been carried out up to 0.9 relative pressure, whereas the full range has been attempted in the second case. It can be observed that the fittings are clearly improved with respect to the single DR or BET cases shown in Figure 7, but the high pressure range cannot be represented, and the calculated isotherms fail at the highest values or relative pressure. Therefore, it can be concluded that eq 1 with n ) 0 is a very good option for representing a very wide range of relative pressure, but fails when trying to fit the overall range (up to 0.995), thus making it necessary to modify the BET term proposed in the present work. Figure 10 shows the fit obtained with the suggested equation using one DR term and n * 0 (see the corresponding root-mean-square of residuals in Table 2). It can be seen that the equation represents the experimental data with a high degree of accuracy. The values of the parameters obtained with this fit are the following: V0 ) 0.251 cm3 liquid N2/g, k ) 0.0109, Wm ) 0.218 g N2/g, C ) 0.135, n ) 0.839, and OF ) 2.31.
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Figure 11. Experimental type IV isotherm and prediction from the DR and the BET equation with the parameters corresponding to the conventional analysis.
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use of n * 0 provides a marginal improvement to the fitting and is not considered necessary. The examples shown in the previous paragraphs illustrate the ability of the proposed procedure to adequately reproduce adsorption isotherms of very different shapes, corresponding to materials with very different textural properties. Obviously, the number of terms of eq 1 to be used for each case must be minimized, and the situation yielding an acceptable fit in the overall relative pressure range with the lower number of parameters must be chosen. As an example, for correlating a type I isotherms, only one DR term is needed, and more terms should not be used. Contrarily, a type IV isotherm would need two DR terms and one modified BET term (either with n ) 0 or with n * 0), depending on the shape of the isotherm. The BET term must be considered when some adsorption occurs in the high relative pressure range, and the value of n * 0 must be taken into account for reproducing the adsorption in the very high relative pressure range. Therefore, an additional term, or an additional parameter, should only be included when the quality of fitting is significantly improved. A variance analysis may be very useful in minimizing the number of terms and for deciding the use of n ) 0 or n * 0. 4. Conclusions
Figure 12. Comparison between the experimental isotherm shown in Figure 11 and that calculated by fitting eq 1 with two DR terms and n ) 0 in the overall relative pressure range. The respective contribution of each term of eq 1 is also shown.
Example 4: IUPAC Type IV Isotherm. Figure 11 shows the experimental N2 adsorption isotherm corresponding to a mesoporous MCM-41 sample synthesized as described elsewhere,27 and the corresponding DR and BET parameters are shown in Table 1. The calculated isotherms obtained from the DR and the BET equations with the parameters corresponding to the conventional analysis are also shown in Figure 11, where the agreement for the DR equation up to relative pressures of around 0.05 can be observed and that of the BET equation in the range where it has been applied, both failing out of these ranges. Figure 12 shows the fitting in the overall relative pressure range to the equation proposed in the present work, by setting n ) 0 and using two DR-type terms. The comparison among the experimental and calculated values through the root-meansquare of residuals can be seen in Table 2. As can be seen, the agreement degree between the experimental and the calculated data is very satisfactory. The corresponding values of the parameters are: V01 ) 0.381 cm3 liquid N2/g, k1 ) 0.0329, V02 ) 0.444 cm3 liquid N2/g, k2 ) 0.666, Wm ) 0.585 g N2/g, C ) 2.85 × 10-5, n ) 0, and OF ) 11.6. Despite that the suggested model must be considered as an empirical model, some information can be extracted from the values of the parameters, and in the present case, where the parameter n has been maintained equal to zero, (i.e., each term of eq 1 maintains the same form than DR and BET equations), the values of the parameters obtained from fitting the experimental adsorption data permit a coarse determination of the micropore (V01) and mesopore (V02) volumes to be obtained. For this isotherm, the
The suggested model, which includes two DR type terms and one modified BET term, in combination with a nonlinear regression procedure, permits the isotherm of any adsorbent in the full relative pressure range to be represented. The number of DR terms required to represent a given isotherm must be analyzed and the second DR term used only when necessary, for example, for type IV isotherms. The modification of the BET term, that is, using the parameter n * 0, avoids the discontinuity of the BET equation at P/P0 ) 1 and allows the fitting of the adsorption data at high relative pressures. Literature Cited (1) Valladares, D. L.; Rodrı´guez-Reinoso, F.; Zgrablich, G. Characterization of Active Carbons: The Influence of the Method in the Determination of the Pore Size Distribution. Carbon 1998, 36, 1491. (2) Sing, K. S. W. Adsorption Methods for the Characterization of Porous Materials. AdV. Colloid Interface Sci. 1998, 76-77, 3. (3) Sing, K. S. W. The Use of Nitrogen Adsorption for the Characterisation of Porous Materials. Colloids Surf., A 2001, 187-188, 3. (4) Sing, K. S. W. Characterization of Porous Materials: Past, Present and Future. Colloids Surf., A 2004, 241, 3. (5) Dubinin, M. M. Progress in Surface and Membrane Science; Academic Press: New York, 1975; Vol. 9. (6) Parfitt, G. D.; Sing, K. S. W.; Urwin, D. The Analysis of the Nitrogen Adsorption Isotherms of Microporous Materials. J. Colloid Interface Sci. 1975, 53, 187. (7) Rouquerol, F.; Rouquerol, J.; Sing, K. Adsorption by Powders and Porous Solids; Academic Press: London, 1999. (8) Lowell, S.; Shields, J. E. Powder Surface Area and Porosity; Chapman and Hall: London, 1991. (9) Hutson, N. D.; Yang, R. T. Theoretical Basis for the DubininRadushkevitch (D-R) Adsorption Isotherm Equation. Adsorption 1997, 3, 189. (10) Yun, J. H.; Duren, T.; Keil, F. J.; Seaton, N. A. Adsorption of Methane, Ethane, and Their Binary Mixtures on MCM-41: Experimental Evaluation of Methods for the Prediction of Adsorption Equilibrium. Langmuir 2002, 18, 2693. (11) Ustinov, E. A.; Do, D. D. Modeling of Adsorption and Nucleation in Infinite Cylindrical Pores by Two-Dimensional Density Functional Theory. J. Phys. Chem. B 2005, 109, 11653. (12) Ustinov, E. A.; Do, D. D. Modeling of Adsorption in Finite Cylindrical Pores by Means of Density Functional Theory. Adsorption 2005, 11, 455.
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ReceiVed for reView June 3, 2009 ReVised manuscript receiVed October 20, 2009 Accepted October 21, 2009 IE900912X
