4474
Langmuir 1996,11, 4474-4479
A Comparison of Methods for the Analysis of Adsorption-Desorption Isotherms of Microporous Solids M. A. Parent and J. B. Moffat" Department of Chemistry and the Guelph-Waterloo Centre for Graduate Work in Chemistry, University of Waterloo, waterloo, Ontario, Canada N2L 3Gl Received May 2, 1995. I n Final Form: August 3, 1995@ Information on the surface areas, pore volumes, and pore size distributions has been obtained from the analyses of the N2 (78K)adsorption-desorption isotherms of two solids by application of a number of currently available methods. The results are employed to compare and contrast the various calculational techniques.
Introduction Porous solids continue to attract the attention ofworkers interested in heterogeneous catalysis and ad~orption.l-~ The presence of a pore structure will, in general, increase the area of the catalyst available for interaction with the molecules of the reactants, shape selectivity may be possible with micropores, and larger pores may be advantageous for trapping undesirable components contained in a feed ~ t r e a m . ~Information %*~ on the pore size distributions is evidently important not only for the characterization of catalysts prior to their use but also for evaluating the morphological changes which such solids display under realistic reaction conditions, as a result of not only specific reaction parameters but also the effects of processes leading to both chemical and physical deactivation. The microporous structures of solids applicable in various adsorption processes are of no less importance. While the morphological properties of a wide variety of these solids have been studied over the years, considerable recent attention has been focused on carbons of various type^.^^^ Although a variety of methods is available for the measurement of pore size distributions under special circumstances, the determination and analysis ofgaseous adsorption data is undoubtedly the most commonly employed for such purpose^.^ It is well-known that in the adsorption of a one-component gas on a porous solid, two processes may take place. Adsorption layer thickening will occur over the entire range of relative pressures, and where micropores are present, these will fill by this process. In addition, capillary condensation may occur, in those 'Author to whom correspondence should be addressed. E-mail:
[email protected]: (519) 8851211 ext 2502.FAX: (519) 746-0435. @
Abstract published inAdvance ACSAbstracts, October 1,1995.
(1)Occelli, M. L. Synthesis ofMicroporous Materials; Van Nostrand-
Rienhold: New York, 1992. (2) Derouane, E. G., Lemos, F., Naccache, C., Ribeiro, N. R., Eds. Zeolite Microporous Solids: Synthesis, Structure and Reactiuity; Kluwer: Dordrecht, 1991. (3)Barthomeuf, D., Derouane, E. G., Htilderich, W., Eds. Guidelines for Mastering the Properties of Molecular Sieues; Plenum: New York, 1990. (4)Occelli, M. L. Zeolite Synthesis; Americn Chemical Society: Washington, DC, 1989. (5) Ruthven, D. M. Principles ofAdsorption andAdsorption Processes; Wiley: New York, 1984. (6) Ternan, M.J. Catal. 1994,146,598. (7)Puziy, A. M. Langmuir 1996,11, 543. (8) Moreno-Castilla, C.; Carrasco-Marin, F.; L6pez-Ram6n, M. V.; Langmuir 1996,11,247. (9) Rouquerol, J.;Avnir, D.; Everett, D. H.; Fairbridge, C.; Haynes, M.; Pernicone, N.; Ramsay, J. D. F.; Sing, K. S. W.; Unger, K. K. In Characterization ofPorous Solids ZZZ; Rouquerol, J.,Rodriguez-Reinoso, F., Sing, K. S. W., Unger, K. K., Eds.; Studies in Surface Science and Catalysis, Vol. 87;Elsevier: Amsterdam, 1994;p 1.
