Langmuir 1988,4, 911-917
911
Characterization of Energetic and Structural Heterogeneities of Activated Carbons M. Jaroniec,* R. Madey, and X. Lu Department of Physics, Kent State University, Kent, Ohio 44242
J. Choma Institute of Chemistry, WAT, 00908 Warsaw, Poland Received September 11, 1987. I n Final Form: March 1, 1988 New quantities are proposed for characterizing energetic and structural heterogeneities of activated carbons, namely, the mean adsorption potential and the dispersion associated with the adsorption potential distribution and the mean micropore dimension and the dispersion associated with the micropore distribution. These quantities are evaluated by means of the adsorption parameters that appear in an isotherm equation derived for gas adsorption on strongly nonuniform microporous solids. This equation represents the experimentaladsorption isotherm for strongly microporous solids better than the Dubinin-Radushkevich equation, which is used frequently for characterizing activated carbons. For illustrative purposes, the quantities listed above are calculated from the benzene adsorption isotherms for eight activated carbons of different structural heterogeneity. and structural heterogeneities of adsorbents. In addition Introduction to the specific surface area and the pore size distribution, General use of carbonaceous adsorbents in science and these distribution functions should become standard technology1 requires their many-sided characterization. quantities used for Characterizing adsorbents. Information about physical and chemical properties of the solids may be obtained directly by means of various This paper is devoted to characterization of activated modern techniques. Although the role of these techniques carbons, which usually possess nonuniform microporous for characterizing adsorbents is still increasing,2 classical structures.'J7 This structural heterogeneity is the source measurements of adsorption and desorption on solids are of the energetic heterogeneity of activated carbons.18 In very popular and still utilized in surface chemistry because this paper we will apply a simple isotherm equation, which they provide information about the behavior of a solid with was proposed by Jaroniec and C h ~ m a , 'to ~ describe benzene adsorption in micropores. This equation gives a respect to an a d ~ o r b a t e . ~ - ~ good representation of benzene adsorption isotherms, The most widely used measurement in adsorption studies is the adsorption-desorption i ~ o t h e r m ,which ~ , ~ ~ ~ which are measured frequently for characterizing microis used generally to calculate the specific surface area3sS1l porous activated carbons.20 The parameters of the Jarand pore size distribution.8-12 The specific surface area oniec-Choma (JC) equation are used to calculate the adand pore size distribution are fundamental quantities sorption potential and the micropore size distributions. recommended by the International Union of Pure and We will show that the mean values and the dispersions of Applied Chemistry (IUPAC)10J2J3 for characterizing adthese distributions are defined by simple equations; we will sorbents. To evaluate the specific surface area, only a small conclude by recommending that these quantities be used part of the adsorption isotherm is used.3 The pore size to characterize the energetic and structural heterogeneities distribution is evaluated from either the multilayer part of microporous activated carbons. of the low-temperature nitrogen adsorption isotherm or porosimetric measurements based on the penetration of pores by mercury; this distribution does not include very (1) Juntgen, H.Carbon 1977,15, 273. fine pores, i.e., micropores.s (2) McGuire, G. E. Anal. Chem. 1987,59, 294. Part of the adsorption isotherm measured at low con(3) Young, D. M.; Crowell, A. D. Physical Adsorption of Gases; Butterworths: London, 1962. centrations has not been explored sufficiently for charac(4) Ross, S.; Olivier, J. P. On Physical Adsorption; Wiley: New Yor, terizing the sorption properties of ad~orbents.~ Adsorption 1964. data at low concentrations are sources of valuable infor(5) Steele, W. A. The Interaction of Gases with Solid Surfaces;Permation about adsorbate-adsorbent interactions5 and engamon: Oxford, 1974. ergetic and structural heterogeneities of ~ o l i d s . ~ ~ ~ ~ ~ J(6)~Jaroniec, - ' ~ M. Adu. Colloid Interface Sci. 1983, 18, 149. (7) Jaroniec, M.; Brauer, P. Surf. Sci. Rep. 1986, 6, 65. Significant progress in the theoretical description of gas (8) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area, and Poadsorption on heterogeneous solids6~7~14J5 provides founrosity; Academic: New York, 1982. dations for utilizing low-pressure adsorption measurements (9) Dollimore, D. Surf. Technol. 1976, 4, 121. to evaluate adsorption potential distribution^.^ These (10) Everett, D. H.; Parfit, G. D.; Sing, K. S. W.; Wilson, R. J. Appl. distributions characterize the energetic heterogeneity of Chem. Biotechnol. 1974, 24, 199. both nonporous and microporous solids. In the case of (11) Cortes, J. Adu. Colloid Interface Sci. 1985, 22, 151. (12) Havard, D. C.; Wilson, R. J. Colloid Interface Sci. 1976,57,276. microporous solids, the low-pressure adsorption mea(13) Sing, K.S. W. et al. Pure Appl. Chem. 1985,57, 603. surements can be used also to calculate the micropore size (14) Zolandz, R. R.; Myers, A. L. Prog. Filtr. Sep. 1979, 1, 1. di~tributi0ns.l~Further studies are required to establish (15) House, W. A. Colloid Sci. 1983, 4, 1. some recommendations for using these adsorption poten(16) Sing, K.S.W. Ber. Bunsen-Ges. Phys. Chem. 1975, 79, 724. ~ surface tial and micropore size distributions t . characterize (17) Dubinin, M. M. Carbon 1985, 23, 373. *Permanent address: Institute of Chemistry, M. Curie-Sklodowska University, 20031 Lublin, Poland.
