3840
Langmuir 1998, 14, 3840-3846
Micropore Volume Filling. A Condensation Approximation Approach as a Foundation to the Dubinin-Astakhov Equation Vladimir Kh. Dobruskin* st. Aiala 21, Beer Yacov, 70300, Israel Received November 5, 1997. In Final Form: February 4, 1998
A volume filling of a single micropore and a two-dimensional (2D) condensation on its walls occur at the same critical pressure, and the local adsorption behavior may be modeled by the condensation approximation. This phenomenon underlies the approach to the micropore volume filling based on the condensation approximation (VFCA) and the Dubinin-Astakhov (DA) equation. The DA equation follows from the VFCA approach, being an approximate form to the general relationship. The physical meanings of the exponent, n, and the characteristic energy, E, with respect to the surface heterogeneity, are given. As a whole, n is determined only by the standard deviation of the micropore widths: the less the heterogeneity, the larger n, approaching to infinity for the homogeneous carbon. The characteristic energy depends on the average micropore sizes and its standard deviation. In the case of the DA equation with n ) 2 or n ) 3, standard deviations are equal to 0.4915 or 0.3493, respectively, and E depends only upon an average micropore size or upon a related adsorption potential. For the individual micropore, E determines the critical pressure of a 2D condensation. In the case of water adsorption on active carbons, the base heterogeneity due to variation in the pore widths, as perceived by water molecules, is negligible. Hence, water adsorption may be considered to be an extension of a 2D-condensation into practically homogeneous micropore volumes. It is shown that empirical relationships for calculating of average micropore sizes and an applicability of the DA equation to the adsorption on nonporous surfaces may be also explained in the framework of the VFCA approach.
Introduction It was shown that a hypothetical monolayer formation on the walls of narrow micropores would enhance the adsorption affinity in the pore core and the ratio of adsorption affinity in the pore core to that in the first layer, R, would be greater than 1.1 The adsorption process in an individual micropore can be described as follows. As the pressure is increased from the zero, adsorption begins to occur on the pore walls. When micropore surface coverage, θ, comes to a critical value at the critical condensation pressure, pc, surface condensation initiates. This process continues at the same pressure until a point is reached when R ≈ 1. At this point, before a 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 A volume condensation starts in a free micropore volume and results in the complete micropore filling. From this point of view, the statistical mechanical theories of adsorption on homogeneous surfaces considering lateral interactions may act as starting point to the description of physical adsorption in micropores. These theories lead to the following relationship between the adsorption energy, *, the critical pressure, and the * E-mail:
[email protected]. (1) Dobruskin, V. Kh. Langmuir, 1998, 14, 3847. (2) Marsh, H. Carbon 1987, 25, 49. (3) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surface; Academic Press: New York, 1992. (4) Lastokie, C., Gubbins, K. E.; Quirke, N. J. Phys. Chem. 1993, 97, 4786-4796. (5) Lopez-Ramon, M. V.; Jagiello, J.; Bandosz, T. J.; Seaton, N. A. Langmuir 1997, 13, 4435. (6) Gusev, V. I.; O’Brien, J. A.; Seaton, N. A. Langmuir 1997, 13, 2815.
