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Figure 7. Constant C of the BET equation and the space requirement of an N2 molecule on partially DMB- and DM.Phcovered surfaces (fullsymbols)and on those of mixtures of densely
covered and untreated Si02/8.0 samples (open symbols) as a function of the relative coverage, 8. Full lines are traces of the evaluation of the isotherm obtained by linear combination of the individual isotherms; dashed lines are drawn by eye.
with exposed methyl groups (DM.H, DM.M, DMB), on slightly polar surfaces with exposed phenyl groups (DM.Ph), on medium polar surfaces with exposed methoxy, dimethylamino, and cyano groups (DMP.MO, DMP.A, DMP.CN), and finally also on ionic surfaces (DMP.QA.Br and Cl).
695
We put now forward the question of U(Nz)on surfaces that are mixtures of the two classes. In a first series of experiments N2 isotherms were determined on mixtures of treated and untreated sio2/8.0 samples. As anticipated, the properties of the isotherms on these mixtures were the same as the properties of isotherms calculated by linear combination of the individual isotherms as shown in Figure 7. In a second series of experiments heterogeneous surfaces were prepared by partidy covering Si02/8.0 samples with DMB or DM.Ph. On such adsorbents intermediate values between U(N2)= 20 and 16 A2 were observed. Kiselev et al.35 found similar values on partially trimethylsilylated surfaces. In conclusion, on homogeneous surfaces the first adsorbed layer of Nz can have two structures. Loose layers are formed on polar or nonpolar organic surfaces as well as on ionic surfaces, whereas dense layers were observed on surfaces where hydroxyl groups are exposed. Intermediate densities were found on heterogeneous surfaces.
Acknowledgment. This paper reports on part of a project financed by the Fonds National Suisse de la Recherche Scientifique. We thank Dr. Ph. Schneider for help in organic syntheses and Dr. F. PB1 for discussions. Registry No. N,, 7727-37-9. (35) Kiselev, A. V.; Korolev, A. Ya.; Petrova, R. S.; Shcherbakova, K.
D.Kolloidn. Zh. 1960,22,671.
Heterogeneous Adsorption on Microporous Carbons? B. McEnaney,* T. J. Mays, and P. D. Causton School of Materials Science, University of Bath, Claverton Down, Bath BA2 7AY, U.K. Received October 21, 1986. I n Final Form: February 23, 1987 The influence of heterogeneity in microporous carbons on adsorption of gases is studied by using the generalized adsorption isotherm (GAI) as a model. Adsorption of Ar at 77 K on two poly(vinylidenechloride) based carbons activated to 28 wt 70 (PVDC-28) and 80 w t % (PVDC-80) burn-off is measured by using a McBain spring-type balance over a pressure range from 6.7 X lo4 to 23 kPa, corresponding to a relative pressure range of (2.3 X 106)-0.79. The GAI comprises three functions: (i) the total isotherm (determined experimentally), (ii) the local isotherm (assumed), and (iii) the adsorption energy distribution (unknown). Problems of computing the energy distribution are considered. Two previously published analytic solutions of the GAI (Sircar's equation and the condensation approximation)and one numerical method (regularization, incorporating a new smoothing algorithm) are applied to the adsorption data. For all three methods the Langmuir equation is assumed for the local isotherm function. For both carbons the energy distributions obtained from the three methods are similar, but the dispersions of the distributions obtained from regularization are wider than those obtained from Sircar's equation. The distribution functions obtained from the condensation approximation are similar to those obtained from regularization, suggesting that it is a good approximation for adsorption of Ar at 77 K on these carbons. When the two carbons are compared, although PVDC-80 gives a slightly wider dispersion than PVDC-28 for all three methods, the energy distributions do not change a great deal with burn-off. The similarity in shape of the energy distributions is probably a reflection of the similarity in shape of the experimentally determined total isotherms. It is concluded that, although activation from 28 wt % to 80 wt % burn-off increases adsorptive capacity substantially, there is no significant increase in the mean width of micropores. 1. Introduction Microporous carbons are disordered solids with a structure consisting of a twisted network of defective carbon layer planes, cross-linked by an array of aliphatic *To whom correspondence should be addressed. Presented at the "Kiselev Memorial Symposium", 60th Colloid and Surface Science Symposium, Atlanta, GA, June 1518,1986; K. S. W.Sing and R. A. Pierotti, Chairmen. 0743-7463/87/2403-0695$01.50/0
bridging groups, and incorporating heteroatoms, for example, €3 and 0.Micropores are the spaces between the layer Planes whose widths are less than 2 nm.' It is highly likely that adsorption on such a material is strongly influenced by the heterogeneity of the structure. It is therefore of interest to explore the application of the (1) Sing, K. S. W.; Everett, D. H.; Haul,R. A. W.; Moscou, L.; Pierotti, R. A.; Rouquerol, J.; Siemieniewska, T. Pure Appl. Chem. 1985,57,603.
