Micropore structure of activated carbons and predictions of adsorption equilibrium

Carbon 40 (2002) 1003–1010

Micropore structure of activated carbons and predictions of adsorption equilibrium Vladimir Kh. Dobruskin* Ajala 21 st., Beer-Yacov, 70300, Israel Received 23 October 2000; accepted 28 July 2001

Abstract A method for calculating micropore size distributions based on a molecular model of adsorption and analytical solution for the adsorption isotherm is presented in this study. Micropore volume filling is considered to be an evolution of two-dimensional condensation, which occurs on micropore walls at the critical condensation pressure. The treatment of adsorption isotherms of propane, propylene, acetylene, ethylene, cyclopentane and benzene shows that the method offers a quantitative approach to determining a reliable carbon texture, which is independent of adsorbate employed and adsorption temperature. The invariant parameters of porous structure coupled with molecular constants of adsorbate provide a prediction of the adsorption equilibrium in a wide range of pressures. Good agreement between experimental and calculated data is demonstrated for adsorption of both gases and vapors.  2002 Elsevier Science Ltd. All rights reserved. Keywords: A. Activated carbon; C. Adsorption; C. Modeling; D. Microporosity

1. Introduction Adsorption behavior of porous sorbents depends largely upon their pore size distributions (PSD). The experimental isotherm is a composite of individual adsorption isotherms of various sizes of pores present in the sorbent [1]. According to the IUPAC classification, a pore system in general consists of micropores and larger meso- and macropores [2]. The micropores with widths less than 2 nm play the most important role in adsorption, especially at low adsorbate concentrations. For active carbons, an individual micropore is usually considered to be a void space between two semi-infinite graphite slabs or sheets, and the solid–fluid interaction potential is described by a 10–4–3 or 10–4 potential function [3,4]. Numerous studies reveal that, due to the sensitivity of the Lennard– Jones potential to the slit-pore size, the PSD thus obtained will locate mainly in the region of one or two times the Lennard–Jones diameter [5]. In order to develop a useful predictive model, it is

*Tel.: 1972-08-922-5686. E-mail address: [email protected] (V. Kh. Dobruskin).

important to establish the PSD of a sorbent. There are several methods to deal with this problem on the basis of physisorption data. Some are based on the empirical Dubinin–Astakhov equation or its modifications [6–9], others are based on statistical mechanical principles using the density functional theory and molecular computer simulations [10–16]. The latter methods still require some assumptions, which limit their application to adsorbates with simple molecular structures. Recently, it has been shown [17–20] that the adsorption process in activated carbon (AC) micropores can be viewed as a two-dimensional (2D) condensation on micropore walls at a critical condensation pressure, pc , which evolves into volume filling. This means that volume filling of a single micropore and a 2D-condensation on its walls occur at the same pressure and statistical mechanical theories of adsorption on non-porous surfaces, which account for lateral interactions and describe a 2D condensation, may be taken as the starting point for discussing the adsorption equilibrium in micropores. These theories provide us with the following relationship for evaluating the condensation pressure on the surface of an ideal micropore in respect to that on the reference graphite surface [17,19–21]

0008-6223 / 02 / $ – see front matter  2002 Elsevier Science Ltd. All rights reserved. PII: S0008-6223( 01 )00232-9

V. Kh. Dobruskin / Carbon 40 (2002) 1003 – 1010

1004 ref

pc RT ln ] 5 ´ * 2 ´ * ref ; D´ * pc

(1)

Here p ref is the critical condensation pressure on the c nonporous surface, ´ * and ´ * ref are the energies of adsorption in the Henry region on micropore walls and on a reference graphite surface, respectively. D´ * ; (´ * 2 ´ * ref ) is the excess energy of adsorption in a micropore in respect to that on the reference surface at the limit of zero amount adsorbed. Adsorption energies are taken to be positive and the asterisk (*) refers to potential energy minima. We use the term ‘ideal micropore’ to denote a micropore with infinite walls for which the contribution of edge effects is negligible. Micropore walls of real microporous solids are bound in size and thus differ from ideal micropores. Due to edge effects, energies of adsorption on surfaces of finite walls decrease, condensation pressures are shifted to higher values [22], and, for micropore walls with typical sizes, p ref approaches the saturation pressure of a bulk adsorbate, c ps . Substituting ps for p ref in Eq. (1), one has: c

