Particle packings and the computation of pore size distributions from

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4866

J . Phys. Chem. 1986, 90, 4866-4874

Particle Packings and the Computation of Pore Size Distributions from Capillary Condensation Hysteresis Bruce D. Adkins and Burtron H. Davis* Kentucky Center f o r Energy Research, Lexington, Kentucky 40512 (Receiued: March 13, 1986)

Many porous materials are made up of particulate “building blocks”, and the “pores” are voids between the particles. This fact is important when one computes pore size distributions for such materials. Our results show that the “preferred” model for packed-particle pore structures is either a simple cubic or body-centered cubic model. The desorption process can be viewed as a Kelvin-type (hemispherical) mechanism operating in the minimum dimensions of the void structure, while adsorption can be viewed as a Kelvin mechanism operating in the largest dimensions of the void structure. Contact zone coalescence (“assistance filling”) seems to be delayed with respect to this instability. This delayed coalescence is associated with a strong dependence of multilayer adsorption on particle size as well as the surface energy barrier for expansion of a spherical shell. Furthermore, the desorption results are consistent with the additional effect of a “pore-blocking” or network mechanism.

Introduction

The adsorption of vapors by finely divided materials has been a subject of scientific investigation for more than two centuries. As early as 18 15 deSaussure defined many aspects of adsorption including heat of adsorption, influence of porosity, reaction enhancement by adsorption, and synergism in adsorption from gas mixtures.’ Langmuir, following deSaussure by about 100 years, provided a great advance in the understanding of adsorption behavior with his monolayer theory of adsorption.* Langmuir was a strong proponent of the monolayer view, and as his comments about Gurvitsch’s paper3 show, he considered condensation in pores as a secondary phenomenon that did not depend on forces acting between the adsorbent and the l i q ~ i d .The ~ investigations by Emmett and Brunauer culminated with the Brunauer, Emmett, and Teller (BET) equation to describe multilayer adsorption and the universally accepted method of calculating surface areas.5 The success of the BET equation led to numerous investigations during the following 50 years that included increasingly sophisticated treatments to calculate pore size distributions from isothermal adsorption data. The development of methods to compute pore size distributions has followed a very model-specific path because the governing thermodynamics are closely tied to pore shape and size. Although ”modelless” algorithms (e.g., ref 6) exist, they are not generalized from the viewpoint of the thermodynamic relationships. In order to test a thermodynamic theory of capillary condensation, equations are written for a fictitious system which somehow is representative of an actual pore structure. From the viewpoint of simplicity, cylindrical pores have been the obvious favorite. In reality, it is difficult to obtain and verify a cylindrical pore structure. The number of attempts to do this has been very limited (e.g., ref 7). Conversely, the preparation of particulate materials is readily accomplished by a variety of methods. Colloid science has provided a startling array of particle shapes and sizes, often with remarkably narrow particle size distributions. The simplest and most ideal of these are spherical particles. These particles can be packed into structures where the “porosity” is essentially limited to void space between the particles. Although particle size distribution and packing irregularity are obvious factors that can add complexity, these materials still rank among the best for model discrimination. The major goal of this paper is to examine the thermodynamic and geometric assumptions in two popular cylindrical pore models, desaussure, T. Ann. Philos. 1815, 6, 241, 331. Langmuir, I. J. A m . Chem.Soc.1916, 38,2221. Gurvitsch, L. G.J. Russ. Phys.-Chem. Sor. 1915, 47,805. Langmuir, I. J. Am. Chem.Soc. 1917, 39, 1848. Brunauer, S.; Emmett. P. H.; Teller, E. J . Am. Chew Soc.1938, 60. 309. (6) Brunauer, S.; Mikhail, R.; Bodor, E. J . Colloid Interface Sei. 1967, 24, 451. ( 7 ) Ihm, S. K . ; Ruckenstein, E. J . Colloid Interface S c i . 1977, 61, 146.

0022-3654/86/2090-4866$01.50/0 , I

,

particularly as they apply to calculating pore size distributions for materials built up from regular nonporous particles. These assumptions are then addressed in a manner specific to spherical particles, and new models are developed which optimize the trade-off between accuracy and simplicity. Evidence for this optimization is taken from the calculated pore size distributions, with spherical nonporous particles which are available commercially in a wide range of particle sizes and with narrow particle size distributions. The results of our modeling treatments are presented for various packing fractions and two particle sizes. Experimental Section

Materials nnd Data. Nitrogen adsorption isotherms were obtained by using a Micromeritics Digisorb 2500 instrument. Samples were typically outgassed at ca. 425 K for 6 h under vacuum of at least mmHg. Hysteresis measurements most frequently consisted of 21 points for the adsorption and 22 points for the desorption branch. Occasionally, 40 or more points were obtained on each branch; also, selected samples were analyzed more than once to verify the reproducibility of the data. Samples for adsorption measurements typically consisted of about 250 mg of material, powdered to an approximate agglomerate size of 100 pm. Equilibration was determined from successive pressure readings at 6-s intervals, with the differences computed to mmHg. Select isotherms were reproduced with an 18-s equilibration time, and the results were similar. The two materials used in this study are commercially available from the Degussa Corporation: Aluminoxide-C and P25 Titania. The spherical shape and narrow particle size distribution are well documented for these materials. As received, these materials yield type I1 isotherms, with only slight evidence of hysteresis at the highest relative pressures. In this study, two treatments were used to transform these materials so that a type IV hysteresis was obtained: (a) calcination at 675 K for 3-4 h, followed by addition of water to the “incipient wetness” point, and then drying in air at 375 K for at least 24 h; (b) calcination as above, followed by external compaction in a hydraulic press under ca. 300-MPa ram pressure. These treatments will be referred to as “water wetting” and “compaction”, respectively. Data Analysis. Nitrogen adsorption hysteresis loops were analyzed with the aid of the models described in the next section. These models were used to compute pore size distributions, based on modifications of the Barrett, Joyner, and Halenda (BJH)* algorithm. To guarantee numerical accuracy, several meshes were constructed consisting of approximately 50 pore size intervals, with no mesh having a pore size less than 1.5-nm radius nor greater than 30-nm radius. For each model, a relative pressure mesh was calculated from the pore size mesh by using the equations in the next section. Each isotherm was then subdivided by interpolating the volume adsorbed at each relative pressure point (in cm3/g of (8) Barrett, E.; Joyner, L.; Halenda, P. J . Am. Chem.SOC. 1951, 73,373.