0743-?463/88/2404-0911$01.50/0
(18) Jaroniec, M.; Piotrowska, J. Monatsh. Chem. 1986, 117, 7. (19) Jaroniec, M.; Choma, J. Mater. Chem. Phys. 1986, 15, 521. (20) Dubinin, M. M. Prog. Surf. Membr. Sci. 1975, 9, 1.
0 1988 American Chemical Society
912 Langmuir, Vol. 4, No. 4 , 1988
Jaroniec et al.
Characteristic Adsorption Curve for Nonuniform Microporous Solids The characteristic adsorption curve &(A) for structurally heterogeneous microporous solids, i.e., for solids with a nonuniform microporous structure, is represented by the following integral:21 jmex~[-B(A/i3)21 0 F(B) dB
(1)
where the adsorption potential A is17 A = RT In ( p s / p )
for p Ip s
Let us discuss a limiting behavior of the characteristic adsorption curve given by eq 5. The symbol 8, which denotes the degree of volume filling of micropores, is defined as 8=
(7)
where a denotes the amount adsorbed in the micropores a t the equilibrium pressure p and absolute temperature T , and a. denotes the maximum amount adsorbed in the micropores. Replacing 8 in eq 5 by a l a o , we can present this equation in the following form:
(2)
Here p is the equilibrium pressure, p s is the saturation vapor pressure, p is the similarity coefficient that depends on the nature of the adsorbate,20T is the absolute temperature, R is the universal gas constant, B is the structural parameter that is related to the micropore size,17and F(B) is the distribution function of the structural parameter B. This distribution satisfies the normalization condition
U / U ~
In a = In a':
- (n + 1) In [ l + (1/q)(A/p)2]
(8)
Because the value of the parameter q is a large number for real adsorption and because the expression (l/q)(A/@ is small compared with unity for small values of A (e.g., A C 2 for q = 200 or A C 5 for q lOOO), we can approximate eq 8: In a = In aOJC- [(n + l)/q](A/p)* = In aJc - B(A//3)2 (9)
p ( B ) dB = 1
(3)
The integral eq 1 is based on the assumption that the Dubinin-Radushkevich (DR) equation represents adsorption in uniform micropores;17J1in other words, local adsorption is described by the characteristic curve given by the DR equation:20 BDR(A,B)= e~p[-B(A//3)~]
(4)
StoecklP derived an equation for 8(A) under the assumption that the function F(B) is represented by a Gaussian distribution of the structural parameter B. Dubinin,17Rozwadowski, and WojszZ2derived equations for 8(A) by assuming other distribution functions for describing the structural heterogeneity of microporous solids. These equations are rather complex and sometimes do not fulfill physical requirement^.^^ On the basis of the theoretical considerations of Jaroniec and Piotrowska,18 Jaroniec and Choma (JC) derived the following simple equation for %(A):19
r
n
1n + l
Here the superscript JC refers to the characteristic curve 8(A) proposed by Jaroniec and Choma,Igand n(> -1) and q(>O) are parameters of the r distribution F(B): ,n+l
Because the lower limit of integration in eq 1 is assumed to be zero, eq 5 describes adsorption on strongly heterogeneous microporous solids with a broad micropore size distribution. The applicability of the isotherm eq 5 as well as other isotherms derived on the basis of the integral eq 1 is limited to the low-pressure region of the isotherm, which reflects adsorption in the micropores. The upper limit of this pressure region depends on the kind of adsorbate and adsorbent; for example, experimental studies showed that a complete filling of the micropores in activated carbons occurs at the relative pressure p l p , , equal to about 0.17 for benzene and about 0.4 for nitrogen.20 (21) Stoeckli, H. F. J. Colloid Interface Sci. 1977, 59, 184. (22) Rozwadowski, M.; Wojsz, R. Carbon 1984, 22, 363. (23) Choma, J.;Jankowska, H.; Piotrowska, J.; Jaroniec, M. Monatsh. Chern. 1987, 118, 315.
