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Langmuir 1999, 15, 526-532
Contribution of a Repulsive Potential in a Micropore Volume Filling. A Surface of Micropore Walls Vladimir Kh. Dobruskin* Aiala 21 st., Beer-Yacov, 70300, Israel Received March 17, 1998. In Final Form: August 10, 1998 In the case of narrow micropores, a volume filling of the single micropore and a two-dimensional condensation on its walls occur at the same condensation pressure, and the local adsorption behavior may be modeled by the condensation approximation. Proceeding from (i) the accepted model of carbon heterogeneity, (ii) the theory of adsorption on homogeneous surfaces considering lateral interactions, and (iii) the 10-4 potential function, the correlation between a micropore filling pressure, p, and a micropore reduced half-width, d/r0, is derived. This approach provides a correct description of equilibrium data in a range of p/ps ≈ 5 × 10-7 to 0.2 at 293 K and leads to reasonable distribution functions of micropore sizes and adsorption energies. The model contains a specific prediction that there is a minimum filling pressure, pmin, that corresponds to the maximum adsorption energy at d/r0 ) 1 and determines the lower boundary of the micropore volume filling. For benzene adsorption at 293 and 423 K, the pmin/ps values are equal to 7.36 × 10-8 and 1.15 × 10-5, respectively. When p < pmin, only a submonolayer adsorption occurs on a micropore surface. The expression for the calculation of a micropore surface is derived, and the underlying assumption is discussed.
Introduction 1-3
Gregg and Sing suggested that a monolayer formation on the pore walls significantly enhances the adsorption affinity in the pore core (the volume of the pore remaining after the adsorbed layers are formed on micropore walls) and, thus, that a complete pore filling occurs. It was shown that this suggestion is valid in the case of narrow micropores for which the ratio of adsorption affinity in the pore core to that in the first layer R is greater than 1.4 Keeping in mind this phenomenon and models of the adsorption behavior with lateral interactions, an adsorption process in an individual micropore can be describe as follows.4,5 As the pressure is increased from zero, adsorption begins to occur on the pore walls. When a micropore surface coverage, θ, comes to the critical value at the critical condensation pressure pc, surface condensation is initiated. This process continues at the same pressure until a point is reached when R ≈ 1. At this point, before completion of the first layer formation, the adsorption process “is energetically as favorable for an adsorbate molecule to exist between the monolayers of adsorbate in the center of the pore, as it is to complete the monolayer coverage.”6 The volume condensation starts and results in complete micropore filling. This means that a volume filling of the single micropore and a two-dimensional (2D) condensation on its walls occur at the same critical pressure, and the actual isotherm for an individual micropore can be modeled by the simpler step function4,5
θ(p) )
{
0 for p < pc 1 for p < pc
(1)
* E-mail:
[email protected]. (1) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity; Academic Press: London, 1982. (2) Carlott, P. J. M.; Roberts, R. A.; Sing K. S. W. Carbon 1987, 25, 59. (3) Sing, K. S. W. Carbon 1987, 25, 155. (4) Dobruskin, V. Kh. Langmuir 1998, 14, 3847-3857. (5) Dobruskin, V. Kh. Langmuir 1998, 14, 3840-3846. (6) Marsh, H. Carbon 1987, 25, 49.
as is accepted in the condensation approximation (CA) method.7-10 The values of pc may be found from the theories of adsorption on homogeneous surfaces considering lateral interactions. To make use of these theories, the adsorption energies should be calculated. Everett and Powl gave11 the following equation (the 10-4 potential) for interactions, , between a molecule and two parallel carbon lattice planes whose nuclei are a distance 2d apart, the molecule being a distance z from the central plane
(z) )
{ [( ) ( ) ] [( ) ( ) ]}
r0 10 1 *0 3 5 d+z
10
+
r0 10 d-z r0 4 r0 1 + 2 d+z d-z
4
(2)
where *0 is the depth of the energy minimum for the interaction of a single solid lattice plane with a single molecule in a gas phase
6 *0 ) πn*cxr20 5
(3)
Here n is the number of interacting centers per unit area of the lattice plane, *cx is the depth of potential energy minimum for the interaction between a single molecule and carbon atom, and r0 is the distance at which cx ) 0. The asterisk (*) denotes the potential energy minimum. According to the agreement, adsorption energies are taken to be positive. An excess of adsorption energy in micropores with respect to an open graphite surface, ∆ ) (* - *0), is given as (7) Roginski, S. Z. Adsorption and Catalysis on Heterogeneous Surface; Izdatelstvo AN SSSR: Moscow, 1949 (Russian). (8) Roginski, S. Z. Dokl. Akad. Nauk SSSR 1944, 45, 61. (9) Rudzinski, W.; Everett D. H. Adsorption of Gases on Heterogeneous Surface; Academic Press: New York, 1992. (10) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, 1988; pp 45-46. (11) Ewerett, D. H.; Powl, J. C. J. Chem. Soc., Faraday Trans. 1976, 72, 619.