solids where mesopores are present, concomitantly with the process of adsorption layer thickening. Methods for analysis of the adsorption-desorption isotherms of a mesoporous solid usually employ the Kelvin equation to relate the effective pore radius to the relative pressure a t which capillary condensation occurs. Where a microporous structure is present it is generally assumed that these pores fill only by adsorption layer thickening, although some doubts have been raised concerning the validity of this assumption, particularly for the larger micropores. For microporous solids several methods have been employed for the analysis of the physisorption data in order to generate information on the pore size distributions. The MP methodlo is frequently used for such purposes although other techniques such as the finite layer Brunauer/Emmett/Teller (BET)" and the DubininRadushkevich (DRY2equations have been employed. The MP procedure is based on the conversion of the data for the quantity adsorbed a t various adsorption equilibrium pressures into the amount adsorbed as a function of the thickness of the adsorbed layer from which the contribution of pores of different sizes to the volume and surface area can be readily evaluated. Although the relationship between the relative pressure and the thickness of the adsorbed layer would ideally be obtained from a solid of the same nonporous composition as the porous solid of interest, the former is generally not available and the empirical Halsey equation is frequently employed for such purposes. However, a method dependent on the C values from the BET equation has been proposed which may offer advantages not found in the Halsey re1ati0n.l~The finite layer BET equation is derived similarly to the usual BET relation but with the summations carried out over n layers rather than an infinite number. Although the finite layer equation cannot be placed in a linear form, the data from the adsorption isotherm can be computationally fitted to the equation. The calculated values of n , the number of layers, can then be converted to the average radius of the pores. The Dubinin-Radushkevich equation is derived from the exponential dependence of the quantity adsorbed on the adsorption potential and can be employed to generate the micropore volume. (10)Mikhail, R. Sh.; Brunauer, S.; Bodor, E. E. J.Colloid Interface Sci. 1968,26,45. (11)Brunauer. S.; Emmett, P. H.: Teller, E. J.Am. Chem. Soc. 1938, 60,309. (12)(a) Leon, C. A,; Leon D. Altamire Notes 1991,September, 1.(b) Dubinin, M. M.; Zaverina, E. D.; Radushkevich, L. V. Zh. Fiz. Khim. 1947,21,1351. (c)Dubinin, M. M.; Radushkevich, L. V. Proc. Acad. Sci. USSR 1947,55,331. (d)Dubinin, M. M. Russ. J.Phys. Chem. 1965,39, 697. (13)Lecloux, A,; F'irard, J. P. J. Colloid Interface Sci. 1979,70, 265.
0743-7463/95/2411-4474$09.00/00 1995 American Chemical Society
Langmuir, Vol. 11, No. 11, 1995 4475
Analysis of Adsorption-Desorption Isotherms More recently a method for pore size analysis based on the potential between the surface and the adsorbed' molecules has been presented by Horvath and Kawazoe (HK) for a carbon surface.14 This procedure is frequently employed as the software in microprocessors supplied with automatic devices for the measurement of adsorption isotherms. The Horvath and Kawazoe mean-field method has very recently been analyzed by Conner and cow o r k e r ~ .This ~ ~ method has recently been extended to apply to pores ofcylindrical symmetry by Saito and Foley.16 In view of the interest in porous solids and methods for the determination of pore structure, it appears to be of both value and interest to compare the various methods available for such purposes. In the present paper the BET, MP, DR, and Horvath-Kawazoe-Saito-Foley (HKSF) methods are applied to the nitrogen adsorptiondesorption isotherms of two thallium salts of 12-heteropoly oxometalates containing the Keggin anion, thallium dodecatungstosilicate, and dodecamolybdophosphate (abbreviated as TlSiW and TlPMo, respectively), to compare and contrast the aforementioned methods and the results obtained therefrom.
Experimental Section Nitrogen adsorption-desorption isotherms were measured on a standard volumetric glass system fitted with a n MKS Baratron Type 590 HA capacitance manometer. The samples were outgassed a t Torr at 473 K for 2 h prior to the analysis, carried out a t 77 K. Approximately 25 and 10 data points were obtained for the adsorption and desorption parts, respectively, of the isotherm.
Calculations The calculations employed in the present work to obtain estimates of the surface area are those based on the BET equation," the t-plot method,1° and the DubininRadushkevich equation.12 The volume of micropores was estimated by the t-plot methodlo and the DubininRadushkevich equation.12 The pore size distribution of the samples was obtained from the MP method,1° the slit model derived by Horvath and Kawazoe,14and Saito and Foley's line- and area-averaged cylindrical models.16 These four models also provided estimates of the average pore radii for each sample examined.