The right-hand side of eq 9 follows from the fact that (n + l)/q is equal to the mean value B. The linear relationship In a vs A2 given by eq 9 reflects the limiting properties of the JC eq 5 in the region of small values of A; this relationship is identical with that obtained for the DR eq 4 with B = B.
Differential Distributions for Nonuniform Microporous Solids It was shown in the previous section that the r distribution F(R) (eq 6) generates the JC eq 5. This distribution may be recalculated to the micropore size distribution J(x), where x denotes the micropore dimension. For carbonaceous adsorbents with slitlike micropores with limited dimensions, x denotes the half-width of these micropores. On the basis of experimental studies, Dubinin17established the following relationship between the structural parameter B and the micropore dimension x : B = cx2 (10) On the basis of this relationship, it is easy to see the relationship between the distribution functions F ( B ) and J(X):~~
J ( x ) = 2cx F ( B ( x ) ) (11) Equations 6 and 11 give the following micropore size distribution J ( x ) : 2 (q*)n+l J ( ~=) ------9n+l exp(-q*x2); with q* = qc (12) r(n+1) If eq 5 represents the characteristic adsorption curve 8(A), the structural heterogeneity of a nonuniform microporous solid is described by the micropore size distribution given by eq 12. This structural heterogeneity is the source of the energetic heterogeneity, which is characterized by the distribution X(A) of the adsorption potential A. This distribution may be calculated in terms of the condensation approximation method:18J9
XJC(A)= -d8(A)/dA = 2(n + 1)q"+'(A/p2)[q+ (A//3)2]-(n+2)(13) Here the superscript JC denotes that eq 13 is associated with eq 5. The distribution X(A) given by eq 13 charac(24) Dubinin, M. M.; Stoeckli, H. F. J. Colloid Interface Sci. 1980, 75,
34.
Characterization of Activated Carbons
Langmuir, Vol. 4, No. 4, 1988 913
Table I. Quantities That Characterize the Differential Distributions F ( B ) (Eq 6), J ( x ) (Eq 12), XJC(A)(Eq 13), and X D R ( A )
(Eq 14)' P
d P )
terizes the energetic heterogeneity of the microporous structure of a solid. It is noteworthy that even a uniform microporous structure is the source of the energetic heterogeneity;ls therefore, the DR eq 4 generates also the adsorption potential distribution X D R ( A ) : X D R ( A )= 2 ( B / P 2 ) Ae~p[-B(A/b)~]
Equation 20 defines the relationships between the differential distributions F(B),J(x),and X ( A ) and the integral distributions F*(B),J*(x),and X * ( A ) ,respectively. The integral distribution F*(B) associated with the differential distribution F(B) (eq 6 ) is
(14)
The superscript DR is used to distinguish the quantities associated with the DR eq 4. The distribution functions F(B),J ( x ) ,and X ( A ) satisfy the conditions lim 4(p) = 0; 4(0) = 0; P--
L-44~) dp = 1
0,
U,,
(15)
for 4 ( p ) F(B),J(x),and X ( A ) . The distributions F(B), J ( x ) ,and X ( A ) are characterized by means of their mean values and dispersion^,^^ which are defined as follows:
(17)
for +(p) F(B),J(x),and X ( A ) . The symbol p denotes the mean value of p, and ap denotes the dispersion. The maximum of the distribution $(p) occurs at the point p = pm,which may be calculated by solving =O
Table I contains expressions for calculating the values of p , ap,and pm for the distribution functions F(B) (eq 6 ) ,J(x) (eq 12), X J C ( A )(eq 13), and X D R ( A )(eq 14); these expressions were derived by means of eq 16, 17, and 18. Note that the function 4(p) = F(B),A x ) , and X ( A ) denotes a differential distribution of the variable p; however, often for the sake of brevity, we will omit the word "differential" when referring to these distributions.
