Improved Horvath—Kawazoe equations including spherical pore models for calculating micropore size distribution

Pergamon

Ckemic~l Engineeriw

Scipnce. Vol. 49, No. 16. pp. 2599-2609, 1994 Copyri@ Q 1594Ehwk Sciena Ltd Prioti in Great Britain. All rights reserved ooc6-2509/94 s7.00 + 0.00

OOO!I-2509(94)EOO54-T

IMPROVED INCLUDING CALCULATING

Department of Chemical

HORVATH-KAWAZOE EQUATIONS SPHERICAL PORE MODELS FOR MICROPORE SIZE DISTRIBUTION

LINDA S. CHENG and RALPH T. YANG’ Engineering, State University of New York at Buffalo, Buffalo, NY 14260, U.S.A.

(Received

20 October

1993; accepted for publication

4 March

1994)

Abstract-Since its publication, the Horvath-Kawazoe (H-K) equation has been rapidly and widely adopted for calculating the micropore size distribution from a single adsorption isotherm measured at a subcritical temperature (e.g. N2 at 77 K or Ar at 87 K). In the H-K formulation, the ideal Henry’s law (linearity) is assumed for the isotherm, even though the actual isotherms invariably follow the typical type I behavior, which is well represented by the Langmuir isotherm. The H-K formulation is modified by including the nonlinearity of the isotherm. Inclusion of nonlinearity results in sharpening of the pore size distribution and shifting of its peak position to a smaller size. Furthermore, the H-K equation is extended to spherical pores, and the improved H-K equation for spherical pores by including isotherm nonlinearity is also given. It is shown that the spherical-pore model is particularly useful for zeolites with cavities. Using the literature isotherm data, the improved H-K equations for three pore geometries (slit shape, cylinder and sphere) are compared with the original H-K equations. Clear improvements are seen in the calculated micropore size distributions by using the improved H-K equations.

INTRODUCTION

The determination of micropore (i.e. less than - 20 A in width) size distribution remains the most important yet difficult problem in the characterization of porous materials. An experimentally convenient but accurate technique is not available. The need for such a technique has become increasingly urgent while we face a rapid development of new microporous materials for gas sorbents, catalysts and other applications. A direct experimental technique is to measure the saturated amounts adsorbed of probe molecules which have different dimensions. However, this technique is experimentally cumbersome, and there is uncertainty about the results due to the networking effects, e.g. a large pore accessible through small pores which are smaller than the probe molecule, thus the large pore is actually not “probed”. Another approach that has been under investigation for over thirty years is to calculate the pore size distribution (PSD) from a single adsorption isotherm (Jaroniec and Madey, 1988). Here the amount adsorbed is the sum of that in pores of all sizes. Thus, it is equal to an integral of the amount adsorbed in a specific pore size multiplied by a pore size distribution function, integrated over all sizes: W(P)

=

m w(L, P).!-(L) sB

dL

(1)

where w(L, P) is the amount adsorbed in pores of size L andf(L) is the pore size distribution function. To solve for the pore size distribution function is an ‘Author to whom correspondence should be addressed.

ill-posed problem unless the form of the function is defined. By assuming a mathematical function, and by relating the amount adsorbed with pore size through an empirical relationship (Stoeckli, 1977), it is possible to calculate the pore size distribution function [e.g. Jaroniec and Madey (1988); Yang and Baksh (1991); Baksh and Yang (1992)]. Although this procedure is experimentally convenient (because adsorption of supercritical gases can be used), it is unlikely to gain acceptability due to the assumptions and empiricism involved. Recently, more sophisticated models along this approach have been published (Seaton et al., 1989; Lastoskie et al., 1993). Individual-pore isotherms (which are dependent on pore size and geometry) are obtained using techniques grounded in statistical mechanics (e.g. Monte Carlo and density functional theory), are used in conjunction with pore size distribution functions with assumed mathematical forms. The advantage of these models are that they are not limited to micropores; they are suitable also for mesopores. The most convenient and desirable approach has been to measure the adsorption isotherm at a temperatre below or equal to the critical temperature, and to calculate the pore size distribution from the isotherm based on a theoretical framework. Nitrogen (at 77 K) has been used traditionally as the sorbate (Gregg and Sing, 1982), although Ar (at 77 or 87 K) is also used. As the pressure of the gas is increased, adsorption progresses by pore filling, starting from the smallest pores. For large pores, i.e. pores larger than approximately 20 A, capillary condensation takes place hence the Kelvin equation is applicable for calculating the corresponding pore sizes. For smaller

2599

2600

and RALPH T.

