Langmuir 1996, 12, 6501-6505
6501
Adsorption Hysteresis and Interparticle Capillary Condensation in a Nonporous Carbon Black William D. Machin* and Richard J. Murdey Department of Chemistry, Memorial University of Newfoundland, St. John’s, Newfoundland A1B 3X7, Canada Received May 24, 1996. In Final Form: September 9, 1996X The adsorptives 2,2-dimethylpropane, trichlorofluoromethane, and trimethylamine, when adsorbed on the nonporous graphitized carbon black Sterling FT-G (2700), all exhibit hysteresis arising from interparticle capillary condensation. The hysteresis loops have been analyzed to obtain the adsorbent particle size distribution. We find that two of the adsorptives, 2,2-dimethylpropane and trichlorofluoromethane, satisfy the following criteria: (i) the distribution derived from the adsorption branch of the hysteresis loop is the same as that derived from the desorption branch, (ii) the calculated distribution is independent of temperature and the adsorptive, and (iii) the particle size distribution derived from adsorption measurements is the same as that derived using electron microscopy. The trimethylamine isotherms do not satisfy these criteria, and reasons for this discrepancy are discussed. Interparticle pore size distributions are readily derived from the isotherms and, with the exception of trimethylamine, also satisfy these criteria.
The phenomenon of adsorption-desorption hysteresis observed with mesoporous adsorbents is commonly used to determine their pore size distributions (PSDs). Traditional analyses, such as those based on the method of Barrett, Joyner, and Halenda1,2 rely on simplified models for pore shape, usually assumed to be cylindrical or slitshaped, and also assume that the adsorbate is liquid-like, having the same properties as bulk adsorptive. Hysteresis arises when the shape of the liquid-vapor interface during adsorption differs from that present during desorption. This model requires that PSD be independent of temperature and the adsorptive. Furthermore, both branches of the hysteresis loop should yield the same PSD. These conditions, unfortunately, are rarely satisfied in practice; nevertheless, the method continues to be widely used. Independent confirmation of PSD derived from this method is difficult, since pore sizes range from ca. 2 to 50 nm and pore shape may be highly irregular. Recently, some of these difficulties have been reduced, if not entirely eliminated, with the development of MCM-41 type adsorbents3-5 which have nonintersecting cylindrical pores of nearly uniform diameter. Few comparisons, however, have been made of PSDs determined from X-ray or electron microscopy with those derived from hysteresis effects.6,7 In the present work we determine, from several adsorption isotherms, the particle size distribution of a well-characterized carbon black and compare it with that obtained independently by electron microscopy. The results suggest some criteria that must be satisfied if reliable PSD analyses are to be obtained from adsorption measurements. * Author to whom correspondence should be addressed. X Abstract published in Advance ACS Abstracts, November 15, 1996. (1) Barrett, E. P.; Joyner, L. G.; Halenda, P. H. J. Am. Chem. Soc. 1951, 73, 373. (2) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity, 2nd ed.; Academic Press: New York, 1982; Chapter 3. (3) Branton, P. J.; Hall, P. G.; Sing, K. S. W.; Reichert, H.; Schuth, F.; Unger, K. K. J. Chem. Soc., Faraday Trans. 1994, 90, 2965. (4) Rathousky, J.; Zukal, A.; Francke, O.; Schulz-Ekloff, G. J. Chem. Soc., Faraday Trans. 1995, 91, 937. (5) Alfredson, V.; Kenny, M.; Monnier, A.; Stuckey, G. D.; Unger, K. K.; Scheuth, F. J. Chem. Soc., Chem Commun. 1994, 921. (6) Luan, Z.; Cheng, C.-F.; Zhou, W.; Klinowski, J. J. Phys. Chem. 1995, 99, 1018. (7) Schmidt, R.; Hansen, E. W.; Stocker, M.; Akporiaye, D.; Ellestad, O. H. J. Am. Chem. Soc. 1995, 117, 4049.
