A comprehensive theoretical description of physical adsorption of

Aug 17, 1988 - of the mesopores. A gamma distribution is selected for representing the structural heterogeneity of the micropores. This distribution g...
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J . Phys. Chem. 1989, 93, 5225-5230

5225

A Comprehensive Theoretical Description of Physical Adsorption of Vapors on Heterogeneous Microporous Solids M. Jaroniect and R. Madey* Department of Physics, Kent State University, Kent, Ohio 44242 (Received: August 17, 1988; In Final Form: February I , 1989)

A comprehensive theoretical description of physical adsorption of a vapor on a heterogeneous microporous solid is presented

in terms of the distribution functions that characterize the structural heterogeneity of the micropores and the surface heterogeneity of the mesopores. A gamma distribution is selected for representing the structural heterogeneity of the micropores. This distribution generates simple equations for the characteristic adsorption curve and other thermodynamic functions. The asmethod is employed to extract the amounts adsorbed in the micropores and on the mesopore surface from the total amount adsorbed,

Introduction The Dubinin-Radushkevich (DR) isotherm equation' occupies a central position in the theory of vapor adsorption on microporous solids. This equation was proposed by Dubinin and Radushkevichl on the basis of the Polanyi potential theory of adsorptiom2 Experimental s t ~ d i e s proved ~,~ its utility for describing the mechanism of micropore filling and for characterizing microporous structures of solids, especially activated carbons. Usually the DR equation is presented as follow^:^^^

= exP[-Bo(A/p)21 = exP[-(A/mo)21

(1)

where A = -AG = R T In ( p o / p )

(2) Here 0 is the fraction of the micropore volume filled with an adsorbate at equilibrium pressure p and absolute temperature T, A is the adsorption potential, defined by eq 2 as the change in the Gibbs free energy AG taken with the minus sign, po is the saturation vapor pressure, Bo is a structural parameter that characterizes the microporous structure of a solid, Eo = 1/Bo'/2 is the so-called characteristic energy of adsorption for a reference vapor (which is usually taken to be benzene3 for activated carbons), and @ is the adsorbate dependent affinity or similarity coefficient defined as the ratio of the characteristic adsorption energies of the test and reference vapors, (Le., /3 = E / E o = (Bo/B)'/2,where E is the characteristic adsorption energy and B is the structural parameter for the vapor-solid system studied). Experimental studies5s6 showed that the DR eq 1 represents vapor adsorption in uniform (homogeneous) micropores. To describe vapor adsorption on heterogeneous microporous activated carbons, Stoeckli7 proposed the following integral equation

where el is the total fraction of the micropore volume filled with an adsorbate for a heterogeneous microporous solid and F(Bo) is the distribution function (normalized to unity) of the structural , ~ - ~ eq 3 by assuming parameter Bo. Stieckli7 and D ~ b i n i n ~studied a Gaussian distribution for representing the function F(Bo). It was shown elsewhere'OJ' that this distribution does not satisfy all physical requirements for microporous solids. Jaroniec et al.I29I3 studied the integral eq 3 for a gamma-type distribution function F(Bo)and obtained a simple isotherm equation, which gives a good representation for many adsorption isotherms on microporous solids. A disadvantage of this theoretical description is that the distribution function F(Bo) in the integral eq 3 gives no direct information about the structural heterogeneity of micropores. To overcome this difficulty, Dubinin and Stoeckli9 and Jaroniec et al.'3q14utilized an empirical relationship between the structural 'Permanent address: Institute of Chemistry, M. Curie-Sklodowska University, 2003 I Lublin, Poland.

0022-3654/89/2093-5225$0 1.50/0

parameter Bo and the half-width x of the slitlike micro pore^^-^^ to convert the distribution function F(Bo)to the micropore-size .'~ the distribution function J ( x ) ; however, D ~ b i n i n ' ~represented micropore-size distribution J ( x ) by a Gaussian distribution, which does not satisfy all physical requirements18 in the same manner as the Gaussian micropore-size distribution F(Bo). In another paper,Ig Dubinin and Kadlec proposed the distribution function of the characteristic adsorption energy Eo for characterizing the heterogeneity of microporous solids. A disadvantage of using the quantities Bo and Eo in a theoretical description is their imprecise definition; in addition, the relationship between Bo and x has an empirical character, and its applicability is limited only to carbonaceous adsorbents with slitlike micropores with limited lateral dimensions.6.20 Even though the theoretical treatments discussed above are useful for characterizing microporous solids in many practical cases,18 there are some applications where the above-mentioned theoretical deficiencies do have a significant effect. In this paper, we will demonstrate how to overcome the above-mentioned theoretical deficiencies. For this purpose, we will use another form of the DR eq 1 expressed in terms of a well-defined average adsorption potential A,19 defined below. It will be shown that the introduction of the distribution function G( I/]) permits a formulation of a comprehensive and general theoretical description of physical adsorption of vapors on heterogeneous microporous solids.