LINDASCHENG

pores where the pore size approaches a few molecular dimensions, however, the adsorbate molecules can no longer be treated as liquid, as the Kelvin equation does. Here the potential energy fields from neighboring surfaces overlap and the total interaction energy with the adsorbate molecule can be substantially enhanced (Everett and Powl, 1976; Kiselev and Du, 1981; Soto et al., 1981). For this reason, the N, isotherms in microporous materials exhibit the type-1 behavior, where the steep rise in adsorption occurs at a very low relative pressure (P/P,) range, typically between 10 6 and LO-*. Isotherm in this low relative pressure range contains a wealth of information on the micropore sizes, although it is not experimentally convenient to achieve such low relative pressures. A theoretical framework was developed by the Horvath and Kawazoe (1983) for calculating micropore size distribution from the steep-rise range of the isotherm. The theory is simple, but it catches the essence of the picture of progressive pore filling. The result, referred to here as the Horvath-Kawazoe (H-K) equation, provides a simple, one-to-one correspondence between the pore size and the relative pressure at which the pore is filled. For the above reasons, the H-K equation is being rapidly and widely accepted [e.g. Seifert and Emig (1987), Venero and Chiou (198X), Davis et a[. (1988, 19X9), Borghard et al. (1991) and Beck et al. (1992)] (and instruments based on the H-K equation are being commercialized). The H-K formulation was derived for slit-shaped pores, and an extension for cylindrical pores was made by Saito and Foley (1991). Baksh and Yang (1991) obtained an integral for the potential energy of adsorbate molecule in a spherical pore; however, an H-K equation for calculating pore size distribution for spherical pores was not given in their work. Since many simplifying assumptions are made in the H-K formulation, in this work we re-examine the H-K derivation and show a better solution with significant improvement while maintaining the simplicity in calculation. Furthermore, the H-K equation for spherical pores as well as the improved solutions for pores of all three shapes are given; the spherical-pore solution is particularly useful for zeolites containing sphericalshaped cavities where adsorption occurs.

YANG

lel lattice planes whose nuclei are at a distance L apart is

where

K = 3.07;

u = (2/5)“6d0

and z is the distance of the adsorbate molecule from a given atom in a surface layer. In the case of the pore being filled with adsorbate, the potential energy minimum, E*, is given by

where the dispersion constants may be calculated according to Kirkwood-Muller formalism by

A, = +(mc’a,x,). Substituting E(Z) =

NJ,

eq. (4) into eq. (2), one obtains

+ NAAA 2a4

x [

-(g-++(~>‘“-(~>4+(&)1°]. (7)

The average interaction

energy can be obtained

&(I5

-

NAAA

2do)

1

Review of slit-pore model For adsorption in ultramicroporous carbon, Everett and Pow1 (1976) developed a slit-like potential model by assuming the micropore to be slit between two graphitized carbon layer planes. The results are summarized below because they are the basis for the H-K equation. Using the Lennard-Jones 6-12 potential, El2

=

4ET2

[(+)I2

- ($>“I_

(2)

Everett and Pow1 (1976) showed that the potential energy between an adsorbate molecule and two paral-

by

Horvath and Kawazoe (1983) related the average potential energy to the free energy change upon adsorption, RTln(P/P,):

N, A, + THEORETICAL

(6)

.

(9)

From the amount adsorbed at P, eq. (9) yields the corresponding slit-pore width, L; thus the pore volume at pore size L is known.