0 1986 American Chemical Society

The Journal of Physical Chemistry, Vol. 90, No. 20, 1986 4867

Particle Packings and Pore Size Distributions condensed phase, where the condensed nitrogen is assumed to have the density of bulk liquid). Interpolation was done with a cubic spline (piecewise polynomial) algorithm, which has the advantage over other interpolating, algorithms in that it maximizes the smoothness in the interpolating functions. Multilayer corrections for unfilled pore groups were handled as a diagonal matrix; in the case of the packed-particle models, these corrections included a full account of the accumulation of condensate in the interparticle contact zones. Cumulative surface area distributions were computed from the differential pore size distributions by using the appropriate pore geometry. For the packed-particle models, a simple geometric allowance was made for the surface area in interparticle contact zones which is inaccessible to a nitrogen molecule with a hardsphere radius of 0.216 nm. An additional test of the accuracy of the models was to integrate the differential pore volume distribution and compare it to the Gurvitsch v01ume.~The Gurvitsch number in this case was taken from the uppermost mesh point on the interpolated isotherm, by using a mesh which gave closure of adsorption and desorption data in the plateau region of the isotherm. BroekhofPSlo and others have pointed out numerical and conceptual errors in extending these matrix calculations to excessively small pores. Broekhoff advocates the use of the point of deviation from linearity of the adsorption data on the t plot as being the lowest point to extend the calculations. However, our work on the spherical-particle “standard isotherms” (ref 1 1 and described subsequently) has identified variations in the “standard isotherm” with particle size and degree of interparticle coordination, so that the point of deviation based on a single t plot becomes less convincing. Instead, we choose a simpler rationale in comparing the integrated pore volume to the Gurvitsch volume. When the differential distributions showed negative regions at the smallest pore sizes (as happened occasionally), this indicated that all available pore volume had been discretized, and further calculations were ignored. This avoided the “self-correcting” tendency of the BJH computational a l g ~ r i t h m . ~ Capillary Condensation Models. The following equations can be considered fundamental in modeling capillary condensation (adsorption and desorption processes) in pores of various shapes:

RT In ( p / p o ) = N(E, - Eo) + yV, dA/dV, 0 = dN(E,

- &)/dV,

+ YV, d2A/dV2

(1)

(2)

Cohan’s Cylinders. In 1938, CohanI2 provided an explanation for hysteresis based on perfect cylinders. For adsorption in the empty pore, eq 1 can be applied to a cylindrical film of liquid of thickness t , adsorbed on the pore wall, for which dA/dV, = -1 /(R- t ) d2A/dV:

= - 1 / { 2 ~ ( R- t ) 3 )

(3)

It follows from eq 2 that such a film cannot be stable unless the adsorption potential is nonzero. Assuming such an adsorption potential exists, the point of instability is given by

RT In ( P / P o ) = dN(Ec - Eo)/dVc - ./Vc/(R - 2)

(4)

In what is referred to as Cohan’s model, however, this is not rigorously applied. Instead, the following simplifications are made: Assumption a. The film is stabilized by an adsorption potential, but it can be treated as being planar, i.e., dA/dV, = 0. Assumption b. At all isopotential surfaces which correspond to possible interfaces for stable adsorbed films, the condensate adopts the properties of bulk liquid, Le., E, = E,. These assumptions merit special attention. Assumption a has been used as the basis for deriving many different “multilayer equations”. Frenkel,I3 Halsey,14 and HilllS published multilayer equations (“FHH equations”) in the late 1940s in which they integrated Lennard-Jones forces over a planar solid to obtain

RT In ( p / p o ) = -a(t/to)-*

(5)

where a and z are characteristic constants of the interaction. Many pore size distribution calculations use eq 5 to describe multilayer adsorption; the most popular values were recommended in 1964 by Dollimore and HealI6 for nitrogen adsorption on silicas and aluminas ( a = 3.25 kJ/mol, to = 0.354 nm, and z = 3). Assumption b reduces eq 1 and 2 to the well-known form of Kelvin’s equation. For adsorption

RT In @ / P o ) = - 7 V , / ( R

- t)

(6)

RT In ( p / p o ) = -2yV,/(R - t )

(7)

and for desorption

where t must be determined simultaneously from eq 5. Broekhoffs Cylinders. In 1969, Broekhofp obtained a rigorous solution of eq 1 and 2 for the case of the perfect cylinder by rejecting assumptions a and b. To obtain a term for N(E, - E,), he considered adsorption data obtained for materials that demonstrated little or no irreversible capillary condensation and had negligible surface curvature. Under these conditions, he obtained a “standard isotherm” which he then combined with his version of eq 1 to obtain a corrected equation for the equilibrium of a film of condensate adsorbed on the pore wall:

Equation 1 is a statement of the thermodynamic equilibrium between a condensed phase (having volume V,, interfacial area A, surface tension y, and molar volume V,) and a vapor phase at pressure p . N is Avogadro’s number, T is temperature, po is the saturation vapor pressure, and R is the gas constant. E, refers to the potential energy of interaction between a single condensate molecule and the adsorbent; Eo is the potential energy of interaction between the same molecule and a pool of its own liquid. In the following developments, N(E, - E,) will be referred to as the “adsorption potential”. Equation 2 is obtained by differentiating eq 1 with respect to V,. This equation, when combined with ( l ) , gives the relative pressure at which the condensed phase cannot be in equilibrium and is unstable. For a derivation of these equations and a discussion of their uses and limitations, the reader should consult Appendix I. Although these equations appear simple, it is an extremely difficult task to apply them rigorously to all but the simplest pore shapes. In this paper, two existing models are used for the case of open-ended cylindrical pores of radius R. In addition, four models are developed for capillary condensation within the void spaces of regular packings of spherical particles. Each model is based on eq 1 and 2, but with varying degrees of approximation.

Two equations were given to accurately cover the entire region of relative pressures in his measurements. Although Broekhoff s equations were written for nitrogen adsorption at 78 K, they can be rewritten for other adsorbates by using other standard isotherms. His equations have an additional term since he chose not to neglect V,(p -po). Also, note that we have reprinted Brcekhoffs equations as published; both pore radius R and film thickness t are in A.

(9) Broekhoff,J. C. P. Ph.D. Thesis, University of Delft, 1969. (10) Broekhoff, J. C. P. In Preparation of Catalysts; Delmon, B., Grange, P., Jacobs, P., Poncelot,G.,Eds.;Elsevier: Amsterdam, 1979; Vol. 11, p 663. (11) Adkins, B.; Russell, S.; Ganesan, P.; Davis, B. In Fundamentals of Adsorption; Meyers, A,, Belfort, G., Eds.; Engineering Foundation: New York, 1984; p 39.