H

D´ * pc 5 ps exp 2 ]] RT

J

where z is the distance of a molecule from the central plane. The energy of interactions inside a micropore depends upon z and an equilibrium value of z* corresponding to the minimum of ´ * is found by numerical calculation. This approach takes into account repulsive forces, which make their appearance in narrow micropores, and, thus, a more accurate description of adsorption is gained. Our objective here is to demonstrate that parameters of porous structure resulting from treatments of adsorption isotherms in the framework of the developed model are invariant for a given adsorbent and can be the foundation for a prediction of adsorption equilibrium. So far the model was applied only to adsorption of benzene [20–22] and the prediction of equilibrium was not discussed. Unlike earlier publications, a simpler derivation of adsorption equations will be given. Here we restrict ourselves to the analytical solutions following from the application of the approximate exp-potential function; the contribution of the repulsive potential and submonolayer adsorption will be discussed in more detail in future publications.

(2) 2. Microporous structure of active carbons and adsorption equilibrium

and the work of compression, p ref c A c 5 RT ln ], pc

2.1. Correlation between micropore widths and condensation pressures

is replaced by the Polanyi potential, A [23]: ps A ; RT ln ] pc

(3)

Therefore, the application of ps and A assures an approximate description of the adsorption equilibrium and we shall henceforth use this method. Eq. (2) relates the filling pressure to the excess adsorption energy of a micropore. There are two approaches to calculating pc by Eq. (2). The first applies the approximate exponential function (exp-potential) for calculating D´ *:

FS

d D´ * 5 ´ * ref exp 4 1 2 ] r0

DG

(4)

where r 0 is the Lennard–Jones parameter and d is the micropore half-width. This function gives correct results for micropores with reduced half-widths d /r 0 . 1, but it neglects the repulsive potential and leads to deviations for d /r 0 , 1. The second, more rigorous approach applies the cumbersome 10–4 potential function for calculating D´ * [19]:

H

U

10 D´ * 5 Minimum ] ´ * ref 3 10 r0 1 1 ]] 2] d2z 2

S

2 ´ * ref

Let us consider the influence of micropore radii on excess energies of adsorption in micropores and on micropore filling pressures. As an example, we shall examine adsorption of cyclopentane proceeding from both the 10–4 potential function (Fig. 1) and exp-potential (Fig. 2). The

r H]51 FS]] D d 1z 0

10

r r 1S]]D G J D G FS]] D d 1z d2z JU (5) 0

4

0

4

Fig. 1. Cyclopentane adsorption. Excess adsorption energies (D´ * /´ * ref ) and decimal logarithms of relative micropore filling pressure (Log [ p /ps ]) as a function of a reduced half-width (d /r 0 ) for the 10–4 potential function. There are two values of d /r 0 generated by each value of an excess energy or a filling pressure. For example, when p /ps 51310 24 (i.e. Log [ p /ps ]5 24), D´ * / ´ 0* 5 0.85, d 1 /r 0 ¯0.95 and d 2 /r 0 ¯1.06 and only micropores in the range 0.95#d /r 0 #1.06 (or D´ * /´ *0 $0.85) are capable of compressing a vapor into a condensate and bringing micropore filling about.