General Considerations The Dubinin-Radushkevich (DR) equation' may be written in a slightly different formZo ( I ) Dubinin, M . M.; Radushkevich, L. V . Dokl. Akad. Nauk SSSR 1947, 55, 331. (2) Polanyi, M. 2. Elektrochem. 1920, 26, 371; 1929, 35, 431. (3) Dubinin, M. M. Prog. Surf. Membr. Sci. 1975, 9, 1. (4) Dubinin, M. M. Chem. Phys. Carbon 1966, 2, 55. (5) Izotova, T. I.; Dubinin, M. M. Zh. Fiz. Khim. 1965, 39, 2796. (6) Dubinin, M . M. In Characterization of Porous Solids; Greeg, S . J., Sing, K. S. W., Stoeckli, H. F., Eds.; Society of Chemical Industry; London, 1979; pp. 1-11. (7) Stoeckli, H. F. J . Colloid Interface Sci. 1977, 59, 184. (8) Dubinin, M . M. Carbon 1979, 17, 505; 1981, 19, 321. (9) Dubinin, M. M.; Stoeckli, H. F. J. Colloid Interface Sci. 1980, 75, 34. ( I O ) Rozwadowski, M.; Wojsz, R. Carbon 1984, 22, 363. ( 1 1) Choma, J.; Jankowska, M.; Piotrowska, J.; Jaroniec, M . Monarsh. Chem. 1987, 118, 315. (1 2) Jaroniec, M . ; Piotrowska, J. Monatsh. Chem. 1986, I 1 7, 7. (13) Jaroniec, M . ; Choma, J. Mater. Chem. Phys. 1986, 15, 521. (14) Jaroniec, M.; Madey, R.; Lu, X.; Choma, J. Langmuir 1988.4, 91 I . (15) Stoeckli, H . F. Chimia 1974, 28, 727. (16) Dubinin. M. M . Carbon 1985, 23, 373. (17) Dubinin, M . M. Carbon 1987, 25, 593. (18) Jaroniec, M.; Lu, X . ; Madey, R. Monatsh. Chem. 1988, 119, 889. (19) Dubinin, M. M . ; Kadlec, 0. Carbon 1987, 25, 321 (20) Jaroniec, M.; Madey, R. Carbon 1988, 26, 107.

0 1989 American Chemical Society

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The Journal of Physical Chemistry, Vol. 93, No. 13, 1989

enthalpy A&"', which is associated with the DR equation. It is easy to that this equation may be obtained from eq 9 by expressing A by eq 5 . The adsorption potential distribution X ( A ) appears also in the expressions that define the differential molar enthalpy AH and the differential molar entropy AS for a vapor adsorbed on a microporous solid:28 AH = -A - a W A ) I X ( A ) (10)

(4)

where the average adsorption potential /1 is associated with the parameter E, as follows:

The expression 1 - B(A) denotes the fractional unoccupied micropore volume. This unoccupied volume is associated with adsorption potentials smaller than A. The expression 1 - o ( ~ represents an integral adsorption potential distribution X*(A):

The differential adsorption potential distribution X(A) associated with the integral distribution X*(A) is

Note that both the DR eq 1 and its modification (viz., eq 4),which describe the volume filling of micropores, are based on the Polanyi potential theory of adsorption.* Because the fraction of the micropore volume 0 filled with an adsorbate is a function of the adsorption potential A, the differential adsorption potential distribution X(A) may be obtained directly from the integral distribution X*(A) = 1 - B(A), which represents the unoccupied volume of micropores with adsorption potentials smaller than A; for example, eq 7 describes the function X(A) associated with the DR eq 4. Although Cerofolini2' used the condensation-approximation (CA) method to obtain an equation for X ( A ) analogous to eq 7 , we will show that the applicability of this method to the DR isotherm is questionable. The essence of the CA method is to find a relationship between the adsorption energy and the equilibrium pressure. This relationship is obtained on the basis of the local isotherm such as the Langmuir isotherm that represents adsorption on an energetically homogeneous surface;22 consequently, the CA method may be applied to the overall isothems obtained by integrating the local isotherm with an energy distribution function. It was shown e l ~ e w h e r e that ~ ~ , the ~ ~ overall and local adsorption isotherms should show the same limiting properties. Because all known isotherm equations used to represent local monolayer adsorption approximate unity at pressures tending to infinity, the overall isotherms generated by these local isotherms also should satisfy this condition. Note that the isotherm equations obtained on the basis of the potential theory of adsorption reach unity at a finite value of the equilibrium pressure and do not satisfy the above-mentioned condition; for example, the fact the DR isotherm reaches unity at p = p s means that application of the CA method to the DR equation is not justified fully. The average adsorption potential d characterizes the adsorption potential distribution X ( A ) given by eq 7 ; it is defined as follows:

d = &=AX(A) dA with &-X(A) dA = 1

(8)

It was shown previously25 that the immersion enthalpy AMmof a microporous solid in a liquid is related to A by the following equation A H m = -(I

+ aT)d

(9)

where a is the negative of the thermal coefficient of the natural logarithm of the maximum amount adsorbed in the micropores.26 Stoeckli and Kraehenbueh12' derived an equation for the immersion ~~~

~

(21) Cerofolini, C. F. Surf. Sci. 1971, 24, 391. (22) Cerofolini, G. F. Thin Solid Films 1974, 23, 306. (23) Jaroniec. M.; Marczweski, A. W. Monatsh. Chem. 1984, 115, 997. (24) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, 1988. (25) Jaroniec, M.; Madey. R. J. Phys. Chem. 1988, 92, 3986. (26) Bering. B. P.; Dubinin, M . M.: Serpinsky, V. V . J . Colloid Interface Sci. 1966, 21, 378.

Jaroniec and Madey

AS = -aB(A)/X(A)

)

Equations I O and 11, which are special cases of more general expressions,28were obtained by assuming temperature invariance of the B(A)-function. Temperature invariance of O(A),which is usually called the characteristic adsorption curve, is observed frequently for adsorption on microporous solid^.^*^ It is shown elsewhere2*that expression of B(A) and X ( A ) by equations resulting from the DR adsorption isotherm leads to the equations derived earlier by Bering et al.26329 The above discussion shows that the adsorption potential distribution X(A) and the average adsorption potential A have clear physical meanings and are related in a simple way to the thermodynamic functions AH, AS,and A P m . These quantities will be used to formulate a theory of physical adsorption of vapors on heterogeneous microporous solids. This theory provides theoretical foundations for the isotherm equation associated with the gamma micropore-size distribution; the above isotherm was found to be useful for representing experimental isotherms of adsorption on heterogeneous microporous s ~ l i d s . ' ~ J ~

Gamma Distribution Function Experimental and theoretical ~ t u d i e s ~ showed ~ ~ J ~ Jthat ~ the DR equation represents adsorption in uniform (homogeneous) micropores. The inverse value of the average adsorption potential d may be used to characterize uniform micropores; for slitlike micropores of relatively large dimensions, this value is proportional to the half-width x of the micropores. Dubinin and Kadlec19 used a Gaussian distribution of the inverse value of the so-called characteristic adsorption energy for describing the structural heterogeneity of microporous solids. The Gaussian distribution is defined in the interval from minus infinity to plus infinity. Since the adsorption isotherm is obtained by integration from zero to plus infinity, the part of the Gaussian distribution from minus infinity to zero is omitted; but then this distribution should be renormalized to unity.l0 Also, because the Gaussian distribution has a nonzero value at the argument equal to zero, it is difficult to interpret this value for a Gaussian distribution of micropore sizes. Also, the Gaussian distribution generates an equation for the adsorption isotherm that contains an error function of the adsorption potential A, which is inconvenient for extracting parameters from the adsorption isotherm. To avoid these problems, we propose a gamma distribution of the inverse value of the average adsorption potential d for characterizing the structural heterogeneity of microporous solids. Initial s t ~ d i e s ~ showed ~,'~-~~ that gamma-type distribution functions satisfy all physical requirements and lead to simple equations for the adsorption isotherm and the thermodynamic functions. Let z denote the inverse value of the average adsorption potential A , i.e., z = I/A (12) The differential gamma distribution G(z) may be written as follows 2qvi2 G(z) = -rzI' ( u / 2 )

exp(-qz2)

with &-G(z) dz = 1

(13)

where q and u - 1 are parameters greater than zero and r(u/2) denotes the gamma function, which is tabulated in mathematical handbooks.30 For z = 0, the value of G(z) is zero. The G(z) (27) Stoeckli, H. F.; Kraehenbuehl, F. Carbon 1981, 19, 353; 1984, 22, 291.