micropore half-width, d
(
ln(* - 0*) ) ln RT ln
)
( )
pc0 d ) ln 0* + k 1 pc r0
(1)
Here 0* is the well depth and pc0 is the critical pressure of 2D condensation on the nonporous surface, r0 is the adjustable Lennard-Jones parameter, and k is the energy parameter, which may be taken to be 4. This equation is valid for micropores with reduced half-widths d/r0 > 1. A random distribution of micropore widths results into the distribution of micropore walls over the well depths and over pc. Proceeding from a model of the surface heterogeneity and the normal distribution of the micropore widths
fN(d,µd,σd) )
{ (
)}
1 1 d - µd exp 2 σd σdx2π
2
(2)
where µd and σd are the mathematical expectation and the standard deviation of d, respectively, the log-normal distributions of the well depths and pc are derived
fLN(* - 0*,µy,σy) ) 1 (* - 0*)σyx2π
{ (
exp -
)}
1 ln(* - 0*) - µy 2 σy
S0743-7463(97)01210-9 CCC: $15.00 © 1998 American Chemical Society Published on Web 06/09/1998
2
(3a)
Micropore Volume Filling
(
fLN RT ln
)
pc0 ,µ ,σ ) pc y y
Langmuir, Vol. 14, No. 14, 1998 3841
{(
1 1 exp pc0 2 RT ln σ x2π pc y
ln RT ln
)}
pc0 - µy pc
σy
Dubinin-Astachov (DA) equation of single-gas adsorption, which is the more general version of the DubininRadushkevich (DR) equation,8-11 θ, expressed in the terms of the limiting adsorption volume, W0, and occupied adsorption volume, W, as follows
2
(3b)
where µy and σy are the parameters of the distribution that are determined from the following relationships
µy ) k + ln 0* -
k µ r0 d
Equations 3a and 3b were obtained on the basis of the transformation theorems of functions of random variables.1 If one introduces the notation
pc0 Ac ) RT ln pc
(6)
eq 3 may be transformed to
{ (
)}
1 1 ln Ac - µy exp 2 σy Acσyx2π
2
(7)
The occupied fractional adsorption volume, θ, at a given outer pressure p is found by means of a cumulative distribution function (CDF) and is expressed as
θ)1-
1 σyx2π
∫0h A1c exp
{ ( -
)}
1 ln Ac - µy 2 σy
2
dA
(8)
where h ) RT ln pc0/p. This integral is readily available after transforming FLN to FN1
1-θ)
∫-∞(ln A - µ )/σ c
y
y
{ }
1 z2 exp dz 2 x2π
(9)
The quantile of the CDF gives the value of (ln Ac - µy)/σy at which (1 - θ) reaches its value. If experimental data are described by eq 8, the plotting of quantiles via ln Ac should result in the straight line q ) (ln Ac - µy)/σy. The slope, 1/σy, and the intercept, -µy/σy, of this line can be found by the least-squares method. It was also shown that for benzene adsorption on active carbons, pc0 ≈ ps, where ps is the saturated pressure of the liquid adsorbate.1 Hence, Ac in eqs 7-9 may be substituted by the adsorption potential, A
A ) RT ln
[ (βEA ) ]
(4) (5)
ps p
(10)
Traditionally, the quantitative study of adsorption by microporous solids is based on the Dubinin theory.3 In recent years, promising statistical approaches to the physical adsorption in micropores have been developed. These studies4-7 provided information about individual micropore isotherms and porous size distributions. But current achievements in this field concern adsorption of simple molecules on porous solids. In the empirical (7) Gusev, V. I.; O’Brien, J. A. Langmuir, 1997, 13, 2822.
(11)
is related to the adsorption potential, parameters E and n, and the affinity or similarity coefficient, β
θ ) exp -
k σ y ) σd r0
fLN(Ac,µy,σy) )
θ ) W/W0
n
(12)
This equation have been used with much success to correlate a large amount of adsorption data: nearly 500 papers have been published on this subject by the Russian group alone.12 The experimental studies have resulted in a number of useful approximations, the Stoeckli relationship12 being the most important
d h (nm) ) 12/E (kJ/mol)
(13)
where d h is the average half-width of the corresponding micropore system. Note that in the notation accepted further in the present paper d h ) µd. It is generally accepted3,13 that experimental adsorption isotherms, θ(,T,p), represent an average over all values of the adsorption energies, , existing on the gas-solid interface
θ(T,p) )
∫Ω θ(,T,p) f() d
(14)
where θ (T, p) is the single-pore (individual or “local”) isotherm for the adsorption sites exhibiting an adsorption energy , f() denotes the density (differential) distribution function of , and Ω is the integration region over all possible adsorption energies. Rudzinski and Everett3 proposed a theoretical basis for the Dubinin theory in the framework of integral transforms by averaging the Langmuir local isotherm. Dubinin, Stoekli, Jaroniec, et al. proposed the attempts to improve the DA equation on the basis of integral transformations.13-15 They developed the methods to determine the pore size (PSD) and energy distributions for microporous sorbents from adsorption experiments. These methods have principal drawbacks: they use either an empirical relationship (eq 13 or its versions) or arbitrarily chosen kernels of the transformations (the DR or the DA equations) or arbitrary energy distribution functions. A purpose of the present study is to show that the DA equation is an approximate form of eq 9. For this reason, the approach to the micropore volume filling based on condensation approximation (VFCA) gives reasonable explanations to many experimental observations related to the empirical DA equation: (i) quantitative correlation between the parameters and experimental facts; (ii) the (8) Dubinin, M. M.; Radushkevich, L. V. Dokl. Akad. Nauk. SSSR 1947, 55, 331. (9) 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). (10) Dubinin, M. M. In Adsorption-Desorption Phenomena; Ricca, F., Ed.; Academic Press: London, 1972; pp 3-18. (11) Astakhov, V. A.; Dubinin, M. M.; Mosharova, L. P.; Romankov, P. G. TOXT, 1972, 6, 343. (12) Stoecli, F. Adsorpt. Sci. Technol., Special Issue 1993, 10, 3-16. (13) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, 1988; pp 45-46. (14) Dubinin, M. M. Dokl. Akad. Nauk SSSR 1984, 275, 1442. (15) Stoeckli, H. F. Carbon 1990, 28, 1.