V. Kh. Dobruskin / Carbon 40 (2002) 1003 – 1010

u ( p) 5

Fig. 2. Cyclopentane adsorption. Excess adsorption energies (D´ * /´ * ref ) and decimal logarithms of micropore filling pressure (Log [ p /ps ]) as a function of a reduced half-width (d /r 0 ) for the exp-potential function (solid lines). In contrast to the 10–4 function (boxes), the exp-potential neglects repulsive forces and provides a correct approximate description of adsorption behavior only for d /r 0 . 1. It erroneously predicts a continuous growth of D´ * for d /r 0 , 1, where the 10–4 function demonstrates a rapid reduction of potential energies. According to the exp-potential, there is only the upper limit of the width of filled micropores, d 2 /r 0 , whereas the low limit of micropore widths tends to minus infinity.

H

0 1

1005

for p , pc for p . pc

(6)

where p is the outer pressure. Here u is the fraction of a micropore occupied by the adsorbate. In doing so, we suggest that a micropore can be only in two states: either free from adsorbate at p , pc or filled with adsorbate at p . pc . Although this suggestion (known as the condensation approximation) seems to be a rather rough simplification for a single micropore, it leads to the exact result for a collection of micropores with a continuous distribution of adsorption energies. To derive an adsorption equation for a microporous solid, one has to introduce some assumptions about a distribution of micropores in a collection. It has been shown [17] that, if we assume that (i) volumes of all individual micropores are identical and (ii) micropore widths obey the normal distribution over d with the mathematical expectation and standard deviation md and sd , respectively, then the distribution of fractional micropore volume, u, over micropore radii also obeys the same normal distribution: 1 fN (d, md , sd ) 5 fN (u, md , sd ) ]] ] sdŒ2p

conclusions derived from this discussion are valid also for other adsorbates. To calculate pc by Eq. (2), one needs to use the energy of adsorption on the reference surface ´ * ref . In our treatment here we shall employ for this purpose experimental values of energies of adsorption on graphitized carbon black [24]. This problem will be discussed further in more detail. A comparison of data in Figs. 1 and 2 shows that both functions predict close results for d /r 0 . 1, but there is a significant difference in the range of small radii: the exp-potential neglects repulsive forces and erroneously predicts a continuous growth of adsorption energies and reduction of filling pressures even for d /r 0 , 1, whereas the 10–4 function demonstrates a decrease of adsorption energies and a rapid rise of pressures in this range. In the case of the 10–4 potential, there is the minimum filling pressure, pmin that corresponds to the maximum adsorption energy at d /r 0 5 1. This value for cyclopentane is equal to pmin /ps ¯ 1.5 3 10 25 at 298 K and, for micropore filling to occur, the outer pressure must exceed pmin . When p , pmin , only submonolayer adsorption takes place on micropore walls. When p . pmin , there are two values of d /r 0 at every value of pressure (see plot of Log [ p /ps ] 5 f(d /r 0 ) in Fig. 1): the low value, d 1 /r 0 , and the upper value, d 2 /r 0 . Such behavior is absent in the case of exp-potential: there is only the upper limit d 2 /r 0 , whereas the low limit tends to 2` (Fig. 2).

In the case of exp-potential, the low limit of the integral is equal to 2`, but there is an analytical expression for its upper limit. Taking into account that A 5 D´ * (Eqs. (1) and (3)), one has from Eq. (4):

2.2. Adsorption isotherm

r0 d 2 ( ] (4 1 ln ´ * ref 2 ln A) 4

For simplicity, an actual isotherm for an individual micropore may be replaced by a simpler step function:

After replacing both limits in Eq. (8), the integral reduces to [17]:

H S

1 d 2 md 3 exp 2 ] ]] sd 2

DJ 2

(7)