Results The nitrogen adsorption-desorption isotherms for the two solids are depicted in Figure 1. The sharp initial rise at the low relative pressures is indicative of the presence of micropores in the solid. As the relative pressure is increased, the amount of nitrogen adsorbed by the sample from each aliquot of gas decreases, although the TlPMo salt does adsorb a large amount of nitrogen in the high relative pressure region. The surface areas for these salts were determined initially by the BET method1' and are summarized in Table 1, denoted as SBET. The linear region of the BET isotherm has been greatly reduced for both salts from that which is typi~allyobserved,~~ with the upper limit of the relative pressure (PIP,) being 0.07 for TlSiW and 0.17 for TlPMo. The relatively large values for the CBETparameters, summarized in Table 1, although subject to considerable error because of their dependence on the values of the (14)Horvath, G.;Kawazoe, K. J. Chem. Eng. Jpn. 1983,16, 470. (15)Kaminsky, R. D.; Maglara, E.; Conner, W . C. Langmuir 1994, 10, 1556. (16)Saito, A.;Foley, H. C. AIChE J. 1991,37,429. (17)Gregg, S.J.;Sing, K. S.W.Adsorption, Surface Area and Porosity; Academic Press Inc. London Ltd.: London, 1982.
:
0
6
,
' 1.40
0.00
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
I
Peq I Po
1
I
Adsorptbn
A Desorption
~
Figure 1. Nz adsorption-desorption isotherms for T14SiW12040 and T13PMo12040.
Table 1. Calculated Surface Areas and CBETConstants for the Thallous Salts, TlSiW and TPMo
PIP, sample
range
TlSiW TlPMo
0.00-0.07 0.00-0.17
SBET St CBET (mz/g) (m2/d
(m2/g)
n
4700 920
106.2 166.3
1.44 2.79
92.7 147.2
96.2 142.0
SDR
intercepts from the BET equation, are also indicative of the presence of micropores. The finite layer BET relationship of
was also applied to each of the two salts. The values of N , and CBETwere determined by the infinite layer relationship while x and N refer to the relative pressure and moles of nitrogen adsorbed, respectively, in the N2 adsorption isotherm. A nonlinear regression was performed to determine n, the number of adsorbed monolayers. Values of less than 3, for n, are indicative of the presence of micropores within the solid. As shown in Table 1, both salts are consistent with these conclusions. It was noted that of the two constants, N,,, and CBET,the value of N,,, has the greatest effect on the value of n. The surface areas (SJ were also calculated from t-plots employing reference isotherms from Lecloux and Pirard13 and are in good agreement with those obtained by the BET method (Table 1). Both t-plots are well behaved in that the extrapolated curves pass through the origin. However, with TlPMo the volume adsorbed did not reach a constant value a t the highest value for t. It should be noted that the reference isotherms of Lecloux and Pirard have alower limit of 0.02 for the range ofrelative pressure. Extrapolation of the isotherm to zero has led to discontinuity in the constructed t-plot, and as a result, data points obtained for relative pressures less than 0.02 have not been included in the t-plots for the salts examined. The volumes (Vhlp) of the micropores can be estimated for both salts by extrapolating from the linear pressure region of 0.4 < PIP, < 0.6 (corresponding to the thickness range of 1.1-1.3 nm for the adsorbed layers) to obtain the y-intercept. However, with the deviation from linearity that the TlPMo salt experiences as the thickness of the adsorbed layer increases, a more representative value is expected from the y-intercept of a tangent line taken at
Parent and Moffat
4476 Langmuir, Vol. 11, No. 11, 1995 Table 2. Calculated Micropore Volumes and Sizes for the Thallous Salts, TlSiW and TlPMo
TlSiW
TlPMo a
3.8 x 5.9 x
3.5 x 4.2 x (3.7 10-3)"
0.783 0.877
1.565 1.754
1.492 1.656
X
n
X
Determined by the tangent method. 009
+ 0.4 0.2-
I
I I A
I
Figure 3. Dubinin-Radushkevich plots for T14SiW12040and
