J. Chem. SOC., Faraday Trans. I , 1984, 80, 1467-1477 Physical Adsorption of Gases on Heterogeneous Surfaces Series Expansion of Isotherms using Central Moments of the Adsorption Energy Distribution BY JAMES A. O’BRIEN AND ALAN L. MYERS* Chemical Engineering Department, University of Pennsylvania, Philadelphia, Pennsylvania 19104, U.S.A. Received 14th July, 1983 An alternative approach to the use of the usual integral equation for heterogeneous adsorption is developed. It provides a means of obtaining an explicit result for arbitrary local isotherms and distributions of energy of adsorption, The approximation is compared with exact numerical integrations, using the Langmuir local isotherm and a normal energy distribution. The series fails to converge for energy distributions that are very wide.Truncation of the expansion after the first-order term yields an equation which is capable of fitting experimental isotherm data. Although the existence of surface heterogeneity has long been recegnized, much of the prcgress in the field has been made in the last 15 years. There are some comprehensive review articles available, of which the most recent is ref. (1). Other reviews of the problem include those by Jaroniec2 and Zolandz and Myers;3 the latter deals primarily with single-component gas adsorption on heterogeneous solids. The basis for most approaches has been to account somehow for a heterogeneous surface as a collection of homogeneous surfaces and to sum the resultant effects. It is obvious that this becomes quite complex as heterogeneity becomes more pronounced.However, the very complexity of the problem leads to simplification in that hetero- geneity may be taken into account by the methods of probability theory. It is generally postulated that a heterogeneous surface is composed of many energetically homogeneous, non-interacting patches, usually referred to as homotattic patches. Furthermore, each patch is considered to be large enough to assume the existence of a so-called local isotherm for that patch and, at the same time, small enough for the surface energies to be characterized by a continuous probability density function. These adsorption energies depend in general on both adsorbate and adsorbent. If the probability of a patch having an energy between E and E+ dE i s f ( E ) dE, then the key equation for heterogeneous adsorption is This is simply the expected value of O,(T,p, E), where E is distributed in the manner specified byf(E) over a domain of energies, D.Eqn (1) is the starting point for most treatments of heterogeneity, but it has been employed in different ways by various authors. Several, including S ~ P S , ~ ~ Misraa3 and Toth,8 assumed that O,( T , p , E ) was given by the Langrnuir isotherm, and they derived the function, f(E), explicitly for certain isotherms O(T,p) by use of the Stieltjes 14671468 ADSORPTION OF GASES ON HETEROGENEOUS SURFACES transform. A similar method was used by Jaroniec and Sokolo~ski,~ who derivedf(E) by means of the Laplace transform, assuming that OL(T,p,E) was given by the Jovanoviclo isotherm and O( T, p ) was the Dubinin-Radushkevich equation.A different type of approach was taken by Myers and 0u,l1 who assumed that OL(T,p, E ) was a step function (the ‘condensation approximation’) and that the energies followed a beta distribution. Eqn ( I ) was then integrated analytically to obtain a new isotherm which is capable of fitting experimental data for heterogeneous adsorption. Still another method of using eqn (1) has been to pick the form of O,( T , p , E ) (for example Langmuir) and, using experimental data for O( T,p), to calculate f(E) numerically. Probably the best-known algorithms for such calculations are CAEDMON by Ross and Morrison12 and HILDA by House and Jay~0ck.l~ Considerable care is necessary in attempting to perform these calculations, as the problem is ill-posed and there are complications associated with numerical stability if the data contain random pertur- bations (which will be the case with all experimental data).In addition, it has been shown by House and c o ~ o r k e r s ~ ~ ~ l5 that the function,f(E), obtained depends strongly on the form of the local isotherm that is chosen. In this paper we present an approach different from those outlined above. We reformulate the right-hand side of eqn (1) as an infinite series whose terms involve (a) derivatives of OL(T,p, E ) with respect to E, evaluated at the mean of the energy distribution, and (b) the central moments of the distribution. This has an important advantage over the previously mentioned methods since the integral in eqn (1) can only be analytically evaluated for a very limited class of combinations of OL(T,p,E) andf(E), beyond which one must resort to numerical integration.With the approach presented here nb integration is necessary. In addition, if