The Normalization of the Micropore-Size Distribution Function in the Polanyi–Dubinin Type of Adsorption Isotherm Equations

Journal of Colloid and Interface Science 227, 482–494 (2000) doi:10.1006/jcis.2000.6875, available online at http://www.idealibrary.com on

The Normalization of the Micropore-Size Distribution Function in the Polanyi–Dubinin Type of Adsorption Isotherm Equations Piotr A. Gauden and Artur P. Terzyk1 N. Copernicus University, Department of Chemistry, Physicochemistry of Carbon Materials Research Group, Gagarin St. 7, 87-100 Toru´n, Poland Received December 6, 1999; accepted March 23, 2000

The problem of the normalization of the micropore-size distribution (MSD) based on the gamma-type function is presented. Three cases of the integration range (widely known in the literature) of MSD, characterizing the geometric heterogeneity of a solid, are considered (val(≡B, E0 , and/or x)) i.e., from zero to infinity, from valmin to infinity, and the finite range from valmin up to valmax —due to the boundary setting of an adsorbate-adsorbent system. The physical meaning of the parameters of the gamma-type function (ρ and ν) is investigated for the mentioned intervals. The behavior and properties of this MSD function are analyzed and compared with the fractal MSD proposed by Pfeifer and Avnir. The general conclusion is that if adsorption proceeds by a micropore filling mechanism and the structural heterogeneity is described in the finite region (valmin , valmax ), for all cases of the possible values of the parameters of the MSD functions, the generated isotherms belong to the first class of the IUPAC classification (i.e., Langmuir-type behavior is observed). For the other cases (val ∈ < 0, ∞) and val ∈ < valmin , ∞)) some erroneous and ambiguous results are obtained. °C 2000 Academic Press Key Words: adsorption; active carbon; fractal dimension; microporosity; potential theory; pore diameter.

1. INTRODUCTION

It is generally accepted that the basic structure of the graphite crystals of the activated carbon is the graphitic-like aromatic microcrystallites which form slit-shaped micropores between graphite layers (pore walls) (1–3). The distance between the opposite sides of walls of the pore, in which the adsorption can occur, is in the range of molecular dimensions and/or a few times larger than its diameter. The dispersive interactions between the adsorbate molecule and both sides of the pore walls cause the enhancement of an adsorption potential (3). Therefore, the adsorption potential is stronger in a fine micropore than in a wide one, and than in the case of an adsorbate molecule interacting with the flat surface. A small variation in the pore size can cause a substantial change in the adsorption affinity (the molecular sieve properties). The most significant contribution to the geo1 To whom correspondence should be addressed at Artur P. Terzyk, N. Copernicus University, Department of Chemistry, Physicochemistry of Carbon Materials Research Group, Gagarin St. 7, 87-100 Toru´n, Poland. E-mail: [email protected].

0021-9797/00 $35.00

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metrical nonuniformity of the porous solid originates from very fine pores of various sizes (called micropores). The existence of those micropores, characterized by different widths, strongly affects the adsorbent-adsorbate interaction (4). Then, the theoretical description of the overall adsorption process is strongly affected by the micropore-size distribution function that characterizes the structural heterogeneity of the adsorbent. There are some methods, proposed in the literature, which allow evaluation of the micropore-size distribution: • • • •

the MP method (5), the theory of volume filling of micropores (TVFM) (6–9), the Horvath-Kawazoe method (HK) (10–12), the density functional theory (DFT) (13, 14), and others (15, 16). Although those methods are very useful, they have limitations and none can be accepted as general. The above-mentioned models have their own specific assumptions, connected to a description of the porous structure of investigated systems, which have a bearing on the obtained results. Hence, the results evaluated for the same adsorbent from those different methods can (17) but need not be compatible with each other (18). This leads to the statement that the evaluation of microporosity, which substantially affects sorption properties, is still a key problem. Many years ago Dubinin and co-workers (19) proposed an equation (called the Dubinin-Astakhov equation (DA)) for the description of the physical adsorption of a single gas on a microporous solid. This equation is one of the most widely used in the adsorption theory field: · µ ¶ ¸ · µ ¶n ¸ A n A = exp −B , θDA = exp − β E0 β

[1]

where E 0 is the characteristic energy of adsorption; β is the similarity coefficient; n is the parameter of the DA equation; A(=RT ln( ps / p)) is defined as −1G, the negative change of the Gibbs’ free energy; T is temperature; R is universal gas constant; ps and p are the saturation and the equilibrium vapor pressure of the adsorbate, respectively; B(=E 0−n ) is the structural parameter. The microporous solids (especially activated carbons) possess a great number of different classes of micropores of

482

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NORMALIZATION OF MICROPORE–PORE DISTRIBUTION FUNCTIONS

different widths. In this case, the overall adsorption isotherm can be obtained by replacing the summation of the local adsorption isotherm (corresponding to different pore sizes) by the integration (if we assume that the number of contributions increases to infinity). Stoeckli was the first (3) to propose the integral form of the overall isotherm for single gas adsorption on heterogeneous microporous solids, Z 2ov (A) = θ(A, B)F(B) d B, [2]

2. THEORETICAL

Some examples of functions used to express the energy and the pore distribution were summarized in Jaroniec and Madey’s book (24). Only several of the functions presented led to simple analytical equations; for others, the integral given by Eq. [2] can be calculated only by the numerical methods. We assume that the starting micropore-size distribution function can be written in the following general form (24),