S0743-7463(96)00511-2 CCC: $12.00
Experimental Section Sterling FT-G (2700) graphitized carbon black is a standard reference adsorbent obtained from the National Physical Laboratory (U.K.). It has a specific surface area of 11.1 ( 0.8 m2 g-1, and its properties are described elsewhere.8 A sample of 1.7815 g from NPL sample 1A/16/9/2 was used for all adsorptives except trimethylamine in a volumetric apparatus.9 The latter adsorptive was used with 5.80 g from NPL sample 1A/4/9/12 in an automated gravimetric apparatus similar to that described by Klevens et al.10 Both sets of apparatus were grease-free, with stainless steel bellows-seal valves used throughout. Pressures were measured with capacitance manometers (MKS Baratron). The error in pressure measurement is 0.05% or ca. (0.001 in relative pressure. Since the pores in this adsorbent are large, capillary condensation occurs close to the saturation pressure (P°). To achieve satisfactory precision in relative pressure, it is necessary to reduce the uncertainty in P° arising from the fluctuation in bath temperature. The fractional dependence of P° on temperature, (1/P°) dP°/dT, is relatively large (ca. 0.12 K-1) for nitrogen at 77 K but is much smaller (ca. 0.04 K-1) at ambient temperatures for the adsorptives used here. The uncertainty in the bath temperature, 0.05 K, gives an error in relative pressure of ca. 0.002. The adsorptives trimethylamine (TMA), trichlorofluoromethane (CFC), and 2,2-dimethylpropane (DMP) have convenient saturation pressures (50-100 kPa) at the experimental temperatures, are quasi-spherical, and with the exception of TMA, are unreactive. Their purification has been described elsewhere.11
Results and Discussion Relevant parameters for the various isotherms are summarized in Table 1, and isotherms are shown in Figures 1 and 3. To facilitate comparison of the isotherms, amounts adsorbed are given as equivalent liquid volume per gram of adsorbent. The adsorbent particle size distribution, i.e. the volume of particles, ∆v/∆D, as a function of the mean particle diameter, D h , is shown in Figure 2. This distribution is derived from the number distribution provided courtesy of the National Physical Laboratory (U.K.).12 Complete details regarding the distribution are provided in the Appendix. (8) Everett, D. J.; Parfitt, G. D.; Sing, K. S. W.; Wilson, R. J. Appl. Chem. Biotechnol. 1974, 24, 199. (9) Machin, W. D. J. Chem. Soc., Faraday Trans. 1 1982, 78, 1591. (10) Klevens, H. B.; Carriel, J. T.; Fries, R. J.; Peterson, A. H. In Second International Congress of Surface Activity II: Solid-Gas Interface; Schulman, J. H., Ed.; Butterworths: London, 1957; p 160. (11) Murdey, R. J.; Machin, W. D. Langmuir 1994, 10, 3842. (12) Wilson, R. Division of Chemical Standards, National Physical Laboratory U. K., private communication.
© 1996 American Chemical Society
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Table 1. Adsorbate Properties and Isotherm Parameters 2,2-dimethylpropane temp/K surface tension/mJ m-2 molar volume/cm3 mol-1
273.15 14.40 117.70
amount adsorbed at P°/mmol g-1 monolayer volume/mm3 g-1 volume adsorbed at P°/cm3 g-1 porositya nb surface area (desorption)c/m2 g-1 surface area (adsorption)c/m2 g-1 pore volume (desorption)d/cm3 g-1 pore volume (adsorption)d/cm3 g-1
2.24 5.91 0.264 0.361 4.06 11.7 11.4 0.264 0.253
trichlorofluoromethane
Properties 279.85 13.66 119.06 Isotherm Parameters 2.18 5.93 0.260 0.357 4.22 11.6 11.6 0.263 0.260
trimethylamine
283.35 20.36 90.92
290.05 19.41 91.87
273.0 16.25 90.11
283.0 15.12 91.79
2.96 5.42 0.269 0.365 4.06 11.4 11.6 0.267 0.259
2.88 5.45 0.265 0.362 4.04 11.2 10.4 0.263 0.250
2.85 5.41 0.257 0.355 3.70 13.7 9.7 0.248 0.237
2.80 5.44 0.257 0.355 3.68 13.3 10.1 0.259 0.241
a Interparticle porosity assuming the adsorbent density is 2.14 g cm-3 (ref 8). b Number of adsorbed layers present at the lower closure point of the hysteresis loop. c Surface area derived from the particle size distribution using the desorption and adsorption branches of the hysteresis loop. Standard surface is 11.1 m2 g-1 (ref 8). d Cumulative pore volume derived from the particle size distribution.