Here fN (d, md , sd ) and fN (u, md , sd ) are the normal density distribution functions of d and u, respectively. Note that u in Eq. (6) refers to a single micropore, whereas u in Eq. (7) relates to a collection of micropores in a carbon sample. According to the condensation approximation (Eq. (6)), all micropores for which pc , p are filled with adsorbate. Hence, at given pressure p micropore filling occurs within the confines of d 1 /r 0 to d 2 /r 0 and u is equal the probability, Pr(d 1 /r 0 , d /r 0 , d 2 /r 0 ), that d /r 0 falls in the range d 1 /r 0 , d /r 0 , d 2 /r 0 :

u 5 Pr(d 1 /r 0 , d /r 0 , d 2 /r 0 ) d2 /r0

5

E

d1 /r0

1 ]]] ] exp (sd /r 0 )Œ2p

H S

1 x 2 md /r 0 2 ] ]]] 2 (sd /r 0 )

DJ 2

dx

(8)

(9)

V. Kh. Dobruskin / Carbon 40 (2002) 1003 – 1010

1006 ln A2 my / sy

1 2u 5

E

H J

1 z2 ]] exp 2 ] ] Œ2p 2

2`

dz

(10)

where 4 my ; 2 ] md 1 4 1 ln u * ref r0

(11)

4 sy ; ] sd r0

(12)

The essential advantage of the application of exp-potential arises from the analytical expressions for adsorption isotherms given by Eqs. (10)–(12). In the case of cumbersome Eq. (5), only numerical evaluations of d 1 /r 0 and d 2 /r 0 and adsorption isotherms are possible. For most commercial active carbons, the volume of pores with d /r 0 , 1 is small and both functions lead to close results. Although integral (10) may be expressed in an explicit form [25,26]:

F

ln A 2 my 1 1 u 5 ] 2 ] erf ]]] ] 2 2 syŒ2

G

is presented in the Appendix. Adsorption data for benzene and cyclopentane are obtained by T. Jacubov [27]. The microporous activated carbon (AC) manufactured from coconut shell by chemical activation (‘Barnebey Cheney’, USA) and employed by Jacubov is denoted as BC AC. The values a 0 for this carbon are determined as adsorption capacities at inflection points on corresponding isotherms ( p /ps 50.17 for benzene and p /ps 50.23 for cyclopentane). Tabulated experimental isotherms of adsorption of propane, propylene, ethylene, and acetylene are given in studies of Lewis and co-workers [28–30]. For adsorption of ethylene and acetylene [30], the activated carbon with surface area of 805 m 2 / g supplied by the Pittsburgh Coke and Chemical Company (PCC) was employed. The limiting volume of micropores of PCC AC is 0.42 cm 3 / g (Fig. 7 in Ref. [28]) and a 0 is estimated proceeding from adsorbate densities, r *. For acetylene the adsorption temperature exceeds its boiling point, T b , and r * is calculated as [31] ln r * 5 ln rb 2 a (T 2 T b )

(14)

(13)

where erf is the error function, we prefer using Eq. (10), which, in our opinion, better illustrates a method of data treatment. The integrand in Eq. (10) is the standard normal distribution, that is, the integrand is given by Eq. (7) with m 5 0 and s 5 1. The upper limit of the integral is equal to the number, ln A 2 my q 5 ]]], sy at which it reaches the given value (1 2 u ). Values of q, which are called quantiles of distribution, are found either from the standard Normal table or from the embedded computer function. The treatment of experimental pairs of values (a, p), where a is the experimental uptake, may be produced as follows. Transforming a to u 5 a /a 0 , where a 0 is the uptake corresponding to complete micropore filling, determines values (1 2 u ) and allows one to find a set of q. If experimental data are described by Eq. (10), the plotting of quantiles via ln A results in the straight line f(ln A) 5 (ln A 2 my ) /sy , the slope, 1 /sy , and the intercept, 2 my /sy , which are found by the least squares method. Proceeding from sy and my , the values of sd , and md are then evaluated by Eqs. (11) and (12).