T13PM012040. larger than the V M obtained ~ from the t-plots. This is noted especially with the TlPMo salt, whose upward deviation is significantly greater than that with the TlSiW salt in the DR plot. In comparing the surface areas Pore radius (nm) calculated by the three methods in Table 1,those obtained I+ nsiw + npMo 1 from the DR equation are consistently larger than those obtained from the BET and t-plot methods. Figure 2. Pore size distributionofTLSiW12040and T13PM012040 A correction to the DR plots has been illustrated20 to by the MP method. account for the presence and filling of mesopores. The resulting corrected curve is loweredin position with respect PIP, = 0.4 (corresponding to a n adsorbed layer of 1.1 to the y-axis (thevolume of nitrogen), and has a decreased nm). This value is denoted in parentheses in Table 2 and slope, apparent even in the low-pressure region. As a is closer to the VMPobtained for the TlSiW salt. result, it appears that extrapolating the VDRvalue from The MP method was employed to generate the micropore only the low-pressure region (PIP, < 0.1) does not size distribution from the data on the t-plots1° (Figure 2). completely reduce the effect of mesopores present in the The TlSiW salt has a typical distribution observed for the monovalent salts of the 12-heteropoly o x ~ m e t a l a t e s , ~ ~ J ~system, causing the value obtained for VDRto be larger than it actually is. This increase is also reflected to the with a maximum in the 0.7-0.8 nm region, while the same extent in the calculated surface area (SDR). TlPMo salt has a wider breadth for its distribution and The DR method uses the assumption of a Gaussian appears to have three maxima. Both have evidence of distribution for the micropore radii.17 With this, the pores existing up to 2.0 nm. The mean micropore radii average micropore width (denoted as LDRin Table 2) can (FMP) for each salt determined from the equation be calculated by
L,, = ($)[(4.45x 106)(-D)lu2 are summarized in Table 2. As expected, since both salts contain the same cation, the mean micropore radii are similar in size, with that of the TlPMo salt being slightly larger, correlating to its larger surface area. The use of the DR equation12to analyze the adsorption isotherm data is based on the volume filling of the micropores, rather than the assumption of layer-by-layer adsorption on the pore walls that the BET method employs. A linear plot (also referred to as a DR plot) (Figure 3) is obtained from the equation
(4)
The upward deviation as the saturation pressure is reached is attributed to multilayer adsorption and capillary condensation in the mesopores.17 As a result, the volume of the micropores (denoted as VD, in Table 2) is obtained by extrapolation from the low-pressure region (PIP, < 0.1) to obtain they-intercept. These values are
where y is the molar volume of the adsorbate ( y = 0.393 for nitrogen).l2" These values, summarized in Table 2, are slightly smaller than twice the value of the mean micropore radius ( h p ) for each salt. Despite the problem of the increased slope of the DR plot due to the presence of mesopores, these average micropore sizes correlate well with the data from the MP method. It appears that the presence ofmesopores has a greater effect on the intercept of the DR plot than on the slope in the low-pressure region. Horvath and Kawazoe developed a model which takes into account the bulk and surface properties of the adsorbents. On the basis ofthe calculated potential energy profiles for atoms adsorbed in slit-like pores and the enhancement of the depth of the energy well over that for adsorption on a flat surface proposed by Everett and Pow1,21Horvath and Kawazoe's model relates the free energy of adsorption to the average potential inside a slitlike pore.14 Their initial study involved the adsorption of nitrogen on molecular-sieve carbon, with the generic
(18) McMonagle, J. B.; Moffat, J. B. J. Colloid Interface Sci. 1984, 101, 479. (19) Taylor, D.B.; McMonagle, J. B.; Moffat, J. B. J . CoZloidInterface Sci. 1985,108, 278.