Figure 1. Sterling FT-G (2700); adsorption isotherms at high relative pressure. (a) Trimethylamine at 283.0 K. (b)Trichlorofluoromethane at 290.05 K. (c) 2,2-Dimethylpropane at 279.85 K. O denotes desorption, and + adsorption. The dotted line represents the Frenkel-Halsey-Hill equation (eq 5) for the multilayer region extrapolated into the hysteresis region. The solid lines represent the adsorption and desorption branches of the hysteresis loop as calculated from the SEM particle size distribution (see text). The estimated error in the relative pressure, ca. 0.001, is less than the size of the plotting symbols. Isotherm a has been offset by 0.10 with respect to relative pressure and volume adsorbed; isotherm b has been offset by 0.05.
The primary adsorbent particles are nonporous, quasispherical polyhedrons of graphitized carbon black. These particles form larger (ca. 1-2 mm) agglomerates, and the interparticle void space forms the pore volume of the adsorbent. Therefore, with appropriate assumptions, it is possible to derive the PSD from the particle size distribution. These assumptions are as follows: (i) Each size group of adsorbent particles consists of uniform spheres of diameter D h and is independent of all other size groups; i.e., pores are formed only from spheres of equal diameter.
Figure 2. Particle size distribution of Sterling FT-G (2700) graphitized carbon black. The solid lines in parts a and b are the particle size distributions obtained from electron microscopy (see Appendix). (a) Combined results for 2,2-dimethylpropane (273.15 and 298.85 K) and trichlorofluoromethane (283.35 and 290.05 K). Each isotherm yields a PSD from the adsorption branch (+) and the desorption branch (b). The dotted line is intended as a guide to the eye. (b) Results for trimethylamine (273.0 and 283.0 K); + denotes the PSD derived from the adsorption branch, and b denotes the PSD derived from the desorption branch of the isotherm. The dashed and dotted lines are intended as a guide to the eye.
(ii) Each size group has the same packing and a porosity, , equal to that of the total sample ( = 0.36-0.37, see Table 1). (iii) The largest throat, or window, providing access to each pore is formed by a square array of equal spheres, the radius of the throat, rw, being equal to the radius of the largest circle that can be inscribed within; i.e., rw ) 0.207D h. (iv) Each pore fills and empties independently of all other pores; i.e., pore blocking or network effects are absent.