(12) Cohan, L. H. J . A m . Chem. SOC.1938, 60, 433. (13) Frenkel, Y . I. In The Kinetic Theory of Liquids;Clarendon: Oxford, 1946. (14) Halsey, G. D. J . Chem. Phys. 1948, 16, 931. ( 1 5) Hill, T. L. J . Chem. Phys. 1949, 17, 590. (16) Dollimore, D.; Heal, G.J . Appl. Chem. 1964, 14, 109.

log @/PO) = -2.051/(R

- t ) - 13.72/t2 + 0.0376 0.0221(1 - p / p o )

= -2.051/(R - t ) - 15.46/t2

( t c 10 A)

+

0.15726e4’l’2’ 0.0221(1 - p / p o ) ( t > 10 A) (8)

4868 The Journal of Physical Chemistry, Vol. 90, No. 20, 1986

Adkins and Davis

TABLE I: Characteristics of Regular Particle Packines coord no. e“ R,:R,:R,~ 0.66 0.48 0.32 0.28

4 6 8 12

Fraction porosity.

1:1.31:0.73 1:0.73:0.41 1:0.29:0.22 1:0.41:0.15

R , = particle radius; R, = “cavity” radius; R,

= “neck” radius.

Figure 2. The dependence of coefficient a(Rp)and exponent z(RJ in eq 16 on particle size R,.

Y

I

Figure 1. The neck in a simple cubic structure, viewed plane, showing the salient geometric features

in

the (100)

This film becomes unstable, and pore filling occurs, at the point given by the simultaneous solution of eq 8 and -2.051/(R - t)’ -2.051/(R

- t)’

+ 27.44/t3 = 0

(t

< 10 A)

+ 30.92/t3 - 0.01748e411’2‘= 0 (t

> 10 A) (9)

Likewise, desorption from a filled pore was predicted to occur at the point given by the simultaneous solution of eq 8 and log @/PO) = -4.102/(R - t ) - 27.44{(R/t) - 1 In (R/t)] + 0.0376(R - t)’ - 0.0221(1 - p / p o ) ( t < 10 A) = -4.102/(R - t ) - 30.92((R/t) - 1 - In ( R / t ) ]+ 2.8274e4 1 1 1 2 r-( rt - 8.993) 25.43e* I l l Z R 0.0221(1 - p / p o ) ( t > 10 A) (10)

+

After the pore empties, a stable multilayer remains in equilibrium as given by eq 8. Packed-Particle Models. Here, we I 1 develop mr -1s based on eq 1 and 2 for four regular packing different interparticle coordination nun,: jr ( n = 4), simple cubic ( n = 6), body-centered CI, face-centered cubic ( n = 12). The characterist these packings are listed in Table I. The terr the interparticle void in the close-packed plant the neck radius R, is defined as the radius of which can be inscribed in the neck passage. le region of in these packings has a radius R, and is defineu interstitial volume which is congruent with the largt. L sphere which can be inscribed in the interstitial passages of the packing. In Figure 1, which is an illustration of the unit cell r simple cubic packing, the neck can clearly be visualized in ti, ( 100) planes, and the cavity is at the geometric center of the cell. For reasons indicated in Appendix I, it is not feasible to rigorously apply eq 1 and 2 to particle packings. Instead, suitable approximations must be made. To illustrate these approximations, it is helpful to consider the role of assumptions ‘. and b above in reducing the exact equations (e.g., Broekhoffs model) to the approximate equations (e.g., Cohan’s model). It is also helpful to use Figure 1, which dissects the structure into discrete regions of interest. The approximations which result are discussed briefly here; fuller details are available in Appendices 11-IV. Desorption will be considered first. Approximation 1. Cohan’s equations totally neglect the adsorption potential in the condensed phase (assumption b), and Broekhoffs include this term. But in the case of the packedparticle system, one cannot merely add the Broekhoff correction,

for reasons which are purely geometric. When Cohan’s equations and Broekhoffs equations are taken as opposite extremes, it is possible to show that a fully rigorous treatment of the adsorption potential term in the case of desorption from particle packings would give a result which in fact is closer (or at least as close) to Cohan’s equation than it would be to Broekhoffs. This argument is presented in Appendix 11. The somewhat surprising result is that eq 7 is a simpler and more accurate approximation than eq 10. Approximation 2. Although the actual shape of the neck in the regular packings is not a volume of revolution, it can be reasonably approximated as one. The reason is that there are a number of points around the periphery of the neck where the adsorption potential gradient is strictly radial with respect to the center line of the neck. At these points, the curvature in the close-packed plane is circular. This, combined with a negligible adsorption potential in all areas marked “zone 1” (Figure l), allows the neck to be considered a volume of revolution with little loss in accuracy. Further details are given in Appendix 111. Approximation 3. The shaded regions in Figure 1 (“zone 2”) cannot be treated successfully by the approximation of zero adsorption potential, because they are localized to an individual particle. Furthermore, eq 5 cannot be used as written for these regions, because a dependence on particle size cannot be neglected. The correct form for this particle size dependence is obtained by taking the expression for N(E, - E,) (the right-hand side in eq 5) and inserting into the more general eq 1, along with the spherical shell expression for dA/dVc, and provision for a particle size dependence in the F H H coefficient and exponent: RT In ( p / p o ) = -a(R,)(t/to)-’(Rp)

+ 27V,/(R, + t )

(1 1)

This particle size dependence was developed in a previous work1’ based on 35 type I1 isotherms obtained for spherical-particle materials. Both a@,) and z(R,) were shown to be monotonically increasing functions of particle size, as shown in Figure 2. These parameters were regressed to a simple inverse exponential function of particle size in order to provide an asymptotic limit for a( m ) and z ( m ) , as would be expected. These values were approximately a ( m ) = 2.7 kJ/rnol and z ( m ) = 3.5. A similar treatment of the adsorption process in the particle packings results in similar approximations, with eq 7 being used to predict the instability, except that the cavity radius R, is used instead of the neck radius R,. Again, eq 5 cannot be used as the second equation, but eq 11 can. One problem with this adsorption mechanism is that a second mechanism is also likely; this mechanism is contact zone coalescence, in which adjacent contact zones touch in the neck and flood the cavity. Since neither mechanism can be eliminated, computations were performed for both cases. Further details appear in Appendix IV. (17) Adkins, B.; Reucroft, P.; Davis, B. H., submitted for publication in Adsorpt. Sci. Technol.