(20) Dubinin, M. M. In Chemistry and Physics of Carbon; Walker, P. L., Ed.; Marcel Dekker Inc.: New York, 1966; Vol. 2, p 51. (21) Everett, D. H.; Powl, J. C . J . Chem. Soc., Faraday Trans. I 1976, 72,619.
log V = log V, - D 10&P/Po)
(3)
Langmuir, Vol. 11, No. 11, 1995 4477
Analysis of Adsorption -Desorption Isotherms Table 3. Parameters Used for the Adsorbent and Adsorbate Components parameter
adsorbent oxide ion
adsorbate nitrogen (N2)
diameter, d (nm) polarizability, a (nm) magnetic susceptibility, (cm3) density, N (molecules/cm2)
0.276" 2.5 x 10-240 1.3 x 1.31 10150
0.36' 1.74 x 10-24d 2 x 6.7 x 10146
x
"Reference 16. Reference 14. %ference 25. Lide, D. R. Handbook of Chemistry and Physics, 71st ed.; CRC Press Inc.: Boca Raton, FL, 1990; Chapter 10, p 199.
L
equation summarized as
0.01
Figure 4. Pore size distribution of T14SiW12040 by HKSF
+
where d = (dA cia). The model can be extended to other adsorbateadsorbent systems, and Table 3 summarizes the data used for the present system. Incorporating these parameters for the oxide as the adsorbent and updating the parameters for nitrogen as the adsorbate, the resulting equation for the slit model is
In
(E)
=
methods.
i
[
21.77 1.847 x 1013 (L - 0.636) (L- C1.318)~
2'540 - 4.981 x (L - 0.318)'
(6)
Saito and Foley modified the slit model to accommodate the cylindrically shaped pores of the zeolite microstructure.16 For the cylindrical model, the line-averaged and area-averaged cases were examined. With substitution of the parameters listed in Table 3 , the resulting equation for the line-averaged case is
In
i:
i
(E)
04
05
1
1
15
25
I
Effedive Pore Diameter (nm)
I+
S I W
+ *rebwsrapd~ o d s l *
I
timawraped MI
Figure 6. Pore size distribution of T13PMo12040 by HKSF methods.
=
Table 4. Summary of Effective Pore Diameter Calculated for the Thallous Salts, TlSiW and TlPMo effective pore diameter (nm) ~~
while that for the area-averaged case is
In
(k)
=
with
(
a, = -4.;
-
Using pressure data from the nitrogen-adsorption isotherms of the two salts, these three equations were
sample
slit model
line-averaged model
TlSiW TlPMo
0.562 0.544
0.729 0.607
~
area-averaged model 0.769 0.618
solved by application of the Secant method22to determine the unknown variable ( L for the slit model and rpfor the cylindrical models) at each data point. The resulting effective pore diameter was then calculated from ( L- 0.276 nm) for the slit model14 and (2rp - 0.276 nm) for the cylindrical models.16 The micropore size distributions for the two salts are then plotted similarly to the MP method, with the differential volume adsorbed with respect to the differential pore size vs the pore size (Figures 4 and 5 ) . These typically result in a Gaussian distribution, with the maxima taken as the average effective pore diameter (Table 4). The diameters from the cylindrical models show the reverse of the trend previously observed with the MP and DR methods. The TlPMo salt has a smaller pore diameter than the TlSiW salt, and both salts have values less than (22) Riggs,J.B. An Introduction to Numerical Methodsfor Chemical Engineers; Texas Tech UniversityPress: Lubbock, TX,1988;pp 48-63.