Review of cylindrical pore model For cylindrical pores, Everett and Pow1 (1976) assumed that the pore wall comprises a single layer of solid atoms (e.g. oxide atoms in the case of zeolites). The potential energy of interaction I between the adsorbate molecule and the inside wall of cylinder is

Improved Horvath-Kawazce expressed

by

(10) where constants l/Z

ali and & are given by l-(-

_

ak

-

/$I

=

l-(-

4.5 1--

I-( -

4.5)

k)l-(k + 1) 1.5)

1.5 - k)l-(k

+ 1).

(11)

2601

equations

quently, only interactions between adsorbate and lattice oxygen atoms are considered (Barrer, 1978). The potential energy of the adsorbate molecule interacting with the atoms in the cavity wall can be calculated by assuming the wall consists of a single layer of atoms. The interactions between a single adsorbate molecule designated 1 and the inside wall of the zeolite cavity, consisting of a single lattice plane (SIP), are obtained by integrating the Lennard-Jones potential inside the spherica cavity:

(12)

The potential energy minimum E* is defined in eq. (4) r is the distance of the gas molecule from the central axis of the cylinder and L is the radius of the cylinder. Assuming that adsorption occurs only on the inside wall of the cylinder in micropore region, and approximating interactions upon adsorption by the interactions between the gas molecules and the oxide ions of the zeolite, Saito and Foley (1991) developed a cylindrical pore model for zeolites. The area average potential function inside the cylinder is defined by

where L is the cavity radius, z is the radial distance of the adsorbate molecule from the center of the cavity and N, is the number of oxygen atoms per unit area on the cavity surface. The integration has been shown by Walker (1966) and Soto et al. (1979) to yield

l-

2n L-do E(r)=

ss ctno s_i0

s(r)rdrdEJ L--cl0 rdr dfl 0

(13)

Using the formalism of Horvath and Kawazoe (1983), the cylindrical pore model is obtained by Saito and Foley (199 1):

N.4,

(16)

+ NAAA where Ni

(14)

Development

of spherical-pore

= 4nL=N,.

(17)

Using the same approach of Horvath and Kawazoe (1983), the potential expression after taking into account the adsorbate-adsorbate interaction results in the following form:

model

A particular feature of SA and faujasite-type molecular sieve zeolites is that the atoms and ions forming the walls of the cavities lie approximately in a spherical shell. Hence, zeolites of these types have been considered to be a collection of identical spherical cavities where adsorption occurs (Breck, 1974). Therefore, a spherical model of potential energy function would be more appropriate for adsorption in 5A and faujasite-type zeolites. The potential energy for the interaction of adsorbate with the zeolite lattice may in principle be estimated by the summation of all the interactions between the gas molecule and the lattice atoms. Due to the tetrahedral structure of alumina and silica groups, the aluminum and silicon atoms of the zeolite framework are obscured by oxygen atoms. Conse-

(18) where N2 = 4x(L

-

dO)2NA

(19)

2602

LINDA S. CHENG

and the minimum potential:

energies are obtained

E&

from the L-J

&f2 =potential

AHad”

=

as (Ross and Olivier, _

4 airr-RT+

The molar integral

A, 4d:’

change of entropy

s0 Integration

f’(z) =

AS=‘=’ + AS”b (22)

41cz2 dz

6(N,cf,

ASads = AP( The free energy change phase pressure:

results in the following:

AC”*” = RTln

+ N,&)L’

where

(2)

‘i;“j‘(, :ido)3

(24)

I

T2=(l / “;&j’-(, +hj

‘;“)‘-(, ; “I”)

(33) from gas-

(34) + A&,).

of Horvath

W/W,,) + AH”=‘.

Since the adsorbed phase is considered properties as the liquid phase,

1TAStr(W/WO)l

Therefore,

(36)

to have similar

e Iq““‘I

(37)

of qd”’ is

_ @rr = U0 + P, - AHYaP.

(26)

(35)

and Kawazoe

(y)(s).]

q din - TAP(

(25)

(32)

_

can be calculated

T(AS”(W/W,)

+ [RT-

and the definition

T3=(l

= AS 0

W/ IV,,) + AS0

As shown in the derivation (1983):

RTln

T1=(,

(31)

(P/P,)

= AHad” (23)

is

Except.for AS”, AS’“’ and ASrib are nearly constant upon adsorption, as in the H-K formulation:

1-(z)47rz2 dz

o L-,jo

(30)

(y)(gjO.