10.1021/la980310j CCC: $18.00 © 1999 American Chemical Society Published on Web 12/29/1998
Repulsive Potential in a Micropore Volume Filling
{ { [( ) ( ) ] [( ) ( ) ]} |
r0 10 1 *0 3 5 d+z r0 4 r0 1 + 2 d+z d-z
* - *0 ) Minimum r0 d-z
10
10
Langmuir, Vol. 15, No. 2, 1999 527
+
}|
4
- *0
(4)
Proceeding from this equation, the following approximate relation between * and d/r0 was introduced
(
ln(* - *0) ) ln *0 + k 1 -
)
Micropore Volume Filling
d r0
(5)
where k is the energy parameter, which may be taken to be 4. Equation 5 is valid for d/r0 > 1.4 To describe adsorption in a real micropore system, the carbon was approximated by the system of independent micropores of identical volumes, each of the micropores being within the confines of ideal graphite planes. The micropore walls were considered to be unisorptic areas, and all of the heterogeneity of the real active carbons was attributed to the normal distribution of micropore widths
fN(d,µd,σd) )
{ (
)}
1 1 d - µd exp 2 σd σdx2π
2
(6)
where µd and σd are the mathematical expectation and the standard deviation of d, respectively.4 For this model, the distribution of fractional adsorption volumes θ is also described by the same normal model
fN(θ,µd,σd) )
{ (
)}
1 1 d - µd exp 2 σd σdx2π
2
(7)
Proceeding from the heterogeneity model employed and eq 5, the occupied fractional adsorption volume is given by the expression4
1-θ)
∫0(ln A - µ )/σ x12π exp{- z2 } dz 2
c
y
y
(8)
Here
Ac ) RT ln
pc0 p
(9)
where pc0 is the 2D condensation pressure on a nonporous surface, p is the outer pressure, and µy and σy are the parameters of the distribution that are determined from the following relationships
µy ) k + ln *0 σy )
k σ r0 d
k µ r0 d
(10) (11)
It was also shown that, for benzene adsorption on active carbons, pc0 ≈ ps4, where ps is the saturated pressure of the liquid adsorbate, and Ac in eq 8 may be substituted by the adsorption potential A
A ) RT ln
ps p
Dubinin-Radushkevich12 and Dubinin-Astahkov13-15 equations. When pressure reduces below p/ps ≈ 1 × 10-5, there are significant deviations between calculated and experimental uptakes induced by the application of eq 5. The purpose of the present study is to describe the adsorption behavior on the basis of the more realistic 10-4 potential function.