0 1987 American Chemical Society
696 Langmuir, Vol. 3, No. 5, 1987
McEnaney et al.
generalized adsorption isotherm (GAI) to adsorption on microporous carbons and, in particular, to assess the influence of systematic changes in the structure of the carbons upon results from the GAI. Extensive review^^-^ show that many analyses of adsorption on heterogeneous solids use the homotattic patch approximation in which is is assumed that the adsorbent surface consists of an array of patches of uniform adsorption energy, q , each of which fills according to a local isotherm, B(p,q), where p is the equilibrium gas-phase pressure. If it is assumed that the size of the patches is small compared with the total adsorbing area, so that there is a large number of patches, then the distribution of adsorption energies, f ( q ) , can be considered to be continuous and the total adsorption isotherm, e@),is given by
This integral equation is the GAI. In most treatments of heterogeneous adsorption the GAI is formulated for submonolayer adsorption on a plane surface. In the case of microporous solids the adsorption energy is enhanced as a result of the proximity of the pore walls, leading to strong adsorption, termed primary or micropore filling, at pressures which are low relative to the saturated vapor pressure of the liquid adsorbate. If it is assumed that each micropore can be assigned a single value of q, then fl(p,q) represents the local isotherm for the filling of micropores of adsorption energy q and 8 ( p ) is the fraction of the micropore volume filled a t p . In this paper the problems of solving the GAI are considered and two previously published analytic solutions, which are used in this work, are briefly described. A numerical technique called regularization is also used to solve the GAI. This method, incorporating a new optimal smoothing algorithm, is presented in detail. The three methods are applied to adsorption of Ar a t 77 K on two microporous carbons activated to different levels of burn-off. 2. Solutions of the GAI The GAI is a Fredholm equation of the first kind.5 It is linear in the function f(q), the form of which is unknown except that it satisfies the requirements of a probability density function, namely, it is nonnegative and it is normalized to unity, that is J
4min
The local isotherm B(p,q) is the kernel of the Fredholm equation and the total isotherm 8 ( p ) is the driving term. The problem of solving the CAI for f ( q ) given B(p,q)and 0(p) is ill-p~sed.~ This means that small perturbations in 8(p) can give rise to large perturbations in f(q). 111posedness also means that very similar driving terms can result in radically different solutions. In practical terms the perturbations in 8(p) are experimental errors. In this case the GAI becomes fi(pi)= ~ q m x O ( p i , q ) f ldq (q+ ) q; i = 1, 2, 3, ..., I
(3)
9"
where fi(pJ is the measured specific amount adsorbed at
prescure p i and is related to the total isotherm by fi(pi) = m 8 ( p i ) . The parameter m is the specific amount adsorbed in the adsorption space; it is incorporated in the energy distribution by using the relation f l ( q )= mf(q).In eq 3 ti is the error associated with the ith measurement and I is the total number of data. The symbol refers to measured values. Methods developed for solving the GAI involve first choosing a local isotherm function and then (i) smoothing the solution, (ii) smoothing the total isotherm data, or (iii) smoothing the solution and the data, so that perburbations in the solution are minimized. For each of the three methods used in this work to solve the GAI, the Langmuir equation was chosen as the local isotherm function. This assumes that adsorption is localized and that there are no adsorbate-adsorbate interactions. The equation of the Langmuir isotherm is bP (4) B(p,q) = 1 bP where the factor b is given by b = bo exp(q/RT) (5) The constant bo is given from simple kinetic theory6 as
-
+
bo =
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where NA is Avogadro's number, u is the area of the adsorption site, ro (exp(q/RT) is the residence time of an adatom on a site of adsorption energy q and at absolute temperature T , M is the atomic weight of an adatom, and R is the gas constant. The area of the adsorption site can be equated to the cross-sectional area of the adatom, which for Ar at 77 K is 0.138 nm2.7 It is convenient to write p L = l/bo, and assuming r0 = s (ref 6) then for Ar at 77 K p L = 152.6 MPa. 2.1. Analytic Solutions of the GAI. A number of analytic methods have been used previously to solve the GAI.899 For example, in the method of direct integration a smooth analytic function is assumed for f(q) which allows direct integration of the GAI to give an analytic expression for 8(p),the parameters of which can be estimated from the data, for example, by regression analysis, and incorporated into f ( q ) to define a solution. The method of integral transforms is the reverse of the method of direct integration: a smooth analytic function is assumed for e@),the parameters of which can be estimated from the data. For particular functions 8(p) and 6(p,q)the GAI is an integral transform which can be inverted to give f ( q ) . The first method used in this work to solve the GAI is a direct integration proposed by Sircar8 where f ( q ) is assumed to be a gamma-type function given by