L-do

T(z) =

1964):

ASPdS = AS” + AS”’ + ASib,

(21)

inside the spherical cavity is

s

tion is expressed

=s A

4dg

The average

and RALPH T. YANG

the following

relation

(38)

is obtained:

(27)

(39) The H-K

equation

for spherical pores is then given

by

(28) where TIPT,

and (41)

are given above. Substituting

Improved H-K formulation An implicit but important assumption made in the H-K formulation is linearity of the isotherm. In what follows, we will present an improved H-K model by considering the nonlinearity and hence nonideality of the adsorption process. The molar integral change of free energy at a given temperature is given by AGad’ = AHad’ _ The molar

The above derivation is taken directly from Horvath and Kawazoe (1983). Horvath and Kawazoe (1983) further assumed the adsorbed phase to be two-dimensional ideal gas, i.e. Henry’s law region adsorpion. Hence the equation of state is

T ASad’,

integral change of enthalpy

(29) upon adsorp-

into eq. (39), the H-K

RTln

5 (

model is obtained:

= U. + P,, j

(42)

where We and P. are, respectively, adsorbate-adsorbent interactions and adsorbate-adsorbate interactions. The sum U. + P. is equal to the right-hand side of eq. (9) eq. (14) or eq. (28), depending on the pore geometry. As mentioned, the isotherms for adsorption in micropores under subcritical conditions (e.g. N, at

Improved Horvath-Kawazoe 77 K and Ar at 87 K) follow the typical type I behavior (Gregg and Sing, 1982). A representative example is shown in Fig. 1. The best mathof such isotherms ematical representation for type I isotherms is the Langmuir isotherm. The assumption of linearity as made by Horvath and Kawazoe (1983) is clearly inadequate. Although only the region of steep rise is important for determining the pore sizes, the use of Henry’s law would have missed the isotherm data in the concave portion of the steep rise. Consequently, we replace the Henry’s law [eq. (40)] by the Langmuir-type equation of state:

kT

Cylindrical RTln

2603

equations pore:

E O[

RT-

+

1

yin-

l-0

1 (47)

Spherical

pore: 1

RT-yin-

l--B

1

1

“=Blnl-@* Taking

the derivative,

one gets = yln&

(y)(g)@ By substituting

RTln

$ O[ 0

.

where

eq. (44) into eq. (39), we have

RI.-_ln----

+

i-e 1=uo+P.. 1

(45) By taking into consideration the correction for the nonlinearity of the isotherm, we have the improved H-K equations for three geometries: Slit-shaped

pore:

RTln

+

$ O[ 0

RT-_ln&

1 CT4 ai0 = NA+&AA N_,

u4(L -

-$+G 0

3(f. - c&,)~ - 9(L - &)g

2d,)

0

1_

(46)

1.01

I

0.6

0.4

03

0.0 o.OOOo 0.0002

0.0004

o.ooM

0.0008

>I

(44)

0.0010

Fig. 1. The region of the adsorption isotherm of Ar (87 K) on ZSM-5 zeolite useful for evaluation of micropore size distribution (Venero and Chiou, 1988).

T,-T,

ADSORmlON

(48)