Integral Distributions for Nonuniform Microporous Solids The integral distribution $*(p) associated with the differential 4(p) is defined as
where the incomplete r function I'(n+l,qB) is defined a P F(n+l,qB) =
j m p n exp(-p) dp qB
Applying the definition of the differential distribution 4(p), viz.,
to the integral distribution F*(B),we have
Equations analogous to eq 21, 22, and 24 may be written for the distribution J*(x):
J*(x)= 1 -
r(n+l,q*x2) r(n+l)
where the incomplete r function I'(n+l,q*x2)is defined as follows: F(n+l,q*x2) =
q'x2
pn
exp(-p) dp
(26)
According to eq 23 and 25, the differential micropore size distribution J ( x ) is dJ*(x) 1 dI'(n+l,q*x2) J ( x ) = -= -~ dx I'(n+l) dx
(27)
The integral distribution J*(x),which denotes the fraction of the micropores of linear dimensions between 0 and x , may be calculated easily because the r function and the incomplete r function are tabulated in mathematical handbooks.26 According to eq 19, the integral adsorption potential distribution X * ( A ) associated with the differential distribution X ( A ) is X * ( A ) = 1 - B(A)
where ~$*(p)denotes the fraction of a quantity p between p = 0 and p = p. Because the differential function 4 ( f ) satisfies the normalization condition (eq 151, eq 19 gives
(22)
(28)
Using eq 5 to express O(A)in eq 28, we obtain the following equation for the integral distribution of the adsorption potential: X * ( A ) = 1 - [ l + ( 1 / ~ ) ( A / P ) 2 ] - ' " + 1 ' (29)
This equation relates to the characteristic adsorption curve (25) Jaroniec, M.; Madey, R.
press.
J. Chem. SOC.,Faraday Trans. 1 , in
(26) Gradshteyn, I. S.;Ryzhik, I. M. Tables of Integrals, Series, and Products; Academic: New York, 1980.
914 Langmuir, Vol. 4 , No. 4, 1988
Jaroniec et al.
Table 11. Activated Carbons Used To Measure the Benzene Adsorption Isotherms at 293 K ref to activated carbon cwz-3
AC-12 NS W AG-5 A-2 T HS-43 BH
source of adsorbent Company of Carbon Electrodes, Raciborz, Poland Chepurnoy, U.S.S.R. Hajnowka, Poland Hajnowka, Poland Hajnowka, Poland laboratory scale Morawskie Chem. Comp., Czechoslovakia Bender-Hobein, AG, Zurich
benzene adsorption isotherm 27, 28
29 27, 28 23 27, 28 27, 28 30 23
8Jc(A) (eq 5). The distribution X*(A) denotes the fraction
of adsorption sites with adsorption potentials between 0 and A; these sites are located in the micropores.
Results and Discussion To illustrate the utility of the mean values and the dispersions associated with the distribution functions F(B), J ( x ) ,and X ( A ) for characterizing the structural and energetic heterogeneities of microporous activated carbons, we used the benzene adsorption isotherms measured on eight different carbons a t 293 K. These adsorption isotherms were presented e l s e ~ h e r e ;Table ~ ~ ~ I1 ~ lists ~ - ~the ~ types of activated carbons and references to the experimental measurements. The measured total adsorbed amount a, of benzene is the sum of the amounts adsorbed in the micropores (a) and on the mesopore surface (a,):17 a, = a
+ a, = a + s y ,
(30)
where a,, a, and a, are the amounts adsorbed per unit mass of the solid, y, is the amount adsorbed on the solid surface per unit surface area, and S is the specific surface area of the mesopores. To evaluate the amount adsorbed on the mesopore surface, a,, Dubinin17suggested that the benzene adsorption isotherm be measured on a nonporous carbon (reference adsorbent); the chemical nature of the surface of this reference adsorbent is assumed to be identical with the nature of the mesopore surface of the activated carbon studied. As the reference adsorbent for the activated carbons listed in Table 11, we used the HAF carbon black obtained from active furnace soot (from the Podkarpacka Refinery, Poland) heated a t 1173 K in an argon atmosphere. The benzene adsorption isotherm on this carbon black at 293 K was measured by Jankowska et al.31 The monolayer capacity evaluated according to the BET equation from the benzene adsorption isotherm is equal to 0.321 mmol/g; this value gives the specific surface area of 79 m2/g for the above carbon black. To calculate the specific surface area from the monolayer capacity, the value 0.41 nm2 was assumed for the cross sectional area of the benzene molecule. The constant C of the BET equation, as evaluated from the benzene adsorption isotherm on the HAF carbon black in the region (27) Choma, J. Description of Porous Structure of Actiuated Carbons; WAT: Warsaw, 1985 (in Polish). (28) Choma, J.; Jaroniec, M.; Piotrowska, J. Carbon, in press. (29) Dubinin, M. M. Carbon 1981, 19, 321. (30) Kadlec, 0.; Choma, J.; Jankowska, H.; Swiatkowski, A. Collect. Czech. Chem. Commun. 1984,49, 2721. (31) Jankowska, H.; Swiatkowski, A.; Zietek, S. Bid. Wojsk. Akad. Tech. 1977, 26, 131.