1B

where 2ov (A) is the so-called “global” (“overall”) adsorption isotherm; θ (A, B) is the local isotherm in micropores of a given class characterized by the structural parameter B (connected with the characteristic energy of adsorption (E 0 ) and/or the widths of micropores (x)); F(B) is the distribution function characterizing the heterogeneity of the investigated microporous solid; 1B is the integration region. In many approaches it has been proposed to represent the local adsorption isotherm θ (A, B) by the Dubinin-Radushkevich equation (DR) (19) (Eq. [1] with parameter n equal to 2) and/or by the DubininAstakhov (DA) (19) equation (Eq. [1]). In the case of structural heterogeneity, the calculations leading to the solution of the integral (2) can be the source of some fundamental problems. The first problem is associated with the assumption, now and again, of the MSD (micropore-size distribution) functions a priori (6–9, 16). Several functions have been considered for the representation of the pore-size distribution (for example, fractal, gamma, and Gaussian). The second difficulty of the direct solution to the problem is the choice of functional space. The detailed discussion of this problem was published by Cerofolini and Rudzi´nski (20). “The functional space in which the distribution function is searched plays an important role on the discussion of the problem. The choice of the functional space in a physical problem is not straightforward and may affect the solution of the problem; because of this fact, this choice must be decided on physical bases.” These words imply that the considered MSD functions should have a clear physical meaning and fulfil the following normalization conditions: χB ≥ 0 (χB —the normalization factor) and ∀B F(B) ≥ 0. F(B) is a smooth (however, not necessarily continuous) function. • The MSD function should be normalized to unity in the same integration range in which the overall adsorption isotherm (Eq. [2]) is integrated. • •

The aim of the current paper is to describe some traps associated with the problem of normalization. These investigations are confined to one MSD function, namely the gamma-type function. It is chosen due to the possibility of generating different shapes (the Pfeifer-Avnir one (21, 22), especially). Moreover, the final section of the paper deals with the comparative analysis of the solution of the overall adsorption isotherm for the three above-mentioned intervals of integration.

y(t) = a(i) (t − t0 )b exp[d(t − t0 )e ]

[3a]

y(t) = a(i) t b exp[dt e ],

[3b]

and/or

where the parameter a(i) is the normalization factor that fulfills the condition described in the introduction; b, d, and e are the parameters of Eqs. [3]; and t0 is the characteristic value of the variable t for the given distribution function. These similar forms of the distribution functions are chosen because they can generate different shapes for the different values of the parameters: b, d, and e. Furthermore, Eqs. [3] can be reduced for some assumption to simpler expressions (24), for example, (I) the decreasing exponential distribution function of t (b = 0, d < 0, and e = 1), (II) the Rayleigh-type distribution function of t (b = 1, d < 0, and e = 2), (III) the Maxwell-Boltzmann distribution function of t (b = 0.5, d < 0, and e = 1), (IV) the Gaussian distribution function of t (b = 0, d < 0, and e = 2), (V) the gamma-type distribution function of t (e = 1) and others. The fourth case was investigated effectively by Stoeckli (6). He applied the Gaussian distribution function of the structural parameter B(=E 0−n ) to represent F(B)GB (see Eq. [2]), F(B)GB

· ¸ (B − B0 )2 = χBG(i) exp − , 212

[4]

and the parameters and the variable of Eq. [4] can be easily transformed to Eqs. [3], respectively. B0 is the average value of B, 1 is the dispersion of the Gaussian distribution function, and χBG(i) is the normalization factor. The distribution function was normalized to unity in the range (−∞, ∞) and Stoeckli obtained the following form of the factor χBG(i) : χBG(−∞,∞) =

1 √ . 1 2π

[5]

On the other hand, he solved the integral given by Eq. [2], where the local adsorption isotherm θ (A, B) was represented

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by the DR isotherm equation (Eq. [1], for n = 2) and F(B) was given by the combination of Eqs. [4] and [5] in the different interval (0, ∞), which was considered for the normalization of F(B)GB . From the formal point of view, and to satisfy the conditions described in the introduction, the distribution function given by Eq. [4] must be normalized in the same interval of integration as in the case of the overall adsorption isotherm (7, 8),

the frequent application, and the possibility of the comparison of the results obtained in the current study with those published in the Jaroniec and Choma’s studies (7). For the gamma-type distribution function F(B)GT , it is easy to show (7) that the combination of Eqs. [7]–[9] gives the following form of the micropore-size distribution J (x)GT ,

2χBG(−∞,∞) 2 ³ ´´ = ³ ³ ´´ , [6] √ ³ B√0 1 2π 1 + erf 1 2 1 + erf 1B√0 2

where χGT(i) is the normalization factor. The MSD function, expressed by Eq. [10], can be compared with the starting and general form—Eq. [3b]. From this evaluation the following relationships between the variables and the parameters of Eqs. [3b] and [10] are derived:

χBG(0,∞) =

and, as a consequence, there is the multiplication of the distribution F(B)GB (Eq. [4]) by the normalization factor (Eq. [6]). Our consideration will be limited only to the fifth case (V) of the above-mentioned cases (i.e., the gamma-type distribution function) due to the interesting properties and quite a lot of interest in the literature (7, 23–26). On the basis of the starting Eqs. [3a] and/or [3b], replacing t by B, assuming B0 as equal to zero (for simplicity and the reduction of the considered parameters), assuming a(i) , b, d, and e as equal to χGT(i) , ν, −ρ, and unity, respectively, the following equation is obtained: ν

F(B)GT = χGT(i) B exp[−ρ B].

[7]

In the case of microporous activated carbons, it is easier to use the micropore size (x) as the parameter characterizing the structural heterogeneity rather than the parameter B(=E 0−n ) or z(=E 0−1 ) describing energetic heterogeneity (8). The distribution function F(B)GT can be recalculated to obtain the distribution function J (x)GT provided the relationship between B and the micropore size x is known. For that reason, F(B)GT and J (x)GT are associated through the following relationship, F(B)GT d B = J (x)GT d x,

[8]

where x(B) must be a monotonic increasing function. The next important step in the evaluation of the microporesize distribution J (x) from adsorption isotherms (based on the theory of volume filling) is an appropriate description of the relationship between B and x. Many authors have investigated the dependence of the adsorption energy E 0 (=B −n ) on the pore size x (for slit-like micropores), both experimentally (27–30) and theoretically (12, 31, 32). In this paper, our considerations are limited to the simplest but frequently applied relationship, derived by Dubinin and co-workers (6), B = E 0−n

xn = n = ζ xn, κ

[9]

where κ(=ζ −1/n ) is a constant in the micropore region; and its value for the reference system (benzene vapor on activated carbon) is about 12 [kJ nm/mol] (6, 8). The reasons why this relationship is chosen are as follows: the simplicity of the expression,

J (x)GT = χGT(i) x n(ν+1)−1 exp[−ρζ x n ],

[10]

χGT(i) = a(i)

[11a]

n

x =t

[11b]

ν=b

[11c]

−ρζ = d.