2.3. Parameters of porous structure and prediction of the adsorption equilibrium This method of data treatment is applied here to adsorption of cyclopentane, benzene, ethylene, acetylene, propane, and propylene on different grades of activated carbons. The mathematical program illustrating the method

where rb is the density at T b and a is the thermal coefficient of the limiting adsorption [32]:

S D

rb ln ] r *crit a 5 ]]] T b 2 T crit * is the critical Here T crit is the critical temperature and p crit * 51 /b, density of adsorbate, which is estimated as p crit where b is the van der Waals constant. In the case of ethylene, T $ T crit 5286.6 K and ps replaced by pcrit (T / T crit )2 , where pcrit is the critical pressure [32]. In addition, data at higher pressures are treated by introducing fugacities in Eq. (3) due to expected appreciable deviations from ideal gas law. The activated carbon with a surface area of 705 m 2 / g supplied by the Godfrey L. Cabot Company of Boston (GLC AC) was involved in adsorption of propane and propylene [29]. The limiting adsorption volume of this carbon is unknown and a 0 is estimated only approximately as 2.60 mmole / g and 2.84 mmole / g, respectively, at 800 mm Hg (Fig. 2 in Ref. [29]). Conclusions following from adsorption of these substances, strictly speaking, relate only to the portions of micropores that are filled at p # 800 mm Hg. The adsorption isotherms in coordinates of (ln A 2 my ) / sy against ln A are plotted in Figs. 3–6. The main conclusion is that experimental data are well described by Eq. (10) in the wide range of equilibrium relative pressures. The slopes and intercepts of the straight lines give values of my and sy , which are expected to be determined only by a porous structure and, thus, independent of temperatures and adsorbates employed. The analysis of adsorption of cyclopentane at 293 K, 313 K, and 328 K (Fig. 6) demonstrates that my and sy are really independent of temperature and, hence, Eq. (10) does

V. Kh. Dobruskin / Carbon 40 (2002) 1003 – 1010

Fig. 3. Adsorption of benzene (triangles) and cyclopentane (diamonds) on BC AC at 298.15 K.

1007

Fig. 6. Adsorption of cyclopentane at 298.15 K (diamonds), 313.15 K (triangles), and 328.15 K (stars).

[33], fs , corresponding to ps . Data of Lewis and coworkers in Table 1 are presented in the original old system of units. The Lennard–Jones parameters of adsorbates, sxx , except for cyclopentane and propylene, are reported by Reid and Sherwood [34]; for propylene sxx is taken from Ref. [35]. The Lennard–Jones diameter of cyclopentane has not been found in the literature and is approximately calculated from its molar volume at the normal boiling point, Vb , by the expression [36]:

sxx 5 1.18V 1b / 3

(16)

r 0 for gas–carbon interactions is evaluated by the Lorentz combining rule: Fig. 4. Adsorption of ethylene (triangles) (diamonds) on PCC AC at 298.15 K.

and

acetylene

correctly describe the temperature dependence of adsorption. This result was earlier evidenced for adsorption of benzene for a wider temperature range [17]. Figs. 3–5 demonstrate the utility of the model of adsorption for both gases and vapors. The parameters of porous structure of carbons and the properties of adsorbates involved are given in Tables 1 and 2. Also given in Table 1 is the fugacity

scc 1 sxx r 0 5 ]]] 2

(17)

where sxx and scc are the Lennard–Jones parameters for adsorbate–adsorbate and carbon–carbon interactions. Following Steele [3], we accept scc 50.340 nm. Distributions of micropore sizes are shown in Fig. 7. Points in Fig. 7 are evaluated by density normal functions parameters which are given in Table 2. For instance, the distribution of micropores arising from acetylene adsorption is described by the following expression: 1 fN (d, 0.4621, 0.05566) 5 ]]]] ] 0.05566Œ2p 1 d 2 0.4621 3 exp 2 ] ]]] 2 0.05566

H S

DJ 2

(18)

Fig. 5. Adsorption of propane (triangles) (diamonds) on GLC AC at 298.15 K.