(28) Jaroniec, M . Langmuir 1987, 3 , 795. (29) Bering, B.P.; Dubinin, M . M.; Serpinsky, V. V. J . Colloid Interface Sci. 1972. 38, 185.

Adsorption of Vapors on Microporous Solids

The Journal of Physical Chemistry, Vol. 93, No. 13, 1989 5221

distribution is a maximum a t the point

Integration of eq 22 gives the following simple equation for B,(A):

The average value f for the differential gamma distribution G(z) given by eq 13 is

Note that one of the main difficulties in studying adsorption on heterogeneous solids is the ill-posed nature of the integral equations used to represent the overall adsorption isotherm 0,.24v3’332 Numerical inversion of the integral eq 21 with respect to the distribution function C ( z ) is especially difficult.32 The isotherm eq 23, which was obtained for the gamma distribution function G(z) (viz., eq 13), is a special analytical solution because it satisfies physical requirements. An advantage of the isotherm eq 23 is its simplicity. An equation analogous to eq 20 was obtained by Jaroniec et al.12s13for the gamma distribution of the DR structural parameter. It was shown13 that this equation gives a good representation of the vapor distribution isotherms measured on microporous activated carbons. Defining O,(A) as the ratio of the amount a, adsorbed on an heterogeneous solid to the maximum amount a: adsorbed in the micropores, Le.,

The dispersion uz for the above distribution is equal to

The quantities f and uz characterize the distribution function G ( z ) . The integral distribution G * ( z ) associated with the differential distribution C ( z ) is defined as follows

O,(A) = at/a,o

we can present eq 23 in the following logarithmic form:

In a, = In a: - (u/2) In [ I where G*(z) denotes the fraction of the micropores characterized by the inverse value of the average adsorption potential d between z = 0 and z = I/d. Equations 13 and 17 give

(24)

+ (7r/4q)A2]

(25)

For a large value of q (which corresponds to small heterogeneity of the microporous structure) and for relatively small values of A, eq 25 may be approximated as follows 7r7r u In a, = In a? - BA2 with B = - z 2 = - 4 4 29

where 7denotes the second moment of z for the distribution G(z). The linear eq 26 is analogous to the DR eq I . where the incomplete gamma function r ( u / 2 , q x 2 ) is defined as follows:30

It is easy to see that the differential distribution G(z) [eq 131 is generated by the incomplete gamma function given by eq 19

Adsorption Potential Distribution In this section, we discuss the adsorption potential distribution X , ( A ) associated with the characteristic adsorption curve B,(A) given by eq 21. The integral adsorption potential distribution & * ( A ) for an heterogeneous microporous solid characterized by the G ( z ) distribution is given by an equation analogous to eq 6 :

For the differential adsorption potential distribution X,(A), eq 27 gives &(A) =

Later, for the sake of brevity, we will omit the word “differential” when referring to the differential distributions X ( A ) and G ( z ) .

Integral Equation for the Characteristic Adsorption Curve For an heterogeneous microporous solid characterized by the distribution G ( z ) ,the overall characteristic adsorption curve B,(A) may be represented by the following integral equation

/a

where z = 1 and the subscript t is used to denote quantities for a heterogeneous microporous solid. An equation analogous to the integral eq 21 was proposed by Dubinin.I6 It is noteworthy that eq 21 is a transformed form of the integral eq 3 introduced by Stoeckli,’ which was studied later by other author^.^.'^^'^,^^ Substitution of the gamma distribution G ( z ) eq 13 into the integral eq 21 gives

U,*(A) ~

dA

-

:A L m z 2exp( - i A 2 z 2 ) G ( z )dz ( 2 8 )

The integral in eq 28 contains the adsorption potential distribution X ( A ) given by eq 7 . The average value of A for the distribution X,(A) [eq 281 is defined by33

For the gamma distribution C ( z ) [eq 131, the integral eq 29 gives

The dispersion

( 3 0 ) Handbook of Tables f o r Probability and Statistics, 2nd ed.; Beyer, W . H., Ed.; CRC Press: Boca Raton. FI, 1986; p 635.

d4(4 = -dA

uA,,for

the X,(A) distribution given by eq 28 is

(31) McEnaney, 9.; Mays, T. J.; Causton, P. Langmuir 1987, 3. 695. (32) Jaroniec, M.; Brauer, P. Surf.Sci. Rep. 1986, 6 , 65. (33) Jaroniec, M.; Madey, R. J . Chem. Sac., Faraday Trans. I 1988,84, 1139.