The Journal of Physical Chemistry, Vol. 90, No. 20, 1986 4869

ff

(a)

EM

w

\

b 0

0 ) 0

0

0

1

PIP0

PIP0

1

PIP0

Figure 3. Nitrogen adsorption isotherms for (a) as received, (b) water-wetted, and (c) compacted Aluminoxide-C. TABLE 11: Parameters Obtained from Nitrogen Adsorption Measurements

material Aluminoxide-C P25 Titania

treatment as received water wetted compacted as received water wetted compacted

BETO 98 118

pore volb d

103

0.74 0.40

60 62 63

0.43 0.23

d

&

d 0.73 0.59 d 0.62 0.47

Bet surface area, m2/g. *Pore volume from “Gurvitsch” plateau, cm3/g. cPore fraction determined from (density X pore volume) e= (1 + density X pore volume) with densities of 3.6 g/cm3 for y-alumina and 3.84 g/cm3 for anatase Ti02. “Not applicable. Pore Radius, mn

Results Nitrogen isotherms are shown in Figure 3a-c for the three samples of Degussa Aluminoxide-C. Isotherms for P25 Titania were of similar shape and are not shown. The BET surface area and the pore volume estimated by the limiting uptake, or “Gurvitsch volume”, of the isotherms are tabulated in Table 11. These pore volumes were used to calculate the pore fraction e, also listed in Table 11. For both materials, compaction results in a more densely packed structure than water wetting, as evidenced by the pore fractions listed in Table 11. A comparison with the regular packing parameters in Table I shows that water wetting yields a packing having roughly the porosity of a diamond cubic ( n = 4) structure while compaction gives a structure having porosity similar to that in a simple cubic ( n = 6 ) structure. The BET surface areas increased with both treatments; based on our previous experience with interparticle contact zone condensation,” this apparent surface area increase is most likely due to enhanced adsorption in the contact zones. The pore size distributions obtained with the Cohan model, eq 5-7, are shown in Figure 4. Results are shown for the waterwetted (Figure 4a,b) and compacted (4c,d) aluminas, as well as the water-wetted (4e,f) and compacted (4g,h) titanias. Pore size distributions are not shown for the as-received materials since their hysteresis loops are not closed at the highest relative pressures. The uppermost graphs (4a,c,e,g) show the pore size distributions; corresponding cumulative surface area computations are plotted immediately underneath (4b,d,f,h). Pore size distributions are plotted as DV/DR; the area bounded by DV/DR(R) and DV/ DR(R+dR) is the volume of pores having radii between R and R + dR. Cumulative surface area distributions are plotted such that C S ( R ) is the surface area in all pores larger than or equal to R.

Pore Radius, MI

Figure 4. Pore size distributions (top; a,c,e,g) and cumulative surface

area distributions (bottom; b,d,f,g) computed with Cohan’s cylinder model, for (a,b) Aluminoxide-C, water wetted; (c,d) Aluminoxide-C, compacted; (e,Q P25 Titania, water wetted; and (g,h) P25 Titania, compacted. Cumulative surface areas from all of the Cohan calculations are larger by 50% or more than the BET surface areas. However, agreement between adsorption and desorption differential distributions is reasonable in all cases. The compacted materials show a greater difference in the sharpness of the adsorption and desorption distributions, as expected from their isotherms. Not shown are the cumulative pore volume distributions obtained from these models. In every case, the Cohan model produced a distribution which overestimated the Gurvitsch volume by 15-20%. The most plausible explanation is associated with the multilayer growth phenomenon in larger pores. Apparently, the Cohan model provides insufficient multilayer growth in larger pores, and small pores are created artificially from the differentiation process. Figure 5 shows the pore size and cumulative surface area distributions computed with Broekhoff s cylindrical model, described in eq 8-10, for the same materials shown in Figure 4. The peak positions of the pore size distribution indicate the pores are larger than those obtained from the Cohan distributions, by 50% or more. The result is that the calculated cumulative surface areas now give excellent agreement with the BET surface areas. In comparison with the Cohan model, approximately the same differences in sharpness of the adsorption and desorption distributions are seen. Although the cumulative volume distributions are not shown, it is significant that values calculated by using Broekhoffs model closely match the Gurvitsch volume in every case. This difference is rationalized in terms of multilayer growth. Broekhoffs multilayer curve, eq 8, yields a greater thickness of adsorbed film than does eq 5.

4870

The Journal of Physical Chemistry, Vol. 90, No. 20 1986

Adkins and Davis similar to the Cohan distributions shown in Figure 4 and (b) the cumulative surface area numbers are now equal to or slightly lower than the BET surface area. Also, the same basic differences between the sharpness of the adsorption and desorption branches are still seen. The cumulative pore volume numbers, which are not shown, in every case matched the Gurvitsch volume.

pore nAdiU&am

Pore bdip., am

Figure 5. Pore size distributions (top; a,c,e,g) and cumulative surface area distributions (bottom; b,d,f,g) computed with Broekhoffs cylinder model for (a,b) Aluminoxide-C, water wetted; (c,d) Aluminoxide-C, compacted; (e,f) P25 Titania, water wetted; and (g,h) P25 Titania, compacted.

I

. W

1 m W

I5

pore Redip., am

Porn Radius, nm

Figure 6. Pore size distributions (top; a,c,e,g) and cumulative surface area distributions (bottom; b,d,f,g) computed with the packed-particle models, for (a,b) Aluminoxide-C, water wetted; (c,d) Aluminoxide-C, compacted; (e,f) P25 Titania, water wetted; and (g,h) P25 Titania, compacted.

Finally, Figure 6 shows pore size distributions computed with the packed-partjcle models described in the last section. For each of these materials, calculations were performed for all four regular packings listed in Table I. Results are shown only for the packing which yielded the least discrepancy between adsorption and desorption differential diktributions as well as between cumulative surface area number and the BET area. Note that these distributions are plotted as the minimum radius of the pore (the neck radius R,,), In every case, either the body-centered cubic or the simple cubic structure was found to be optimum. It is significant to note that these optimum structures (Table I) contained, in every base, lower porosity than the actual porosities listed in Table 11. Figure 6 shows the adsorption distributions computed under the assumptions of cavity instability. The results obtained under the assumed mechanism of contact mne coalescence are not shown simply because these results inevitably were in total disagreement with those shown; specifically, the adsorption distributions so computed had peak pore sizes which were half as small as the desorption branch or even smaller. This is consistent with the delayed coalescence associated with the spherical-particle multilayer equation in eq l l; for a given cavity and neck size, fixed by the packing, the cavity instability occurs a t a lower relative pressure than that a t which the contact zones coalesce. From Figure 6, two things are immediately noticeable: (a) that the distributions, when plotted against the neck size, are very