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These assumptions are sufficient to derive a PSD based on throat radii, but we also wish to express the PSD in terms of the cavity radius rc. This is defined as the radius of the largest sphere that can be inscribed within the cavity formed by the regular packing of equal spheres. Calculation of the cavity radius depends on the choice of an appropriate packing model. In the present case the experimental porosity, ) 0.36-0.37, is close to that obtained for primitive hexagonal packing of equal spheres, ) 0.395. The cavity radius for this packing is rc ) 0.264D. However, since the experimental porosity is slightly less than this ideal value, we have reduced the cavity radius by the factor (0.365/0.395)1/3, which results in rc ) 0.257D. For this packing of uniform spheres of diameter D it is readily shown that the pore volume, vp, is related to the particle volume, vs, as
vp ) vs
(1 - )
(1)
and that the surface area within these pores, Sp, is given by
Sp )
6vp 1 - D
(
)
(2)
Adsorption within this pore structure proceeds by multilayer film growth on the walls of empty pores until the point is reached, for a given set of pores, where each pore fills by capillary condensation at a relative pressure, P/P°, governed by the Kelvin equation; i.e.,
rc - t )
-2γV cos Θ ) rK RT ln P/P°
(3)
where rK is the Kelvin pore radius, γ and V are the surface tension and molar volume of the capillary liquid respectively, Θ is its contact angle with the adsorbent, and t is the thickness of the multilayer film at P/P°. Pore condensation continues as relative pressure is increased until all pores are filled at saturation pressure. When relative pressure is subsequently decreased, desorption proceeds through capillary evaporation when
rw - t )
-2γV cos Θ ) rK RT ln P/P°
(4)
Changes in thickness, of exposed multilayer films as relative pressure changes, also contribute to the changes in the total amount adsorbed. When all pores have been emptied of capillary liquid, the desorption branch of the hysteresis loop rejoins the adsorption branch at the lower closure point of the hysteresis loop. At this point the remaining adsorbate is present as a liquid-like multilayer film on the pore walls. Derivation of the PSD from the isotherm requires the following assumptions about the properties of the capillary condensate: (i) The surface tension, γ, and molar volume, V, are the same as those of the bulk liquid. (ii) The adsorbates wet the adsorbent; i.e., the contact angle is equal to zero and hence cos Θ ) 1 in eqs 3 and 4. If these assumptions are valid and if our model for the structure of the adsorbent is reasonable, then the PSD derived from the various isotherms should be invariant with respect to temperature and adsorptive and should agree with the PSD derived from the particle size distribution. Following Roberts,2,13 the isotherms are plotted as liquid volume adsorbed, vL, against Kelvin radius, rK ) -2γV/ RT ln(P/P°), as in Figure 3. Along the adsorption branch (13) Roberts, B. F. J. Colloid Interface Sci. 1967, 23, 266.
Figure 3. Adsorption isotherms on Sterling FT-G (2700) graphitized carbon black. These isotherms are plotted as liquid volume adsorbed against Kelvin radius, rK (eqs 3 and 4). (a) Trimethylamine at 273.0 K. (b) Trichlorofluoromethane at 283.35 K. (c) 2,2-Dimethylpropane at 273.15 K. O denotes desorption, and + adsorption. The dotted line represents the Frenkel-Halsey-Hill equation (eq 5) for the multilayer region extrapolated into the hysteresis region. The dashed lines correspond to the adsorption and desorption branches of the hysteresis loop as calculated from the SEM particle size distribution (see text). Error bars correspond to an error of (0.001 in relative pressure, but at rK < ca. 70 nm they are smaller than the plotting symbols. Isotherm a has been offset by 0.30 with respect to volume adsorbed; isotherm b has been offset by 0.15.
of the hysteresis loop rK corresponds to rc - t (eq 3) while it corresponds to rw - t along the desorption branch (eq 4). The multilayer thickness, t, is obtained from the Frenkel-Halsey-Hill (FHH) equation
ln(P/P°) ) -b/(vL)n
(5)