are given by eqs (24)-(27). ISOTHERMS

AND

PHYSICAL

PARAMETERS

The adsorption isotherms used in this work were taken from the literature. According to the geometric characteristics of the microporous sorbents, isotherms on three groups of sorbents were included: N2 on carbon molecular sieve (with slit-shape pores), Ar on channel type zeolites (with cylindrical pores) and Ar on cavity-type zeolites (with spherical pores). To test the improved H-K equation for slit-shape pores, the isotherm data of Horvath and Kawazoe (1983) on N, adsorption at 77 K on carbon molecular sieve HGS-638 (made by Takeda Chemical Company) were employed. For the cylindrical-pore model, isotherms of Ar at 87 K on aluminosilicate ZSM-5 zeolite by Venero and Chiou (1988), on aluninophosphate AlPO,-5, AlPO,-11 and VPI-5 zeolites by Davis et al. (1989) were used. For spherical model, isotherms of Ar at 87 K on faujasite by Borghard et al. (1991), and that on zeolite 5A by Venero and Chiou (1988) were tested. The peak position of the curve has been shown to be pore size distribution affected by the choice of physical parameters for the cylindrical-pore model on zeolite type Y (Saito and Foley, 1991). In applying the H-K equation to calculate the pore size distributions in zeolites, some have chosen to use the parameters for carbon surface while treating certain other parameters in the gas-solid systems as fitting parameters (Borghard et al., 1991; Davis et al., 1988). The need for using fitting parameters also arose from the geometric problem in applying the slit-pore model to zeolites or the cylindrical-pore model to zeolites with cavities. In this work, physical parameters from the published literature were used: diameter of oxide ion (Broussand and Shoemaker, 1960), diameter of argon and polarizability of oxide ion (Taniguchi and Takaishi, 1986), polarizability and magnetic susceptibility of argon (Sams et al., 1960), and polarizability of oxide ion (Barrer, 1978). The densities per unit area for argon and oxide ion were estimated from argon

2604

LINDA S. CHENG

and

solid density and oxide ionic diameter for aluminosilicate (Saito and Foley, 1991). The oxygen diameter for aluminophosphate was given by Davis et al. (1991) as 2.6 8, and thus the estimated density was 1.48 x lOI molecules/cm’. The oxygen density inside the cavity walls of faujasite and SA zeolite, however, was taken from Barrer (1978). This value was lower than the value used by Saito and Foley (1991), due to the fact that there are six apertures in each cavity, and hence, when modeled as a continuous spherical cavity wall the density on the surface should be lower. Details are shown in Table 1. All physical parameters for the N,/carbon molecular sieve system were taken directly from Horvath and Kawazoe (1983). RESULTS

AND DISCUSSION

Analysis of the improved H-K equations The linear isotherm is assumed in the original H-K formulation. Nonlinearity of -the isotherm is considered in this work, resulting in the coverage (@dependent term: RT- (RT/B)ln[l/(l - 8)]. The meaning of this term and its effects on the pore size distribution are discussed next. The integral molar enthalpy change upon adsorption is expressed as (Young and Crowell, 1962): AHad’ = - q”’ + r(S’ - 6)

(49)

and (50) where S” is the molar entropy of adsorbate, and p is the differential molar entropy of adsorbate which can be obtained by:

where M is number of moles. The distinction between differential and integral entropies has been illustrated and discussed by Hill

RALPH T. YANG

(1952), using the BET theory and isotherms of various systems. The integral and differential entropies have different functional dependence on 0 from both statistical mechanics derivation and experimental data,. and as a result, S” - s’ varies with surface coverage or degree of filling in the case of micropore adsorption. In the H-K model, S’ - k is assumed to be equal to gas constant R, throughout the whole range of 8. However, such assumption is valid only for two-dimensional ideal gas or when 8 approaches zero. It can be estimated, assuming localized adsorption, that when 8 equals to 0.5, 1S’ - 6 1 will be 39% off the ideal case. Of course, the influence of the term RT - (RTjB) In [l/(1 - e)] and thus 8 also depends on P/PO where the adsorption occurs and the shape of the isotherm. In the initial part of the adsorption isotherm, 19is small and RT - (RT/O) In [l/(1 - e)] approaches zero, the H-K model gives a good approximation. As 0 is increased, the difference between S’ and ?? becomes significant especially in the region of upward turn in the isotherm, and so the pore filling term RT - (RT/B) In [l/( 1 - 0)] becomes more negative. In the meantime, as the relative pressure is increased, the free energy term R T In (P/P,) increases. The increase of the free energy term is partly offset by the pore filling term, RT - (RT/B) In [l/(1 - e)]. so the LHS of the H-K equation [eqs (46)-(48)] is increased at a slower rate as compared with the original H-K equation. Consequently, the calculated pore size is increased at a slower rate, resulting in sharpening of the pore size distribution. In addition, the peak of the distribution curve is shifted towards a smaller size as compared to the original H-K model. Adsorption isotherms Isotherms for microporous sorhents with pores that can be approximated by three shapes were collected from the literature and are shown in Figs 2-4. The isotherm of N,/carbon molecular sieve (77 K) is given in Fig. 2; that of Ar on four channel-type zeolites (at 87 K) are given in Figs 2 and 3; that of Ar on two