Summary A number of features which are pertinent to the calculation of pore size distributions for materials which consist of spherical particles have been considered in this paper. Although the models are still a t an early stage of development so that approximate, rather than analytical or numerical, solutions are used, and many questions remain unanswered, several significant observations have emerged from this analysis which should be useful as starting points for future work. Some of these observations are summarized below. The desorption process in particle packings appears to be adequately described by the instability of a Kelvin-type meniscus in the neck of the structure. When the Kelvin desorption mechanism is used with a cylindrical pore model, i.e., in Cohan’s model, the resultant cumulative surface area and integrated pore volume are inconsistent with the more reliable BET area and Gurvitsch volume, respectively. When the desorption mechanism for these cylinders is modeled more accurately (Broekhoffs model), these inconsistencies are eliminated. If the pore structure is then approximated as the interstices in a packing of spheres, and not as cylinders, a semiquantitative theoretical analysis shows that the mechanism which was more accurate for the case of the cylinder is now no more accurate than the Kelvin mechanism. When pore size distributions are recomputed with a Kelvin desorption mechanism in the neck of the void structure (and providing for certain particle size dependences), the inconsistencies are again eliminated. Since these materials are certainly particulate, we take this as supporting evidence of the validity of the neck radius and the Kelvin desorption mechanism. The desorption measurements inevitably lead to a sharper distribution than the adsorption branch. At present, we consider this to be consistent with a “pore-blocking” effect. The adsorption process is consistent with the instability occurring in the material adsorbed on the walls of the cavity and with the delay of an instability which is associated with the coalescence of contact zone menisci. Although the approximation of zero adsorption potential seems more or less justified at most locations along a vapor-liquid interface within a packed-particle pore structure, the approximation is least appropriate at points in the structure where the adsorption potential field is dominated by a single particle. In these regions, an adsorption potential which is dependent upon particle size should be used. This dependence, along with the energy barrier for growth of a spherical shell, can have the effect of delaying the coalescence of contact zones to the point where the adsorption instability is governed by the stability of an essentially spherical film adsorbed in the cavity. Pore size distribution calculations based on both contact zone coalescence and cavity instability further suggest that the cavity instability is indeed occurring. The regular packed-particle models, although providing most consistency of any models examined, fail to describe these materials from the viewpoint of porosity. Without exception, the materials are more porous than the models which provide the best fit. It is very likely, however, that vacancies could be introduced into the regular structures without changing the adsorption-desorption properties, and the appropriate porosities could be achieved. Note that the cylindrical models permit no estimate of void fraction at all. The benefits of the regular packing models are (1) they provide a defined relationship between cavity radius and neck radius and (2) they provide “bookkeeping” for quantifying the differences in cylindrical pore walls and walls made up of particles. These two aspects are not overly specific to a single model; for example, similar results were obtained in this study by using the diamond cubic, simple cubic, and body-centered cubic structures. Packing

The Journal of Physical Chemistry, Vol. 90, No. 20, 1986 4871

Particle Packings and Pore Size Distributions of real particles are probably not regular, but they may include geometric features which are evident, and easily handled, in regular packings. Appendix I. Model Development Based an Polanyi’s Potential Theory

The potential theory is one of the older theories of physical adsorption and is based on thermodynamic equilibrium. According to this theory, condensate and vapor have the same chemical potential. At any vapor chemical potential, the total quantity of condensate is simply the volume enclosed by a surface having the same chemical potential (the isopotential surface). Polanyi’* defined an “adsorption potential”, -RT In @ / p o ) ,which he suggested should be independent of temperature at constant volume adsorbed. (Later work19 and our eq 1 and 2 show that this is not necessarily true.) In the derivation which follows, our use of the term adsorption potential is restricted to the part of (-RTln (p/po)) which is due to electromagnetic interactions between the solid and the condensed adsorbate, that is, N(E, - Eo). A second contribution to the total potential (here, called “pressure energy”) will also be introduced to allow for mechanical equilibrium at the interface. The use of this term is in keeping with the views of Br~ekhoff,~ and other members of the “deBoer School”, as well as Dubinid’ and DerjaguhZ2 With this, the theory can be described in terms of the following: p, is the molar chemical potential of the condensed phase, pc,ais the molar adsorption potential energy, and pc,p.is the molar pressure energy. The statement of equilibrium is

=

c~v

~ r = c pc,a

+ c~c~p

(12)

where pv is the chemical potential of the vapor. This viewpoint has been discussed extensively by Everett and H a y n e ~ . * ~ To proceed, a Gibbs-Duhem equation can be written for each term and integrated between states. Isothermal processes are assumed. For a vapor considered to be ideal, one obtains wo - pV= -RT In

@/PO)

(13) where po is the chemical potential of the vapor at saturation pressure po. For the adsorption potential, the following equation can be written: w0,a

- @c,a = -N(Ec - EO)

(14)

Expressions for E, must come from a “law of interaction”; the 6-12 relationship integrated over all atoms in all “infinite” solid is most often used to model this term. Note that po,ais the contribution to the chemical potential of the bulk liquid due to EO. Finally, the pressure energy contribution must be obtained. This is most easily done by balancing forces on (assumedly) a perfectly elastic interface and using Laplace’s equation to obtain M0.p

- Mc,p = -vc(P - Po) - YVC dA/dVC

(15)

where ko,pis the pressure energy contribution to the chemical potential of the bulk liquid. The derivative dA/dV, is also equal to the harmonic mean of the two radii of curvature of the interface. The equation of equilibrium is now taken from eq 12-1 5. The term V,(p - p o ) ,which can be shown to be small for,many gases, can be neglected. Finally, po = pO,a+ po,p,so that one can easily obtain eq 1. Equation 1 applies to an isopotential interface which is in equilibrium with vapor at pressure p . This equation can now be used to illustrate one explanation for capillary condensation (18) Polanyi, M. Verh. Dfsch Phys. Ges. 1916, 18, 5 5 . (19) Bering, B. P.; Dubinin, M. M.; Serpinsky, V. V. J . Colloid Interface Sci. 1966, 21, 378.

(20) Broekhoff, J. C. P.; Linsen, B. G. In Physical and Chemical Aspects of Adsorbents and Catalysts; Linsen, B. G . , Ed.; Academic: London, 1970. (21) Dubinin, M. M.; Kataeva, L. I. Izu. Akad. Nauk. SSSR., Ser. Khim. 1977, 516. (22) Derjaguin, B. In Proceedings of the 2nd International Conference of Surface Actiuify; Butterworths: London, 1957; Vol. 11, p 154. (23) Everett, D. H.; Haynes, J. M. Spec. Period. Rep.: Colloid Sci. 1973, 1, 123.