5228 r A , ,=

The Journal of Physical Chemistry, Vol. 93, No. 13, 1989

[J - ( A

Jaroniec and Madey amount a, adsorbed in the micropores and the amount a,,, adsorbed on the mesopore surface, i.e.,

- d , ) 2 X , ( A )dA]1'2 =

= at + (44) Here the subscript T is used to distinguish quantities for a solid with micropores and mesopores, and the subscript t refers to the quantities defined either for heterogeneous micropores or for the heterogeneous surface of the mesopores. Equation 44 may be rewritten as follows

Equations 13, 30, and 31 give

aT,t

Equation 32 defines the dispersion rA,,for the adsorption potential distribution X,(A) [eq 281 associated with the gamma distribution G ( z ) [eq 131. An analytical form of this adsorption potential distribution X , ( A ) is

+

X , ( A ) = ( i r / 4 ) ~ q ' / ~ A [ q ( T / ~ ) A ~ ] - ( ' + " / ~ ) (33)

The adsorption potential distribution &(A) given by eq 33 is associated with the characteristic adsorption curve 8,(A) [eq 231; this distribution may be obtained either by differentiation of eq 23 with respect to A [see eq 281 or by calculating the integral in eq 28 for the gamma distribution G ( z ) given by eq 13. The X , ( A ) distribution given by eq 33 is a maximum a t the point: (34) The quantities A,, A,,,,,, and bution.

rA,tcharacterize

the & ( A ) distri-

Thermodynamic Functions for Adsorption on Heterogeneous Microporous Solids Thermodynamic functions for adsorption on heterogeneous microporous solids are defined by equations analogous to those presented in the introduction AH," = -(I

+ CYT)A,

(35)

AH, = -A - aTO,(A)/X,(A)

(36)

AS, = CY^,( A) / X , ( A )

(37)

AH,lm = -yq1i2(1

+ UT)

AH, = -A - (aT/uA)(4q/x AS, = -( a/uA)(4q/*

+ A2)

+ A*)

(38) (39)

A = 2(4/*)1'2(

I - e12/~)1/*/811/~

(41)

Expressing in eq 39 and 40 the adsorption potential A by eq 41, we obtain

(43) Equations 42 and 43 permit calculation of the differential functions AH, and AS, against 8,

Microporous Solids with a Large Surface Area of the Mesopores The equations discussed in the previous sections are valid for microporous solids where the amount adsorbed on the mesopore surface is negligible in comparison to the amount adsorbed in the micropores. According to Dubinin,I6 the total amount aT,,adsorbed on an heterogeneous microporous solid is the sum of the

(45)

A, = ii,/Li: (46) For the same valugs of the relative pressures p / p o , the surface coverages 8,,, and 8, are equal because we assumed that physicochemical nature of the mesopore surface of the solid studied and the surface of the reference adsorbent are identical. The above assumption permits presentation of eq 45 in the following form: (47) aT,, = a, + a,,:& = a,%, + a,,:& W e can present eq 47 in terms of the ( r , - m e t h ~ d ; according ~~-~~ to this method, the quantity a, is defined as the ratio of the amount ii, adsorbed on the surface of the reference adsorbent to the amount iio4 adsorbed on this same adsorbent a t the value of the relative pressure p / p o = 0.4, Le., CY, = iit/ii04 (48) Equations 46, 47, and 48 give as,:20.4

= a,O8, + a, 2,o

(49)

In the case of a microporous solid, the micropores are filled a t relatively small values of p / p o ; and for large values of p / p o , adsorption occurs on the mesopore surface. Assuming such a stepwise mechanism of adsorption, we have 8, = 1 for large values of p / p o ; then eq 49 assumes a linear form with respect to the reduced standard adsorption a,:37

(40)

The thermodynamic functions AH, and AS, may be expressed in terms of 8,. Equation 23 gives

+ as,?Os,i

Here at0 denotes the maximum amounb adsorbed in the micropores, and a,,: denotes the monolayer capacity of the mesopore surface. T o evaluate the amount as,, adsorbed on the mesopore surface, DubininI6 propose! the use of a nonporous reference adsorbent as a standard. Let 8, denote the relative surface coverage for the standard isotherm, defined as the ratio of the amount 2, adsorbed on the surface of the reference adsorbent to the monolayer capacity ,:i i.e.