3842 Langmuir, Vol. 14, No. 14, 1998
Dobruskin
peculiarity of water adsorption on microporous carbons; (iii) empirical relationships for calculating of the average micropore size; (iv) the applicability of the DA equation to the adsorption on nonporous surfaces. Correlation between the DA and the VFCA Equations Derivation of the DA Equation. Astakhov was the first9 who in 1969 drew attention to the fact that a fraction of unoccupied adsorption volumes, (1 - θ), expressed in terms of eq 12
[ (βEA ) ] n
1 - θ ) 1 - exp -
χ σ
λ
[ ( )]
[ (βEA ) ] n
(19) (20)
The random variables in eqs 19 and 20 may be either ln(* - 0*) or ln(RT ln(pc0/pc)) (eq 1). The appropriate values of the Weibull model are following16
A n An-1 exp n βE (βE)
2 1 + Γ2 1 + n n
[(
)
σW2 ) (βE)2 Γ 1 +
(
µW ) βEΓ 1 +
(
)]
(21)
1 n
)
(22)
where Γ is the gamma function. Hence, one has the system of equations for the two unknown values
exp{2µy + σy2}(exp{σy2} - 1) )
[(
(βE)2 Γ 1 +
{
exp µy +
}
2 1 + Γ2 1 + n n
)
(
( )
σy2 1 ) βEΓ 1 + 2 n
)] (23) (24)
(17)
This function is associated with the Weibull density function16 fW (A, E, n):
fW(A,E,n) )
}
σy2 2
σLN2 ) exp{2µy + σy2}(exp{σy2} - 1)
(16)
where χ is the random variable and σ and λ are the scale and shape parameters,16 respectively. From a mathematical point of view, eqs 15 and 16 are equivalent if one accepts that n ) λ, βE ) σ, and A ) χ. For this reason, Astakhov came to the conclusion that eq 12 is based on the Weibull distribution of fractional adsorption volumes over the adsorption potential with parameters of n and (βE)
FW(A,E,n) ) 1 - exp -
{
µLN ) exp µy +
(15)
is reminiscent of the Weibull cumulative distribution function, FW
FW(χ,σ,λ) ) 1 - exp -
The expected value, µLN, and the variance, σ2LN , of the LN random variable are given by16
n
[ ( )]
The solution of this system provides the relation between the parameters of two models. If experimental data have been treated by the DA equation (the W model), the appropriate parameters of eq 8 are given by the solution of eqs 23 and 24
(18) 1 ( n) µ ) ln 2 Γ (1 + ) n 2 Γ (1 + ) n σ ) ln 1 Γ (1 + ) n EΓ2 1 +
To distinguish different families of distributions, the symbols F and f are used; for example, fW denotes the Weibull density distribution function. The parameters of the distribution functions are calculated with respect to the standard gas with β0 ) 1, which may be considered to be a probe of energy heterogeneity that provides information about adsorption sites. Despite the apparent discrepancy of eqs 8 and 12, there is the intrinsic correlation between these expressions. In the statistic theory terms, eqs 8 and 12 are based on the log-normal (LN) model and the Weibull (W) model, respectively. The comprehensive review of the statistical models is given by Bury.16 It is known that the central portions of the number models (log-normal, gamma, Weibull, Rayleigh, normal, etc.) may approximate one another. In particular, under certain conditions the LN model may be well approximated by the W model. These models are extraordinarily flexible and capable of modeling a wide range of random phenomena. The inferences based on these models are sensitive only to the values assigned to the parameters. To approximate models, the appropriate two parameters determining each of the models should be chosen. For example, expected values and variances could be assigned to be equal, and then the parameters of the models would be determined by the system of these two equations. We shall apply this method to the lognormal and Weibull models.