[11d]

In the field of the theoretical description of adsorption still more and more investigations are being carried on as far as the problems of adsorption on fractals are concerned (33–35). It leads to the conclusion that fractal geometry is one of the most popular means of taking the structural heterogeneity of solids into account. The fractal analogs of BET (36, 37), Henry’s (38), or FHH (39, 40) adsorption isotherms were developed; however, these isotherms can be applied only for an approximate description of the influence of an adsorbent heterogeneity on adsorption in microporous systems. Therefore some attempts have been made to develop the theory of adsorption on fractal microporous solids and the fractal analogs of DR and DA equations were derived (41) (as well as corresponding thermodynamic relationships (42)) and analyzed extensively (43, 44). Fractal analogs of DR (FRDR) and DA (FRDA) equations were developed assuming that the local adsorption isotherm in Eq. [2] is represented by Eq. [1] and the MSD by the function proposed by Pfeifer and Avnir (21, 22). These authors showed that in the case of an adsorption process on and/or in the solids that are treated as fractal objects, the pore-size distribution can be presented by J (x)PA = χPA(i) x 2−D ,

[12]

where the geometric heterogeneity of the microporous solids is described by the structural parameter D, the fractal dimension; χPA(i) is the normalization factor. Moreover, Eq. [12] has been also recommended by IUPAC for the evaluation of the fractal dimension (45). Finally, from the above discussion it is seen that Eq. [3b] can be regarded as the generalized form of Eqs. [4], [7], [10], and [12]. Moreover, under some conditions the parameters of those

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NORMALIZATION OF MICROPORE–PORE DISTRIBUTION FUNCTIONS

four equations simplify to the parameters a, b, d, and e of the starting distribution function Eq. [3b]. 3. CALCULATION RESULTS AND DISCUSSION

All the distribution functions considered (Eqs. [7], [10], and [12]) should satisfy the following normalization condition (see introduction): Z

Z y(t) dt =

1= 1t

Z F(B) d B =

1B

J (x) d x.

[13]

1x

On the basis of the postulate that the so-called global adsorption isotherm equation, which describes the adsorption process in the microporous solids with a great number of micropores of different sizes characterized by the micropore-size distribution function of the half-width x (Eqs. [10] and/or [12]), Eq. [2]) can be rewritten in the well-known form (7, 8): Z 2ov (A) =

θ(A, x)J (x) d x.

[14]

1x

One of the most important problems is the choice of the normalization (the integration) range. One can make the assumption that the mathematical description of the adsorption on the microporous solids can be represented by the integral (Eq. [14]) in the integration limit from zero to infinity (6, 7, 9, 24). However, it can be concluded from consideration of some theoretical investigations on energetic and structural heterogeneity of activated carbons (and due to the boundary conditions of the adsorbate-adsorbent system) that the integration limit should be restricted to a finite range (8, 12, 15, 26, 42, 43, 46–49). That implies taking into account the slit-like model of micropores of an active carbon, the description of the adsorption mechanism in the microporous solid by the MSD function in the finite limit (valmin , valmax ) of the parameter characterizing the geometric heterogeneity of the investigated system (val(≡B, E 0 , and/or x)). Then, the lower limit (valmin ) is connected with the minimum slit-like pore half-width accessible to the adsorbate

molecule (in which adsorption occurs by the micropore filling mechanism proposed by Dubinin and co-workers). On the other hand, the upper limit (valmax ) is the maximal slit half-width of the micropores, which is equal to 1 nm according to the IUPAC classification (50). It was assumed that the effect of maximum accessible pore size is negligible in the overall adsorption isotherm calculations. This leads to the extension of the integration range to infinity (valmax → ∞). Therefore, it has been postulated to integrate the 2ov in the interval of integration (valmin , ∞) (16, 51). However, the most popular presumption is the extension of the integration limit from zero to infinity. Hence, the calculation is limited to 1val ∈ (0, ∞). This assumption is justified by a small and/or negligible error that can arise. Therefore, we want to indicate the basic differences in the approach of integration of the overall adsorption isotherm and the normalization of the gamma-type MSD function, Eqs. [3b] and/or [10]. This function can be normalized within the three different ranges noted above and provided in the literature. The calculated factors are listed in Table 1. In this table the results of χPA(i) obtained for the Pfeifer-Avnir function (Eq. [12]) are also presented. It is seen, from this table, that the first function (y(t), Eq. [3b], e = 1) and the second function (J (x)GT , Eq. [10]) can be normalized in all the ranges, contrary to the third function (J (x)PA , Eq. [12]). This case cannot be normalized for the integration range (0, ∞) due to the properties of the function (the hyperbolic type). It was shown by us previously (43, 44, 52) that the Pfeifer-Avnir MSD can be normalized from the xmin to infinity and the derived factor is given by the equation: 3−D . However, for the accepted values χPA(xmin ,∞) = (D − 3)/xmin of the fractal dimension: 2 ≤ D < 3, it is impossible to obtain sensible results because the integral (x 2−D ) does not converge. The result is indeterminate. It is very important to keep the same interval of the integration for the overall isotherm of adsorption on heterogeneous microporous solids and for the micropore-size distribution as well, because they were integrated with respect to the same investigated system. It was shown previously (43, 52) that the unfulfilment of the above-mentioned condition could lead to surprising results. Therefore, using the same interval of integration (three cases of the normalization factor tabulated in Table 1) and solving the

TABLE 1 The Normalization Factors Obtained from the Integration of the Gamma-Type MSD Function (Eqs. [3b] and/or [10]) and the Pfeifer-Avnir Function (Eq. [11]) in Three Various Ranges, Used Widely in the Literature 1val (val: t and/or x) (0, ∞)

y(t) Eq. [3b] e = 1 a(0,∞) =

(−d)b+1 0(b + 1) (−d)b + 1 0(b + 1) − γ (b + 1,−dtmin )

(valmin ,∞)

a(tmin, ∞) =

(valmin , Valmax )

a(tmin ,tmax ) =

(−d)b+1 γ (b + 1,−dtmax ) − γ (b + 1,−dtmin )

J (x)GT Eq. [10] χGT(0,∞) =

J (x)PA Eq. [12]

n(ρζ )ν+1 0(ν + 1)

χGT(xmin ,∞) =

—

n(ρζ )ν+1 n 0(ν + 1) − γ (ν + 1,ζρxmin ) ν+1

) χGT(xmin ,xmax ) = γ (ν + 1,ζρx nn(ρζ n max ) − γ (ν + 1,ζρx min )

χPA(xmin ,xmax ) = χPA(xmin ,xmax ) =

D−3 3−D xmin 3− D 3−D 3−D xmax − xmin

Note. 0 is the Euler gamma function, γ is the incomplete gamma function, valmin and valmax are the lower and upper limits of t and/or x. The other parameters are the same as in the text.