and

propylene

The most important result is the following: parameters of porous structure do not depend on adsorbates and temperatures and, hence, are intrinsic properties of the adsorbent. This fact opens the possibility of predicting adsorption isotherms from minimum data. Proceeding from values sd and md for the reference adsorbate, s ref and m dref paramed ters of Eq. (10) related to a target adsorbate, s ytr and m ytr are calculated as follows:

V. Kh. Dobruskin / Carbon 40 (2002) 1003 – 1010

1008

Table 1 Limiting adsorption in micropores and properties of adsorbates at 298 K Adsorbate

Carbon

a0 [mmole / g]

´ * ref [kJ / mol]

sxx [nm]

r0 [nm]

ps [Torr]

r* [g / cm 3 ]

fs

Propane Propylene Acetylene Ethylene Benzene CP a

GLC GLC PCC PCC BC BC

2.60 2.84 7.75 6.81 4.20 3.83

20.5 21.7 14.2 16.3 40.0 27.5

0.5061 0.4678 0.4221 0.4232 0.5270 0.5610

0.423 0.404 0.382 0.382 0.428 0.450

7095 8603 36890 42537 b 95.2 318.9

0.5500 0.5538 0.4795 0.4540 c 0.8890 0.7454

0.85 0.70 0.68 0.72 1 1

a

CP, cyclopentane. Value of pcrit (T /T crit )2 . c Value of r *crit 51 /b. b

4 ref m 5 2] m 1 4 1 ln ´ * tr r 0tr d

(19)

4 ref s try 5 ] s r 0tr d

(20)

tr y

For example, if sd and md are known from adsorption of ethylene, for acetylene adsorption we have m try 5 2(43 0.4592) / 0.382141ln 14.251.8449 and s ytr 5(43 0.05238) / 0.38250.5485 and adsorption of acetylene is described as follows: ln A21.8449 / 0.5485

1 2u 5

E

1 ]] exp Œ] 2p

2`

1 1 5 ] 1 ] erf 2 2

F

H J G z2 2] 2

was about 11–13 m 2 / g, which on the molecular level may be considered as consisting of infinite basal planes. The thickness and diameters of walls for a microporous carbon with s ¯ 1000 m 2 / g are of the order of a nanometer. Factors associated with edge effects decrease adsorption energies on surfaces of finite particles; and for particles with diameters about 1.4 and 1.8 nm average adsorption

dz

ln A 2 1.8449 ]]]] ] 0.5485 Œ2

(21)

From sd and md for benzene, s ytr and m ytr for cyclopentane are estimated as m ytr 52.78904 and s ytr 50.56164. Predicted isotherms for acetylene and cyclopentane are in a good agreement with experimental data (Figs. 8–9).

2.4. Energies of adsorption on the reference nonporous surface In our treatment here we employed experimental values of the energy of adsorption on the nonporous surface as ´ * ref . These values were determined in experiments with a graphitized carbon black whose specific surface area, s, Table 2 Parameters of adsorption on activated carbons Adsorbate

Carbon

sy

my

md [nm]

sd [nm]

Propane Propylene Acetylene Ethylene Benzene CP

GLC GLC PCC PCC BC BC

0.4406 0.4482 0.5828 0.5485 0.5905 0.5728

2.4092 2.3626 1.8141 1.9823 2.9311 2.8118

0.4876 0.4762 0.4621 0.4592 0.5091 0.5065

0.04559 0.04527 0.05566 0.05238 0.06318 0.06443

Fig. 7. Micropore size distributions. (1) PCC AC from adsorption of ethylene (triangles) and acetylene (diamonds); (2) GLC AC from adsorption of propane (triangles) and propylene (diamonds); and (3) BC AC from adsorption of benzene (triangles) and cyclopropane (diamonds).