fitted to data that lie below the onset of hysteresis (ca. 0.93P°) and above the monolayer region. All isotherms are well described by this equation with b ) -3.90 × 10-5 and n ) 2.0. The t curve for each isotherm, extrapolated into the hysteresis region, is shown for each isotherm in Figures 1 and 3. At the onset of capillary condensation, near the lower closure point of the hysteresis loop, the adsorbed film thickness corresponds to several monolayers (see Table 1) and increases relatively slowly as the pores fill. The volumes adsorbed at equal increments in rK are determined, and the difference, ∆vL, between successive radii, rn and rn+1, is equal to the pore core volume, ∆vc, over that interval plus a contribution from the change in multilayer thickness, ∆t, in empty pores; i.e., n
) ∆vc|n+1 + ∆t|n+1 ∆vL|n+1 n n n
∑0 ∆s
(6)
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where ∑∆s is the total surface area in empty pores. The pore volume, ∆vp, between rn and rn+1 is n+1 ∆vP|n+1 ) ∆vc|n+1 + ht |n+1 n n n ∆s|n
(7)
where ht is the average film thickness and ∆s is the surface area in those pores emptied between rn and rn+1. This area is calculated using eq 2. The corresponding adsorbent volume, vs, is calculated from eq 1. Results obtained for CFC and DMP, each at two temperatures, are shown in Figure 2a, plotted as particle volume over a given size interval, ∆vs/∆D, against the average particle diameter, D h . Note that each of the four isotherms yields two independent results, since the distribution is calculated from the adsorption branch as well as the desorption branch of each isotherm. More significant, however, is the fact that all eight calculations yield essentially the same result, a particle size distribution in good agreement with that determined from electron microscopy. With the exception of TMA, the total surface area and pore volume derived from the particle size distributions are in good agreement with the BET (N2) area and the Gurvitch volume (see Table 1). The principal differences between the two methods lie at either extreme of the particle size distribution. PSD analysis based on adsorption isotherms indicates larger volumes of very small particles (D h < 100 nm) and very large particles (D h > 350 nm) than found by electron microscopy. In part, these discrepancies may arise from differences in the sample size related to each method. Adsorption samples, ca. 1-5 g, represent ca. 5 × 1013 particles, whereas a total of 3125 particles were counted on electron micrographs with only 49 at D h < 100 nm and 21 at D h > 350 nm (see Appendix). The results from adsorption measurements suggest that particles larger than ca. 400 nm are present. If so, they are relatively few in number and may have been missed by direct counting in electron micrographs.14 As well, our model for adsorbent structure assumes that all pores are formed from the packing of spheres of equal diameter. While this may be reasonable for particle groups clustered close to the median particle diameter, it is less valid for those groups that contribute only a small fraction of the total sample, i.e. those groups at either extreme of the particle size distribution. For the sample as a whole, however, our results validate both our model for the adsorbent structure and the Kelvin equation as a description of pore condensation and evaporation when, as in the present example, the pores are large and a relatively thick multilayer film is present. For these adsorptives, our results also show that their properties as capillary liquids are the same as bulk properties and their contact angles on the adsorbent are zero. The same cannot be true for TMA, however. It is clear from the results in Figure 2b that the PSDs calculated from these isotherms lack self-consistency, since the PSD derived from the adsorption branch of the hysteresis loop differs from that derived from the desorption branch and, further, both differ from the distribution obtained from electron microscopy. If the properties (surface tension, molar volume) of TMA as a capillary liquid differ from bulk properties, a satisfactory PSD may not be obtained. If the contact angle is finite, both adsorption and desorption branches of the hysteresis loop will be shifted to higher relative pressures and the corresponding PSD will be shifted to larger pore radii (or larger particle diameters). If irreversible thin film-thick film transitions also occur within the hysteresis region, this may act to shift the PSD to smaller pore radii. However, none of these possibilities, (14) Arnell, J. C.; Hennebury, G. O. Can. J. Res. 1948, 26A, 29.