N(E, -E,)

”+

0

”v,

O =

RT In(P pol

dVc

Figure 7. A schematic representation of (a) adsorption potential energy, (b) surface energy, and (c) chemical potential plotted against quantity condensed to illustrate a n explanation for capillary condensation hysteresis based on the potential theory.

hysteresis which is based entirely on equilibrium. Figure 7 provides an example of one possible relationship between the components of eq 1 and the quantity of condensed phase, N,. This plot is constructed for the case where (a) both contributions to chemical potential are constant at all points on the interface and (b) the harmonic mean radius of curvature of the interface is negative (using the convention that positive radii of curvature have origins lying inside the condensed phase). Figure 7 , a and b, show the molar adsorption potential and pressure energy, respectively; in Figure 7c, the summation of these two contributions is plotted. Conceivably, if one could independently adjust the chemical potential of the adsorbed material, the path DABC could be traced experimentally. However, since physical adsorption experiments use. excess quantities of vapor, R T In (p/po) is constrained and cannot decrease spontaneously. Thus, when the relative pressure is increased to point A, the experimental observation will be stepwise pore filling. Similarly, stepwise “pore emptying” will occur on desorption along line segment BD. The net effect is an experimental hysteresis over the path ACBD. The positions of the points of instability, A and B, are easily obtained by differentiation of eq 1 with respect to N, or V,. Equation 2 is the result. Equations 1 and 2 are the fundamental statements of the potential theory of capillary condensation. Identical equations can be obtained by considering the Gibbs free energy, G, of the combined condensate-vapor system and obtaining expressions for the minimum Gibbs free energy. This corresponds to dG/dVc = 0 and d2G/dV2 L 0; eq 2 then gives the point where d2G/dV,2 = 0, and equilibrium of the vapor-adsorbate system is not possible. For the details of the derivation, the reader is referred to Broekh~ff.~ In reality, there are at least four problems with using eq 1 and 2 and Figure 7 to model capillary condensation hysteresis. These problems are important enough to be discussed in further detail. Consider the following: Problem 1. For realistic pore shapes, it is reasonable to expect isopotential surfaces where the terms on the right-hand side of eq 1 vary with position; only their sum must be constant by this theory. In the most general case, each point on the isopotential surface would be considered individually, and the appropriate adsorption potential and surface energy terms inserted in eq 1 . The surface terms will not be obtained in explicit form but will be coupled through conditions of continuity. Furthermore, each p o i n t on the interface will have a point of instability as given by eq 2, so that the instabilities may be localized to discrete regions or even points. For “real-world” pores, analytical expressions based on eq 1 and 2 which describe the entire interface will be extremely complex. Conceivably, an approach could be developed in which the equations are stated numerically at discrete points on the interface; it might then be possible to link the interface through a numerical discretization of a system of ordinary differential equations. Such a numerical approach has been used frequently in problems involving surface tensions; the numerical solutions of Larkin24for

4872

The Journal of Physical Chemistry, Vol. 90, No. 20, 1986

the problem of bubble shape on inclined plates under uniform gravitational attraction is one example. However, formidable difficulties will be encountered by using this approach for the system given in eq 1 and 2. An approximation which is much easier to use is to represent the interface as being made up of a small number of distinct regions. This method sacrifices some accuracy to obtain workability. In each of these regions, the adsorption potential and surface curvature are taken as invariants, and a single statement of eq 1 and 2 can be written. Simple boundary conditions can then be used to link these regions, and the local maxima and minima can be obtained with eq 2 for each region. These ”piecewise approximations” are used extensively in the packedparticle models. Problem 2. The irreversibility in the condensation-evaporation process is depicted as arising from a ”microdomain” assumption, Le., that the chemical potential of the condensate is dictated by that of a much larger quantity of vapor. The argument assumes that a single path AB does exist, but in fact the reality of physical states which correspond to this path is questionable. It is possible that a second path lies between points A and C which is not accessible to the equilibrium adsorption states, perhaps due to an energy barrier associated with reconfiguring a meniscus, but which is accessible to the filled pore and the desorption process. This point does not, however, present problems in translating the theory into mathematical descriptions. A description of the hysteresis does not require that any path be constructed between A and B. Furthermore, the path DA is constructed assuming that the pore is originally empty (and hence the usual starting point of multilayer adsorption under the assumption of zero contact angle); likewise, path CB is constructed from the physical viewpoint of a filled pore. Thus, in practice, different forms of eq 1 and 2 are used for adsorption and desorption, and the paths DA and CB are not forced to connect smoothly in the region AB. Problem 3. The behavior of a domain of given shape and size may be totally amenable to this theory when the domain is completely isolated, and yet its behavior may be completely altered when it is part of a larger domain. Viewpoints on this aspect are varied and range from (a) the “pore-blocking” picture of the desorption p r o c e s ~to~ ~(b), ~“assisted ~ filling processes” on ad~orption.~’ This problem is an important one and cannot be thoroughly treated here. Suffice it to point out that there are at least two ways in which this phenomenon can be rationalized. The major difference in the two viewpoints is the way in which “domain” is defined. First, take the domain to be the smallest fraction of the void space which is somehow representative of the entire void space in the same way that a unit cell is representative of a periodic lattice structure. For example, in a pore structure consisting of many nonintersecting cylindrical pores, each of these pores is a separate domain. Each domain has (in theory) a curve such as in Figure 7c which describes its adsorption behavior in equilibrium. But this equilibrium condition requires that each domain have access to the vapor phase. If connectivity does not allow certain domains access to the vapor phase, then we observe ”pore-blocked” behavior and conclude that the process is a nonequilibrium or metastable one. If we define the domain in a less restricted manner, then a different viewpoint can be presented. If the entire void space is taken as the domain, then one can assert that an extremely complex version of the curve shown in Figure 7c is again characteristic of the domain. Under this definition, the concept of the individual “pore”, and the pore size distribution, becomes meaningless. The curve may have many points corresponding to the maxima or minima shown in eq 2, so that numerous stepwise transitions may occur. The difference is that a nonequilibrium (24) Larkin, B. J. Colloid Interface Sci. 1967, 23, 305. (25) Kanellopoulos, N. K.; Petrou, J. K.; Petropoulos. J. H. J . Colloid Interface Sci.1983, 96, 101. (26) Mason, G. J . Colloid Interface Sci. 1982, 88,36. (27) Mayagoitia, V.; Rojas, F.; Kornhauser, I. J . Chem. Sor., Faraday Trans. 1 198s. 91,2931.