(12)
The model describes benzene adsorption in the range p/ps ≈ 1 × 10-5 to 1 × 10-1, leads to reasonable values of micropore sizes, and provides the foundation for the
Micropore Filling Pressure. In the simplest FowlerGuggenheim (FG) approach, the lateral interaction is taken into account by assuming that the total interaction energy is the same for all configurations of atoms on the adsorption sites and is equal to cw, where c is the average number of nearest-neighbor sites and w is the lateral interaction energy. The condensation pressure for a nonporous surface is obtained from the FG isotherm as follows9
pc ) K exp[(-* + 2kTc)/kT]
(13)
where Tc ) |cw/4k| is the critical temperature for a 2D condensation. The constant K has the following detailed form
K)
(2πm)3/2(kT)5/2qg(T) h3ga(T)
(14)
Here m is the mass of the adsorbate molecule, h is Planck’s constant, and gg and ga are the partition functions for the internal degrees of freedom of the gas and of the adsorbed species, respectively. If one substitutes in eq 13 the adsorption energy *0 for a hypothetical ideal graphite surface, this equation will define the critical pressure for a graphite surface pc0
pc0 ) K exp[(-*0 + 2kTc)/kT]
(15)
Since micropore walls are considered to be unisorptic areas, a lateral interaction energy may be regarded to be identical for all walls; that is, cw ) constant. For a carbon micropore with a half-width d, * in eq 13 is a function of only d/r0. Combining eqs 13 and 15, after some algebra, one arrives at
ln
pc0 * - *0 ) pc RT
(16)
The adsorption energy in eq 16 is expressed in units that refer to one mole of a gas. This equation gives the relation between a micropore filling pressure (MFP) and ∆ ) (* - *0). For the benzene-active carbon system, *0 ) 40 kJ/mol and can be calculated theoretically, as was demonstrated by Kiselev and co-workers.16 If one denotes (12) Dubinin, M. M.; Radushkevich, L. V. Dokl. Akad. Nauk. SSSR 1947, 55, 331. (13) Astakhov, V. A.; Dubinin, M. M.; Romankov, P. G. In Adsorbents, their Preparation, Properties and Application; Dubinin, M M., Plachenov, T. G., Eds.; Nauka: Leningrad, 1971; p 92 (Russian). (14) Dubinin, M. M. In Adsorption-Desorption Phenomena; Ricca, F., Ed.; Academic Press: London, 1972; pp 3-18. (15) Astakhov, V. A.; Dubinin, M. M.; Mosharova, L. P.; Romankov, P. G. TOXT 1972, 6, 343. (16) Avgul, N. N.; Kiselev, A. V.; Poshkus, D. P. Adsorption of Gases and Vapors on the Homogeneous Surfaces; Khimia: Moscow, 1975 (Russian).
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Figure 1. Dependence of the adsorption energies (*) upon the reduced micropore half-widths (solid line, the 10-4 model; dashed line, eq 5). * ) 0 at d/r0 ) 0.858, and * ) *0 at d/r0 ) 0.889.
Figure 2. Dependence of the excess adsorption energy upon the reduced half-width. There are two values of reduced halfwidths (d1/r0 and d2/r0) corresponding to every value of ∆*. The excess adsorption energy ∆* ) 0 at d/r0 ) 0.889.
{ { [( ) ( ) ] [( ) ( ) ]} }| |
r0 10 10 1 + 3 5 d+z 4 r0 r0 4 1 + 2 d+z d-z
φ(d/r0) ≡ Minimum r0 d-z
10
-1
(17)
eqs 4 and 16 can be rewritten as
(* - *0) ) *0 φ (d/r0) ln
pc0 *0φ(d/r0) ) pc RT
(18) (19)
In the case of approximate relationships, eq 19 reduces to
ln
pc0 *0 exp[k(1 - d/r0)] ) pc RT
(20)
Figure 1 shows the plots of *(d/r0) corresponding to eqs 4 and 5. For the 10-4 potential, the repulsive parts of the opposing wall potentials begin to increase as the pore width is reduced below d/r0 < 1, until the entire solid-fluid slit potential becomes repulsive. The values of d/r0 ) 0.858 and d/r0 ) 0.889 correspond to * ) 0 and * ) *0, respectively. This means that, for micropores with d/r0 < 0.858, the pore space is inaccessible for adsorbate and no adsorption occurs. A volume filling of micropores with d/r0 ) 0.889 occurs at the same pressure pc0 as the surface condensation on an ideal graphite surface (eq 15). A filling of pores with 0.858 < d/r0 < 0.889 occurs at pc0 < p < ps and cannot be described by the proposed condensation
Figure 3. Decimal logarithm of the micropore filling pressure as a function of reduced half-widths. The curve is calculated on the basis of the 10-4 model at T ) 293 K. When log[p/ps] ) -3.5, d1/r0 ) 0.91 and d2/r0 ) 1.18. Log[pmin/ps] is equal to 7.36 × 10-8.