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overall adsorption isotherm given by Eq. [14] (using Eqs. [1], [9], and [10]), the following analytical forms of adsorption isotherms are obtained: µ 2GT(0,∞) =

¶−(ν+1) An +1 (0 < x < ∞) βnρ

[15]

(xmin < x < ∞) [16] 2GT(xmin ,xmax ) ³

An βn ρ

+1

´−(ν + 1) h ³ ´ ´ ³ n ´ ³ ³ n ´i n n γ ν + 1, βAn + ρ ζ xmax − γ ν + 1, βAn + ρ ζ xmin £ ¡ ¢ ¡ ¢¤ n n γ ν + 1, ζρxmax − γ ν + 1, ζρxmin (xmin < x < xmax ),

The adsorbate is benzene (β = 1). The temperature is taken as 293 K. • The adsorbent is a strictly microporous activated carbon. Therefore, the value of the maximal slit half-width (xmax ) is assumed as equal to 1 nm (the upper limit of micropore diameters recommended by IUPAC (54)). The adsorbate has a molecular size and this parameter is connected with the minimal slit halfwidth (x min ) (for benzene xmin = 0.2295 nm (55)). • The adsorption isotherms were generated at the relative pressure range from 1 × 10−7 to 0.999 p/ ps (331 points altogether). • The parameter n is equal to 2. This assumption is associated with the simplification of the calculation but does not influence qualitatively the obtained results. • •

2GT(xmin ,∞) ³ n ´−(ν+1) h ´ ³ ³ n ´i A A n + 1 + ρ ζ x 0(ν + 1) − γ ν + 1, n n min β ρ β £ ¡ ¢¤ = n 0(ν + 1) − γ ν + 1, ζρxmin

=

an adsorbate and the parameters describing the studied system. Thus:

[17]

where 0 is the Euler gamma function, γ is the incomplete gamma function, and the meaning of the other parameters is described above. The first of the solutions of the global adsorption isotherm (Eq. [15]) is the well-known Jaroniec-Choma equation (7). The properties of the Pfeifer-Avnir MSD function were extensively investigated by us (43, 44, 52), previously. Avnir and Jaroniec (44, 52, 53) postulated to split up the global integral adsorption isotherm (Eq. [14]), presuming the following integration range 1x ∈ hxmin , xmax i, into three integrals and distinguish the intervals of integration: from 0 to infinity, 0 to xmin , and xmax to infinity. Next, they neglected the second and third integral and the consideration was limited only to the first one. Then, they derived the adsorption equation for 1x ∈ h0, ∞). Nevertheless, the integration range of the global equation (1x ∈ < 0, ∞) and the interval of normalization of the pore-size distribution function— J (x)PA (1x ∈ hxmin , xmax i) were not equivalent. Two kinds of the solution, complete (FRDA equation—(xmin < x < xmax )) and approximate (the Avnir-Jaroniec one (0 < x < ∞)), were compared with each other based on the percentage relative absolute error given by %2GT(error) ¯ ¡ ¢¯ ¯2GT(x ,x ) − 2GT(0,∞) and/or 2GT(x ,∞) ¯ min max min = × 100%. 2GT(xmin ,xmax ) [18] It is assumed that the complete solution (the adsorption isotherm (2GT(xmin ,xmax ) ) has been treated as a reference isotherm (see (44)). The properties of the MSD gamma-type function J (x)GT , given by Eq. [10], and the adsorption isotherms described by Eqs. [15]–[17] can be investigated in the following way: using Eqs. [10] and [15]–[17] adsorption isotherms were generated numerically. To achieve this it is necessary to assume the type of

In the current study we investigate only the selected cases of the possible shapes of the gamma-type distribution function (J (x)GT ) given by Eq. [10] (Fig. 1). Six examples are considered. These cases are chosen because, in our opinion, they are the most interesting and acceptable for the description of the structural heterogeneity of the microporous activated carbons. Others are presented in Bronsztejn and Siemiendiajew’s book (56). As was shown under Theoretical, there exist the relationships between the starting distribution function y(t) (Eq. [3b]) and J (x)GT (Eqs. [11]). Therefore, the general plots of y(t) are generated based on the values of the parameters (d and b) listed in Table 2. These values are not chosen randomly but in order to provide the readers with some nuances of the gammatype distribution function. Moreover, for the curves I–IV (the solid lines), the values of the normalization factors—a(0,∞) — (applied to the generation of these curves) are calculated for the integration range (0, ∞) (Table 2). For this interval of integration, the changes in the shape of the distribution function are clearly visible. The solid circles are attributed to the micropore range (tmin , tmax ). These points are limited by the values of upper and lower cutoffs—the micropore range (tmin , tmax ). It is obvious that the different values of the normalization factors— a(tmin ,tmax ) —are calculated (Table 2). To mark off the part of the curves responsible for the micropore range (points), the constancy of a(i) (=a(0,∞) ) is assumed for these two intervals of normalization. For the cases V and VI the points are plotted for a(tmin ,tmax ) due to the impossibility of the computation of a(0,∞) (Tables 1 and 2). Below, the properties of 2GT and J (x) for all the cases presented in Fig. 1 are considered in detail. (I) (d(≡−ρζ ) > 0 and b(≡ν) > 1). The first MSD plot is an increase to infinity function (Fig. 1). All the relationships describing the normalization factors (a(i) and/or χGT(i) —Table 1) include the exponential expressions ((−d)b+1 and/or (ρζ )ν+1 )). The sign of these expressions as well as the sign of all the normalization factors are determined by the value of the parameter b. Therefore, this case diverges into three subclasses:

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NORMALIZATION OF MICROPORE–PORE DISTRIBUTION FUNCTIONS

FIG. 1. The possible shapes of the gamma-type distribution function (y(t)—Eq. [3b]). The curves are plotted based on the values of the parameters (d and b) listed in Table 2. The solid points are attributed to the micropore range (tmin , tmax ) but the solid line to (0, ∞). The normalization factors for cases I–IV are the same for these two intervals of normalization and are equal to a(0,∞) (Table 2) and for V–VI are equal to a(tmin ,tmax ) (Table 2).