where c = l/pL[exp(q/RT) - 11 (8) and a > 0 and n = 0, 1, 2, ... are parameters of the distribution. Integration of the GAI gives N P )= m[l - ( a / p ) exp((a/p) + (a/PL))En+l((a/P) + ( ~ / P L ) ) I (9)
~
(2) Zolandz, R. R.; Gyers, A. L. In Progress in Filtration and Sepa~ ration; Wakeman, R. J., Ed.; Elsevier: Oxford, 1979; Vol. l , .l. (3) Jaroniec, M.; Patrykiejew, A.; Borowko, M. In Progress zn Surface and Membrane Science; Cadenhead, D. A., Danielli, J. F., Eds.; Academic: London, 1981; Vol. 14, p 1. (4) Jaroniec, M. Adu. Colloid Interface Sci. 1983, 18, 149. (5) Miller, G. F. In Numerical Solution of Integral Equations; Delves L. M., Walsh, J., Eds.; Clarendon: Oxford, 1974; Chapter 13, p 175.
(6) Adamson, A. W. Physical Chemistry of Surfaces, 2nd ed.; Interscience: London, 1967; Chapter 8, pp 571-572. (7) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity, 2nd ed.; Academic: London, 1982; Chapter 2, p 75. (8) Sircar, S. J. Colloid Interface Sci. 1984, 98, 306. (9) Cerofolini, G. F. Thin Solid Films 1974, 23, 129.
Langmuir, Vol. 3, No. 5, 1987 697
Heterogeneous Adsorption on Microporous Carbons where Ek(z)is the kth order exponential integral defined as
which has been used beforel' in regularized solutions of the GAI. Accordingly, the problem now is to minimize with respect to the f l ( q j ) the following functional
G+yS; y 3 0 Values of the three parameters a,n, and m are estimated by a nonlinear least-squares fit of the data to eq 9. The second method used here to solve the GAI is the condensation approximation in which the local isotherm is represented by a step function and the total isotherm is given by the Dubinin-Radushkevich, DR, equation
N p ) = m exp[-D2 In2 ( p 0 / p ) I
(11)
where D is the DR constant and p o is the saturated vapor pressure of the liquid adsorbate. For supercooled, liquid Ar a t 77 K, p o = 28.9 kPa.'O The values of the two parameters D and m are estimated by a nonlinear leastsquares fit of the data to the DR equation. When the Langmuir isotherm is represented by the step isotherm, differentiation of the DR equation gives
where q A = RT/D and q D = RT ln (pL/po)= 5.49 kJ mol-'. This method, described in detail by C e r ~ f o l i n i ,is~ a modification of the method of integral transforms where the problem of inverting the GAI to give f ( q ) is replaced by the simpler one of differentiating the total isotherm. 2.2. A Numerical Solution of the GAI: Regularization. The third method used in this work to solve the GAI is regularization in which the solution is smoothed numerically. The method is as follows. A measure of the goodness of fit of the model to the data in eq 3 is the sum of squared errors which, when the integral is approximated by a quadrature (numerical integration formula), is given by J G = C[fl(pJ- Ce(pi, qj)fl(qj)6qj12; j = 1, 2, 3, -9
where 6q, is the jth weight in the J-point quadrature. The least-squares solution to eq 3, that is, the minimum of G with respect to f l ( q j ) , j = 1,2, 3, ...,J, is usually a wildly oscillating function which does not satisfy the nonnegativity constraint of a probably density function. The instability of the solution is reduced by incorporating the nonnegativity constraint and the additional constraint that f(q) should be zero a t qminand (Imax, that is, fl((71)
= f'(SJ) = 0
fl(qj) > 0; j = 2, 3, 4,
..., (J- 1)
(14)
However, even with these constraints the least-squares solution often remains uneven and irregular. Without additional information there is no reason for rejecting any solution, irregular or otherwise. The method of regularization provides additional information by assuming that the solution is smooth in some sense. Regularization requires that the solution should minimize jointly the goodness-of-fitfunctional, eq 13, and a functional S which smooths by causing perturbations in the solution to be small. The smoothing functional used in this work is
(10)Clark, A. M.; Din, F.; Robb, J.; Michels, A.; Wassenaar, T.; Zwietering, Th. Physica 1951,17, 876.