Adkins and Davis interpretation is not required, and the “pore-blocking” process can then be viewed as an increase in the multiplicity of equilibrium surfaces with increasing domain size. (Note in Figure 7 that, for even the simplest case, there are in many places three surfaces which can exist at the same energy. These surfaces are separated by barriers corresponding to the evaporation or condensation of large amounts of condensate.) Unfortunately, neither viewpoint facilitates the calculation of an unbiased pore size distribution. At present, this problem is best treated qualitatively. “Assisted filling” processes seem to have the essential feature of the interaction of menisci at junction points. As an example, for the cylindrical pore, the theory simply says that a number of filled pores which intersect an empty pore can change this pore from “open-ended” to “half-closed”, so that the point of instability is changed. Two things can be qualitatively judged about this phenomenon: (a) it will sharpen the adsorption pore size distribution at the smaller pore sizes, and (b) it is connectivity dependent, but not necessarily dependent upon domain size. For “pore blocking”, a variation upon domain size is predicted; the effect should be more pronounced in larger domains. Like “assisted filling”, it should result in a sharpening of the distribution at smaller pore sizes, this time for the desorption branch. Problem 4. The concept of an isopotential adsorbed phase requires that the pressure vary internally, since E, will vary with position. For a vapor-liquid interface with a negative harmonic mean radius of curvature the pressure is lower inside the adsorbed phase than in the vapor. Furthermore, this internal pressure decreases with distance from the adsorbent surface because the effect of the attractive field of the solid is to offset this pressure decrease somewhat.23 Thus there are two offsetting effects in small pores. As the pore size decreases, the radii of curvature of the hemispherical meniscus become smaller and the internal pressure is lowered. At the same time, the center line of the pore is closer to the adsorbent surface, so that the pressure drop is offset somewhat. In spite of the offset the adsorbate experiences the greater tensile stresses in the smallest pores. A logical extension is that the adsorbate in pores below a critical size will be placed under a tensile stress which is greater than the tensile strength and will rupture before the instability given by eq 2 is reached. This frequently is the explanation for the lower limit of hysteresis which seems to be an adsorbate property;28 e.g., for nitrogen, hysteresis is rarely reported below p / p o = 0.4. For very small packed-particle pore systems the situation is worsened by the geometry, since a very small meniscus can exist above fluid which is reasonably far from the adsorbent. A reasonable assertion is that hysteresis should be limited to even higher relative pressures for packed-particle materials. For the general case of desorption instability, however, the tensile stresses are not considered here. Appendix 11. The Correction for Adsorption Potential for Packed-Particle Models

The shape of the meniscus can be obtained from a boundary value solution of eq 1 written in terms of curvature derivatives. One boundary condition depends upon the equation of multilayer equilibrium, i.e., the thickness of stable adsorbed material which remains after the pore empties. This boundary condition, which depends strongly upon whether or not assumption a is used as well as the nature of the solid-adsorbate interaction, essentially specifies a ucore’’ radius or dimension, Rk,as shown in the illustrations of Figure 8a. By use of this core radius, the following argument is made independent of whether or not assumption a is used. The concept of contact angle is also required; the usual choice is a contact angle of zero. The second boundary condition exists at the center line of a pore having rotational symmetry. This boundary condition is intimately connected with the use of assumption b, Le., whether a zero or a continuous, nonzero adsorption potential is used. In Figure 8a, the meniscus is drawn under the assumption of zero adsorption potential, as shown in Figure 8b. From eq 1 and this (28) Evertt. D. H.; Burgess, C. J . Colloid Interface Sci. 1970, 33, 61 1 .

Particle Packings and Pore Size Distributions

The Journal of Physical Chemistry, Val 90. No. 20. 1986 4873 pore size distributions from hysteresis measurements. This complication is well-known (e&, BroekhofP in his discussion of ink bottle type cavities), but it has escaped analytical treatment, for example, in the usual assumption of a ‘sufficiently long neck” in ink bottle models. For particle packings, Figure 8 shows that it cannot be ignored.