(a) b(≡ν) is an odd integer, (b) b(≡ν) is an even integer, (c) b(≡ν) is a decimal fraction. For the subclass where b is equal to 3.00, the normalization factors are listed in Table 2. They give the positive values for all the integration ranges considered. However, only for one interval of integration (xmin , xmax ), the generated adsorption isotherm based on the equation 2GT(xmin ,xmax ) belongs to type I of the IUPAC classification (i.e., Langmuir) (50) (Fig. 2). On the other hand, for 2GT(0,∞) and 2GT(xmin ,∞) , the erroneous results are obtained; i.e., the degree of pore filling is always much

greater than unity (Table 2). In Fig. 2 and in the next figures, the adsorption isotherms will be plotted in the logarithmic scale (log ( p/ ps )) to emphasize the differences between one another, especially the inconsiderable ones (in all figures “log” marks the logarithmic scale). In order to compare these two classes of isotherms (those that generate values greater than unity and those that do not), the relative absolute error (%2GT(error) —Eq. [18]) is evaluated (Fig. 3). Some values of %2GT(error) (between 0.001 and 0.01 p/ ps ) are enormous (the open diamond symbol (for 2GT(0,∞) ) and the dotted line (for 2GT(xmin ,∞) )). The results confirm the existence of the considerable differences between these two groups of

TABLE 2 The Values of the Normalization Factors (a(i) and/or χGT(i) ) Calculated Based on the Equations Listed in Table 1 (for the Gamma-Type MSD Function (Eqs. [3b] and/or [10]) in Three Various Intervals of Integration) y(t) Eq. [3b] e = 1, t = x n

J (x)GT Eq. [10] χGT(i)

a(i)

I II III IV V VI

2GT Eqs. [15]–[17]

B

d

(0, ∞)

(tmin , ∞)

(tmin , tmax )

ν

ρ

(0, ∞)

(xmin , ∞)

(xmin , xmax )

(0, ∞)

(xmin , ∞)

(xmin , xmax )

3.0 2.6 0.7 −0.8 −0.8 0

1.3 −2.3 −2.3 −2.3 0 0

0.4760 5.3954 4.5347 0.2573 — —

0.4760 5.3956 4.6112 0.8580 — —

1.3845 19.8831 6.2034 0.8840 0.4495 1.0556

3.0 2.6 0.7 −0.8 0 0

−187.2 331.2 331.2 331.2 331.2 0

0.9520 10.7908 9.0693 0.5146 — —

0.9250 10.7912 9.2223 1.7160 — —

2.7691 39.7663 12.4068 1.7680 0.8989 2.1112

− + + + − −

− + + + − −

+ + + + + +

Note. The calculation are done assuming six cases of the values of the parameters (ν (and/or b) and ρ ( and/or d))—see text. The results of the calculations of 2GT are presented also (xmin = 0.2295 [nm]; xmax = 1 [nm]; β = 1.; T = 293 K; n = 2.0 and the relative pressure range from 1 × 10−7 to 0.999 p/ ps (331 points altogether)). Plus represents the case in which the adsorption isotherms belong to the Langmuir type and minus do not.

488

GAUDEN AND TERZYK

FIG. 2. The comparison of benzene adsorption isotherms (T = 293 K), generated by Eq. [17], for case (I) (d(≡−ρζ ) > 0 and b(≡ν) > 1), for ρ (=−187.2) and ν = 3.00, 5.55, and 10.00 (n = 2, xmin = 0.2295 nm, and xmax = 1 nm) and the gamma-type MSDs, generated by Eq. [10] in the micropore range, for the same values of ρ and ν.

2GT . It implies that whether part of Eq. [14] (i.e., in the range (0, xmin )) is taken into account in the calculation or not leads to insignificant differences between 2GT(0,∞) and 2GT(xmin ,∞) and to considerable ones in comparison with 2GT(xmin ,xmax ) .

FIG. 3. The percentage relative absolute error calculated using Eq. [18], for the same value of ρ (=−187.2) and for the relative pressure ranges 1 × 10−7 – 0.999. The open symbols represent the plots generated for 2GT(xmin ,∞) and the lines for 2GT(0,∞) .

In the case where b is an even integer (≡ν = 10), the values of the pore-size distribution functions are smaller than unity for the following ranges, (0, ∞) and (xmin , ∞) due to the expressions ((−d)b+1 and/or (ρζ )ν+1 )) that give negative values (the denominator of χGT(xmin ,xmax ) and 2GT(xmin ,xmax ) gives the negative value also for 1x ∈ hxmin , xmax i). Furthermore, for 2GT(0,∞) and 2GT(xmin ,∞) , the negative values are computed for the low values of the relative pressures (up to 0.0035 p/ ps ) and the positive ones for the upper values of the relative pressure (and greater than unity), but, unfortunately, they are decreasing and tending to 2GT(0,∞) = 1. The third case (b(≡ν) = 5.55) is different from the other two. Based on Eq. [10] it is impossible to calculate the values of the MSD function. With reference to the starting condition (d > 0), of course, the expressions (−d) and/or (ρζ ) give negative values (equal to −1.3). These terms are raised to the power equal to (b + 1) and/or (ν + 1), respectively. Because the values of exponents are fractions, the numerical procedures for the integration of Eq. [10] in the range (xmin , xmax ) (the calculation of the normalization factors) should be applied (52). For the same reason (i.e., the fractional exponent), it is also impossible to calculate the values of the analytical solution of the overall adsorption isotherm given by Eqs. [15]–[17]. Therefore, the procedures of numerical integration are applied once again. It should be noted now that the calculation of the values of the generalized incomplete gamma function given by the following general form of the integral, Zz1 γ (a, z 0 ) − γ (a, z 1 ) =