(16)
subject to the constraints in eq 14. The Lagrange multiplier y in eq 16 is called the smoothing parameter; it represents the relative weighting of the two functionals. As y increases the solution is smoother but the goodness of fit is worse and vice versa. Optimal smoothing in regularization is in theory achieved by finding the y which minimizes the difference between the approximate solution f l y ( q )and the true solution f l ( q ) . However, this method is hypothetical since the true solution in practical cases is unknown. Accordingly, without further information on the solution, optimal smoothing methods are semiemprical, relying more or less equally on statistical and subjective criteria. The smoothing algorithm used in this work is as follows. The constraints on f ( q ) , eq 14, are relaxed so that the minimumum of eq 16 is found simply by solving a set of linear simultaneous equations in the unknowns fi(qj),j = 1, 2, 3, ..., J. With q1 = qmin = 0, it is found that for a given y there is maximum qmaxwhich satisfies the constraints in eq 14. It is also found that there is a minimum y for which a maximum qmaxcan be determined: this critical y is considered to be the optimal value of the smoothing parameter. Accordingly, this smoothing criterion supplies the least-smoothed solution with the widest range. I t is considered that this criterion gives the maximum information on the solution subject to the smoothing and nonnegativity constraints. Once a solution is obtained it can be integrated to give the adsorption capacity by using the relation fl(q) = mf(q)and the normalization condition, eq 2. The quadrature used in this work is a 33-point Simpson's rule. Application of Sircar's equation and the method of regularization to adsorption data enables analytic and numerical solutions of the GAI to be compared. Comparison of the solution obtained by using the condensation isotherm with the solutions from the other two methods allows the validity of this approximation to be assessed. Finally, the statistical analyses of the goodness of fit of the models to the experimental data allow the appropriateness of the Langmur kernel to be tested. 3. Experimental Details The carbons used in this investigation were prepared from a high molecular weight vinylidene chloride, 20 wt 70acrylonitrile copolymer (British Drug Houses) by carbonization in Nzat 1273 K; the material is termed PVDC carbon. Activated samples were prepared from the PVDC carbon by reaction to 28 wt % and 80 w t % burn-off in a H20/H2/Ar gas mixture at atmospheric pressure and at 1233 K. The two activated carbons are termed PVDC-28 and PVDC-80. After outgassing at less than 13 mPa for 12 h at 523 K, adsorption of Ar at 77 K was measured by using a McBain spring-type adsorption balance over a pressure range from 6.7 X to 23 kPa, corresponding to a relative pressure range of (2.3 X 10-5)-0.79. 4. Results and Discussion
The adsorption isotherms for the two carbons, Figure 1, are typical of microporous carbons. They rise steeply at low relative pressures, below about 0.05, with compar(11)Britten, J. A.; Travis, B. J.; Brown, L. F. AIChE Symp. Ser. 1983, 79 (230), 7.