Figure 4. Micropore filling pressure as a function of d/r0 (solid line, the 10-4 model at 423 K; points, the 10-4 model at 293 K; dashed line, the approximate relationship at 293 K).
mechanism. Equation 5 with k ) 4 gives a good fit to the 10-4 function only for d/r0 > 1 and leads to the wrong results for d/r0 < 1. Figure 2 plots (* - *0)/*0 via reduced half-width for d/r0 > 0.889. According to eq 4, there are two values of reduced sizes, d1/r0 and d2/r0, corresponding to every value of energy. Consequently, the inverse equation, d/r0 ) φ-1(*), has two roots d1/r0 and d2/r0 for each *. Such behavior is absent from the approximate eq 5. The MFPs for benzene adsorption as a function of d/r0, p ) φ1(d/r0) corresponding to eqs 19 and 20 are plotted in Figures 3 and 4. As the pore widths are reduced, eq 20 shows a continuous decreasing of the MFP, whereas the 10-4 potential predicts a rapid rise in filling pressures for micropores in the range 0.889 < d/r0 < 1 (Figure 4). As a result, there is the minimum filling pressure pmin (Figure 3) corresponding to the maximum adsorption energy at d/r0 ) 1. For benzene adsorption at 293 and 423 K, the pmin/ps values are equal to 7.36 × 10-8 and 1.15 × 10-5, respectively. Physically this condition merely means that the outer pressure must exceed pmin for micropore volume filling to occur. Just as in the case of adsorption energy, for the 10-4 model there are two values of d/r0 corresponding to every value of filling pressure, and the inverse equation, d/r0 ) φ1-1(p), has two roots d1/r0 and d2/r0 for each p (Figure 3). Adsorption Isotherm. According to the CA (eq 1), all micropores for which pc < p are filled with adsorbate, where p is the outer pressure. Hence, we come to the conclusion (Figure 3) that the volume filling occurs within the confines of the lower boundary, d1/r0, and the upper boundary, d2/r0, of micropores. Therefore, the fraction of
Repulsive Potential in a Micropore Volume Filling
Langmuir, Vol. 15, No. 2, 1999 529
and must be calculated by means of the cumulative distribution function (CDF).17 Since the random variables d and d/r0 are related by the linear function and d is distributed according to a normal model (eq 6), d/r0 is also distributed according to a normal model with the expected value and the standard deviation17
The excess adsorption energy is a function of the random variable d. The above-mentioned approach provides the general way for calculation of the CDF of such functions. If an expression for adsorption energies is given by the simple monotone function for which the inverse function can be found, the distribution of ∆ can be obtained in an analytically closed form. In such a case, the relationship between distribution functions is established by the following theorem:17 if y is a monotone function of random variable x, y ) φ(x), and a distribution of x is described by the density function f(x), a density function of y, g(y), is given by the expression
µd/r0 ) µd/r0; σd/r0 ) σd/r0
g(y) ) f(φ-1(y))|(φ-1(y))′|
occupied micropore as a function of d/r0 is equal the probability, Pr(d1/r0 < d/r0 < d2/r0), that d/r0 falls in the range
d1/r0 < d/r0 < d2/r0
(21)
(22)
Hence, the CDF of the normally distributed random variable d/r0 is given by the expression
Fd/r0(d/r0,µd/r0,σd/r0) )
{ (
∫
dx (23)
)}
dx (24)
2
and
θ ) Pr(d1/r0 < d/r0 < d2/r0) )
∫dd/r/r (σ /r1)x2π exp 2 0
1 0
d
0
{ ( -
1 x - µd/r0 2 (σd/r0)
2
The lower and upper limits of the integral are calculated by means of the inverse function d/r0 ) φ1-1(p). Equation 24 describes the dependence of θ upon pressure, that is, the adsorption isotherm. When the outer pressure tends to the minimum filling pressure pmin (Figure 3), the lower and the upper limits of the integral converge and θ tends to zero. As the pressure is reduced below pmin, only a submonolayer adsorption occurs on micropore walls. Distributions of Adsorption Energies. In addition to the adsorption isotherm, the distribution of adsorption energies is of interest. Now if we return to Figure 2, we note that the probability of the occurrence of energy values ∆ ) (* - *0) < is equal to the probability, Pr(0.889 < d/r0 e d1/r0 ∪ d2/r0 e d/r0 < ∞), that d/r0 falls in the ranges
0.889 < d/r0 e d1/r0 and d2/r0 e d/r0 < ∞
(25)
Proceeding just as in the case of deriving the isotherm equation, we obtain the CDF for excess adsorption energies, F∆
F∆0(∆,µd/r0,σd/r0) ) d /r 1 exp ∫0.889 (σ /r )x2π 1 0
d
0
{ ( )} { ( )} -
∫d∞/r (σ /r1)x2π exp 2 0
d
0
1 x - µd/r0 2 (σd/r0) -
2
1 x - µd/r0 2 (σd/r0)
where x ) φ-1(y) is the inverse function. This procedure was used in a previous publication,4 proceeding from the simple monotone energy function given by eq 5. Experimental Results
)}
d/r0 1 x - µd/r0 1 exp -∞ 2 (σd/r0) (σd/r0)x2π
(27)
dx + 2
dx (26)
where d1/r0 and d2/r0 are the real roots of the equation d/r0 ) φ-1(*). Since the inverse function cannot be expressed exactly in a symbolic form, we have to resort afresh to the numerical calculations. The density distribution function f∆ is calculated by numerical differentiation of eq 26. (17) Ventzel, E. S. Probability Theory; Physico-Mathematic Literature Press: Moscow, 1958 (Russian).