k a−1 exp[−k] dk,

[19]

z0

is also based on the well-known numerical procedures (52, 57). Then, from the formal point of view, the directly numerical integration of the gamma-type MSD function given by Eq. [10] (for 1x ∈hxmin , xmax i) and the calculation of the 2GT(xmin ,xmax ) values (Eq. [17]), that include the numerical computation of the incomplete gamma function (52, 57), are equivalent (with some comparable precision). Moreover, the procedures published by Press and co-workers (57) are used in this paper, generalized and extended to the cases: z 0 < 0, z 1 < 0, and/or a < 0 (Eq. [19]). For (b(≡ν) = 5.55), the adsorption isotherm 2GT(xmin ,xmax ) is numerically generated based on Eq. [14]. In Fig. 2 the comparative plots of the adsorption isotherm obtained for the same interval of integration (from to xmin up to xmax ) and the value of ρ equal to −187.2 but the different values of the parameter ν (3.00, 5.55, and 10.00) are presented. As can be expected, the plots of the adsorption isotherm and the micropore-size distribution plots generated for ν equal to 5.55 (from the numerical integration) lie between isotherms calculated for the lower and higher values of b, respectively (from the numerical computation of the generalized incomplete gamma function). The properties of 2GT of J (x)GT computed for various intervals of integration and the various b values are similar (Table 2, Figs. 2 and 3).

489

NORMALIZATION OF MICROPORE–PORE DISTRIBUTION FUNCTIONS

As was shown previously (43, 44), Eq. [14] (1x ∈ hxmin , xmax i) can be split up into three integrals:

2GT(xmin ,xmax )  ≡

Zxmax = θ (A, x)J (x) d x xmin

Zxmax

Z∞ B1 d x =

xmin

Zxmin B2 d x −

0

Z∞ B3 d x −

0

 B4 d x  ,

xmax

[20]

FIG. 4. The comparison of benzene adsorption isotherms (T = 293 K), generated from 2GT , Eqs. [15]–[17], for case (II) (d(≡−ρζ ) < 0 and b(≡ν) > 1), for ρ (=331.2) and ν (=2.6) (n = 2, xmin = 0.2295 nm, and xmax = 1 nm) and the gamma-type MSDs, generated by the combination of Eq. [10] and the normalization factors listed in Table 2, for the same values of ρ and ν (presented in the micropore range).

(II) (d(≡−ρζ ) < 0 and b(≡ν) > 1). The second type of the MSD plots is an asymmetrical bell-shaped function (Fig. 1) possessing two points of inflexion (56). The results of the calculation of the normalization factors for b = 2.6 and d = −2.3 are summarized in Table 2. For all the intervals of integration (Table 1), the normalized gamma-type distribution function gives positive numbers (Fig. 4). In Fig. 4 the plots of the adsorption isotherms, generated based on Eqs. [15]–[17] and also for the parameters listed in Table 2 are presented. From this figure, a general tendency is seen that for a fixed value of the relative pressure ( p/ ps ), all the values of the degree of pore filling for 2GT(xmin ,xmax ) are larger than for 2GT(xmin ,∞) and/or 2GT(0,∞) , respectively. Moreover, the shapes of the plots of 2GT generated based on Eqs. [15] and [16] are very similar. The behavior of the adsorption isotherms obtained for the different intervals of integration is associated with the dependence of the shape of the pore-size distribution J (x)GT (Figs. 1 and 4). Figure 4 presents the plots of three cases of J (x)GT in the range of the lower and upper cut-offs (xmin , xmax ) (the values of the parameters and of the normalization factors are tabulated in Table 2). For an illustrative purpose, in Fig. 1 (the range considered (0, ∞)—lines) are the segments, marked by solid points, that correspond to the same interval as in Fig. 4 (1x ∈ hxmin , xmax i).

where Bi are the integrands. These terms for the different subscripts (i) (for a fixed value of p/ ps ) can be equal to each other, providing the identical values of the normalization factors are assumed (43, 44). The assumption of the equality of normalization factors was used by Avnir and Jaroniec (53) for the Pfeifer and Avnir pore-size distribution function (J (x)PA —Eq. [12]). The state of equality of Bi ’s is the inadmissible approximation due to the various integration ranges for every integral. On the other hand, for the gamma-type distribution function, the normalization factors obtained for the three considered intervals are not equal to each other (for the same values of the parameters of MSD function (Table 2)). Therefore, it is easy to show that the values of integrands are different (B1 6= B2 6= B3 6= B4 ). It is impossible to explain the behavior of 2GT based on the above equation directly, but it can be done based on the investigation of the influence of the parameters of the micropore-size distribution and of the range of the integration on the shape of the MSD function. From the comparison of J (x)GT presented in Figs. 1 and 4 (ρ = 331.2), it is seen that the difference between J (x)GT(0,∞) and J (x)GT(xmin ,∞) is negligible (very similar values of χGT — Table 2).R This result is caused by the small contribution of the x integral 0 min B3 d x. On the other hand, from the analysis of two subclasses of the integration ranges (xmin ,Rxmax ) and ((0, xmin ) ∞ and/or (0, ∞)), it is visible that the integral xmax B4 d x gives the largest contribution. Then, this segment of Eq. [20] determines the difference between these subclasses of 2GT , respectively. The confirmation of this fact is the shape of the relative absolute error (%2GT(error) ) shown in Fig. 5 (the solid line and the open triangle). Each of the adsorption formulas, given by Eqs. [16] and [17], includes the Jaroniec and Choma equation (Eq. [15]) multiplied by the fraction, where the numerators among other factors are a function of the relative pressure. The denominators are the part of the normalization factors (Table 1). These ratios also determine the mutual location of the adsorption isotherms. The behavior of the adsorption isotherms and of the percentage relative absolute errors is associated with the dependence of the parameters ν and ρ on the shape of the micropore-size distributions. One example for this case (b = 2.6 and d = −2.3) has been analyzed so far (Fig. 4). The influence of the parameter ρ on the shape of the MSD plots can easily be observed for the following case: for the fixed value of ν (=2.6), the increase of ρ

