Sphere interstices

The discrepancy between the pore area or the core area on the one hand and the BET area on the other is proportionately larger with silica than with alumina, particularly at the higher degrees of compaction. The fact that silica is a softer material than alumina, and the marked reduction In the BET area of the compact as compared with that of the loose material, indicates a considerable distortion of the particles, with consequent departure of the pore shape from the ideal of interstices between spheres. The factor R for cylinders (p. 171), used in the conversion to pore area in the absence of a better alternative, is therefore at best a crude approximation. 

The limits of pore size corresponding to each process will, of course, depend both on the pore geometry and the size of the adsorbate molecule. For slit-shaped pores the primary process will be expected to be limited to widths below la, and the secondary to widths between 2a and 5ff. For more complicated shapes such as interstices between small spheres, the equivalent diameter will be somewhat higher, because of the more effective overlap of adsorption fields from neighbouring parts of the pore walls. The tertiary process—the reversible capillary condensation—will not be able to occur at all in slits if the walls are exactly parallel in other pores, this condensation will take place in the region between 5hysteresis loop and in a pore system containing a variety of pore shapes, reversible capillary condensation occurs in such pores as have a suitable shape alongside the irreversible condensation in the main body of pores. 

Even in a close-packed structure, hard spheres do not fill all the space in a crystal. The gaps the interstices—between the atoms are called holes. To determine just how much space is occupied, we need to calculate the fraction of the total volume occupied by the spheres. 

FIGURE 14.9 In. i metallic hydride, the tiny hydrogen atoms, (the small spheres) occupy gaps—called interstices— between the larger metal atoms (the large spheres). 

As surface area and pore structure are properties of key importance for any catalyst or support material, we will first describe how these properties can be measured. First, it is useful to draw a clear borderline between roughness and porosity. If most features on a surface are deeper than they are wide, then we call the surface porous (Fig. 5.16). Although it is convenient to think about pores in terms of hollow cylinders, one should realize that pores may have all kinds of shapes. The pore system of zeolites consists of microporous channels and cages, whereas the pores of a silica gel support are formed by the interstices between spheres. Alumina and carbon black, on the other hand, have platelet structures, resulting in slit-shaped pores. All support materials may contain micro, meso and macropores (see text box for definitions). 

The number of octahedral holes in the unit cell can be deduced from Fig. 17.1(c) two differently oriented octahedra alternate in direction c, i.e. it takes two octahedra until the pattern is repeated. Flence there are two octahedral interstices per unit cell. Fig. 17.1(b) shows the presence of two spheres in the unit cell, one each in the layers A and B. The number of spheres and of octahedral interstices are thus the same, i.e. there is exactly one octahedral interstice per sphere. 

The size of the octahedral interstices follows from the construction of Fig. 7.2 (p. 53). There, it is assumed that the spheres are in contact with one another just as in a packing of spheres. A sphere with radius 0.414 can be accommodated in the hole between six octahedrally arranged spheres with radius 1. 

From Fig. 17.1 we realize another fact. The octahedron centers are arranged in planes parallel to the a-b plane, half-way between the layers of spheres. The position of the octahedron centers corresponds to the position C which does not occur in the stacking sequence ABAB... of the spheres. We designate octahedral interstices in this position in the following sections by y. By analogy, we will designate octahedral interstices in the positions A and B by a and j3, respectively. 

As can be seen from Fig. 17.2(b), there is one tetrahedral interstice above and one below every sphere, i.e. there are two tetrahedral interstices per sphere. 

There are four spheres, four octahedral interstices and eight tetrahedral interstices per unit cell. Therefore, their numerical relations are the same as for hexagonal closest-packing, as well as for any other stacking variant of closest-packings one octahedral and two tetrahedral interstices per sphere. Moreover, the sizes of these interstices are the same in all closest-packings of spheres. 

Structure Types with Occupied Octahedral Interstices in Closest-packings of Spheres... 

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