4478 Langmuir, Vol. 11, No. 11, 1995
the mean micropore radii ( F M ~ calculated ) by the MP method and less than half of the average pore width (LDR) calculated by the DR method. Saito and Foley varied each ofthe parameters listed in Table 3 by f30% and concluded that the diameter of the oxide ion a t the surface had the greatest influence on the calculated pore size, while the other parameters had only a moderate to a small effect.16 In the present work some error may be introduced by the assumption that the diameter of the oxide ion in a zeolite Y system would be comparable to the Keggin anion in the thallous salt. Discussion While no rigorous definition of the surface of a solid is currently available, there is general agreement that the interaction of nonreacting molecules with a solid surface can provide operational definitions of the surface areag. The IUPAC approved method for generating surface area data applies the infinite layer BET theory to the experimentally measured adsorption isotherm. Concave to the abscissa departures of the experimental data from the BET equation a t the relative pressures have been ascribed to the presence of physisorption sites on the surface with higher energies than the expected BET values for the molecules adsorbing immediately adjacent to the solid surface. Such higher energies may result from the existence of a n approximately planar surface which is energetically heterogeneous or from the existence of micropores, which as a result of the close proximity of the adsorbing molecules to a larger fraction of the surface generate enhanced global potential surfaces. With solids containing micropores the reduced range of linearity of the BET plots introduces errors in the determination of both the slopes and the intercepts. In obtaining morphological information, the finite layer BET and the DR theories suffer from their inabilities to yield pore size distribution. The HKand HKSF methods, while taking into account the presence of potential surfaces within the pores which are dissimilar from those on the peripheral surfaces, have the disadvantage that either two-body or global potentials for solid surfaces are difficult to construct for real catalytic solids and the simplest adsorbate molecules. In contrast the MP method avoids any requirement for the calculation of a potential surface but necessitates the availability of relationships between the thickness of the adsorbed layer and the equilibrium relative pressure in the adsorption process. Although in principle the latter can be obtained from the adsorption isotherm of a nonporous analogue of the porous solid of interest, in practice this is difficult to achieve. It is clear that the MP and the HK (and HKSF) methods each have their advantages and disadvantages. In the present work the former technique has been employed together with the Lecloux and Pirard procedure for generating t vs P I P , data. While the latter has been ~ r i t i c i z e d ,comparisons ~~,~~ between the results obtained therefrom and those obtained using nonporous solids of similar, but not identical, composition as those of the porous solids of interest provide evidence for the relative validity of the LP procedure.lsJ9 However, the absence of specificreference isotherms for the larger values of CBET imposes a limitation on this approach. The deviation from linearity that is evident in t-plots for solids with high (23) Parfitt, G. D.; Sing, K. S. W.; Urwin, D. J. Colloid Interface Sci. 1975,53, 187. (24) Dubinin, M. M. In Surface Area Determination; Everett, D. H. Ottewill, R. H., Eds.; Butterworth and Co. (Publishers) Ltd.: London, 1970; p 123.
Parent and Moffat surface areas also contributes to errors in the determination of the micropore volume (VMP). The DR approach, not unexpectedly, also has its advantages and disadvantages. The existence of linearity in the DR plots is not sufficient to guarantee micropore filling or the validity of vDR.25 This is particularly evident from the observation of linearity a t relative pressures approaching unity.17 In addition the presence of mesopores may, even with relative pressures less than 0.1, result in inflated values for the micropore volume.20 Although a number of techniques have been proposed to correct for the presence of mesopores, none of them is universally applicable.26 Although a consensus has not been a c h i e ~ e d it~ ~ has - ~been ~ suggested that the BET and DR equations should provide results in good agreement where the solids have the preponderance of available surfaces located in pores of molecular dimension^.