490

GAUDEN AND TERZYK

Figure 7 presents the plots of the relative absolute error calculated for two subclasses of the value of the parameter ρ, 331.2 and 11563.2, respectively (ν is equal to 0.7). The increase in the value of ρ results in the decrease of %2GT(error) to zero for 2GT(xmin ,∞) (similar to case (II)). This situation (for ρ equal to 11563.2) is the result of the negligible contribution of integral R xthe R∞ min B d x and the increase of the participation of B 3 d x. xmax 4 0 (IV) (d(≡−ρν < 0 and b(≡ν) < 0). The fourth MSD plot is a decreasing to zero function (Fig. 1). All the relationships describing the normalization factors (a(i) and/or χGT(i) —Table 1) include the exponential expressions ((−d)b+1 and/or (ρζ )ν+1 )). The possibility of the calculation of these factors is dependent on the value of the parameter b and/or ν. Therefore, this case splits into the following five subclasses: (a) −1 < b(≡ν) < 0 ⇔ (b + 1(≡ν + 1) is a positive fraction), (b) b(≡ν) = −1 ⇔ (b + 1(≡ν + 1) is equal to zero), (c) b(≡ν) < −1 ⇔ (b + 1(≡ν + 1) ∈ (ofr , efr ) is a negative fraction), (d) b(≡ν) < −1 ⇔ (b + 1(≡ν + 1) is a negative integer), (e) b(≡ν) < −1 ⇔ (b + 1(≡ν + 1) ∈ (efr , ofr ) is negative fraction), where ofr and efr is odd and even integer, respectively.

FIG. 5. The percentage relative absolute error calculated using Eq. [18], for the same value of ν (=2.6) and for the same p/ ps ranges as in Fig. 3. The symbols represent the plots generated for 2GT(xmin ,∞) and the lines for 2GT(0,∞) .

from 331.2 to 11563.2 brings about the reduction of the fraction of micropores possessing the half-widths close to xmax . In other words, for ρ tending to infinity, 2GT(xmin ,xmax ) tends to 2GT(xmin ,∞) and the values of absolute error between them go to zero (Fig. 5). (III) (d(≡−ρζ ) < 0 and 0 < b(≡ν) ≤ 1). The third example of MSD plots is also an asymmetrical bell-shaped function. The function possesses only one point of inflexion (56) in contrast to the second case (II)—characterized by two points. Despite this difference, the properties of J (x)GT and 2GT are very similar due to the same (bell-shaped) shape. The results of the calculations of the normalization factors (a(i) and/or χGT(i) ) are presented in Table 2 for the same value of parameter d(=−2.3) as in the case (II) but for the different value of the parameter b equal to 0.7. Contrary to the second case, in Fig. 6 for a fixed value of p/ ps (especially, for the values of the relative pressure smaller than 1 × 10−3 ), the plot of 2GT(0,∞) is situated between 2GT(xmin ,xmax ) and 2GT(xmin ,∞) . Once again, this situation can be explained based on the behavior of the gamma-type distribution function. The difference between the values of the degree of pore filling (for a fixed value of p/ ps ) calculated from REqs. [15]–[17] xmin is R ∞caused by the contribution of the integrals: 0 B3 d x and xmax B4 d x , respectively.

FIG. 6. The comparison of benzene adsorption isotherms (T = 293 K), generated from 2GT , Eqs. [15]–[17], for case (III) (d(≡−ρζ ) < 0 and 0 < b(≡ν) ≤ 1), for ρ (=331.2) and ν (=0.7) (n = 2, xmin = 0.2295 nm, and xmax = 1 nm) and the gamma-type MSDs, generated by the combination of Eq. [10] and the normalization factors listed in Table 2, for the same values of ρ and ν (presented in the micropore range).

NORMALIZATION OF MICROPORE–PORE DISTRIBUTION FUNCTIONS

491

(the value of d is fixed and equal to −2.3 (ρ = 331.2)) is presented in Fig. 9. For all the subclasses, the generated adsorption isotherms give 2GT(xmin ,xmax ) the values of the degree of pore filling smaller than unity (Figs. 8 and 9). The influence of b on 2GT(0,∞) is shown in Fig. 10. From this figure it is seen that for only one example out of from the considered subclasses, b = −0.8 (presented also in Fig. 8), the obtained isotherm belongs to type I of IUPAC classification (i.e., Langmuir type) (50). The analysis of Eq. [15] shows that for b equal to −1, the values of the degree of pore filling are constant and equal to unity. For the next cases (c–e), values of 2GT(0,∞) are greater than unity (Fig. 10). Moreover, for (b) and (d) it is not possible to calculate any values of the gamma-type distribution function. On the other hand, for cases (a) and (c), the calculation provides positive values in contrast to case (e) (the negative values). Finally, the comparative analysis of 2GT(0,∞) and 2GT(xmin ,xmax ) is applied. In Fig. 11 the plots of the absolute relative error for five values of parameter (b) are presented. The evaluated values of %2GT(error) , for the adsorption isotherms presented in Fig. 9 (2GT(xmin ,xmax ) ) and Fig. 10 (2GT(0,∞) ), decrease from the appreciable (10000%) to zero percentage. It confirms the

FIG. 7. The percentage relative absolute error calculated using Eq. [18], for the same value of ν (=0.7) and for the same p/ ps ranges as in Fig. 3. The symbols represent the plots generated for 2GT(xmin ,∞) and the lines for 2GT(0,∞) .