The treatment of a set of experimental pairs of values (a, p), where a is the experimental uptake, is produced as follows. Excess adsorption energies ∆ are found from eq 16 after substitution of pc by the experimental values of p. As was mentioned, for benzene adsorption pc0 ≈ ps. The lower boundary d1/r0 and the upper boundary d2/r0 of reduced half-widths are calculated by the numerical solutions of eq 4 or 19. After transformation of a to θ ) a/a0, where a0 is the uptake corresponding to the complete micropore filling, a set of values (θ, d1/r0, d2/r0) is found. Mathematically, θ is expressed by eq 24. The problem now is to construct an integrand from information about the integral. This can be done numerically starting from the specified point (µ, σ) and progressively trying to get closer to the solution. In our treatment we assign the values of µd and σd found from eq 8 to be a starting point. The procedure for finding these values is described in a previous publication.4 This approach was applied to four benzene adsorption isotherms. The active carbons employed were microporous carbons with small mesopore surfaces.4 The parameters of pore size distributions corresponding to eqs 8 and 24 are given in Table 1. It is shown that both of the equations give practically equal values of µd, whereas the 10-4 function leads to the lower value of σd. Experimental and calculated values of adsorption uptakes are presented in Table 2. For ART active carbon, also given in Table 2 are the lower and upper boundaries of reduced half-widths within the confines of which the volume filling occurs. The 10-4 model provides a significantly better representation of the experimental data, especially in a lowpressure region, and eq 24 is valid in the range p/ps ≈ 5 × 10-7 to 0.2. Due to a repulsive potential, the smallest micropore can be filled only at high p. Since eq 5 neglects the repulsive potential for d/r0 < 1, it predicts the wrong adsorption behavior of the smallest micropores. As a result, eq 8 overestimates the contribution of the smallest micropores at the lowest p and underestimates their contribution at high pressures. The distributions of fractional volumes over reduced pore widths are illustrated by Figure 5. The 10-4 model predicts the narrower distributions of adsorption volumes and shows that micropores with d/r0 < 0.889 do not take part in the adsorption process. The cumulative distribution functions F∆* corresponding to the 10-4 model for the same species of ACs are given in Figure 6. In particular, it demonstrates that the probability of the occurrence of the adsorption volume for which ∆* ) *0
530 Langmuir, Vol. 15, No. 2, 1999
Dobruskin
Table 1. Parameters of Micropore Size Distributions S0 (m2/g)
active carbons
µd/r0 eq 8
σd/r0 eq 8
µd/r0 eq 24
σd/r0 eq 24
W0 (cm3/g)
Ea (kJ/mol)
d (nm)
eq 30
eq 31
eq 32
AG-3P ART SKT-3 SAU
1.34 1.17 1.17 1.14
0.16 0.15 0.14 0.10
1.298 1.167 1.168 1.144
0.117 0.091 0.085 0.077
0.455 0.459 0.495 0.398
13.27 24.43 24.73 28.46
0.904 0.491 0.485 0.422
862 960 1040 853
503 935 1020 943
855 959 1034 849
a
Taken from Table 5 in ref 4. Table 2. Benzene Adsorption on Active Carbons at 293 K ART
SAU
P/Ps
θexp
d1/r0
d2/r0
θcal (eq 24)
θcal (eq 8)
P/Ps
θexp
θcal (eq 24)
θcal (eq 8)
4.75 × 10-7 4.86 × 10-6 1.09 × 10-5 3.03 × 10-5 6.84 × 10-5 0.000 148 0.000 536 0.000 638 0.001 31 0.002 66 0.0124 0.0638 0.126
0.126 0.245 0.287 0.366 0.418 0.494 0.613 0.628 0.703 0.791 0.885 0.960 0.990
0.910 0.910 0.910 0.910 0.910 0.910 0.910 0.910 0.910 0.910 0.910 0.905 0.902
1.065 1.105 1.119 1.135 1.148 1.162 1.189 1.193 1.211 1.232 1.289 1.388 1.453
0.128 0.247 0.297 0.360 0.416 0.477 0.593 0.611 0.685 0.760 0.908 0.990 0.997
