Surface, equations from simple shapes

Rate equations expressing the a—time variations resulting from the inward advance of a reaction interface from the existing surfaces of other reactant shapes follow directly by the application of simple geometric considerations. The approach can also include quantitative allowance for any... 

In fact, the molecular shape determines only the local shape of the interface, i.e. the interfacial curvatures. This can be demonstrated using simple equations from differential geometry describing the area of parallel surfaces. (Parallel surfaces are discussed in section 1.13.) If we trace out surface patcties that are parallel to a patch of area [Pg.145]

It is at once evident that there is a remarkable degree of similarity between the shapes of L-type solute isotherms and Type I physisorption isotherms. However, this similarity is misleading since the adsorption mechanisms involved are likely to be quite different. We have seen already that Type I physisorption isotherms for gas-solid systems are normally associated with micropore filling. In contrast, the plateau of an L-type solute isotherm usually corresponds to monolayer completion. In this respect, solute adsorption appears to correspond more closely to the classical Langmuir mechanism. If this is indeed the case it would seem to be possible to calculate the surface area from nj by the application of a simple equation of the same form as Eq. (2). 

Though the case of constant matrix elements and the example investigated by Hite are the only situations for which Che stoichiometric relations have been fully established in pellets of arbitrary shape, it is worth mentioning situations in which these relations are known not to hold. When the composition and pressure at the surface of the pellet may vary in an arbitrary way from point to point it seems unlikely on intuitive grounds that equations (11.3) will be satisfied, and Hite and Jackson [77] confirmed by direct computation that there are, indeed, simple situations in which they are violated. Less obviously, direct computation [75] has also shown them to be violated even when the pressure and composition of the environment are the same everywhere, in the case where finite resistances to mass transfer exist at the surface of Che pellet. 

Fig. 2.18 A cross-section of a much-quoted model (following Freeze and Cherry, 1979, who cited Hubbert, 1940). The surface is described as undulating in a mode that can be expressed by a simple mathematical equation, and the water table is assumed to follow topography in a fixed mode. The stippled section describes a water system from a low-order divide to a nearby low-order valley the thick lines mark there impermeable planes that are an intrinsic part of the U-shape flow paths model, enlarged in Fig. 2.19. The cross-section emphasizes topographic undulations and disregards the location of the terminal base of drainage and the location of the main water divide.

The extent of adsorption can have a profound effect on the rate of the surface reaction. Equilibrium isotherms of many kinds have been reported for adsorption from solution and have been classified by Giles et al. [24-27], The shapes of these adsorption curves often furnish qualitative information on the nature of the solute-surface interactions. Several of the types of isotherm observed in dilute solution are represented reasonably well by three simple and popular isotherm equations, those of Henry, Langmuir, and Freundlich. Their shapes are illustrated in Fig. 1. Each of these isotherms relates the surface concentrations cads (mol m"2) to the bulk equilibrium concentration c of the solute species in question. When few surface sites are occupied, Henry s law adsorption... 

The way in which Lippens and de Boer (1965) made use of the universal /-curve is simple. The experimental isotherm is transformed into a /-plot in the following manner the amount adsorbed, n, is replotted against /, the standard multilayer thickness on the reference non-porous material at the corresponding p/p°. Any difference in shape between the experimental isotherm and the standard /-curve is thus revealed as a non-linear region of the /-plot and/or a finite (positive or negative) intercept of the extrapolated /-plot (i.e. at / = 0). By this method a specific surface area, denoted a(t), can be calculated from the slope, s, = nft, of a linear section. From Equations (6.10), (6.11) and (6.15) we then get ... 

Equation 4.14 makes use of the fact that the scattered intensity is proportional to the amount of a particular phase, e.g. see Eqs. 2.17 to 2.19 in Chapter 2, with a correction to account for different absorption of x-rays by two components in the mixture. Since the ratio of intensities fi om a pure phase and a mixture is employed, diffraction patterns from both the pure material and from the analyzed mixture must be measured at identical instrumental settings, in addition to identical sample characteristics such as preparation, shape, amount, packing density, surface roughness, etc. King s equation becomes a simple intensity ratio when two phases have identical absorption coefficients, i.e. when p/pa = p/pb. We note that the composition of the second phase (or a mixture of all other phases) should be known in order to determine its mass absorption coefficient. Otherwise, mass absorption should be determined experimentally. When absorption effects are ignored, the accuracy of quantitative analysis may be lowered drastically. 

In this equation A, and A2 are the areas of the appropriate surfaces 1 and 2. Angles 0, and 02 are the angles defined by the shortest distance r between the surfaces and the appropriate normal vector to each surface. There are many different ways of solving the view factor equation. The solution depends on the shape and orientation of the surfaces. Comprehensive listings of view factors for commonly encountered configurations can be found in References 2 and 18 through 20. The application of view factors can be illustrated with a simple example. Suppose we have surface 1 with temperature 7 , and emissivity e and a black surface 2 (e = 1) with temperature T2 then the total radiant heat flux from surface 1 to surface 2 is... 

In this case, the Wulff shape is truncated at the interface by an amount Ah, which is proportional to the adhesion energy [3. The latter represents the work to separate the supported crystal from the substrate at an infinite distance, hg and 7s are the central distance and the surface energy of the facet parallel to the interface, respectively. In particular, this theorem shows that the stronger the particle-substrate interaction (given by / ) is the flatter is the supported particle. Equation (3.7) offers a simple way for determining the adhesion energy of a supported crystal from TEM pictures of supported particles observed in a profile view [47]. 

In general, a drop is formed and care must be taken to avoid any disturbance of its shape. Then the dimensions of the drop are measured, e.g. from a photograph. Usually a rather large drop is formed, because only one radius of curvature, that in the plane of drawing, is considered. For this very simple method, the contact angle is not required and only the distance between the equatorial plane and the apex is measured (Figure 6.15). Surface tension is calculated from the equation... 

Porous catalysts are used in a variety of shapes, including spheres, cylinders, rings, irregular particles, and thin coatings on tubes or flat surfaces. Diffusion plus reaction in a flat slab is a simple case to analyze, since the area for diffusion does not change with distance from the external surface. The equation is similar to that for diffusion and reaction in a straight... 

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