For all the calculations the value of the parameter d is arbitrarily assumed. It is equal to −2.3. For the first subclass (b = −0.8), the normalization factors are listed in Table 2. They give the positive values for all the investigated intervals of integration. The generated adsorption isotherms based on Eqs. [15]–[17] belong to type I of IUPAC classification (i.e., Langmuir type) (50) (Fig. 8). For the next subclasses (b–e), the problem of the calculation of the values of the normalization factors (Table 1) and 2GT (Eqs. [15]–[17]) becomes more complicated than, for example, in the case where b is equal to −0.8. J (x)GT and 2GT include the gamma and the incomplete gamma function, respectively. The behavior of these two functions (for the fixed value of the parameter (d) is very similar and is determined by the value of the expressions: (b + 1) and/or (ν + 1). Therefore, for the second and the fourth subclass (b is equal to −1.0 and −3.0, respectively), for both gamma and incomplete gamma functions, infinity is obtained. For (c) (b = −2.6) and (e) (b = −3.6), these functions give positive and negative values, respectively. Although, it is impossible to calculate directly the values of the normalization factors for the above-mentioned integration ranges, based on the equations listed in Table 1 (b = −1.0 and b = −3.0), the procedures for numerical integration (in the range (xmin , xmax )) can be applied. The comparison of the microporesize distribution plots for different considered values of b(≡ν)

FIG. 8. The comparison of benzene adsorption isotherms (T = 293 K), generated from 2GT , Eqs. [15]–[17] for case (IV) (d(≡−ρζ ) < 0 and b(≡ν) < 0), for ρ (=331.2) and ν (=−0.8) (n = 2, xmin = 0.2295 nm, and xmax = 1 nm) and the gamma-type MSDs, generated by the combination of Eq. [10] and the normalization factors collected in Table 2, for the same values of ρ and ν (presented in the micropore range).

492

GAUDEN AND TERZYK

The normalization of the gamma-type distribution function in the integration range (tmin , ∞) is impossible for b(≡ν) ≥ −1 due to the fact that integrals: t b and/or x n(ν+1) −1 do not converge. The result of this integration is indeterminate. For the values of b lower than minus one, the factors (a(tmin ,∞) and/or χGT(xmin ,∞) ) can be evaluated from the following relationships: a(tmin ,∞) =

b+1 b+1 −tmin

and/or χGT(xmin ,∞) =

n(ν + 1) n(ν+1) −xmin

.

[22]

The assumption of equality to zero of the parameter d(≡ −ρζ ) leads to the concept that, for some conditions, the gamma-type distribution function is reduced to the Pfeifer and Avnir one. The problem of the normalization of J (x)PA was discussed by us previously in detail (42–44, 58). Moreover, in this paper we only point out the existence of the connection between these two functions. The detailed analysis of the problem of the simplification of J (x)GT to J (x)PA will be discussed in the future.

FIG. 9. The comparison of benzene adsorption isotherms (T = 293 K), generated by Eq. [17], for the same case as in Fig. 8, for ρ (=331.2) and ν = −0.8, −1.0, −2.6, −3.0, and −3.6 (n = 2, xmin = 0.2295 nm, and xmax = 1 nm) and the gamma-type MSDs, generated by Eq. [10] in the micropore range, for the same values of ρ and ν.

behavior of these isotherms that tend to unity for p/ ps tending to unity without the interval of integration. (V) (d(≡−ρζ ) = 0 and b(≡ν) 6= 0) and (VI) (d(≡−ρζ ) = 0 and b(≡ν) = 0). The fifth type of the MSD plots is a decreasing function and the sixth one gives only constant values (Fig. 1). For these two cases (d is assumed to be equal to zero), the values of the pore-size distribution function, for all the values of the parameter b (≡ν) 6= −1, are greater than zero in the case of the interval of integration from tmin up to tmax (Table 1), only. The normalization factors can be expressed in the following equivalent forms: a(tmin ,tmax ) =

b+1 b+1 tmax

b+1 − tmin

and/or χGT(xmin ,xmax ) =

n(ν + 1)

. n(ν+1)

n(ν+1) xmax − xmin

[21]

FIG. 10. The comparison of benzene adsorption isotherms generated by Eq. [15] assuming the same values of parameters as in Fig. 9 (symbols as in Fig. 9).

NORMALIZATION OF MICROPORE–PORE DISTRIBUTION FUNCTIONS

493

finite cutoffs (the lower and upper limits) that made the calculations independent from the properties of the MSD function (for example, the hyperbolic shape). For the other intervals of integration, the generated adsorption isotherms belong to type I of IUPAC classification (50) only for some selected cases. In order to compare the various behavior of the two groups of 2GT (i.e., (2GT(xmin ,xmax ) and (2GT(0,∞) or 2GT(xmin ,∞) ), the relative absolute errors (%2GT(error) ) are calculated; the achieved results suggest the existence of appreciable differences between them (for example, Fig. 3.) It is very interesting that the assumption of the equality to zero of the parameter b (and/or ρ) of the gamma-type distribution function J (x)GT leads to the simplification of this function to the Pfeifer and Avnir one—J (x)PA . Based on this fact, it can be presumed that J (x)PA is the special case of J (x)GT . ACKNOWLEDGMENTS P.A.G. gratefully acknowledges the financial support from the Foundation for Polish Science and from KBN Grant 3T09A 150 18. A.P.T. gratefully acknowledges the financial support from KBN Grant 3T09A 005 18.

REFERENCES

FIG. 11. The percentage relative absolute error calculated on the basis of Eq. [18], for adsorption isotherms from Figs. 9 and 10 (symbols as in Fig. 9).

4. CONCLUSIONS

According to the theory of volume filling of micropores, both the application of the micropore-size distribution function (J (x)GT ) and the assumption of a strictly microporous adsorbent permit the calculations based on Eq. [14]. The analytical solution of this global adsorption isotherm equation for three intervals of integration, (xmin , xmax ), (xmin , ∞), and/or (0, ∞), leads to Eqs. [15]–[17], respectively. These adsorption isotherm equations are obtained assuming the MSD function as a gamma type. The normalization factors are summarized in Table 1. It is very important that both the integration of the overall adsorption isotherm and the normalization of MSD are connected with the same integration range. The gamma-type distribution function (Eq. [10]) can generate some various types of shapes (56). The most interesting and representative cases are shown in Fig. 1. In this figure the influence of parameters b (and/or ν) and d (and/or ρ) on the generated shape of the investigating function is also presented. From the results of the calculations, summarized in Table 2, it is seen that it is not possible in all cases to obtain the values of 2GT smaller than unity (the Langmuir-type adsorption isotherms). It is proved that it is possible only for the gammatype distribution function in the case of the integration range (xmin , xmax ). This result is expected due to the existence of the

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