For an ordinary active carbon with E ≈ 20 kJ/mol and n ) 2, the parameters of eq 8 are equal to µy ) 2.7542 and σy ) 0.4915, and the characteristic curves treated by eqs 8 and 12 demonstrate the reasonable convergence (Figure 1).
(16) Bury, K. V. Statistical Models in Applied Science; John Wiley: New York, 1975.
(17) Batuner, L. M.; Pozin, M. E. Mathematical Methods in Chemistry; Goschimizdat: Leningrad, 1960; pp 194-196 (Russian).
y
(25)
1/2
2
1/2
y
(26)
2
The empirical DR and DA equations differ only by the value assigned to n: in the DR equation n is taken to be 2, while in the more general DA equation, n can be any positive integer. For n ) 2, substitution of the gamma function values17 in eqs 25 and 26 leads to
Eπ 4
(27)
xln π4
(28)
µy ) ln σy )
Micropore Volume Filling
Langmuir, Vol. 14, No. 14, 1998 3843
Figure 1. Characteristic curves for the Weibull models (dashed lines: curve 1, E ) 20 kJ/mol and n ) 2; curve 2, E ) 14.372 kJ/mol and n ) 2.5763) and for the log-normal models (solid lines: curve 1, µy ) 2.7542 and σy ) 0.4915; curve 2, µy ) 2.4364 and σy ) 0.4000).
Figure 3. Characterictic curves of the benzene adsorption on SKT-3. Points denote experimental data; dashed and solid lines are calculated by the DA and the VFCA equations.
Figure 4. Characterictic curve of the adsorption of benzene on SAU. Points denote experimental data; dashed and solid lines are calculated by the DA and the VFCA equations, respectively.
Figure 2. Adsorption of benzene on ART. A linear representation in the VFCA (a) and the DA (b) coordinates.
For n ) 3, eqs 25 and 26 give
µy ) ln(0.8405E);
σy ) 0.3493
(29)
If experimental data have been treated by eq 8 (LN model), the appropriate parameters of the DA equation should be calculated by the numerical solution of eqs 25 and 26. For example, if parameters of eq 8 are equal µy ) 2.4364 and σy ) 0.4000, the numerical solution of the system of eqs 23 and 24 gives the values of E ) 14.372 kJ/mol and n ) 2.5763, and eqs 8 and 12 afresh demonstrate the reasonable convergence (Figure 1). It should be mentioned that Dubinin considered n to be an integer.18 But there was another point of view that n may take any positive values.19 When deriving the relation between the parameters of the both models, we assigned the first and the second moments of the distributions to be equal. In terms of the third central moments, one can also see the similarity of the models: all log-normal densities and the Weibull densities with n < 3.6 are skewed to the right.16 Since linear representations of eqs 8 and 12 are carried out in the different coordinate systems (Figure 2), a visual comparison of the plotted data is more convenient in terms (18) Dubinin, M. M. In Adsorption in Micropores; Eds.; Dubinin, M. M., Serpinski, V. V., Ed.; Nauka: Moscow, 1983; p 186 (Russian). (19) Kadlez, O. See ref 18 p 75.
Figure 5. Characteristic curve of the adsorption of benzene on AG-3P. Points denote experimental data; dashed and solid lines are calculated by the DA and the VFCA equations, respectively.
of a characteristic curve. The experimental uptakes for benzene adsorption on active carbons (AC) are plotted in Figures 3-5 in the coordinate system of a characteristic curve, that is, θ via A. The AC samples employed are follows: AG-3P produced by the vapor-gas activation from coal feedstock; ART and SKT-3 prepared by the chemical activation of peat; SAU produced by the thermal decomposition of polymer starting material. Figures 3-5 illustrate an accuracy of representation of the experimental data by the DA and the VFCA equations. If we compare the results of treatments, as a whole, eq 8 gives the more explicit description of experimental isotherms. The relative deviations for eq 8 do not exceed 1.5% for AG-3P and 5% for SKT-3 and ART; in the case of the DA equation, the deviation extends to 5-7% for SKT-3 and up to 10-15% for AG-3P. For adsorption on SAU AC,
3844 Langmuir, Vol. 14, No. 14, 1998
Dobruskin
Figure 7. Dependence of the ratio E/exp(µy) upon n.