UT,,

where the subscript t distinguishes quantities for an heterogeneous microporous solid. Expressing, respectively, A,, O,(A),and X , ( A ) by eq 30, 23, and 33, we obtain

=

where

as.10ii0.4

(50) = a,O + ii,o For large values of p / p o (up to about 0.8), it is noteworthy that a multilayer is formed after the monolayer fills the mesopore surface; in this pressure region, the dependence aT,,vs a, is linear even for a reference nonporous adsorbent with a surface of a slightly different nature than that of the solid studied because multilayer formation on a nonporous solid is almost independent of its surface proper tie^.^^ If capillary condensation in the mesopores occurs a t high relative pressures, then the a,-plot deviates from a straight line.38 Because the values of i: and iio,4 are calculated from the standard adsorption isotherm, eq 50 permits evaluation of the monolayer capacity a,,: for the mesopore surface of the adsorbent studied, and calculation of the maximum amount a: adsorbed UT,^

(34) Sing, K. S. W . In Surface Area Determination; Everett, D. H.. Butterworths: London, 1970; p 25. (35) Carrot, P. J. M.; Roberts, R. A,; Sing, K. S . W. Carbon 1987, 25, 59. (36) Roberts, R. A.; Sing, K. S . W . Langmuir 1987, 3 , 331. ( 3 7 ) Jaroniec, M.; Madey, R.; Choma, J.; McEnaney, B.; Mays, T. Carbon 1989, 27, 77.

(38) Gregg, S . J.; Sing, K . S . W. Adsorption, Surface Area and Porosity, 2nd ed.; Academic Press: Yew York, 1986.

The Journal of Physical Chemistry, Vol. 93, No. 13, 1989 5229

Adsorption of Vapors on Microporous Solids

in the micropores (viz., the micropore capacity). Based on the assumption that the physicochemical properties of the mesopore surface of the solid studied and the surface of the reference nonporous adsorbent are identical, we can extract from the total amount adsorbed aT,,[eq 471 either the relative amount 8, adsorbed in the micropores or the absolute amount a, adsorbed in these micropores. In the first case, eq 47 gives Note that the relative amount 8, adsorbed in the micropores was calculated with respect to a? evaluated according to the asmethod [eq 501. The extracted data 8, vs p / p o may be described by eq 23, which contains only two parameters q and u; these parameters permit calculations of the thermodynamic quantities that were discussed in the previous sections. In the second case, we can calculate from eq 47 the absolute amount a, adsorbed in the micropores: at

=

(52)

-

In this case, we do not take into account the value a:, which is provided by the a,-method [eq 501. The at vs p / p o adsorption curve may be described also by eq 23 in order to evaluate a?, q, and u. The value of a: obtained by means of eq 23 may be compared with that obtained by the asmethod. Here, we would like to mention that the amount a, adsorbed in the micropores may be evaluated also by other methods such as the t which is analogous to the c ~ ~ - m e t h o d and , ~ I the preadsorption meth~d.~*.~~ The total maximum amount aT,: is equal to

+

= a? a,,? Let ft and A,, denote the following fractions:

(53)

ft = a?/aT,?;

(54)

UT,:

f,,t

=

as,:/aT,?

Figure 1. a,-plot for benzene adsorbed on the type RKD-4 microporous activated carbon at 293 K. ,-.

E

v

4

-1

5

10

15

20

2 I

Adsorption P o t e n t i a l , A(kJ/mole)

Figure 2. Comparison of the experimental points for benzene adsorbed in the micropores of the type RKD-4 activated carbon with the theoretical (solid line) adsorption isotherm calculated according to eq 23.

On the basis of eq 54, we can rewrite eq 45 as follows:

=Let+fs,t8s,t

with eT,t = aT,t/aT,? (55) In a previous paper," we showed that the total adsorption potential distribution XT,,(A) may be expressed in terms of the & ( A ) distribution, which characterizes the structural heterogeneity of the micropores, and the & , ( A ) distribution, which characterizes the surface heterogeneity of the mesopores. This total distribution XT,,(A) is given by 6T,t

