Journal Of Colloid And Interface Science Volume 21 Issue 4 1966 [doi 10.1016%2f0095-8522%2866%2990006-7] J.h. De Boer; B.c. Lippens; B.g. Linsen; J.c.p. Broekhoff; A. Va -- Thet-curve Of Multimolecula.pdf

JOURN&IJ OF COLLOID AND INTERFAC~ SCIENCE 21, 405-414 (1966)

THE t-CURVE OF MULTIMOLECULAR N2-ADSORPTION J. H. de Boer, B. C. Lippens, B. G. Linsen, J. C. P. Broekhoff, A. van den Heuvel, and Th. J. Osinga (Technological University of Delft, The Netherlands) Received April 7, 1965

During his many years of editorship of this Journal, Victor K. La Mer has certainly had the opportunity to witness major and minor changes in theoretical conceptions. In colloid science, as in all other branches of science, theoretical conceptions lead often to equations which fit experimental data satisfactorily. Even if later such theoretical conceptions may be proved to be wrong, the equations may serve a practical purpose. In this special issue of the Journal of Colloid Science, dedicated to the parting Editor-in-Chief, he may be interested to learn how the study of multimolecular adsorption, though approached theoretically from various angles, is still largely based on empirical or semiempirical equations, which, however, enable us to derive many useful data concerning the texture of microporous substances, such as adsorbents, catalysts, or catalyst carriers. I . T H E t - C u R v E , A ~([ASTER CURVE FOR MULTIMOLECULAR

ADSORPTION OF N2 Ever since the Brunauer, Emmett, and Teller method for the estimation of surface areas was introduced, a vast literature about the BET equation has been published. The originators of the BET method knew that their equation did not fit the experimental data for all relative pressures, and they recommended rightly, for practical purposes, restricting its use to the range of relative pressures between 0.05 and 0.35. For other estimations, such as the sizes, the shapes, and the distribution of capillaries, however, the multimolecular adsorption curve over its whole range of pressures is required. If we restrict ourselves to one adsorbate (nitrogen) and to one temperature (liquid nitrogen temperature) it is a fortunate fact that, for a wide variety of adsorbents, the multimolecular adsorption curve proves to be identical, provided no capillary condensation occurs and no narrow pores are put out of action in the course of progressing adsorption. These two restrictions, just mentioned, demand a careful choice of the adsorbents for the determination of the universal multimolecular adsorption curve which is to be used as a reference curve for texture 4O5

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determinations. Recent experimental work (1) enables us to recommend for this purpose, up to a relative pressure of 0.75, the data which we published early in 1964 (2). We hope to extend the curve by further careful measurements. The data are expressed in terms of an average thickness of the adsorbed layer in Angstrom units (hence the name t-curve). In doing so we assume the adsorbed layer to behave as a normal liquid nitrogen layer with its proper density at the given temperature, and we also assume a hexagonal dense packing (2). In Fig. 1 we show this master curve (curve A) together with the BET-curve (curve B) fitted to give the same surface area. As the thickness of one statistical layer is 3.54 A, a value of n X 3.54 A indicates an adsorption of n = Va/V,~ layers, where Va is the adsorbed volume and Vm the monolayer capacity, both in milliliters STP per gram of adsorbent. II. PRACTICAL APPLICATIONS

A practical way to use this master curve (curve A in Fig. 1) is to plot experimental adsorption curves as a function of the t-values;hence V = f ( t ) instead of V = f ( p / p o ) . In any normal case of multimolecular adsorption the experimental points should then fall on a straight line through the origin. The slope of this line gives the specific surface area, S t , in m.~/g., by means of the equation (3): St = 15.47(Va)/t.