0.177 0.265 0.305 0.364 0.417 0.474 0.581 0.597 0.664 0.731 0.871 0.973 0.992
4.81 × 10-7 3.44 × 10-6 7.29 × 10-6 1.63 × 10-5 3.81 × 10-5 6.03 × 10-5 9.09 × 10-5 0.000 128 0.000 189 0.000 257 0.000 605 0.001 45 0.002 25 0.005 67 0.0113 0.0181 0.0396 0.0765
0.156 0.269 0.330 0.395 0.471 0.518 0.559 0.600 0.643 0.680 0.755 0.835 0.864 0.912 0.940 0.955 0.978 0.996
0.152 0.283 0.338 0.402 0.472 0.510 0.547 0.579 0.616 0.646 0.732 0.818 0.858 0.929 0.966 0.982 0.996 0.999
0.218 0.312 0.358 0.410 0.473 0.510 0.544 0.573 0.610 0.636 0.714 0.792 0.830 0.900 0.941 0.963 0.987 0.997
) 40 kJ/mol is equal to unity and, consequently, micropores with * > 2*0 are absent in carbon species. On the contrary, the approximate equation predicts the occurrence of unrealistic energy values > 2*0 (see Figure 7 in ref 4). Further Discussion Accuracy of the Model. Equation 24 is applicable for a benzene adsorption in the range p/ps ≈ 5 × 10-7 to 0.2 at 293 K. According to the underlying CA model, the micropore and its walls may only be in two states: either free from adsorbate at p < pc or filled with adsorbate at p > pc. Hence, in the lowest pressure region, eq 24 does not take into account the submonolayer adsorption and does not converge to the Henry isotherm limit. To describe adsorption in a submonolayer region, the surface area S0 of micropore walls and its distribution over adsorption energies should be determined. One can visualize the heterogeneous surface corresponding to a micropore carbon as follows.5 If one (i) disassembles a set of micropores with different sizes, (ii) puts together the walls of all individual micropores, and (iii) attributes to each pair of opposing walls the same energy which they had inside a micropore, a patchwise heterogeneous surface19 will be formed, each patch being composed of two walls. The overall isotherm in a submonolayer region may be found by integration of the local isotherm over all patches.9,10,18,19 Surface of Micropore Walls. Proceeding from a normal distribution of micropore widths, the number of micropores with half-widths from x to x + dx, dN, is equal to (18) Ross, S.; Olivier, J. P. On Physical Adsorption; Interscience: New York, 1964. (19) Steele, W. A. The Interaction of Gases with Solid Surface; Pergamon Press: Oxford, 1974.
dN ) NfN(x,µd,σd) dx
(28)
where N is the total number of micropores in a carbon sample. The patch area s (that is, the surface area of both opposing walls) is equal to v/x, where v is the volume of the single micropore. The surface area ds of these dN patches is found as
v ds ) NfN(x,µd,σd) dx x
(29)
and the total geometric surface of the micropore walls is given by the expression
S0 )
vNfN(x,µd,σd) dx ) 0 x ∞ 1 x - µd 1 exp W0 0.889r 0 2 σd xσdx2π
∞ ∫0.889r
∫
{ ( )} 2
dx (30)
Here W0 ) vN is the micropore volume. The derivation of eq 30 implies that an adsorbent consists of slitlike micropores of identical volumes but different widths. The values of W0 and r0 exert a strong influence on S0. In the present paper we consider only experimental data for benzene adsorption. Different values of r0 for benzene do appear in different papers. Wang and Do accept r0 ) 0.37 nm;20 Matsumoto et al. for a single-center model take r0 ) 0.43 nm.21 When estimating S0 and the average micropore half-width d, Stoeckli22 takes the critical dimension of benzene to be equal to 0.41 nm, and we shall accept this value for our further calculations. The micropore volume is usually found from the adsorption isotherm as W0 ) a0/F, where a0 is the uptake corre(20) Wang, K.; Do, D. D. Langmuir 1997, 13, 6226. (21) Matsumoto, A.; Zhao, J.-X.; Tsutsumi, T. Langmiur 1997, 13, 496. (22) Stoecli, F. Adsorpt. Sci. Technol., Special Issue 1993, 10, 3.