the infinite series converges rapidly one can evaluate O( T, p ) with knowledge of only the first few central moments of the energy distribution. This can also provide information on how the shape of the distribution affects the generated isotherm O(T,p), since the central moments of a distribution are related to its shape. DEVELOPMENT OF EXPANSION PROCEDURE Consider a general case of eqn (1) where we take the expectation of a function of a random variable, g(x), with respect to its probability density function, f ( x ) , valid on the domain D of the random variable, x: Suppose that the mean offlx) is p and expand g(x) in a Taylor series about this point.This requires that g(x) have an infinite number of derivatives with respect to x at this point and also that it be continuous there. Thus where, for compactness, dng g y x ) = -(x) dxn g q x ) = g(x).J. A. O’BRIEN AND A. L. MYERS 1469 Inserting eqn (3) into eqn (2) and interchanging the order of integration (note that the subtleties and implications of this operation for convergence have been overlooked for the present), we obtain We now recognize that the quantity in square brackets in eqn (4) is the nth central moment off@), which is normally denoted by pn = E [ ( X - ~ ) ~ ] = (x-p)nf(x)dx. ( 5 ) D Note that ,u is not one of the p n : it is, however, the symbol most commonly employed to denote the mean of the distribution.Inserting the definition eqn (5) into eqn (4) All that now remains is to transcribe eqn (6) to the usual notation of heterogeneous adsorption. From here on it is convenient to work not with the energy, E, but with the dimensionless quantity E-Emin Z T RT * (7) All other energy variables referred to later are also made dimensionless by dividing through by RT. We also note that the local adsorption isotherm e,(T,p, z ) must possess the properties demanded of g(x) in the above treatment. The result is: Eqn (8) is the basis for the remainder of this work. Note that we have replaced the problem of integrating eqn (1) by that of differentiation and summation. Nothing has been said, up to this point, about the convergence properties of eqn (8).However, if we regard eqn (8) as a perturbation expansion on OL(T,p,p) it seems intuitive that convergence may be expected when the moments p n are sufficiently small. In general, by definition, Po = 1, Pl = o Now, p2 is more commonly denoted by 02, the variance of the distribution. In addition one can define the parameterslG a,=? P a, G1” o3 ’ 0, (9) where a3 is the ‘coefficient of skewness’ and is a measure of the asymmetry off(z), and a4 is the ‘kurtosis’ of f(z) and is proportional to the ‘peakedness’ of the distribution. Thus both the normal and the uniform distribution have a3 = 0. For the uniform distribution a, = 1.8 and for the normal distribution a, = 3. Eqn (8’) may be rewritten as1470 ADSORPTION OF GASES ON HETEROGENEOUS SURFACES where the second, third and fourth terms may be identified with the effects of spread, asymmetry and peakedness of the distribution, respectively.We note from eqn (10) that if the series is converging rapidly the effect of the width of the distribution is more pronounced than that of its detailed shape. IMPLICATIONS FOR HENRY’S CONSTANT The above derivation has an interesting implication for the effect of heterogeneity on the Henry’s constant (or surface second virial coefficient), defined as de e p+odP p+oP H E lim - = lim -. Consider eqn (6), where we can insert HF)(p) for g ( n ) ( p ) and H for Y. HL(z) is the Henry’s constant for the local isotherm O,( T,p, z) on a patch of energy z. Performing this substitution and noticing that since for all local isotherms, H(Ln)(p) = H&) (for all n) HL(z) cc ez we arrive at This is a general result, applying to adsorption on a surface of arbitrary adsorption- energy distribution.By picking various distribution functions, JTz), with their associ- ated central moments, we can now obtain an interesting insight into the dependence of H on heterogeneity. For a uniform distribution of energies with mean p and variance 02, it can be shown that Similarly, for a Gaussian distribution of energies, given by it can be shown that H = H,(p)exp - (3 where o2 and p have the same meanings as in eqn (14). In the derivation of eqn (16) it is assumed that the probability of negative energies can be neglected, a physically realistic situation. This last result is essentially the same as that derived by Pierotti and Th0mas.l’ Both eqn (14) and eqn (16) illustrate the point that the variance or spread of the energy distribution strongly influences the Henry’s constant of the heterogeneous isotherm, with a functionality that is exponential in character. It can also be seen that modest values of o will produce isotherms which have high values of slope at zero coverage, an observation which is typical in experimental data for adsorption on heterogeneous surfaces.In addition, these equations demonstrate that H does not depend merely on the energy of the highest-energy patch on the surface.J. A. O'BRIEN AND A. L. MYERS 1471 TESTING OF THE SERIES EXPANSION: A MODEL STUDY The energy distribution employed here is the Gaussian form given in eqn (1 5), under the condition that negative energies can be neglected.For such a density function it can be shown15 that ( n - l)(n-3) ...