[1]

In Fig. 2 curve A gives the t-plot of an alumina prepared by heating well-crystallized boehmite at 750°C. ; the isotherm of this specimen is described in No. II of our series on pore systems in catalysts (4). A pore-size distribution analysis of this alumina (5) showed a set of wide (>25 A) slit-shaped pores, open on all sides, giving no capillary condensation during the adsorption process, until high relative pressures ( > 0.75) were reached; the desorption branch shows a wide hysteresis loop, closing at p i p e = 0.48. During adsorption, therefore, the multimolecular adsorption develops, unhindered by capillary condensation or by closing of pores. Curve B shows the t-plot of an alumina made by heating bayerite on 580°C. At a t-value of 4.65 A (piPe) = 0.24) the experimental curve deviates from the straight line, indicating that more nitrogen is taken up than corresponding with multimolecular adsorption. Capillary condensation has set in at this point. At an average t-value of about 9 A (p/po N 0.72) the pores responsible for this phenomenon are filled, and a new straight line shows the continuation of the multimolecular adsorption on the remaining surface (outer surface and/or surface in wide pores); the slope indicates this remaining surface area to be about 20 m.2/g., whereas the original slope of the t-plot indicated a surface area of 243 m.2/g. (S•ET = 245 m.2/g.). Curve C gives the t-plot of an aluminum oxide (hydroxide) prepared by

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FIO. 2. V~-t-plots of some aluminas as described in the text heating the same bayerite, as above, at 250°C. From the slope of the first part of the t-plot follows St = 483 m.2/g. (SBET ---- 489 m.2/g.). Between the t-values of 4 A and 5 A all narrow slit-shaped pores, which are responsible for this high value of surface area, are apparently filled; the new slope indicates a surface area of about 20 m.2/g, for the surface area of wider pores and the outer surface. These examples may suffice to show the significance of the deviations from the straight t-plot. Other combinations may be found; in many cases a downward deviation indicating a closing of capillaries which, because of their size and shape, do not give capillary condensation in the adsorption branch (slit-shaped pores with parallel walls) may later be followed by an upward deviation indicating capillary condensation in wider pores (in the same elementary particles or as "stacking pores" between those particles), etc. Together with the shape and the position of a hysteresis loop, the t-method may give a rather useful picture of the pore texture of microporous substances. In some cases one may find pores of such narrow dimensions that a normal BET-plot gives erroneous results. A t-plot does not give a straight line through the origin in such cases, but yields lines cutting the zero relative pressure at positive values. The slope of the line drawn from the origin to the first experimental point at the lowest relative pressure gives then a lowest value of the surface area. This method yields extremely useful results with regard to the study of the texture of carbon blacks (6)

THE t-CURYE OF MULTIMOLECULAR N2-ADSORI:'TION

409

III. EQUATIONS DESCRIBING THE t-CuRvE Figure 1 shows clearly that the B E T equation cannot describe the experimental master curve of multimolecular nitrogen adsorption. 1V[any modifications and alterations have been suggested to improve the B E T equation or to adapt it to special cases. One of these suggested modifications, the equation given by R. B. Anderson (7), can be made to fit our experimental curve A of Fig. 1. As shown in Fig. 3 the Anderson equation Va

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[2]

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(1 -- k x ) ( 1 +

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where x = p/po gives, when c = 53 and k = 0.76, a good fit with the experimental curve. (The value of V,~ derived from an Anderson plot is always higher than the corresponding figure from a BET-plot; in our case it is a factor 1.16.) This result, however, has no physical significance. In Anderson's conception k < 1 means that the heat of adsorption of the second and higher layers is less than the heat of liquefaction--a condition which is even less favorable for multimolecular adsorption than the B E T conception itself, according to which the heat of adsorption of the second and higher layer equals the heat of liquefaction. In the B E T theory a picture is chosen which gives a high figure of combinatorial entropy because attraction forces between neighbor molecules in the various layers are excluded. As 5/IcMillan and Teller (8) have shown this means that the surface tension of the liquid is neglected; when the real surface tension is introduced hardly any multimolecular adsorption occurs when the heat of adsorption of the second and higher layers is not higher than the heat of liquefaction (9). Starting from the conception of the condensed (liquid) character of the mu]timolecular adsorbed film and from an empirical equation of state for condensed films, Harkins and Jura (10) derived in 1944 a simple equation log x = B -

A (Va)----~,

[3]

where again x = p/po, Va is the adsorbed volume, and A and B are constants. Our experimental curve A of Fig. 1 can excellently be described by this equation for all values of x between 0.1 and 0.75, as is shown in Fig. 4, where we compare the experimental values of t with those calculated with the equation: 13.99 log x = 0.034 -- t--V-

[3a]

As t -~ 3.54Va/Vm (see Section I), the equation may also be written: log x -- 0.034

13.99(V~) 2 (3.54 Va) 2 - 0.034

1.116(Vm) 2 (Va) 2

[3b]

410

DE BOER ET AL.

