Distribution of pore size

The inner surface area of solid catalysts is mainly distributed in the pores and channels of crystal-particles, and furthermore the diffusion and mass transfer dm-ing reaction process is directly dependent on the pore structure. Hence, the pore size and pore volume are sometimes more important than the surface area of the pore as structural information. 

The pore size of the catalyst relates to the mass transfer process of catalytic reaction. When reactions proceed in an inner diffusion area, the mass transfer rate is relatively slow, and the pore size relates to the surface utilization ratio of the catalyst in the reaction. If the target product is an unstable intermediate, the pore size will affect the selectivity of reaction. Therefore, for a given reaction condition and composition of the catalyst, a catalyst should have the uniform pore size distribution in order to develop of a good catalyst. 

The analysis of the pore distribution needs to measure adsorption isotherms. In addition, it is mainly based on gas-liquid equilibrium theory in thermodynamics to study the characteristics of adsorption isotherm, and use the different pore models to calculate the distribution of pore. In the experimental methods for the pore structure determination, steam physical adsorption and pressed mercury ways are the two key technologies. These technologies correlate with the rationalization and continual development in theory and in a variety of simulation technologies of physical adsorption, while ensuring that the experimental equipment are easy-to-automate, small and bear good facilitation. 

It can see from the above-mentioned discussion that capillary cohesion is closely related to the curved liquid surface. The pressme boimdary causes capillary cohesion — the critical vapor pressure relates to the cmvatme radius of liquid surface. Kelvin equation has been derived from thermodynamics, where the curvature radius (rjs) of the meniscus of hemispherical (concave) liquid and the equilibrium vapor pressure (p) has the following relationships  

It can be seen from Eq. (7.63) that p/po 1, and the smaller is the r, the smaller is p/po. This means that for very small pore, capillary cohesion can occur when the steam pressure is lower than the saturated vapor pressure po at the adsorption temperature. Furthermore, rj, means the critical pore diameter, and that is in the equilibrium vapor pressure p. When the pore radius of solid is smaller than or equal to the given value of rj, in Eq. (7.63), capillary cohesion takes place while when the pore radius of solid is larger than the given value rk in Eq. (7.63) cohesion would not happen, only casing the multi-layer film on the pore wall. Therefore, if 

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A procedure that is more suitable for obtaining the actual distribution of pore sizes involves the use of a nonwetting liquid such as mercury—the contact angle on glass being about 140° (Table X-2) (but note Ref. 31). If all pores are equally accessible, only those will be filled for which... 

Below the critical temperature of the adsorbate, adsorption is generally multilayer in type, and the presence of pores may have the effect not only of limiting the possible number of layers of adsorbate (see Eq. XVII-65) but also of introducing capillary condensation phenomena. A wide range of porous adsorbents is now involved and usually having a broad distribution of pore sizes and shapes, unlike the zeolites. The most general characteristic of such adsorption systems is that of hysteresis as illustrated in Fig. XVII-27 and, more gener-... 

As pointed out earlier (Section 3.5), certain shapes of hysteresis loops are associated with specific pore structures. Thus, type HI loops are often obtained with agglomerates or compacts of spheroidal particles of fairly uniform size and array. Some corpuscular systems (e.g. certain silica gels) tend to give H2 loops, but in these cases the distribution of pore size and shape is not well defined. Types H3 and H4 have been obtained with adsorbents having slit-shaped pores or plate-like particles (in the case of H3). The Type I isotherm character associated with H4 is, of course, indicative of microporosity. 

In a typical amorphous adsorbent the distribution of pore size may be very wide, spanning the range from a few nanometers to perhaps one micrometer. Siace different phenomena dominate the adsorptive behavior ia different pore size ranges, lUPAC has suggested the foUowiag classification ... 

Fig. 26. Screen filters contain pores of a uniform size and retain all particulates greater than the pore diameter at the surface of the membrane. Depth filters contain a distribution of pore sizes. Particulates entering the membrane are trapped at constrictions within the membrane. Both types of filters are rated 10...

There are many complications with interpreting MWCO data. First, UF membranes have a distribution of pore sizes. In spite of decades of effort to narrow the distribution, most commercial membranes are not notably sharp. What little is known about pore-size distribution in commercial UF membranes fits the Poisson distribution or log-normal distribution. Some pore-size distributions may be polydisperse. 

A question of practical interest is the amount of electrolyte adsorbed into nanostructures and how this depends on various surface and solution parameters. The equilibrium concentration of ions inside porous structures will affect the applications, such as ion exchange resins and membranes, containment of nuclear wastes [67], and battery materials [68]. Experimental studies of electrosorption studies on a single planar electrode were reported [69]. Studies on porous structures are difficult, since most structures are ill defined with a wide distribution of pore sizes and surface charges. Only rough estimates of the average number of fixed charges and pore sizes were reported [70-73]. Molecular simulations of nonelectrolyte adsorption into nanopores were widely reported [58]. The confinement effect can lead to abnormalities of lowered critical points and compressed two-phase envelope [74]. 

Natural rocks seldom have a single pore size but rather a distribution of pore sizes. If all pores are in the fast-diffusion limit, have the same surface relaxivity and have no diffirsional coupling, then the pores will relax in parallel with a distribution of relaxation times that corresponds to the distribution of the pore sizes. The magnetization will decay as a sum of the exponentials as described by Eq. (3.6.4). 

The relaxation time for each pore will still be expressed by Eq. (3.6.3) where each pore has a different surface/volume ratio. Calibration to estimate the surface relaxivity is more challenging because now a measurement is needed for a rock sample with a distribution of pore sizes or a distribution of surface/volume ratios. The mercury-air or water-air capillary pressure curve is usually used as an estimator of the cumulative pore size distribution. Assuming that all pores have the same surface relaxivity and ratio of pore body/pore throat radius, the surface relaxivity is estimated by overlaying the normalized cumulative relaxation time distribution on the capillary pressure curve [18, 25], An example of this process is illustrated in Figure 3.6.5. The relationship between the capillary pressure curve and the relaxation time distribution with the pore radii, assuming cylindrical pores is expressed by Eq. (3.6.5). 

The desorption isotherm approach is the second generally accepted method for determining the distribution of pore sizes. In principle either a desorption or adsorption isotherm would suffice but, in practice, the desorption isotherm is much more widely used when hysteresis effects are observed. The basis of this approach is the fact that capillary condensation occurs in narrow pores at pressures less than the saturation vapor pressure of the adsorbate. The smaller the radius of the capillary, the greater is the lowering of the vapor pressure. Hence, in very small pores, vapor will condense to liquid at pressures considerably below the normal vapor pressure. Mathematical details of the analysis have been presented by Cranston and Inkley (16) and need not concern us here. 

Where there is a wide distribution of pore sizes and, possibly, quite separately developed pore systems, a mean size is not a sufficient measure. There are two methods of finding such distributions. In one a porosimeter is used, and in the other the hysteresis branch of an adsorption isotherm is utilised. Both require an understanding of the mechanism of capillary condensation. 

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