as 5-6.3,9 In the limit of n f ∞, the porous system approaches a homogeneous one; the DA equation reduces to a Dirac delta distribution and describes the phase transition (condensation) that occurs at A/E ) 1 Figure 6. Adsorption of benzene on AG-3P in the DA coordinates: (a) n ) 2; (b) E ) 13.2713 and n ) 1.6410.
both these equations describe the experimental data with approximately the same precision. Physical Meaning of the DA Parameters. In general, it follows from eqs 25-29 that n must not be the integer. If one assigns only integer values of n, it sometimes leads to significant deviations of the linear dependence of ln θ via An. For example, benzene adsorption on the AG-3P carbon with widening porous distribution is well described by the DA equation with n ) 1.6410, whereas the plot of the same data treated by the DR equation demonstrates a large deviation from the straight line and looks like a combination of two straight lines.20 (Figure 6) This fact was also observed by Kadletz.19 It follows from eqs 5 and 26 that
σd )
r0 k
x
2 n ln 1 Γ2 1 + n
( (
Γ1+
) )
(30)
A relationship n ) f (σd) may be found by the numerical solution of eq 30. Equation 30 shows that n does not depend on the pore dimensions and is entirely determined only by the standard deviation of micropore sizes. Hence, n characterizes the surface heterogeneity: the more surface heterogeneity there is, the smaller the value of n (eqs 26 and 30). When n f ∞, Γ(1 + (1/n)) and Γ(1 + (2/n)) tend to the unity and, consequently, σd f 0. The latter is valid only for the homogeneous system with constant d. The confirmation of this dependence may be obtained from the observation of adsorption properties of active carbons and zeolites.3,21 In general it appears that 1 < n < 2 refers to carbons with a widening of micropore size distribution, which are prepared by a large burning-off of a starting material. For usual microporous active carbons, the recommended value of n is equal to 2; polymer-based active carbons are likely to be more homogeneous that those from the coal-, wood-, or peat-starting materials, and for polymer-based active carbons, the exponent is often taken to be 3. For the more homogeneous crystalline porous structures typified by the zeolites, n can be as high (20) Izotova, T. I.; Dubinin, M. M. Zh. Fiz. Khim. 1965, 39, 27962803. (21) Stoeckli, H. F. Carbon, 1981, 19, 325.
θ(E) )
{
[ (EA) ] ) 0 for EA > 1 A A limit exp[-( ) ] ) 1 for 8, E practically does not depend on n. In the DA approximation for which σ is constant, E is considered to be determined only by the average micropore size or by the related average adsorption potential. From
Micropore Volume Filling
Langmuir, Vol. 14, No. 14, 1998 3845
eqs 25 and 26 one has
for n ) 2: E ) 4/π exp(µy) ) 1.2732 exp(µy)
µd ) (35)
(
)
r0 0* π - ln k + ln k E 4
(37)
For the given adsorbate, eq 37 takes the form and
µd ) a - b ln E for n ) 3: E ) 1.1898 exp(µy)
(36)
Hence, if for usual ACs one assigns n in a range of 2 < n < 3, E may differ within 6.5%. The limiting adsorption volume, W0, in the DRA equations is considered to be the constant parameter of the equations. In fact, as pointed out by Sircar,22 the DA equation gave varying micropore volumes for different gases on the same adsorbent. It was shown in the VFCA approach that there is the limit slit width, dlimit, above which a 2D condensation is not followed by the condensation in a free micropore volume. For argon adsorption, dlimit was found to be ≈0.7 nm, whereas for xenon this value is equal to 0.8 nm. In general, the limit slit width is defined by the fluid-solid and fluid-fluid interaction parameters. Hence, W0 depends on adsorbates employed. This value may be kept constant only for adsorbates with the close Lennard-Jones parameters. Water Adsorption. The peculiarity of water adsorption on active carbons may be also explained in light of the present approach. Stoeckli at al.23,24 demonstrated that the water adsorption isotherm on active carbons can be decomposed into two contributions, each of which may be treated as the DA isotherms. The initial section suggests the presence of sites with characteristic energies in the range of 5-8 kJ mol-1 and n ≈ 1-1.6; the second part of the isotherm suggests the presence of sites with E in the range of 1-2.4 kJ mol-1 and may be described by eq 12 in which n may be as much as 6-8. Stoeckli and