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Table I. Parameters Determined from Three Methods Used To Solve the Generalized Adsorption Isotherm for Adsorption of A r at 77 K on Two Microporous Carbons Activated to 28 and 80 wt % Burn-Off burn-off method 28 wt % 80 wt % Sircar's (eq 7-9) m = 224.8 mg g-' m = 440.4 mg-' n=O n=O a = 3.36 Pa a = 4.87 Pa condensation approx m = 238.1 mg g-' m = 471.3 mg g-l q A = 6.38 kJ mol-' q A = 5.93 kJ mol-' (eq 11, 12) regularization m = 235.2 mg g-' m = 461.1 mg g-' y = 5.1 X = 4.4 x 10-3 (eq 13-16) qmax= 15.3 kJ mol-' qmax= 18.4 kJ mol-'
Figure 1. Total isotherms for adsorption of Ar at 77 K (saturated vapor pressure, p o = 28.9 kPa) on two microporous carbons: (0) activated to 28 wt % burn-off; (0) activated to 80 wt % burn-off. 320
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Figure 2. Adsorption energy distributions obtained from three methods used to solve the generalized adsorption isotherm for adsorption of Ar at 77 K on a microporous carbon activated to 28 wt % burn-off: (-) Sircar's method, eq 7 , 8 (---) condensation regularization. approximation, eq 12; (0) atively little adsorption taking place a t higher relative pressures. The shapes of the two isotherms are very similar, despite the large difference in burn-off, which suggests qualitatively that both carbons are highly microporous. The three methods for solving the GAI were applied to the isotherms in Figure 1 and the resulting solutions for PVDC-28 are in Figure 2; very similar distributions were obtained for PVDC-80. Parameters of the Sircar equation and the DR equation and the optimal smoothing parameters from regularization for both carbons are in Table I. For all three methods the means of the energy distributions are similar; however, the f(q) obtained from regularization is wider that the f(q) obtained from the Sircar equation. The f ( q ) obtained from the condensation approximation is similar to that from regularization but with slightly greater dispersion. Thus it appears that the condensation approximation is a good one for adsorption of Ar at 77 K on these carbons. However, the condensation approximation is only applicable at low temperatures and it cannot be assumed that it applies at higher temperatures. Calculations of adsorption potentials in model micropores12show that the maximum value of q in micropores is about twice the value of q on the plane surface. This suggests that qmaK= 2qmin.For Sircar's equation the ratio qmax/qmtn = 1.8,but for the condensation approximation and the method of regularization qmax/qmin= 4.0. For the condensation approximation it is known that the approx~ regulariimate f ( q ) is wider than the accurate f ( ~ 7 ) . For (12)Everett, D.H.; Powl, J. C. J. Chem. SOC.,Faraday Trans. I 1976, 72, 619.
Figure 3. Adsorption energy distributions obtained by using the method of regularization to solve the generalized adsorption isotherm for the adsorption of Ar at 77 K on two microporous carbons: (0) activated to 28 wt % burn-off; (0) activated to 80 wt % burn-off. zation the smoothing algorithm is designed to give the least smoothed solution with the widest range in an attempt to give the maximum information on the energy distribution. Additional constraints could be included in the smoothing algorithm, for example, by setting the values of qmaxand qmin so tht qmax/qmin= 2; this idea will be considered in a future publication. The distribution functions obtained from regularization for the two carbons are compared in Figure 3. These curves show that although PVDC-80 gives a slightly wider dispersion of f ( q ) than PVDC-28, the energy distribution does not change a great deal with burn-off. Similar results are obtained using Sircar's equation and the condensation approximation, but the solutions obtained from these methods are less sensitive to burn-off. This is at first sight a surprising observation, since the process of activation from 28 to 80 w t 5% burn-off has increased the adsorptive capacity substantially. For example, the micropore capacities for PVDC-28 and PVDC-80 estimated using the condensation approximation are 238 and 471 mg g-l, respectively. The increase in adsorptive capacity which follows from activating carbons is usually accompanied by a widening of micropore sizes, sometimes extending them into the mesopore size range (widths 2-50 nm'). Such a process might be expected to result in a decrease in the mean of f ( q ) and an increase in its dispersion, since calculations of adsorption potentials in model micropores show that adsorption energies decrease rapidly with increasing pore size.12 In the present case it appears that activation increases adsorptive capacity without significantly increasing micropore size since the shapes of the adsorption isotherms for the two carbons are very similar. Possibly the widening of existing micropores is accompanied by the creation of new, small micropores. Thus the similaricy in shape of the distribution functions in Figure
Langmuir 1987, 3, 699-703 2 is probably a reflection of the similarity in shape of the experimental isotherms in Figure 1. When fitting models to experimental data, it is necessary to ensure that the residuals are not biased, that is, they do not vary systematically with the independent variables. Inspection of the present results shows that for all three methods the residuals are biased, the model tending to overpredict the amount adsorbed at low pressures and to underpredict at high pressures. It is probable that this indicates a limitation of the Langmuir equation as the local isotherm for this system. The choice of the Langmuir equation for the local isotherm implies that the total isotherm has the same limits, that is, tending to Henry’s law at low pressures and to saturation at high pressures. Neither of these limits is observed over the range of pressures for this work. The wide range of q for regularization resulting in qma/qmin = 4 may also be due in part to the bias resulting from the use of the Langmuir kernel. Better results may be obtained in future work either by extending the range of pressures or by choosing a local isotherm whose dependence on pressure is similar to that of the total isotherm.
