WILLIAMH. WADE
322
The Coordination Number of Small Spheres
by William H. Wade Department of Chemistry, University of Tezaa, Austin, Pew8
78716
(Received August $7, 1964)
Water adsorption isotherms were obtained for uncompressed and compressed AI2O3spheres of 75 A. radius. The isotherms for the compressed samples reach a saturation value for a given porosity sample. Nitrogen surface areas were obtained for these samples both bare and with water preadsorbed. The analysis of these data ultimately yields the coordination numbers of the powder samples which were found to vary from 3 to 10.
Introduction I n a previous publication,’ experimental data were given for the variation of the area of the exposed surface of water films on a substrate as a function of the total amount of adsorbate. Vada measurements were performed on powdered samples with areas (2) varying from 2.72 to 220 m.2/g. The high specific area samples, 65 6 2 6 220, all exhibited a rapid loss of 2 for small volumes adsorbed, Vads, followed by a more gradual loss of area with increasing V a d s . The sole low Z sample (2.72 m.2/g.) showed no rapid initial drop but only a gradual diminution. These data were interpreted in terms of a dual adsorption model blending uniform film adsorption and capillary condensation a t the contact points of the adsorbent particles. The adsorbent particles were assumed to be uniform spheres of radius R, packed to a uniform coordination number, n. The mathematical formulation of this model correlated qualitatively but only semiquantitatively with the experimental data. In the initial study, the coordination number, n, was treated as an adjustable parameter. No attempt was made to evaluate or vary this parameter independently for a given sample. The best fit values of 3 n 6 5 for the high Z samples of low bulk density appeared reasonable as did the value of n = 9 for the 2.72 m.2/g. sample. Kiselev2 has attempted to estimate n for silica gel samples by gas adsorption techniques but has not attempted to vary n for a given sample. From geometrical considerations, Manegold3 has evaluated the bulk porosity of agglomerates of spheres with given coordination numbers. Porosity is defined as
rN2, and the surface areas as a function of pressure will be given by eq. 2 combined with eq. 4a and 3. Figure 2 illustrates this variation of 2 with P for .n = 2 to 12 using R = 75.4 and 6 = 3.60. The points a t zero pressure were calculated from eq. 5. Calculations were not extended below a pressure of 3 mm. for a t this point rm > r N 2 . At lower water pressures there is a transition region where neither eq. 2 nor Ei is valid. These calculated curves for the 2 variation assure no particular packing geometry and for their validity assume that adjacent pendular rings do not overlap. As discussed by Kruyer16 and the pressure a t which overlap occurs marks the onset of capillary condensation and the point a t which eq. 1 and 2 become invalid. I n ref. l and 15, such critical pressures
are listed for simple geometric packings assuming t = 0. Using Kruyer's definition of #I as the angle between the center line of regularly packed spheres and the line from the center of a sphere to the center of the regular array forming the opening between spheres, it can easily be shown that a t the instant of merging of adjacent pendular rings t
+
T,
=
R(sec. 4 - 1)
(6 1
+
From eq. 3 and 4a ( t r,) as a function of P can be obtained assuming R is known. For cubic packing (#I = 45") of spheres with R = 75.4 i., the critical pressure P, = 12.4 mm., and for hexagonal close packing (h.c.p.; #I = 30°), P, = 20.6. These values are noted in Figure 2 and niark the limit of validity for the two curves where n = 6 and 12, respectively. For cubic packing, a t P, the isotherm should discontinuously rise to its maximum value and become horizontal a t higher pressures, and the surface area should simultaneously drop to zero. For n = 12, Kiselevll has shown that a more complicated behavior exists and that capillary condensation for this particular sample should occur in two or three stages. The absence of such behavior in the experimental results indicates that the samples here produced are not geonietrically simple but have a variety of geometrical configurations which the measurements average. In addition, there is a spectrum of R values, the 75.4 A. being an average value. These two effects produce adsorption isotherms of finite slope during capillary condensation and wash out any fine structure due to h.c.p. regimes. Some indication of such fine structure exists for samples C and D, for a t -18 mm. small inflection points are observed. Due to these complications, filni area measurements were not extended to thick water films where capillary condensation would have invalidated the coniparisons to the calculated values. I n addition, 2 values of very thin water filnis were not obtained because of the probable invalidity of the Kelvin equation where rm is of molecular dimensions. The experimental film areas for samples A through D are plotted in Figure 2. As observed previously, the unconipressed sample A fits n = 5-6 a t low pressures and n = 4 a t high pressures. This lack of agreement over the entire range of film coverages has been discussed.' The coiiipressed saniples B-D closely follow the theoretical curves for n = 8,10, and 12, respectively, and are so listed in Table I. For saniple B, the fit is
(15)
s. Kruyer,
Trans. Faraday Soc., 54, 1758 (1958)
Volume 69, Yumber 1 January 1066
326
particularly good, which probably coincides with the fact that all equilibration pressures are