Jakubov24 showed that a plot of W/W0 via RT ln P0/P leads to the unique characteristic curve for carbons, as was found in the case of organic vapors. It is generally accepted the initial section due to the formation of hydrogen bonds between the oxygen complexes on the surface and between the adsorbed molecules themselves.25 It is apparent that the energy of hydrogen bonds does not depend on porous widths. In the case of water adsorption on active carbons, dispersion interactions are weak and can be neglected to the first approximation. For this reason, the base heterogeneity due to variation in the pore width, as perceived by water molecules, is negligible; that is, from a “water point of view”, the micropore spaces are almost homogeneous regardless of their widths. The second part of the isotherm containing up to 95% of total adsorption amount is caused by the extension of 2D condensation into almost “homogeneous” micropore volumes. The filling of the main volume of carbon micropores with water occurs at high relative pressures, p/ps, which may be estimated by eq 33. For carbons with E in the range of 1-2.4 kJ mol-1, pc/p0 falls in the range of 0.38-0.67 with good fits to Stoeckli’s experimental data.23,24 Micropore Average Half-Width. In the case of benzene adsorption (β ) 1) and the DA equation with n ) 2, an average micropore half-width may be found by combining eqs 4 and 27
(38)
where a ) (r0/k)(k + ln 0* - ln(π/4)) and b ) r0/k. It should be emphasized that these expressions are valid only in the range of d < dlimit (about 0.8 nm in the case of xenon adsorption). The eq 38 is reminiscent to the empirical relationship found by McEnaney26 for a wider range of µd
µd )3.72 - 0.89 ln E
(39)
To calculate µd, the values of 0*, k, and r0 must be assigned. Proceeding from the energy of benzene interaction with the graphite surface at the limit of zero amount adsorbed,27 we accept 0* ) 40 kJ/mol. The parameter of k may be taken to be 4 as found for monatomic gases.1 Equation 37 shows that r0 exerts strong influence on µd. This parameter was introduced to describe the dependence of the well depth upon the micropore half-width in the case of adsorption of monatomic gases. In the case of nonspherical polyatomic molecules, probably, it is better to accept the experimental value of r0. When estimating the average micropore sizes, Stoeckli12 takes the critical dimension of benzene to be equal to 0.41 nm. If we accept r0 ) 0.41 nm, the calculated average half-widths of SAU and SKT-3 are found to be 0.46 and 0.48 nm, in good agreement with the values of 0.43 and 0.49 nm calculated from eqs 13 and 37. Monolayer Adsorption on Heterogeneous Surface. We can visualize the heterogeneous surface corresponding to a micropore carbon as follows. If we (i) disassemble a set of micropores with different sizes, (ii) put together the walls of all individual micropores, (iii) attribute to each pair of opposing walls the same energy that they had inside a micropore, the patchwise heterogeneous surface28 will be formed, each patch being composed of two walls. For this surface and an origin micropore system, the distributions of the surface area and adsorption volumes over the adsorption energies will be the same. Regardless of the underlying reasons, let us assume that a nonporous patchwise heterogeneous surface may be also characterized by the log-normal model of the adsorption energy distribution
fLN(*,µy,σy) )
{ (
)}
1 1 ln * - µy exp 2 σy *σyx2π
2
(40)
Proceeding just as in the cases of the VFCA approach, we shall arrive at the conclusion that the fraction of the surface occupied by the condensed gas will be described by the equations that are analogous to eqs 8 and 9
θ)1-
1 σyx2π
∫0h A1 exp
{ ( -
)}
1 ln Ac - µy 2 σy
2
dA (41)
and (22) Sircar, S. Carbon 1987, 25, 48. (23) Stoeckli, F.; Currit, L.; Laederach, A.; Centeno, T. A. Chem. Soc., Faraday Trans. 1994, 90 (24), 3689-3691. (24) Stoeckli, F.; Jakubov, T. J. Chem. Soc., Faraday Trans. 1994, 90 (5), 783-786. (25) Steele, W. A. The Interaction of Gases with Solid Surface; Pergamon Press: Oxford, 1974; pp 212-215.
(26) McEnaney, B. Carbon 1987, 25, 69. (27) Avgul, N. N.; Kiselev, A. V.; Poshkus, D. P. Adsorption of Gases and Vapors on the Homogeneous Surfaces; Chemistry: Moscow, 1975; pp 130-131 (Russian). (28) Ross, S.; Olivier, J. P. On Physical Adsorption, Interscience: New York, 1964.