of relative pressures from 0.06 to 0.36, is equal to 39. The amount of benzene adsorbed per unit surface area of the HAF carbon black was identified with y,,which denotes the amount of benzene adsorbed per unit surface area of the mesopores. The specific surface area of the mesopores was estimated according to the integral equation used by D ~ b i n i n : ~ ~
where u, is the surface energy, ai is the amount adsorbed a t the initial point of the hysteresis loop of the adsorption-desorption isotherm, and a, is the limiting adsorption at a relative pressure of unity. The values of the specific surface area of mesopores for activated carbons studied are summarized in Table 111. These values and the adsorption isotherm y,for benzene on the HAF carbon black were used to calculate a, (see eq 30) and then to evaluate the amount adsorbed in the micropores: a = a, - a,
(32) The experimental isotherm a@)that represents adsorption in the micropores was described by the isotherm equation obtained from the JC eq 5
The parameters aoJC,q, and n of the JC eq 33 for the adsorption systems studied are summarized in Table 111. The similarity coefficient for benzene is equal to unity.17 For the purpose of comparison, the parameters of the DR isotherm were calculated according to the following equation: a b ) = aODRexp[-(B/P)[RT In ( P , / P ) ~ ~ ] (34)
The DR parameters are given in Table 111. This table contains also the ratio rJc DR of the standard deviations for the JC eq 33 and the D k eq 34. The fact that this ratio is smaller than unity for six of the eight systems studied means that the JC eq 33 gives a better representation than the DR eq 34 for these systems. The values of the parameters n and q listed in Table I11 were used to calculate, according to the expressions given in Table I, the mean value AJc and the dispersion cAJCthat characterize the JC adsorption potential distribution X ( A ) (eq 13; see Table IV). The values of yA,which are needed to evaluate AJC and uAJC, were calculated according to the equation given in the Appendix; values of yAcalculated for the parameter n from 0 to 3 are given in Table VI. For the activated carbons studied, the value of n does not exceed 3. In order to compare the values AJc and eAJc with those obtained for the DR adsorption potential distribution (eq 14), we calculated the values ADRand eADR (see Table IV). For all activated carbons studied, Table IV shows that AmJC,the maximum of the JC adsorption potential distribution, occurs a t a lower value than AmDR,the maximum of the DR adsorption potential distribution; however, the values of AJc and uAJCare greater than the corresponding values of ADRand aADR (see Table IV). The values of ADRand uADRfor the XDR(A)distribution (eq 14) satisfy the condition oADR/ADR = 0.523,%which means that the average value ADRdetermines automatically the disFor uniform microporous solids, this conpersion eADR. dition is fulfilled; for nonuniform microporous solids, the relationship between AJc and eAJcis more complicated and (32) Dubinin, M. M. Carbon 1983, 21, 359.
Langmuir, Vol. 4, No. 4, 1988 915
Characterization of Activated Carbons
Table 111. Parameters of the DR Eq 34 and the JC Eq 33 for Adsorption of Benzene in the Micropores of Various Activated Carbons mesopore standard DR eq 34 JC eq 33 deviation ratio surface area, activated carbon m2/g ~JCJDR aoDR, mmol/g 103B, (mol/kJ)' a t C ,mmol/g n q, (kJ/mol)' CWZ-3 163 0.32 5.89 2.22 6.06 2.50 1277 AC-12 81 0.95 4.35 2.25 4.47 2.01 1085 NSW 107 2.30 4.83 2.04 4.93 1.04 793 AG-5 96 0.30 3.52 2.30 3.63 1.36 751 A-2 115 0.19 3.47 2.35 3.64 1.21 662 T 63 1.71 5.30 3.51 5.75 0.77 308 HS-43 76 0.21 5.98 3.57 6.85 0.14 151 BH 320 0.24 2.23 5.65 2.67 0.21 97 h
Table IV. Mean Values and Dispersions for the JC and DR Distributions of the Adsorption Potential" activated distribution XDR(A) distribution XJC(A) carbon AmDR AmDR aADR AmJC AJC aAJC CWZ-3 15.0 18.8 9.8 12.6 19.1 12.1 9.8 12.4 19.4 12.8 14.9 18.7 AC-12 19.6 10.3 12.5 21.8 17.0 NSW 15.7 AG-5 14.7 18.5 9.6 11.5 19.0 13.8 A-2 14.6 18.3 9.5 11.1 18.7 14.0 T 11.9 15.0 7.8 8.2 15.2 13.0 14.8 HS-43 11.8 7.8 6.8 16.3 28.5 BH 9.4 11.8 6.2 5.3 12.2 17.7
2
2 4
v
X
B
4
uAJc/AJc = (4/ayA2 - 1)"'
(35)
where y A is a function of the parameter n (see Table I). For small values of n,the values of differ significantly from unity (see Appendix) and the values of the dispersion uAJCare larger than the value AJc (cf. the values in Table IV for HS-43 and BH activated carbons). For values of n greater than about 2, the values of YA are close to unity; then eq 35 may be approximated by the same relationship that is valid for the X D R ( A )distribution, i.e. uAJC/AJC
%
uADR/ADR = 0.523
a
.AJc
= (@/2)(7rq/n)1/2 = (p/2)(7r/Bm)1/2
0000 0 10 20 30 40 50 60 Adsorption Potential A, (kJ/mole)
Fi ure 1. Com arison of the adsorption potential distributions X (A)and X D2 (A)for benzene on CWZ3 activated carbon. The solid line denotes XJC(A)(eq 13) for n and q taken from Table I11 for CWZ-3 carbon. The dashed line denotes XDR(A)(eq 14) for the B value obtained from the DR eq 4 for CWZ-3 carbon,
E
whereas the dotted line denotes the same distribution calculated for B = B, = n/q.