Repulsive Potential in a Micropore Volume Filling
Langmuir, Vol. 15, No. 2, 1999 531
Figure 6. Cumulative distribution functions of adsorption volumes over the excess adsorption energies ∆*. F∆* is the probability of the occurrence of adsorption volumes for which ∆* is less than or equal to any particular value. For example, F∆*(∆* e 10 kJ/mol) ≈ 0 for SAU, and F∆*(∆* e 10 kJ/mol) ≈ 0.5 for AG-3P.
For the arbitrary system of slitlike micropores that is characterized by random values of micropore volumes and half-widths, the total micropore surface is given by the expression N
vi
∑ i)1 d
S0 )
(33)
i
It is obvious that eq 31 (or eq 32) may be transformed as follows N
S0 )
{
W0
N
∑ i)1
)
) N
d
N-1
N
vi
di ∑ i)1
vi ∑ i)1
N
(34)
di ∑ i)1
and one may see that Figure 5. Micropore size distributions calculated on the basis of the 10-4 model (solid lines) and the approximate relationship (dashed lines).
sponding to complete micropore filling and F is the density of the liquid adsorbate. Although there are, strictly speaking, several uncertainties connected with calculation of W0 from this relationship, we shall use this value in our calculations. At present, S0 of carbon slitlike micropores is estimated as follows
S0 ) W0/d
(31)
where the average micropore half-width is found as d (nm) ) 12/E (kJ/mol).22 Here, E is the characteristic adsorption energy. The comparison of S0 calculated by numerical solution of eq 30 with that resulting from eq 31 (Table 1) shows that the values of S0 are close for ART, SKT-3, and SAU, whereas there is significant deviation in the case of AG-3P. On the other hand, it is obvious that d is equal to the mathematical expectation of d and, consequently,
N
N
vi
∑ i)1 d
S0 )
N )
vi ∑ i)1
N
di ∑ i)1
i
when di ≡ d (35)
N
N
vi
∑ i)1 d
i
N *
vi ∑ i)1
N
di ∑ i)1
when di * d
(32)
Hence, eqs 31 and 32, strictly speaking, are valid only in the case of micropores with identical widths. Special consideration should be given to the fact that both the eqs 30 and 32, based on such different underlying assumptions, lead to practically identical results. We recall here that eq 30 is valid for slitlike micropores of identical volumes. For the normal distribution with the mean µd/r0 and standard deviation σd/r0, 68.3% of d/r0 values deviate less than σd/r0 and 95.5% of d/r0 values deviate less than 2σd/ r0 from µd/r0.23 The parameters of micropore active carbons
Substitution of µd in eq 32 leads to the coincidence of S0 calculated by eqs 30 and 32.
(23) Burington, R. S.; May, D. C. Handbook of Probability and Statistics with Tables; McGraw-Hill: New York, 1970.
S0 ) W0/µd
532 Langmuir, Vol. 15, No. 2, 1999
(Table 1) are such that, on average, ≈70% of micropores fall in the range of widths µd ( 10% and ≈96% of micropores fall in the range of sizes µd ( 20%. The close values of S0 following from eqs 30 and 32 indicate that not only a dispersion of micropore widths but also that of micropore volumes correspond to the narrow distribution function, and one may consider the volumes of all individual micropores to be approximately identical. This simplified model of a real adsorbent was introduced in a previous publication.4 The good agreement between results predicted on the basis of this approximation and the experimental data4,5 seems to justify it. It is worth mentioning here that eq 30 bears more information than eqs 31 and 32 and could be used for calculations of the distribution of a micropore surface over adsorption energies.
Dobruskin
As a whole, the conclusions resulting from the present approach should be regarded as relating to an effective porous material, in which all of the heterogeneity of the real AC is attributed to a normal distribution of micropore sizes. This effective porous material is approximated by the system of independent micropores of identical volumes, each micropore being within the confines of ideal graphite planes. Acknowledgment. The author is grateful to Dr. S. D. Kolosentzev and Dr. Yu. Ustinov of the Technological Institute of St. Petersburg for kindly supplying experimental adsorption data. LA980310J