V. Kh. Dobruskin / Carbon 40 (2002) 1003 – 1010

1009

standard deviation of this value, sy 5 sln A ´ * . Note that the average values of D´ *, mD ´ * , and its deviation, sD´ * are given as follow [26]:

H

s 2y ] mD ´ * 5 exp my 1 2

J

s 2D ´ * 5 exph2my 1 s 2y j(exphs 2y j 2 1)

(22) (23)

For instance, mD ´ * for acetylene is equal to exp(1.8141 1 0.5828 2 / 2) 5 7.27 kJ / mol. For usual active carbons, mD ´ * is close to a value of characteristic energy of the Dubinin– Astakhov equation (Table 5 in Ref. [17]). Fig. 8. Adsorption of acetylene on PCC AC at 298.15 K. Experimental points (triangles); predicted points (stars).

3. Conclusions ref

ref

energies decrease to ¯ 0.72´ * and ¯ 0.81´ * [22]. For particles with diameters about 3.2 nm, the appropriate estimation leads to ¯ 0.88´ * ref . If we accepted that energies of adsorption on nonporous walls are equal to 0.80´ * ref then calculated average micropore half-widths would reduce by (r 0 / 4)3ln 0.8¯0.02 nm. Strictly speaking, sizes of micropore walls in activated carbons usually are unknown and further studies are desirable for a justified optimal choice of ´ * ref .

The method of micropore size distribution analysis presented in this study offers a quantitative approach to determining reliable parameters of carbon porous structure, which are independent of the adsorbate employed and the working conditions. Being based on a molecular description of adsorption, the model provides predictions of adsorption equilibrium in a wide range of relative pressures.

2.5. Parameters of the equations used in the present study

Acknowledgements

The physical meaning of parameters of the normal distribution of micropore sizes, md and sd , is plain to see: md is the average micropore half-width and sd is the standard deviation of half-widths. It has been shown [17] that, if micropore sizes are distributed according to the normal law and the exp-potential is used as a model of interactions, the resulting distribution of D´ * obeys the lognormal law. In this case, my and sy are parameters of the lognormal distribution: my is the average value of the logarithm of the excess energy, my 5 mln D ´ * and sy is the

The author is grateful to Prof Ustinov E of the Technological Institute of St. Petersburg and Prof Jacubov T of Royal Melbourne Institute of Technology (RMIT University) for kindly supplying experimental adsorption data.

Appendix A.1. Mathematical program for calculation of adsorption of ethylene and acetylene (See printed file Appendix nb (Mathematica), 3 pages) A.2. Program comments In [4] In [5]

In [6] In [7] and In [8]

Fig. 9. Adsorption of cyclopropane on BC AC at 298.15 K in the coordinate system u –ln A. Experimental values (polygons); predicted points (crosses).

In [9] In [10] In [11]–In[14]

Parameters of ethylene adsorption 14 experimental points in the form of h p [mm Hg], a [mmole / g]j (see Table 2 in Ref. Lewis WK et al., J Am Chem Soc 1950;72:1158) Transformation of h p, aj into h f/fs , uj Transformation of h f/fs , u j into hln A, q ; quantile of (1 2 u )j Plot of q 5 f(ln A) versus ln A Fitting parameters of the straight line f(ln A) 5 (ln A 2 my /sy ) to experiments Yield values of my , sy , md and sd

1010

In [15] In [16]

In [27]–In [28]

In [29] In [30] In [31] In [32]–In [33] In [34] In [35] In [36]

V. Kh. Dobruskin / Carbon 40 (2002) 1003 – 1010

Plot of the straight line f(ln A) 5 (ln A 2 my /sy ) Demonstration that experimental points fall on the straight line in coordinates (q, ln A) Give values of my and sy for acetylene proceeding from known values of md and sd for ethylene Write down experimental values of ln A Give predicted values of quantiles at ln A taken from Out [29] Yield predicted values of u Give a predicted isotherm in the form h f/fs , u j Definitions Compare experimental and predicted isotherms Take 85 values of half-widths ranging from 0.21 to 0.84 nm, calculate values of density distribution functions at these points for ethylene and propylene proceeding from estimated values of md and sd and yield plots of both functions.

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