3846 Langmuir, Vol. 14, No. 14, 1998
1-θ)
∫-∞(ln A - µ )/σ c
y
y
{ }
z2 1 exp dz 2 x2π
Dobruskin
(42)
The variable in eq 41 changes from zero to infinity. The lower limit ) 0 corresponds to the hypothetical surface patch with the zero adsorption energy. It is clear that for such a patch pc0 ) ps and RT ln ps/p should be used instead of RT ln pc0/p. Since eqs 41 and 42 may be approximated by the DA equation, the latter will also describe an adsorption on the heterogeneous surface. In fact, only Jaroniec and Madey had reviewed 34 examples of great utility of the DR equation for describing gas adsorption on nonporous and macroporous solids.13 The underlying reason of the applicability of the DA equation both to a monolayer adsorption on wall surfaces and to a condensation in a micropore volume is a one-to-one correspondence between these phenomena. Correlation between the VFCA and the DA Approaches. Equation 31 establishes the same individual isotherm that is accepted in the CA approach. In terms of eq 14, the physical meanings of the VFCA and the DA equations are following: both equations are associated with the local isotherms given by the step function; but the first is based on the well-founded log-normal distribution of adsorption energies (eq 3), whereas the DA equation is associated with the well-adjusted Weibull distribution (eq 18). In fact, the DA equation may be considered to be the approximate form of the VFCA equation. Therefore, the principles underlying both approaches are significantly the same. We recall the main principal propositions justified by calculations and assumptions that lead to both equations: (i) for micropores with d < dlimit, a micropore volume filling occurs at the same pressure as the 2D condensation on its walls; (ii) the well depth for a micropore surface at the limit of zero amount adsorbed may be approximated by eq 1; (iii) a distribution of micropore widths follows the normal model; (iv) volumes of all individual slitlike micropores are equal to one another; (v) the total micropore surface is considered to be composed of micropore bases (walls) that are regarded unisorptic areas, totally independent of each other; (vi) adsorbate lateral interactions are allowed only between molecules on the same base. An applicability and an accuracy of eq 8 and 12 are determined by the same factors; the reasons of the observed deviations in the lowest and high-pressure regions were discussed in the previous publication. According to the CA model, the micropore and their walls may be only in two states: either to be free from adsorbate at p < pc or to be filled with adsorbate at p > pc. Hence, in the low-pressure region the DA equation does not take
into account the submonolayer adsorption and does not converge to the Henry isotherm limit. Nomenclature A, adsorption potential Ac, defined in eq 6 a, defined in eq 38 b, defined in eq 38 d, micropore half-width dlimit, limit slit width d h ≡ µd, average half-width of the micropore system E, parameter in eq 12 f(), density (differential) distribution function of in eq 14 fN, fLN, fW, normal, log-normal, and the Weibull density functions, respectively FN, FLN, FW, normal, log-normal, and the Weibull cumulative distribution functions, respectively h ) RTlnpc0/p in eq 8 k, energy parameter in eq 1 n, parameter in eq 12 p, pressure pc, critical pressure of a 2D condensation pc0, critical pressure of a 2D condensation on the nonporous surface ps, saturated pressure of the liquid adsorbate q ) (ln Ac - µy)/σy quantile of CDF R, gas constant r0, the Lennard-Jones parameter T, absolute temperature W, occupied adsorption volume W0, limiting adsorption volume β, affinity or similarity coefficient χ, random variable *, adsorption energy 0*, well depth for the nonporous surface Γ, gamma function λ, shape parameters in eq 16 µd, mathematical expectation of d µLN, µW, mathematical expectations of log-normal and the Weibull distributions, respectively µy, mathematical expectation of Y defined in eq 4 θ, surface coverage, occupied fractional adsorption volume σ, scale parameters in eq 16 σd, standard deviation of d σLN, σW, standard deviations of LN and the Weibull random variables, respectively σy, standard deviation of Y defined in eq 5 Ω, integration region over all possible adsorption energies Acknowledgment. The author is grateful to Professor David Avnir of the Herbrew University of Jerusalem for fruitful discussion of results and to Dr. S. D. Kolosentzev and Dr. Yu. Ustinov of the Technological Institute of St. Petersburg for kindly supplying experimental adsorption data. LA9712101