( 5)(3) (l)on,neven pn=( 0, n odd. The disappearance of all central moments of odd order is a direct result of the symmetry of the density function about the mean. All that now remains is to choose a local isotherm. A preliminary investigation of the Hill-de Boer equation for a two-dimensional van der Waals fluid has been undertaken because it accounts for adsorbate-adsorbate interactions. This equation may be written p = 1 -eL ~,e-Z(*)exp[(*)--~,~~] I -eL where the significance of Kl and K , are well understood.'* However, the differentiations required for substitution into eqn (8') quickly become unmanageable owing to the fact that eqn (18) is implicit in 8,.Therefore we did not carry the expansion beyond the second derivative in eqn (1 0). Another suitable choice, although physically less realistic than eqn (18) at least in terms of its neglect of intermolecular interactions, is the Langmuir isotherm: Note that temperature is implicitly contained in z owing to the method of non- dimensionalizing energy variables in use here. Unlike eqn (18), eqn (19) does not exhibit so-called cooperative effects, but it has, nevertheless, been frequently used as a local isotherm in studies of heterogeneity. It also avoids the problem of distinguishing between the patchwise and random models of adsorption heterogeneity. There is a further, more pragmatic, reason for its use here, and that is to avoid problems associated with taking an arbitrary number of derivatives.Rewriting eqn (19) in the form where we have defined x by it can be shown that the general form of the nth derivative is (20) x = CpeZ (21) X &(x) = - x+ 1 n A recursion relation may be derived which gives the coefficients an+,, in terms of the an, as follows: - an+,,, - -%.,I an+,, n+l - an, n = 1. a,,,, i+l = (n - i+ l)a,, - ( i + l)an, i+l (i = 1,2, . . . , n- 1) -1472 ADSORPTION OF GASES ON HETEROGENEOUS SURFACES The procedure may be initiated by noting that so that al, = 1 . By recursion, as many z-derivatives of the Langmuir equation as desired may be calculated. We note that the expansion will consist of terms involving the product of a central moment of the distribution and a rational function of p.Eqn (20)-(22c) together with eqn (17) may now be easily combined in an algorithm for generating approximations to any desired order, which we implemented as a short program in BASIC computer language. For purposes of illustration, the first few terms of the expansion, eqn (S'), are presented using the Langmuir local isotherm with x evaluated at the mean energy z = p. For an arbitrary density function This technique may also be applied to the Jovanoviclo equation or, rewritten in terms of x from eqn (21), &(x) = 1 -eP5. The general form of the nth derivative here is d"8 n L = e-5 an,ixi dzn i - 1 I I I I I I I I 1 Z Fig. 1. Normal distribution of energies used to calculate adsorption isotherms.Mean energy p = 10. (1) o/p = 0.075, (2) o/p = 0.125.J. A. O’BRIEN AND A. L. MYERS 0.8 I 1 I I I 0.7 - 0.6 - 0.5 - 1473 3 P Fig. 2. Exact numerical integration of eqn (1) using Langmuir local isotherm and distributions shown in fig. 1 . (1) o/p = 0, (2) g/p = 0.075, (3) g/p = 0.125. and we have, once again, a set of recursion relationships - an+,, 1 - an, 1 = 1 an+l,i+l = (i+ l)an,i+l-an,i %+I, n+1 - -an, n - (i = 1, . . . , n - l ) - Once again, the procedure is started by noting that so that al, = 1. To evaluate the effectiveness of the expansion exact results were necessary. These were obtained by numerical integration of eqn (1) using eqn (1 5) and (19) asf(z) and 8,( T,p, z), respectively. The algorithm employed was that of Romberg integration.19 It was necessary to choose specific values of the constants p, C and B before proceeding.We set p = 10, which corresponds to an energy of 24.94 kJ mol-1 at 300 K. This is the correct order of magnitude for a typical heat of adsorption. The value of C = e-l0 was chosen so that 8 = 0.5 atp = 1 .O. Since the product (Cp) is dimensionless, the units of p are insignificant. Finally, the parameter 0 was varied to investigate the effect of heterogeneity. RESULTS OF MODEL STUDY The density function,flz), is plotted on fig. 1 for values of B studied. Fig. 2 presents the exact, numerically integrated isotherms for each B. The corresponding results,1474 ADSORPTION OF GASES ON HETEROGENEOUS SURFACES 0 0.5 1.0 1.5 2.0 2 5 3.0 P Fig. 3. Convergence of eqn (8’): percentage error as a function of pressure and number of terms in series.Labels 0, 1, 2 and 3 refer to the number of correction terms added to O,(T,p,p). Table 1. Parameters of eqn (28) for CO, adsorption data20 m RTP RTa C T/K /mmol g-l /kJ mol-l /kJ mol-l /kPa-l A 212.7 1 1.005 19.371 1.8 6.933 x 0.0188 260.2 10.500 18.848 2.1035 6.933 x 0.006 301.4 9.2 197 19.003 2.2153 6.933 x 0.0084 obtained using the approximation eqn (S’), are presented in terms of their percentage deviation from the exact results: ) x 100. exact - approx. Fig. 3 gives a plot of 6 against p for approximations with varying numbers df terms, using G = 0.75. It is apparent that the series gives an oscillating convergence. The maximum deviation after four correction terms is 0.05%. After only one correction term, the deviation is < 1.5%.It should be mentioned in this connection that the addition of m correction terms, when using the Gaussian distribution, involves terms up to the 2mth derivative owing to the fact that every second term in the expansion, eqn (S’), disappears because of eqn (17). A similar calculation of 6 against p for 0 = 1.25 yielded oscillatory, increasing deviations after four correction terms. If the series eventually converges for CT greater than unity, too many terms are necessary for practical use. Although no rigorous mathematical analysis of the convergence properties of eqn (8’) has been undertaken here, it has been shown that rapid convergence may be expected for values of G < 1. Recalling that the dimensionless value of CT under discussion is obtained by dividing it by RT, the series converges for ratios of o/p < 0.1,J.A. O’BRIEN AND A. L. MYERS 1475 10.0 8.0 7 6.0 00 c.l E 3 4.0 2.0 Fig. 4(a) and (b). For legend see page 1476. or at room temperature for values of CT < ca. 2 kJ mol-l. A similar study using the uniform distribution fails to converge eventually, but is convergent for values of o / p > 0.1. Thus it would appear that the Gaussian distribution gives a ‘worst-case’ estimate for convergence. These results might be phrased in the alternative form that the expansion procedure is convergent for ‘moderately’ heterogeneous surfaces. For such surfaces the method produces an approximate (although slightly unwieldy) version of eqn (I) which is explicit and, at least in principle, may be used with an arbitrary local isotherm and distribution of energies.APPLICATION TO EXPERIMENTAL DATA Although the series expansion, eqn (S’), has not converged after one correction term, truncation at this point should give an equation which can be used to correlate experimental isotherm data. Thus the first-order approximation, using the Langmuir1476 ADSORPTION OF GASES ON HETEROGENEOUS SURFACES P m a Fig. 4. Comparison of eqn (28) with CO, adsorption data for various temperatures. (a) 212.7, (b) 260.2 and (c) 301.4 K. local isotherm, has been fitted to the experimental data of Reich et aL20 for the adsorption of CO, on an activated carbon at three temperatures. This equation is written as where x is given, as before, by eqn (21). The resulting parameters, as well as the objective function for the fit are given in table 1 .Fig. 4(a)-(c) show the comparison between the equation and the data points, and the agreement is seen to be good. Thus this modification of the Langmuir equation has endowed it with enough heterogeneous character to describe adsorption on an activated carbon. These results indicate the possibility of modifying any of the usual homogeneous isotherm equations to account for heterogeneous surfaces. Even implicit isotherms, such as eqn (18), may be used in this way since it is possible, although tedious, to calculate d26/dz2, the only derivative necessary in this case. However, only the Langmuir case has been investigated in this work.J. A. O’BRIEN AND A. L. MYERS 1477 NOMENCLATURE an, i C D E f H HL rn n n c a l ~ nexptl P R T X Z a3 a4 6 A e 8, P P n 0 coefficients in eqn (22a) and (26a) constact in Langmuir isotherm, eqn (19) domain of energy distribution energy of adsorption probability density function for energy Henry’s constant, eqn (1 1) Henry’s constant on local patch adsorption saturation capacity number of moles adsorbed per gram of adsorbent calculated value of n from eqn (28) experimental value of n pressure gas constant temperature dimensionless group, eqn (21) dimensionless energy of adsorption coefficient of skewness off(z), eqn (9) kurtosis off(z), eqn (9) percentage deviation, eqn (24) objective function minimized in data fit, eqn (29) fractional coverage, entire surface fractional coverage, local patch mean of energy distribution nth central moment of energy distribution standard deviation of energy distribution This work was supported by National Science Foundation Grant CPE-8 1 17 188.REFERENCES W. A. 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