S t a n d a r d Adsorption, as

the mesopore surface. To extract the adsorbed amount of a, from the total adsorbed amount aT,,, we used the a,-method (viz., eq 50) and eq 52. The standard adsorption a,was calculated from the benzene isotherm measured on a reference (type HAF) carbon black a t 293 K.46 This reference carbon black (from the Podkarpacka Refinery, Poland) was obtained from active furnace soot heated at 1173 K in an argon atmosphere. The parameters of this standard adsorption isotherm were published elsewhere. The a,-plot in Figure 1 was prepared according to eq 50 for benzene adsorbed on type RKD-4 activated carbon at 293 K. This a,-plot permits evaluation of the maximum amount a: adsorbed in the micropores and the monolayer capacity a,,: of the mesopore surface: a? = 4.53 mmol/g and a,,: = 0.58 mmol/g. Conversion of the a,,: value for benzene adsorbed on the mesopore surface of the RKD-4 carbon gives the specific area of this surface equal to 140 m2/g. Using the a,,: value and the standard benzene isotherm 8, in eq 52, we extracted the amount a, adsorbed in the micropores of the type RKD-4 activated carbon from the total adsorbed amount uT,,. The adsorbed amount a, was described next by the isotherm eq 23 in order to evaluate the parameters q and v : the fitting of eq 23 to the experimental data a, vs A gives q = 1628 (kJ/mol)2 and u = 9.9. The experimental points for benzene adsorbed in the micropores of type RKD-4 activated carbon are compared in Figure 2 with the theoretical (solid line) isotherm calculated according to eq 23. This comparison shows that eq 23 gives an excellent representation of the experimental adsorption of benzene on the type RKD-4 activated carbon. The evaluated parameters q and v were used to calculate the distribution function C ( z ) (viz., eq 13) that characterizes the heterogeneity of the microporous structure of type RKD-4 carbon; this distribution is shown in Figure 3. Because the variable z = 1 is proportional to the half-width of the slitlike micropores,19 the distribution G(z) provides almost the same information as the micropore-size dis14937346

Because we assumed that the physicochemical properties of the mesopore surface of the adsorbent studied and the surface of the reference no_nporous solid are identical, the adsorption potential distribution & ( A ) evaluated from the standard adsorption isotherm may be introduced into eq 56 instead of the X,,,(A) distribution: xT,t(A) =

+

(57)

Equation 57 permits the calculation of the total adsorption potential distribution from both the adsorption isotherm studied and the standard isotherm measured for the reference nonporous solid.

Illustrative Example To illustrate the utility of eq 23 for describing vapor adsorption on heterogeneous microporous solids, we used the benzene adsorption isotherm measured on type RKD-4 activated carbon at 293 K.45 Type RKD-4 activated carbon is a commercial adsorbent from the Norit Co., The Netherlands. From eq 44, the measured total adsorbed amount aT,,is the sum of the amounts a, adsorbed in the micropores and the amount a,,, adsorbed on Lippens, B. C.; de Boer, J. H. J . Catal. 1965, 4, 319. Rand, B.; Marsh, H. J . Colloid Interface Sci. 1972, 40, 478. Sing, K. S. W. Ber. Bunsen-Ges. Phys. Chem. 1975, 79, 124. Ali, S.; McEnaney, B. J . Colloid Interface Sci. 1985, 107, 355. Martinecz-Martinez, J. M.; Rodriguez-Reinoso, F.; Molina-Sabio, M.; McEnaney, B. Carbon 1986, 24, 255. (44) Jaroniec, M.; Madey, R. Carbon 1987, 25, 579. (45) Choma, J. Sc.D. Thesis, WAT, Warsaw, 1985. (39) (40) (41) (42) (43)

/a

(46) Jankowska, H.; Swiatkowski, A,; Zietek, S. B i d . Wojsk. Akad. Tech. (Warsaw) 1977, 26, 13 1.

5230

J . Phys. Chem. 1989, 93, 5230-5237

2

Parameter z = 1 / A , (mole/kJ)

Figure 3. Distribution function G ( z ) , calculated according to eq 13, for the type RKD-4 microporous activated carbon.

tribution function. Equations 14, 15, and 16 permit the calculation of the parameters,,,z, Z, and uz that characterize quantitatively the distribution function G(z); these calculations give zmax= 0.052 mol/kJ, = 0.054 mol/kJ, and u, = 0.012 mol/kJ. Assuming that type RKD-4 activated carbon possesses slitlike micropores Z, and u, with a half-width X, we can convert the quantities,,,,z to xmax,X, and ux. According to experimental studiesI3J6J9 of adsorption on microporous activated carbons, z (= 1/2)is proportional to the half-width x with the proportionality constant k = 0.094/@ mol/(kJ-nm)

z = l/2 = kx

(58)

Because the similarity coefficient (3 for benzene is assumed to be equal to ~ n i t y ,the ~ . ~constant k = 0.094 mol/(kJ-nm) was used to convert,,,,z Z, and uz to x,,, X, and 0,; this conversion gives x,, = 0.55 nm, X = 0.57 nm, and ox = 0.13 nm. These parameters characterize the structural heterogeneity of the micropores of the type RKD-4 activated carbon.