taken alone, will account for the observed shift (Figure 2b) of the adsorption PSD to larger pore radii and the desorption PSD to smaller pore radii. The simplest possible explanation may be that the effect is an experimental artifact. The TMA measurements were carried out on an automated gravimetric system which required a large (ca. 5 g) sample of adsorbent, and the criteria for equilibrium were constant mass and constant equilibrium pressure. However, if the adsorption/desorption equilibrium is very slow, an apparent equilibrium may be established that is not true equilibrium. Failure to attain true equilibrium would shift the adsorption branch to higher relative pressures and the desorption branch to lower relative pressures, as observed. We plan to investigate this possibility using a volumetric apparatus and a smaller mass of adsorbent. Finally, we note that the converse process, calculation of the adsorption and desorption branches of the hysteresis loop from the electron micrograph particle size distribution, is readily carried out. The results of these calculations are shown in Figures 1 and 3. As expected, agreement between experimental and calculated isotherms is good for all adsorptives except TMA. Conclusions When the assumptions concerning the properties of the adsorptive and adsorbent are valid, an accurate PSD can be derived from adsorption isotherms that exhibit hysteresis arising from capillary condensation/evaporation. While the model used here yields consistent results, other systems may fail to do so. For example, if network effects are important, if thick liquid-like multilayers are not present during capillary condensation, or if other conditions are not satisfied, then consistent, accurate, pore size distributions may not be obtained from the analysis of hysteresis. In the absence of independent information about the pore structure of an adsorbent, we suggest that the following criteria may be applied to assess the validity of PSD analysis based on adsorption hysteresis: (i) The PSD derived from each branch of the hysteresis loop should be identical. (ii) The PSD should be independent of both temperature and adsorptive. (iii) The surface area derived from the PSD should agree with the BET (N2) area, and the pore volume derived from the PSD should agree with the Gurvitsch volume. Acknowledgment. The authors thank Professor P. D. Golding for his comments and advice. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada. Appendix I. Adsorbent Particle Size Distribution from Electron Microscopy. A histogram showing the number of adsorbent particles within a given size interval was provided courtesy of Dr. R. Wilson of the National Physical Laboratory (U.K.). The average particle diameter, D h i, size interval, ∆D, and particle count, ni, for each size interval as read from the histogram are recorded in columns 1-3 in Table 2. When normalized to 1 g of adsorbent, the volume within each size interval, ∆vi, is given by 26
h i)3/ ∆vi ) 0.4673[ni(D
∑1 ni(Dh i)3] cm3 g-1
(A-1)
where 0.4673 is the specific volume of the adsorbent. The h i is particle size distribution, ∆vi/∆Di, as a function of D shown in Figure 2. The surface area for each size interval,
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Table 2. Particle Size Distribution of Sterling FT-G (2700) from Electron Microscopya D hi
∆D/ nm
ni
104∆vi/ cm3 g-1
104∆vi/ ∆D
∆si/ m2 g-1
403 390 377 363 349 335 322 309 295 281 268 255 241 227 213 200 186 172 158 145 131 117 104 91 77 63
13 13 14 14 14 13 13 14 14 13 13 14 14 14 13 14 14 14 13 14 14 13 13 14 14 14
4 9 2 6 11 24 32 38 52 85 102 132 166 192 246 292 363 340 318 302 198 98 64 35 10 4
43.15 87.99 17.66 47.30 77.07 148.7 176.1 184.8 220.0 310.9 323.6 360.8 383.0 370.2 391.8 385.0 385.0 285.2 206.7 151.7 73.37 25.87 11.87 4.347 0.753 0.165
3.319 6.769 1.262 3.379 5.505 11.44 13.55 13.20 15.72 23.91 24.89 25.77 27.36 26.44 30.14 27.50 27.50 20.37 15.90 10.84 5.240 1.990 0.913 0.311 0.0537 0.0118
0.0642 0.1354 0.0281 0.0782 0.1325 0.2664 0.3281 0.3588 0.4475 0.6637 0.7245 0.8488 0.9535 0.9784 1.1037 1.1551 1.2419 0.9947 0.7851 0.6279 0.3360 0.1327 0.0685 0.0287 0.0059 0.0016
a The total particle count (column 3) is 3125, and the total surface area (column 6) is 12.48 mg2 g-1.