Table 1. Physical parametersfor micropore size distribution calculation Adsorbent Oxide Ion

Parameter Diameter, d, nm Polarizability, q cm3 Magnetic susceptibility, x, cm3 Density, N, molecule/cm2 Density (cavity)**, N, molecule/cm2 Density (cavity)t*, N, molecule/em2 ‘Brossard and Shoemaker (1960). ‘Davis et al. (1989).

*Taniguchi and Takaishi (1986). “Sam6 et al. (1960). “Saito and Foley (1991). TtFaujasite cavity. **Barrer (1978).

lP5A cavity.

Aluminosilicate 0.276’ 2.5 x 10-z+* 1.3 x 10-297 1.31 x 10’57 8.48 x 10’4** 8.73 x 1Ol4 **

Aluminophosphate 0.260’ 2.5 x lo-‘+ 1.3 x 10-29 1.48 x 10’5

Adsorbate Argon 0.336 1.63 x lO-z*” 3.24 x lo- 29” 8.52 x la’47

Improved HorvathLKawazoe

2605

equations

0.6

0.6

O-4

0.2

0.0

IO”

10.’ 10”

10-S 10”

1o-3 10.’ 10-l

P/PO Fig. 2. Adsorption isotherms of nitrogen on carbon molecular sieve HGS 638 at 77 K ( O)(Horvath and Kawazoe, 1983) and of argon on ZSM-5 zeolite at 87 K ( rl): (Venero and Chiou, 1988)

Fig. 4. Adsorption isotherms of argon on SA (Venero and Chiou, 1988) and faujasite zeolites (Borghard et al., 1991) at 87 K: (0) 5A zeolite; (0) faujasite.

Comparisons

of

the H-K

and the improved

H-K

equations

0.8

D

0.6

$0.4

0.2

0.0

10” 10” 10” 1o.5 10’

1o-3 1o-Z 10-l

ppo Fig. 3. Adsorption isotherms of argon on AIPO,-5, AIPOa11 and VPI-S akminophosphates at 87 K (Davis et al., 1988); (0) AIPO,- 11; (0) AlPO,-5; (0) VPI-5.

cavity-type zeolites (at 87 K) are shown in Fig. 4. N,/zeolite data were not used to avoid the strong quadrupolar interactions, which are not considered in the H-K formulation. All isotherms, as plotted in the semi-log forms, exhibit the sigmoidal behavior. However, if plotted in the linear fashion, as done in Fig. 1, they can be well represented by the Langmuir isotherm. Thus, the Langmuir isotherm, as used in this work, is a good approximation. Other isotherm forms can also be expressions of the pore filling term, Used; RT - (T/3/0) (NI/aT),, for some isotherms are given by Ross and Olivier (1964). CES

49: 16-E

The isotherms provide the W(P) relationships. The H-K equations relate Pore size, L, with pressure. A one-to-one correspondence between the amount adsorbed, W, and the pore size is then obtained. The pore size distribution function is taken as d W/dL (or d W/dr), as a function of L. We shall then compare the pore size distributions calculated from the original H-K equations, eqs (9), (14) and (28), and that calculated from the improved H-K equations, eqs (46), (47) and (48), using the same parameters for each system. The slit-pore models are compared for two cases: N,/carbon molecular sieve and Ar/faujasite, shown in Figs 5 and 6, respectively. The pores in carbon may be considered as slit shaped. Faujasite contains cavities of nearly spherical shape, in which adsorbate molecules reside. However, the Ar/faujasite data were treated by using the original H-K slit-pore model (Borghard et al., 1991). Figure 6 illustrates the differences between the H-K model and the improved H-K model, both assuming slit-shaped pores. In both Figs 5 and 6, sharpening and shifting towards smaller sizes in the pore size distribution curves are seen as caused by considering the adsorption nonidealing, as done in the improved H-K model. The effects are particularly strong for the faujasite (Fig. 6), where the improved model yields a very narrow distribution. This is the correct picture because of the crystalline structure of faujasite which actually has a uniform cavity size. The actual cavity size, given in Table 2, is considerably larger than the values shown in Fig. 6. This error is caused by using the slit-pore model for the spherical pores and the spherical-pore model results will be given shortly. For the carbon molecular sieve (Fig. 5), the effect of the improved model is small, because of the narrow pores as well as the slow rise in adsorption