The energetic heterogeneity of these micropores may be characterized by the quantities d, (eq 30) and uA,I (eq 32) that are associated with the adsorption potential distribution function X , ( A ) (viz., eq 33). Calculation of these quantities gives d, = 19.7 kJ/mol and uA,, = 11.8 kJ/mol. In our previous paper,25 we showed that the geometric surface area of micropores is from this proportional relationship, we obtain proportional to 2,; a value for the geometric surface area of the micropores of type RKD-4 carbon equal to 810 m2/g. The addition of this value to the mesopore specific surface area obtained by the as-method (140 m2/g) gives the total specific surface area of the RKD-4 carbon equal to 950 m2/g. This analysis of the benzene adsorption isotherm on type RKD-4 microporous activated carbon illustrates the applicability of our theoretical description for characterizing microporous solids.

Conclusions A complete description of physical adsorption of a vapor on an heterogeneous microporous solid is presented in terms of the distribution functions that characterize the structural and energetic heterogeneities of this solid. It is shown that characterization of the structural heterogeneity of the micropores of a solid by the gamma distribution, viz., eq 13, leads to simple equations for the characteristic adsorption curve, viz., eq 23, the adsorption potential distribution, viz., eq 33, and the thermodynamic functions, viz., eq 38,42, and 43. Another important advantage of the proposed description is its association with the asmethod in order to extract the amounts adsorbed in the micropores and on the mesopore surface. Analysis of these extracted quantities permits evaluating (through eq 57) the adsorption potential distributions that characterize the structural heterogeneity of the micropores and the surface heterogeneity of the mesopores. Acknowledgment. This work was supported in part by the National Science Foundation under Grant No. CBT-872 1495. We thank Dr. J. Choma and Mr. X . Lu for providing figures.

A Study of the Mechanism of the Partial Oxidation of Methane over Rare Earth Oxide Catalysts Using Isotope Transient Techniquest Alfred Ekstrom* and Jacek A. Lapszewicz CSIRO Division of Fuel Technology, Lucas Heights Research Laboratories, Private Mail Bag 7 , Menai, N S W, 2234, Australia (Received: September 26, 1988; I n Final Form: February 8 , 1989)

The mechanism of the partial oxidation reaction of methane over three catalysts (Sm203,Li/Sm2O3, and Pr6OI1)having a range of activities and product selectivities has been studied by using isotope transient techniques. The most important conclusions from this study are ( 1 ) that large amounts of CH4 are adsorbed on all working catalysts, (2) that the reaction takes place on a small number of very active catalyst sites and does not involve the adsorbed CHI, (3) that gas-phase oxygen exchanges rapidly with the lattice oxygen atoms of the working catalysts, (4) that the rate of CH4 conversion is dependent on the rate of lattice oxygen exchange, and ( 5 ) that the carbon oxides are substantially formed by the secondary oxidation of the reaction products. A mechanism based on the formation and reactions of [O-]species is proposed for the C2+products, but a different form of activated oxygen appears to be responsible for the formation of the carbon oxides.

Introduction The catalyzed reaction of methane with oxygen under "fuel rich" (i.e., CH4 >> 0,) conditions appears to have been first studied by Boomer and Thomas,l who found that small amounts of ethane were formed when methane/oxygen mixtures were reacted over a copper catalyst a t high pressures. More recently, Keller and Bhasin2 reported that the atmospheric-pressure reaction was ' A partial account of this work was presented at the Symposium on Direct Methane Conversion, held by the Division of Petroleum Chemistry, at the American Chemical Society Los Angeles meeting, Sept 25-30, 1988.

0022-3654/89/2093-5230$01 S O / O

catalyzed by a variety of metal oxides, which gave C2 selectivities3 up to 60%. Lunsford et aL4 showed that use of a Li2C03-promoted MgO catalyst resulted in high C2 selectivities and relatively high methane conversions. These observations lead to a resurgence ( I ) Boomer, E. H.; Thomas, V. Can. J . Res. Sect. B 1937, 15, 401. (2) Keller, G.E.; Bhasin, M. J . Carol. 1982, 73, 9. (3) The C2selectivity is defined as 2(rate of C2H6 C2H4formation)/(rate of total CHI conversion). (4) Ito. T.: Wang, J.-X.; Lin, C-H.; Lunsford, J. H . J . Am. Chem. SOC. 1985. 107. 5062.

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6 1989 American Chemical Society