c 0 3
(36)
P
In this case, the quantities AJc and uAJCare expressed by the approximate equations AJC
0030
4
"All values in kJ/mol.
depends on the degree of the structural heterogeneity of a solid. For nonuniform microporous solids, the ratio of gAJcto AJc is given by
0050
4
6
0 040 \ \
4
(37)
5
B
p[(1 - ~ / 4 ) ( q / n ) ] l / ' = p[(1 - ~ / 4 ) / B , ] l / ~ (38)
4
0 000 0
10
20
30
40
50
I
Lk
Equations 37 and 38 are identical with those obtained from the X D R ( A )distribution (see Table I) calculated for B = B,. To illustrate this fact, we compare the distribution functions X J C ( A )(eq 13) and X D R ( A )(eq 14) for the benzene adsorbed on CWA-3 activated carbon (see Figure 1); the values of the parameter n for this activated carbon are the highest of the eight activated carbons studied (see Table 111). The solid line in Figure 1denotes the X J C ( A ) distribution for CWZ-3 activated carbon. The dashed and dotted lines in this figure denote, respectively, the XDR(A) distributions (eq 14) calculated for the B value obtained for CWZ-3 carbon by means of the DR eq 4 and for the value B = B, = n / q . It follows from Figure 1that these distributions are not much different. Figure 2 shows the adsorption potential distributions for benzene adsorbed on BH activated carbon; the value of n for BH carbon is equal to 0.21 and is about 12 times smaller than that for CWZ3 carbon. Only HS-43 activated carbon has a smaller value of n than that for BH carbon (see-Table 111); the values of n for the other five activated carbons studied are considerably larger than those for BH and HS-43 carbons.
Adsorption Potential A , (kJ/mole)
Fi ure 2. Com arison of the adsorption potential distributions X (A)and X D (A)for benzene on BH activated carbon. The solid line denotes XJC(A)(eq 13) for n and q taken from Table I11 for BH carbon. The dashed line denotes XDR(A)(eq 14) for the B value obtained from the DR eq 4 for BH carbon, whereas the dotted line denotes the same distribution calculated for B = B, = n / q .
5
II
Because the value YA for n = 0.21 differs from unity = 0.64), eq 36 is not fulfilled; consequently, the distributions X J c ( Z )and X D R ( A )for BH carbon have different shapes and values of A,, A, and nA (see Figure 2 and Table IV). Figure 3 illustrates the adsorption potential distribution XJC(A)for CWZ-3 (solid line) and BH (dotted line) activated carbons. Comparison of both distribution curves in Figure 3 suggests that BH activated carbon shows a smaller energetic heterogeneity than CWZ-3 activated carbon; however, the value of the dispersion uAJCfor BH carbon is larger than that for CWZ-3 carbon (see Table IV). Comparison of the numerical values of both distribution
Jaroniec et al.
916 Langmuir, Vol. 4, No. 4 , 1988 h
3
E
0 O 120
v h
.=
v
I
0 loot
X
4
2
0040
c2 4
5
4
0000 O 0 o 10
20Z 30
40 o
50
L60
Structural Parameter B, (mole/kJ)'
1
Adsorption Potential A, (kJ/rnole)
Fi ure 3. Comparison of the adsorption potential distribution X E ( A ) (eq 13) for CWZ-3 (solid line) and BH (dotted line) activated carbons.
Figure 4. Comparison of the distribution F(B) (eq 6) for CWZ-3 (solid line) and BH (dotted line) activated carbons.
