same volume as a sphere of radius r, specifically, 4^/3 = idCP-L/A, which gives It is easy to show that die specific surface area S(L/D) from Eq. (10.5) is given by

This expression S(L/D), which has a minimum S^ = 1.1465spb for the ratio (L/D) = 1, is plotted in Fig. 10.7, normalized relative to S^. The normalization factor S^ was chosen because a sphere has the smallest surface area of any object with a particular volume. Figure 10.7 shows how the surface area increases when a sphere is distorted into the shape of a disk with a particular L/D ratio, without changing in its volume. This figure demonstrates that nanostructures of a particular mass or of a particular volume have much higher surface areas S when they are flat or elongated in shape, and further distortions from a regular shape will increase the area even more.

10.2.3. Porous Materials

In die previous section we saw that an efficient way to increase the surface area of a material is to decrease its grain size or its particle size. Another way to increase the surface area is to fill the material with voids or empty spaces. Some substances such as zeolites, which are discussed in Sections 6.2.3 and 8.4, crystallize in structures in which there are regularly spaced cavities where atoms or small molecules can lodge, or they can move in and out during changes in the environmental conditions. A molecular sieve, which is a material suitable for filtering out molecules of particular sizes, ordinarily has a controlled narrow range of pore diameters. There

Figure 10.7. Dependence of the surface area S(L/D) of a cylinder on its length: diameter ratio L/D. The surface area is normalized relative to that of a sphere S^ = 3/pr with the same volume.

Figure 10.7. Dependence of the surface area S(L/D) of a cylinder on its length: diameter ratio L/D. The surface area is normalized relative to that of a sphere S^ = 3/pr with the same volume.

are also other materials such as silicas and aluminas which can be prepared so that they have a porous structure of a more or less random type; that is, they serve as sponges on a mesoscopic or micrometer scale. It is quite common for these materials to have pores with diameters in the nanometer range. Pore surface areas are sometimes determined by the Brunauer-Emmett-Teller (BET) adsorption isotherm method in which measurements are made of the uptake of a gas such as nitrogen (N2) by the pores.

Most commercial heterogeneous catalysts have a very porous structure, with surface areas of several hundred square meters per gram. Ordinarily an heterogeneous catalyst consists of a high-surface-area material that serves as a catalyst support or substrate, and die surface linings of its pores contain a dispersed active component, such as acid sites or platinum atoms, which bring about or accelerate the catalytic reaction. Examples of substrates are the oxides silica (Si02), gamma-alumina (y-Al203), titania (Ti02 in its tetragonal anatase form), and zirconia (Zi02). Mixed oxides are also in common use, such as high-surface-area silica-alumina. A porous material ordinarily has a range of pore sizes, and this is illustrated by the upper right spectrum in Fig. 10.8 for the organosilicate molecular sieve MCM-41, which has a mean pore diameter of 3.94 nm (39.4 A). The introduction of relatively large trimethylsilyl groups (Cl^Si to replace protons of silanols SiH3OH in the pores occludes the pore volume, and shifts the distribution of pores to a smaller range of sizes, as shown in the lower left spectrum of the figure. The detection of the nuclear magnetic resonance (NMR) signal from the 29Si isotope of the trimethylsilyl groups in these molecular sieves, with its + 12ppm chemical shift shown in Fig. 10.9, confirmed its presence in the pores after the trimethylsilation treatment.

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