LINDASCHENG

and RALPHT.YANG

0.8

0.4 0.5 0.6 0.7 Effective Pore Size (MI)

Fig. 5. Micropore size distributions of carbon molecular sieve HGS 638 calculated from slit-pore models: ( l ) improved model; (0 ) H-K model.

with pressure. The carbon sieve does have a wide pore size distribution, which is properly reflected in Fig. 5. The pore size distribution of the HGS-638 carbon was also measured directly by molecusieve lar probing (Horvath and Kawazoe, 1983). The probed PSD was indeed both sharper and smaller than that calculated by the H-K equation. So the improved model does seem to give better results. The structures of the aluminosilicate zeolite ZSM-5 and the aluminosphosphate AlPOd-5, AlPOd-1 1 and VPI-S consist of channels (ZSM-S with three-dimensional channels and the aluminophosphates with onedimensional channels). Since adsorption takes place inside these channels, cylindrical-pore models are used for the comparison. The PSD results using eqs (14) and (47) are displayed and compared in Figs 7-10. In all cases, the PSD curves are substantially sharpened, consistent with the crystalline nature of the zeolites. The peak position of the PSD, however, is known to be sensitive to the physical parameters used

in the calculation [e.g. Saito and Foley (1991)]. For the two zeolites with elliptical-shaped pores, ZSM-S (Fig. 7) and AlPOd- I1 (Fig. 9), the peak positions calculated by the improved H-K equation are closer to the values from the crystallographic data (Table 2). For AIP04-5 (Fig. 8) and VPI-5 (Fig. lo), however, the peak positions from the original H-K equation seem to be closer. No adjustable physical parameters are used in this work. The most uncertainty in the physical parameters (listed in Table 1) is the magnetic susceptibility of the oxide ion on the zeolite surface. The value of 1.3 x lO_” cm3 taken from Saito and Foley (1991) is used here. The values given in the earlier literature are higher: 1.66 x 10dz9 cm3 from Kiselev and Du (1981)

40

6 3 P * x 20

n -0.6

and 2.09 x 10mz9 cm3 from Barrer (1978). If these higher values were used in our calculations, the peak positions would have shifted substantially toward 0.8 0.7 0.9 Effective Pore Size (urn)

Fig. 6. Micropore size distributions of faujasite zeolite calculated from slit pore models: ( l ) improved model; (0) H-K model.

Table 2. Molecular sieve crystallographic and/or cavity size Molecular sieve ZSM-5 AlPO,-5 AIPOL- 11 VPI-5 Ca-A Faujasite

Crystallographic channel size (nm) 0.56 x 0.53’ 0.73’ 0.63 x 0.39’ - 1.20’ - 0.49’ 0.14+

‘M&r and Olson (1987). *Davis et al. (1989). ‘Dyer (1988). “Breck (1983). qUppal er ai. (1988).

channel

Cavity size (nm) l.lQ” 1.377

0.6 OS 0.7 Effective Pore Size (mu)

0.8

Fig. 7. Micropore size distributions of ZMS-S zeolite calculated from cylindrical pore models: ( l ) improved model; (0) H-K model.

improved

2607

Horvath-Kawazoe equations

30

10

8

a ‘a z 3 a’4

0

0.5

0.6 0.7 Effective Pore Size (nm)

0.8

Fig. 8. Micropore size distributions of AlPO,-5 calculated from cylindrical pore models: (+) improved model; (0) H-K model.

6

0.6

0.8 1.0 1.2 Effective Pore Size (mn)

Fig. 10. Micropore size distributions of VPI-5 calculated from cylindrical nore models: f*) imoroved model:., f 0)I H-K mod& .