Table V. Mean Values and Dispersions for the Distributions F(B)(Eq6) and J ( x ) (Eq 12) distribution F(BA activated carbon cwz-3 AC-12 NSW AG-5 A-2
T HS-43
BH
(mol/kJ)*
io3& 1.96 1.85 1.31 1.81 1.83 2.50 0.93 2.17
103B 2.77 2.78 2.57 3.14 3.34 5.75 7.55 12.47
10%~ 1.48 1.60 1.80 2.05 2.25 4.32 7.07 11.34
distribution A x ) . nm x,
I
0.58 0.61 0.58 0.60 0.53 0.57 0.60 0.64 0.61 0.66 0.77 0.85 0.78 0.94 1.03 1.21
ux
xDR, nm
0.17 0.18 0.21 0.21 0.23 0.33 0.46 0.57
0.57 0.57 0.54 0.58 0.58 0.71 0.72 0.90
curves in the interval of A from 0 to 150 kJ/mol shows that the XJc(A)distribution for BH carbon is decreasing slowly and for A > 94 kJ/mol its values are larger than those for the distribution that characterizes CWZ-3 carbon. For the BH carbon, the contribution of adsorption sites with the highest adsorption energies to the values of AJC and uAJc is significant with the result that these values are larger than those obtained for the CWZ-3 carbon (see Table IV); consequently, the energetic heterogeneity of the BH carbon is larger than that for the CWZ-3 carbon. This discussion indicates that comparison of the values of the dispersion aAJC for various adsorbents provides valuable information about the energetic heterogeneity of these adsorbents. As Figure 3 illustrates, sometimes a simple comparison of the distribution curves for various adsorbents can lead to incorrect conclusions about the energetic heterogeneity of these solids; therefore, the mean value AJC and the dispersion aAJC are recommended for characterizing the energetic heterogeneities of microporous solids. The nonuniform microporous structure of solids may be characterized also by the distribution functions F(B) and J(x). The values of B,, A, and UB that characterize the distribution F(B)were calculated for the activated carbons studied according to expressions given in Table I; these values are summarized in Table V. Figure 4 presents a comparison of the F(B) distributions (eq 6) for CWZ-3 (solid line) and BH (dotted line) carbons. The broader distribution for the BH carbon indicates that the structural heterogeneity of the microporous structure for this carbon is greater than that for the CWZ-3 carbon. Because the structural parameter B is related through eq 10 to the micropore dimension x , it is better to characterize the structural heterogeneity of microporous solids by the micropore size distribution J(x) (eq 12). The values x,, f , and a, that characterize the micropore size distribution J(x) were calculated for the activated carbons studied according to the equations given in Table I. According to
22
o O o o l - - - ~ -
-
~
0000 1250 2 500 Micropore Dimenslon x ( n m )
Figure 5. Comparison of the micropore size distribution J(r) (eq 12) for CWZ-3 (solid line) and BH (dotted line) activated carbons. Dubinin,I7the constant c in eq 10, established for benzene on activated carbons, is equal to about 0.00694 (mol/ (kJ-nm)*). The values x,, f , and a, are summarized in Table V. This table contains also the values of x that were evaluated from the B values obtained for the DR eq 4,i.e., by fitting the DR eq 34 for the benzene adsorption isotherms. It follows from Table V that for strongly heterogeneous microporous solids, e.g., HS-43 and BH activated carbons, the DR eq 34 leads to smaller values of x than the values of x , and f predicted by the JC eq 33. In comparison to the DR equation, the J C eq 33 provides values of x,, f , and a, that characterize more specifically the microporous structure of solids. In Table V we classified the activated carbons by means of the value of the dispersion a,. The smallest value of a, (i.e., the smallest structural heterogeneity) was obtained for CWZ-3 carbon, whereas the largest value of a, is for BH carbon. Figure 5 shows a comparison of the micropore size distributions J(x) for CWZ-3 (solid line) and BH (dashed line) carbons. This comparison shows that BH and CWZ-3 differ considerably; the BH carbon possesses a strongly nonuniform microporous structure, whereas the CWZ-3 carbon is more homogeneous with respect to the micropore sizes. Table V shows also that values of x , and f are valuable for characterizing activated carbons, e.g., the smallest value of f and x, was obtained for NSW carbon although its dispersion was larger than that for CWZ-3 carbon. Conclusions Theoretical considerations showed that the JC equation, viz., eq 5 , is associated with an adsorption potential distribution (eq 13) and a micropore size distribution (eq 12) that represent energetic and structural heterogeneities of microporous activated carbons. The equations that define
Langmuir 1988, 4 , 917-920 Table VI. Values of Y~ and 7%Calculated According to Eq AI and A2, Respectively, for the Values of n from the Interval (0,3) with the Step 0.05 n YA Y X n YA Yx 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50