^^ The approach of Horvath and Kawazoe14 appears to correct the problems associated with the BET method by taking into account the nature of the adsorbent, with an average potential function determined for the inside of the slit-like pores. However,as noted earlier in this report, the determination of a realistic global potential inevitably necessitates the introduction of assumptions and simplifications. However, the application of a global potential accommodates the enhanced adsorption which may occur in micropores and that the BET and t-plot methods do not explicitly take into account. Saito and Foley confirm that useful information about the micropore structures can be derived from the nitrogen and argon isotherms, in terms of CBET,t-plots, and the DR model.16 However, as Saito and Foley note these methods do not provide a straight forward relationship between the logarithm of the relative pressure and the pore size16such as found in the Kelvin equation. However the applicability of the Kelvin equation is dependent upon the formation of a liquid like meniscus which is not found in micro pore^.^^ The application of the slit and cylindrical models provides a method permitting such a direct relationship and appears to accommodate successfully different trends which may occur in the isotherm. Literature reports of the use of the models developed by Horvath and Kawazoe and Saito and Foley have been successful in describing different systems, correlating well with other method^,^^,^^ although the limitations of each, especially the slit model, have been re~0gnized.l~ The present authors conclude that in view of the deficiencies in each ofthe presently available methods for the determination of micropore distributions from adsorption-desorption isotherms, prudent workers may elect to compare the results obtained from the application of a variety of these techniques. Acknowledgment. The financial support of the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. (25) Carrott, P. J. M.; Roberts, R. A,; Sing, K. S. W. In Characterization of Porous Solids; Unger, K. K., Rouguerol, J., Sing, K. S. W., Kral, H., Eds.; Studies in Surface Science and Catalysis Vol. 39; Elsevier: Amsterdam, 1988; p 89. (26) Dubinin, M. M. Carbon 1987,25,593. (27) (a)Klemperer, D. F. In Surface Area Determination; Everett, D. H., Ottewill, R. H., Eds.; Buttenvorth and Co. (Publishers)Ltd.: London, 1970; p 55. (b) Gottwald, B. A. In Surface Area Determination; Everett, D. H., Ottewill, R. H., Eds.; Butterworth and Co. (Publishers) Ltd.: London, 1970; p 59. (28) Granville, A,; Hall, P. G.; Hope, C. J. Chem. Ind. 1970,435. (29) Walker, P. L.; Patel, R. L. Fuel 1970,49, 91. (30) Fisher, L. R.; Israelachvili, J. N. J.Colloid Interface Sci. 1981, 80, 528. (31) Gil, A,; Montes, M. Langmuir 1994,10, 291. (32)Ackerman, W. C.; Smith,D. M.;Huling, J. C.;Kim,Y.-W.;Bailey, J. K.; Brinker, C. J. Langmuir 1993,9, 1051.
Langmuir, Vol. 11, No. 11, 1995 4479
Analysis of Adsorption -Desorption Isotherms
Nomenclature aK = constant A, = constant in Lennard-Jones potential (J/molecule) AA= constant in Lennard-Jones potential (J/molecule) PK = constant CBET = constant from the infinite layer BET relationship d,= diameter of adsorbent atom (nm) d A = diameter of adsorbate molecule (nm) D = slope of log V versus logz(P/Po) y = molar volume of the adsorbate relative to a standard (benzene) K = Avogadro’s number (molecules/mol) L = distance between nuclei of two layers (nm) LDR= average micropore width determined by the DR method (nm) n = number of adsorbed monolayers N = moles of nitrogen adsorbed from Nz adsorption isotherm (mol) N , = number of atoms per unit area of adsorbent (atom/ cm2) NA = number of molecules per unit area of adsorbate (molecules/cmz) N , = moles of Nz(g) required to form a monolayer on the surface, determined by the infinite layer BET relationship (mol) P = pressure of Nz(g) (Pa) Po= saturation pressure of Nz(g) (Pa) PIP, = relative pressure of Nz(g)
r = pore radius (nm) R = gas constant (J/(mol K)) k p = mean micropore radii (nm) rp = radius of micropore cylinder (nm) Ar = change in the mean radius between two data points in a t-plot (nm) 0 = distance between a gas atom and the nuclei of the surface a t zero interaction energy (nm) SBET = surface area determined by the infinite layer BET method (mz/g) SDR = surface area determined by the DR method (mz/g) St = surface area determined from the t-plot (mz/g) t = statistical thickness of the adsorbed layer of Nz(g) from the Lecloux and Pirard reference isotherm (nm) T = temperature (K) TlPMo =thallium dodecamolybdophosphateT13PM012040 TlSiW = thallium dodecatungstosilicate T14SiW12040 V = volume adsorbed a t temperature T and relative pressure PlP, (mug) VDR= total volume of micropores extrapolated from the DR plot (=V,) (mug) VMP= volume of micropores extrapolated from the t-plot (mug) V, = total volume of micropores (mug) AV= change in volume adsorbed between two data points in a t-plot (mug) I = relative pressure from Nz adsorption isotherm LA950344G