40 r

30

20

* ‘a 3 2o 3 z

3 a 2 3 3 10

0

0.4

0.5 Effective

0.6

0.7

0.8

Pore size (rim)

Fig. 9. Micropore size distributions of AlPO,-1 1 calculated from cylindrical pore models: ( l) improved model; ( 0) H-K model.

pore size values, and consequently the values from the improved H-K equation would be closer to the real values in all cases. In short, the goodness of the improved model cannot be judged from the peak positions because of the sensitivity of the position on the physical parameters and the uncertainty of these parameters. Figure 10 shows two peaks for VPI-5, a major one at 8.6 A and a minor one at 11.7 A. These peaks arise from the two inflection points in the isotherm. The free diameter for VPI-5 should be 12.1 A (Davis er al., 1988). However, based on the Ar isotherm and their analysis employing the original slit-pore model of H-K, the pore size was only 10.5 A. The discrepancy was attributed to the Ar packing arrangements. Using the O2 isotherm, the calculated major peak was also

1.1 1.3 1.5 Effective Pore Size (nm) Fig. Il. Micropore size distribution of faujasite zeolite calculated from spherical Pores models: (+) improved model; ( 0) H-K model.

higher

10.5 A, with a small shoulder at 12 A. A further study was conducted by Hathaway and Davis (1990) on VPIJ as well as other zeolites. No definitive conclusion seems to have been drawn as to which peak (in Fig. LO)represents the pore dimension in VPI-5. For type A and faujasite (X and Y) zeolites, the sorbate molecules reside within the cavities. For this reason, the spherical-pore model should be used. The PSDs calculated from the improved H-K equation [eq. (48)] and the original H-K equation [eq. (2X)] using the Ar isotherm data are compared in Fig. 11 for faujasite and in Fig. 12 for 5A. In both cases, the sharpening is substantial. Figure 11 shows that not only the shape but also the peak position fcr the PSD in faujasite are correctly predicted by the improved

2608

LINDA S. CHENG and RALPH T. YANG

20

entropy of adsorption molar entropy of adsorbate differential molar entropy of adsorbate absolute temperature adsorbate-adsorbent interactions amount adsorbed saturated amount adsorbed distance of adsorbate molecule from a surface atom in the slit layer for slit-pore model or from the cavity center for spherical-pore model

1

3 ‘a 3 IO 5 x

d Effective Pore Size (mn) Fig. 12. Micropore size distributions of 5A zeolite calculated from spherical pore models: (+) improved model; (0) H-K model.

spherical-pore model. It is also interesting to compare the results if the slit-pore model is used, shown in Fig. 6. The slit-pore model underestimates the pore size by about 5 A. This comparison illustrates the importance of selecting the model with the correct pore geometry.

Greek letters polarizability u uk

;k IE

e

l-I cr 4 x

constant, defined in eq. (10) constant constant, defined in eq. (10) potential energy of interaction in zeolite cavity potential energy of interaction degree of void filling or angle in spherical coordinate system, 0 $ B < rc, in eq. (15) spreading pressure distance from an atom in the surface layer at zero interaction energy angle in spherical coordinate system, OG4ld2n magnetic susceptibility

Subscripts Acknowledgements-This work under Grant CTS-9212279.

was supported

by NSF

NOTATION

dispersion constant speed of light diameter of atom arithmetic mean of diameters of adsorbate and adsorbent atoms free energy enthalpy of adsorption enthalpy of vaporization distance between nuclei of the parallel layers for slit-shaped pores, likewise defined as the radius (not diameter) for cylindrical and spherical pores density per unit area Avogadro’s number number of oxygen atoms on the wall surface of cavity number of argon atoms inside a cavity pressure adsorbate-adsorbate interactions saturate vapor pressure of adsorbate differential heat of adsorption isosteric heat of adsorption distance between gas molecule and cylinder’s central axis (cylindrical model) or gas molecule and oxygen atom in the spherical wall (spherical model) gas constant

1 2 :

adsorbent adsorbate adsorbent adsorbate

atom molecule

(or atom)

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