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[4]

With N2 as an adsorbent at liquid ni£rogen temperature, and expressing

411

THE t-CURVE OF MULTIMOLECULAR N2-ADSORPTION

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DE BOE~ ET AL.

Va in milliliters STP per gram and S in square meters per gram, they found k = 4.06. When we combine Eqs. [3b] and [4] with [1], we obtain: k = 4.14. In a study about the relationship between the B E T equation and the equation of ttarkins and Jura, Livingston (11) concluded that, in a range of relative pressures between 0.07 and 0.72, both equations may fit experimental data equally well, if the value of c in the B E T equation is chosen between c = 50 and c = 100. Later in his paper, however, it is clearly stated that Emmett's (12) calculations, showing that both equations may cover an experimental plot between p/po = 0.11 and p/Po = 0.40 (for c-values between 25 and 250), are correct. In the present section of this article we come to the conclusion that the Anderson equation (one more constant than the B E T equation) and the Harkins and Jura equation cover our experimental master curve equally well for a long range of relative pressures. IV. THE HIGH ~ELATIVE PRESSURE END OF THE CURVE The Frenkel-Halsey-Hill equation (13) In x = -- - -

C

(Va) r'

[5]

where C is a constant and x and V~ have the same meaning as above, assumes an adsorbed multilayer of liquid character with the same density and packing as the normal liquid, the heat of evaporation being higher than the normal one due to the extra van der Waals' forces exercised b y the adsorbent on the adsorbed liquid layer. If only London-dispersion forces are present r - 3. In many cases of adsorption on surfaces of oxides, salts, or metals, however, polar contributions are also present, and these contributions to the van der Waals' forces tend to lower the value of r. In the old conception of de Boer and Zwikker (14) (1928) such polar contributions were introduced as an addition to the normal heat of evaporation. Since London introduced the dispersion forces (1930) we know that polar contributions cannot explain the normal magnitude of the van der Waals' forces; they may, however, lower the value of r from 3 to values in the neighborhood of 2, as is often found. In the old conception of 1928, however, it was also suggested that a small deviation of the liquid density (liquid packing) might lead to an intersection of the adsorption curve with the vertical axis at p/po = 1, instead of gNing an asymptotic approach. Our experimental curve (curve A in Fig. 1) shows the tendency to lead to a finite amount of adsorption at

THE t-CURVE OF MULTIMOLECULAR N2-ADSORPTION

413

saturation pressure. The extrapolation of Eq. [3] gives Va/V,~ = 5.73 (t = 20.3 A) at p/po = 1. The Frenkel-Halsey-Hill equation, of course, leads to infinite adsorption at saturation pressure. As already stated in section I we hope to extend the experimental work and to avoid the contributions from capillary condensation. V. THE CURVE AT THE