5. Conclusions Two analytic solutions to the GAI (Sircar’s equation and the condensation approximation) and one numerical me-
699
thod (regularization incorporating a new smoothing algorithm) were applied to adsorption of Ar at 77 K on two microporous carbons activated to 28 wt % and 80 w t % burn-off. For both carbons the energy distribution functions obtained from the three methods were similar but the dispersions of the distributions obtained from regularization were higher than those obtained from Sircar’s equation. The distribution functions obtained from the condensation approximation are similar to those obtained from regularization, suggesting that it is a good approximation for adsorption of Ar at 77 K on these carbons. Comparing the two carbons, although PVDC-80 gives a slightly wider dispersion than the PVDC-28 for all three methods, the energy distribution does not change a great deal with burn-off. It is concluded that, although activation from 28 to 80 wt % burn-off increases adsorptive capacity substantially, there is no significant increase in the mean width of micropores.
Acknowledgment. We thank the Science and Engineering Research Council of the UK and the British Gas Corporation for financial support. B.McE. also thanks the American Chemical Society for a travel grant which enabled him to present this paper at the Kiselev Memorial Symposium. Registry No. Ar, 7440-37-1; C, 7440-44-0.
Computer-Controlled Vacuum Microbalance Techniques for Surface Area and Porosity Measurements? K. A. Thompson* and E. L. Fuller, Jr. Plant Laboratory, Oak Ridge Y-12 Plant,t Martin Marietta Energy Systems, Inc., Oak Ridge, Tennessee 37831 Received November 24, 1986. I n Final Form: February 24, 1987 A versatile, automated, high-vacuum microbalance system has been constructed and evaluated for obtaining extensive and precise physical adsorption and desorption data. A dedicated minicomputer (LSI 11/23) is used to monitor and control the relevant parameters (temperature, pressure, mass, etc.) so that detailed kinetics and mechanisms can be evaluated. In this manner, each value of the adsorption isotherm is constructed as the composite result of a rapid isochoric pressure change followed by a kinetically controlled isobaric approach to the final steady-state mass at the fixed temperature and pressure of interest. With the aid of the computer, it is often possible to calculate the steady-statemass before equilibrium by monitoring the rate of mass change. However, the system was designed to be as versatile as possible, not depending on any single-order kinetics but using it to full advantage when required. The test sample used for initial testing was a commercially available silica-supportedalumina catalyst (A1203) for which nitrogen and argon sorption isotherms were acquired. True hysteresis is shown to prevail by virtue of the isobaric kinetic accountability that cannot be determined volumetrically. Other comparisons are given, and the merits of the automated high-vacuum microbalance become quite evident when nonroutine, detailed exploratory studies are required.
Introduction Physisorption measurements are a common and often preferred method of determining surface area and pore-size distributions. Since the advent of the Brunauer-Emmett-Teller (BET) theory of “multilayer adsorption”,’ Presented a t the “Kiselev Memorial Symposium”, 60th Colloid and Surface Science Symposium, Atlanta, GA, June 15-18,1986; K. S. W. Sing and R. A. Pierotti, Chairmen. *Operated for the US.Department of Energy by Martin Marietta Energy Systems, Inc., under Contract DE-AC05-840R21400.
0743-7463/87/2403-0699$01.50/0
sorption isotherm analysis for surface area has become commonplace for industrial use. For this application, the absolute value of surface area or porosity is not required and is used only as a relative parameter. In cases in which absolute values are needed, more detailed analyses are required, along with further improvements upon the theory. Such attempts have been made in the past, such as the potential theory,2 as,or t method^.^,^ For further (1) Brunauer, S.; Emmett, P. H.; Teller, E. 60. 309.
0 1987 American Chemical Society
J.Am. Chem. SOC.1938,