0

1; I( n z20p, i

:i

~~~

\

Appendix 111. Effect of Approximating the Neck as a Volume of Revolution The actual shape of the neck in the simple cubic or other packing is not a volume of revolution. As shown in Figure 1, where the “neck” is viewed in the (100) plane, the minimum dimension Rk*is exceeded in directions oriented tangential to interparticle contact points, Le., in the direction of interparticle contact zones (the unshaded areas labeled zone I).However, these interparticle contact zones are regions somewhat analogous to the center line of a symmetrical pore, at least in the overlapping potential fields that exist. At points equidistant from both particles in contact (the points which lie along the lines directed tangential to the interparticle contact points), the gradients in adsorption potential are directed radially from the center line of the neck, and the situation is very similar to that at the center line of an axially symmetrical pore (see Appendix 11 and Figure 8). At the point where adjacent zones I are touching, the adsorption potential essentially emanates from a single particle, and once again the potential gradient is radial. Because of this, we can expect that the circumference of the meniscus in the neck should be locally circular a t eight points, four of which are at along the lines tangential to interparticle contact points and the other four being those most localized to the individual particles. It is also not unreasonable to assume that the adsorption potential in the entire zone 1 region is zero, since the interface is on the whole farther removed from the solid than in the case of zone 2. This allows us the simplification of bounding zone 1 by a torus of revolution, and since a tangency condition must exist with zone 2, the net effect at the point where the zone I regions touch is a meniscus having a circular periphery in the close-packed plane. This circle has the core radius R,*. Even under the condition of zero adsorption potential, the zone Iinterface is not really a torus of revolution; however, it is very closely approximated by The radii of curvature and volume of rotation of the region bounded by two particles and the torus of revolution can be found in the l i t e r a t ~ r e . ~Insertion ~ of these radii of curvature into eq I at zero adsorption potential allows the size of the contact zone to be related to vapor relative pressure; zones 1 and 2 are linked through the appropriate tangency condition. W e used this procedure to calculate the multilayer corrections for each of the packings.

Appendix IV. Contact Zone Coalescence as an Adsorption Mechanism A potential complication arises here which is essentially an “assisted filling” problem. Kiselev and Karhaukhov” have considered the growth of the zone 1 regions up to the point where all zones coalesced. For packing structures like those considered in this paper, they predicted that the entire void structure would be filled at the point where these contact zones coalesced; the exception is the close-packed structure, for which they predicted certain peculiarities. Their prediction is based on a model identical with the one just developed in Figure I,with the important exception that “the strengthening or weakening of the adsorption potential as a function of the geometrical factors is not considered here”. This differs from our analysis which suggests that these effects are most important in zone 2. Since the thickness of zone 2 is a boundary condition for the growth of zone I,its particle size dependence can therefore exert an inhibiting influence on contact wne coalescence. For this reason, we decided to investigate ~~

(29) Kruyer. S . Tram. Faraday Soc. 1958.34, 1758. (30)Wade, W. H. J. Phys. Chem. 1965.69, 322. (31) Aristov. B.; Karnaukhov. A.; Kisclcv. A. R u s J. ~ Phys. Chem. (Engl. Tmnsl.) 1962, 36. I 1 59.

J . Phys. Chem. 1986, 90, 4874-4877

4874

two mechanisms for the adsorption process. In the first, contact zone coalescence is used. This condition was calculated conditionally; Le., the computer determined the instability point simply by testing the results of the equations describing the growth of the contact zone. In the second, we consider this coalescence to be delayed sufficiently so that the adsorbed film in the “cavity” of the structure becomes unstable in exactly the same way that the instability occurs on the interior of a perfectly spherical cavity. For reasons

similar to those described for the desorption process in the neck, we have chosen to approximate the meniscus as being spherical at the point of instability and having two equal and negative radii of curvature which are equal to the minimum dimension of the cavity. With this, the instability condition is again given by eq 7 and 11, except that the cavity radius R, is used.

Acknowledgment. This work was supported by the Commonwealth of Kentucky through the Kentucky Energy Cabinet.

OTHER SYSTEMS Ammonia Synthesis on Molybdenum Nltride Leo Volpe and M. Boudart* Department of Chemical Engineering, Stanford University. Stanford, California 94305 (Received: June 18, 1985; In Final Form: November 13, 1985)

Ammonia synthesis on unsupported Mo2N powders was studied at atmospheric pressure between 557 and 717 K. Turnover rates based on titration of sites by CO decreased by a factor of 25 as the particle size of MozN went from 12 to 3 nm. This confirms the structure sensitivity of the rate of ammonia synthesis. Strong inhibition of the rate by ammonia was observed. It is attributed to large values of the binding energy of nitrogen to the catalyst surface.

Introduction

TABLE I: Values of the N, BET Specific Surface Area

Molybdenum is one of the more active catalysts for ammonia synthesis.l During synthesis on Mo powders at atmospheric pressure and temperatures up to 900 K, dimolybdenum nitride yMo,N is formed as the stable phase. This paper reports data for ammonia synthesis on y-Mo2Npowders of high and medium specific surface area, S,. Since surface structure of a metal depends on particle size, when this dimension is less than ca. 10 nm: which in the case of Mo2N corresponds to S, = 60 m2 g-l, new Mo2Ncatalysts with S, values between 10 and 200 m2 g-I give us the opportunity of investigating on Mo2Nthe structure sensitivity of NH, synthesis, a phenomenon first observed on small supported iron particles4 and confirmed directly on large single crystals of FeS and Re.6 Previous studies of ammonia synthesis were performed on nitrided molybdenum samples that had low or unspecified S,. Kiperman and Ternkin’ thoroughly analyzed the synthesis kinetics on Mo2N but did not report the S, of their catalyst. Hillis et al.* studied the reaction on molybdenum dioxide reduced in H2 and nitrided with N2. Aika and Ozaki’ carried out isotopic tracer investigations of NH, synthesis at very low conversion on Mo,N

sample

$I-,

(m g )

Mo~N-HI2 190 Mo~N-M” 50 M o ~ N - L ~ ~ ’ ” 12

dpl

co uptake/ (pmol g-])

loi4cm-2

3

882

12 63

104 25

2.8 1.2

nm

ncol

1.3

dcl

nm 6 10 10

with S, = 13 m2 g-I. They concluded that the reaction’s ratedetermining step (RDS) was the same as on iron catalysts: dissociative adsorption of N2. More recently, Oyama and Boudarti0studied NH, synthesis on three molybdenum-based catalysts: Mo, Mo2C, and MOO& Despite differences in composition and crystallography, site-time yields on these materials reached similar steady-state values, when referred to the number of sites titrated by CO. Steady state was achieved only after the catalysts absorbed small amounts of N corresponding to less than three atomic layers. Thus, the catalytic activity of these materials appeared to be determined by surface molybdenum nitride layers, regardless of the structure or composition of the bulk. In the present study, we focus our attention on the effect of Mo2N particle size on the rate of ammonia synthesis. Experiment and Results

(1) Ozaki, A.; Aika, K. In A Treatise on Dinitrogen Fixation; Hardy, R . W. F. et al., Eds.; Wiley: New York, 1979; p 169. ( 2 ) Van Hardeveld, R.: Hartog, F. Surf. Sci. 1969, I S , 189. (3) Volpe, L.; Oyarna, S. T.; Boudart, M . In Preparation ofcatalysts III; Poncelet, G., Grange, P., Jacobs, P. A,, Eds.; Elsevier: Amsterdam, 1983: p 147.

(4) Dumesic, J. A.; Topsae, H.; Boudart, M. J . Catal. 1975, 37, 513. (5) Spencer, N. D.; Schoonmaker, R. C.; Somorjai, G.A. J . Catal. 1982. 7 4 , 129. ( 6 ) Asscher, M.; Somorjai, G. A. Sur/. Sci. 1984, 143, L389. ( 7 ) Kiperman, L.; Temkin, M. Acta Physicochim. URSS 1946, 21, 267. (8) Hillis, M . R.; Kemball, C.; Roberts, M . W. Trans. Faraday SOC.1966, 62, 3570. (9) Aika, K.; Ozaki, A. J . Catal. 1969, 14, 31 I .

0022-3654/86/2090-4874$01.50/0

Of the three samples with high, medium, and low S,, denoted here as Mo2N-H, -M, and -L, the latter two have been described previously.l0,’’ Details of the Mo,N-H preparation and characterization are given el~ewhere,~*’~ and only the properties relevant to this study will be repeated here. Table I lists the values of the N, BET specific surface area (S& the corresponding particle size ~~~~

(10) Boudart, M.; Oya,ma, S. T.; Leclercq, L. Proc. Int. Congr. Catal., 7th Seiyama, T., Tanabe, K., Eds.; Elsevier: Amsterdam, 1981; p 578. (1 I ) Oyama, S. T., Ph.D. Dissertation, Stanford University, Stanford, CA, 1981. (12) Volpe, L.: Boudart, M. J . Solid State Chem. 1985, 59, 332

0 1986 American Chemical Society