LOWER END OF RELATIVE PRESSURES

Our experimental curve starts at p/po = 0.08, where Va/V,~ = 0.99; this means that the statistical monolayer of nitrogen is practically complete. If the adsorbing surface were entirely homogeneous, a two-dimensional gas (the adsorbed layer) would be in a two-dimensional condensed state if the temperature of the experiment were lower than the two-dimensional critical temperature. If the nitrogen molecules are still rotating in the adsorbed state, or if they adsorbed in random positions or in flat positions, their two-dimensional critical temperature would be well below 77°K., which is the experimentally used temperature. There would, therefore, not be a two-dimensional condensation, unless the nitrogen molecules were adsorbed in an upright position (15). Experimentally no two-dimensional condensation was found in the monolayer region at 77°K. with nitrogen on homogeneous surfaces (16). It is quite possible that in multimolecular adsorption condensation to an adsorbed liquid layer may occur. On a heterogeneous surface (all experimental evidence and all equations in this article refer to heterogeneous surfaces) adsorption on top of a monolayer, hence multimolecular adsorption, sets in at various relative pressures, depending on the strength of the adsorption forces. These relative pressures may be, partially, far lower than p/po ~ 0.1. In our experiments, therefore, we may expect a large number of liquid patches in the adsorption layer, at p/po N 0.1 when the statistical monolayer is nearly complete. This is the only reason for using the liquid density for the adsorbed layer at such low relative pressure. From the practical point of view one may, of course, add, as further evidence, the facts that the surface area per nitrogen molecule derived from the liquid density (viz. 16.27 A 2) agrees well with the estimation of surface areas by the "absolute method" of Harkins and Jura (17) and with the surface area estimations by means of lauric acid (18). It may be remarked here that this agreement does not hold when the Anderson equation is used instead of the BET equation. SUMMARY

ExperimentMly an identical multimoleeular adsorption curve of nitrogen at liquid nitrogen temperature is found for a wide variety of adsorbents, provided no capillary condensation occurs and no narrow pores are put out of action in the course of progressing adsorption. This curve, at present

414

DE ~OER ET AL.

known for relative pressures between 0.10 and 0.75, can be expressed in terms of an average thickness of the adsorbed layer (the t-curve). When experimental adsorption data on microporous adsorbents are plotted as a function of this t-curve, the t-plot obtained gives direct information about the specific area, the capillary condensation, the width of pores, etc. Two equations m a y describe the t-curve more or less adequately; the theoretical nonsignificance or significance of this is discussed.

REFERENCES 1. 2. 3. 4. 5.

6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

DE BOER, J. H., LINSEN, B. G., AND OSINGA,TH. J., J. Catalysis, 4, 643 (1965). LIPPE~S, B. C., LINSEN, B. G., AND DE BOER, J. H., J. Catalysis 3, 32 (1964). LIPPENS, B. C., AND DE BOER, J. H., J. Catalysis 4,319 (1965). DE BO~.R, J. H., AND LIPPENS, B. C., J. Catalysis 3, 38 (1964). LIPPENS, B. C., AND DE BOER, J. H., J. Catalysis 3, 44 (1964). DE BOER, ft. H., LINSEN, B. G., VAN DER PLAS, TH., ~ND ZOI~DERVAN,G. J., J. Catalysis, 4, 649 (1965). ANDERSON,R. B., J. Am. Chem. Soc. 68,686 (1946). McMILLAN, W. G., AND TELLER, E., J. Phys. & Colloid Chem. 65, 17 (1951); see also ttlLL, T. L., Advan. Catalysis 4,236-242 (1952). HALSE:f, G. D., Advan. Catalysis 4,263,264 (1952). HARKINS,W. D., AND JURA, G., J. Am. Chem. Soc. 66, 1366 (1944). LIVINGSTON,H. K., J. Chem. Phys. 15,617 (1947). EMMETT, P. H., J. Am. Chem. Soc. 68, 1784 (1946). HIT.L, T. L., Advan. Catalysis 4,236-242 (1952). nE BOER, J. It., Proc. Roy. Acad. Amsterdam 31,906 (1928). ])E BOER, J. H., AND ZWI•KER, C., Z. Physik. Chem. (Frankfurt) 3B, 407 (1929). DE BOER, J. H., "The Dynamical Character of Adsorption," pp. 153, 154. Clarendon Press, Oxford, 1953. Ross, S., AND OLIVlER, J. P., "On Physical Adsorption," pp. 227-231. Interscience Publishers, New York, 1964. HARKINS,W. D., AND JURA, G., J. Am. Chem. Soc. 66, 1362 (1944). DE BOER, J. H., HOUBEN, G. M. M., LIPPENS, B. C., MEYS, W. I-I., ANDWALRAVE, W. •. A., J. Catalysis 1, 1 (1962).