0.3712 0.4950 0.5748 0.6323 0.6760 0.7105 0.7368 0.7617 0.7813 0.7979 0.8122 0.8248 0.8357 0.8454 0.8541 0.8619 0.8689 0.8753 0.8810 0.8862 0.8909 0.8954 0.8996 0.9033 0.9070 0.9102 0.9132 0.9161 0.9188 0.9213
0.8909 0.8954 0.8996 0.9033 0.9070 0.9102 0.9132 0.9161 0.9188 0.9213 0.9239 0.9259 0.9280 0.9300 0.9319 0.9337 0.9354 0.9370 0.9385 0.9400 0.9413 0.9427 0.9440 0.9452 0.9464 0.9475 0.9485 0.9496 0.9506 0.9516
1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00 2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00
0.9239 0.9259 0.9280 0.9300 0.9319 0.9337 0.9354 0.6370 0.9385 0.9400 0.9413 0.9427 0.9440 0.9452 0.9464 0.9457 0.9485 0.9496 0.9506 0.9516 0.9525 0.9534 0.9542 0.9550 0.9558 0.9565 0.9573 0.9580 0.9587 0.9594
0.9525 0.9534 0.9542 0.9550 0.9558 0.9565 0.9573 9.9580 0.9587 0.9594 0.9598 0.9606 0.9613 0.9618 0.9624 0.9630 0.9635 0.9640 0.9645 0.9650 0.9657 0.9660 0.9664 0.9669 0.9673 0.9677 0.9681 0.9685 0.9689 0.9693
the mean values and the dispersion for these differential distributions are simple and useful for characterizing activated carbons. The mean values of the adsorption potential and the micropore dimension as well as the dispersions of these quantities are recommended for char-
917
acterizing the energetic and structural heterogeneities of microporous activated carbons. This paper shows also that the JC equation generates simple equations for the differential and integral distributions of the adsorption potential and the micropore dimension. The differential enthalpy and the differential entropy of adsorption predicted by the JC equation were discussed in a previous paper.33 These theoretical studies indicate that the JC equation has many advantages in comparison to other equations (e.g., the DR equation); these advantages indicate a favorable forecast for its use in describing physical adsorption of vapors on microporous activated carbons and other heterogeneous microporous solids.
Acknowledgment. This work was supported in part by the Division of Chemical Sciences, Office of Basic Energy Sciences, Department of Energy. Appendix According to Table I the quantities yA and y x depend on the parameter n and are defined as follows: YA
= r(n+f/2)/[n1/2r(n)l
(AI)
Table VI contains the values of YA and y x calculated for values of n from 0 to 3 with a step of 0.05. Registry No. Carbon, 7440-44-0. ~~
(33) Jaroniec, M. Langnuir 1987, 3, 673.
Bonding Studies of Chromium-Nitrogen Molecules E. E. Mola,* E. Corone1,t Y. Joly,t and J. L. Vicente Divisibn Qulmica Tebrica, INIFTA, Sucursal4, Casilla de Correo 16, 1900 La Plata, Argentina Received September 9, 1987. I n Final Form: March 9, 2988 The Cr(lOO)-(lXl)N superstructure is naturally formed during the cleaning cycles of a Cr(100) sample. According to the annealing temperature the nitrogen present in the bulk of the substrate segregates to form the (1x1) structure. The analysis of a low-energy electron diffraction spectrum of such a surface remains unsatisfactory, and more sophisticated model calculations are required to describe properly the bonding strength and, eventually, charge transfer between nitrogen and the chromium cluster, in order to improve the agreement between experiment and theory. For those reasons, we present in this paper a theoretical study of NCr, Cr2,NCr2, Cr5, and NCr6 molecules. For this purpose we have employed the modified neglect of diatomic overlap (MNDO) method. From the present calculations it may be concluded that nitrogen binds very strongly to the chromium cluster with a net charge transfer to the metal. A metal work function reduction upon nitrogen chemisorption may be expected. This calculation also predicts a slight increase in the distance between chromium atoms induced by the adsorbate and larger values of the phase shifts than those obtained from the muffin tin approximation.
Introduction Low-energy electron diffraction (LEED) is a well-established method for studying well-ordered surfaces.l To determine adatom positions or first layer relaxations, the
analysis of diffracted beam intensities is indispensable.2 For this a theory-experiment comparison technique is necessary. Then we have to presume various geometries and make the diffracted beam intensity calculation for each geometry. The best agreement between experiment
* Author to whom correspondence should be addressed. 'On leave of absence from Universidad Mayor de San AndrBs, La Paz, Bolivia. *Onleave of absence from Laboratoire de Spectrometrie Physique de Grenoble, France.
(1) Pendry,J. B. Low Energy Electron Diffraction; Academic: London, 1974. (2) Van Hove, M. A.;Tong, S. Y. Surface Cristallography by L E E D Springer: Berlin, 1979.
0743-1463f 88f 240~-0911$01.50 f 0 0 1988 American Chemical Society