∆si, is given by
∆si ) 6∆vi/D hi
rK/ Vt/ t/ nm cm3 g-1 nm
(A-3)
and
D/ nm
150
0.265
5.50 751.2
140
0.264
5.32 702.0
130
0.263
5.14 652.8
120
0.262
4.95 603.6
110
0.260
4.77 554.5
100
0.254
4.59 505.3
90
0.248
4.32 455.7
80
0.239
4.14 406.5
70
0.224
3.87 356.9
60
0.199
3.60 307.3
50
0.160
3.33 257.6
40
0.102
2.97 207.6
30
0.055
2.61 157.5
20
0.031
2.16 107.1
10
0.017
1.53
D h/ nm
∆D/ Σvs/ 104∆vs/ Σs/ nm cm3 g-1 ∆D m2 g-1
726.6 49.2 0.0019
0.380
0.01
677.4 49.2 0.0037
0.380
0.03
628.2 49.2 0.0056
0.380
0.05
579.1 49.2 0.0094
0.764
0.08
529.9 49.2 0.0207
2.31
0.20
480.5 49.6 0.0320
2.28
0.33
431.1 49.2 0.0491
3.47
0.55
381.7 49.6 0.0776
5.74
0.96
332.1 49.6 0.1254
9.63
1.74
282.5 49.6 0.2004
15.1
3.17
232.6 50.1 0.3119
22.3
5.75
182.6 50.1 0.4012
17.8
8.38
132.3 50.5 0.4423
8.15
10.16
81.4 51.4 0.4587
3.19
11.66
55.7
∆vc/cm3 g-1 ) v1 - v2 ) 0.001
(A-2)
and these quantities ∆vi, ∆vi/∆D, and ∆si are given in columns 4-6 in Table 2. II. Example Calculation of the Particle Size Distribution from an Adsorption Isotherm. The isotherm is plotted as liquid volume adsorbed against Kelvin radius, rK (eqs 3 and 4), as in Figure 3c. At 10 nm intervals from r1 ) 150 nm to r15 ) 10 nm, the volumes v1 to v15 along each branch of the hysteresis loop are determined. The multilayer thicknesses t1 to t15 are calculated (eq 5) at each rK from
ln P/P° ) -3.90 × 10-5/vt2.0
Table 3. Calculation of the Particle Size Distribution of Sterling FT-G (2700) from the Desorption Branch for 2,2-Dimethylpropane Adsorbed at 273.15 K
the average film thickness, ht, is
ht /nm ) (t1 + t2)/2 ) 5.41
(A-8)
and the surface area, ∆s, within this group of pores is given by eq 2; i.e.
∆s/m2 g-1 ) (6∆vc/D h)
(1 - ) ) 10438∆v /Dh ) 0.0140 c
(A-9) The total volume, ∆vp, in this first group of pores, is given by eq 7; i.e.
∆vp/cm3 g-1 ) ∆vc + ht ∆s ) 1.07 × 10-3 t/nm ) 1000vt/11.1 cm3
(A-4)
g-1
and the specific surface where vt is expressed in area of the adsorbent is 11.1 m2 g-1. These quantities, rK, vt, and t, are given in the first three columns of Table 3. Along the desorption branch of the hysteresis loop, the particle diameter, D, corresponding to each rK is given by
D/nm ) (rK + t)0.207
(A-5)
and along the adsorption branch
D/nm ) (rK + t)/0.257
(A-6)
The average particle diameter, D h , and the interval width, ∆D, are calculated from successive values of D. These quantities are given in columns 4-6 of Table 3. We assume that there are no pores where rK > 150 nm. Hence, over the interval r1 to r2 (150-140 nm) along the desorption branch of the isotherm the core volume, ∆vc, is given by
(A-7)
(A-10)
and the corresponding adsorbent volume, ∆vs (eq 1), is given in column 7 of Table 3, where
∆vs/cm3 g-1 ) ∆vp
(1 - ) ) 1.74∆v
p
) 0.0019 (A-11)
Column 8 gives ∆vs/∆D for each group of pores, and Figure 2 shows this quantity plotted vs D h . Finally, column 9 shows the cumulative surface area, ∑∆s, as the calculation progresses. This quantity is required to calculate ∆vc for other pore groups; i.e. n
∑0 ∆s
∆vc/cm3 g-1 ) (vn - vn+1) - ∆t
(A-12)
where ∆t ) tn - tn+1. The calculation then returns to eqs A-8 through A-12 for successive pore groups. The numbers in Table 3 are for the isotherm for 2,2-dimethylpropane at 273.15 K. LA960511S
