Geometrical partitioning

It should be clear that most of the methods discussed in the present section extend readily to three dimensions—the random geometric partitioning of a volume. For example, the linear exponential expression corresponding to (8.59) is... 

Despite claims by Spry and Sawyer (1975) of analytical measurements verifying asphaltene molecular sizes in the 100 A range at ambient conditions, it is unlikely that molecules this bulky exist at reaction conditions. The good predictive capability of the model may therefore result from a compensation effect. Electrostatic and adsorption interactions between solute molecules and the pore walls not explicitly accounted for with the purely geometric partition coefficient may result in the diffusing molecules appearing larger than they are at reaction conditions. 

These three types of surface species—inner-sphere complex, outer-sphere complex, and diffuse-layer—represent three modes of adsorption of small aqueous ions that contribute to the formation of the electrochemical double layer on clay mineral surfaces. No inference of special planes containing adsorbed ions is required by these surface chemical speciation concepts, nor is detailed molecular structure implied, other than the general notions of surface complexes and vicinal dissociated ions. It is sometimes convenient, although not necessary, to group the two types of surface complex into a Stern layer to distinguish them from diffuse-layer ions [18]. This geometric partitioning of surface species, however, should not be taken to mean that diffuse-layer ions necessarily approach a particle surface less closely than do Stern-layer ions. 

In the earlier mentioned Multiple-Scattering Xa Method [6], the molecule was geometrically partitioned into three fundamental types of region, Fig. 4 ... 

There are some items of interest here. A few basic properties are the basis for separation in two different types of basic separation processes. For example, condensability of a vapor/gas species is useful for vapor absorption as well as for membrane gas separation geometrical partitioning (or partitioning by other means between a pore and an external solution) is useful both in adsorp-tion/chromatography as well as in the membrane processes of dialysis and ultrafiltration, etc. Further, there are many cases where chemical reactions are extraordinarily useful for separation these are not identified here since chemical reactions can enhance separation only if the basic mechanism for separation exists, especially in phase equilibrium based separations. However, there are a few cases where chemical reactions, especially complexations, provide the fundamental basis for separation, as in affinity chromatography, metal extractions and isotope exchange reactions. 

Geometrical partitioning of a solute between pore and a solution... 

When the solute dimensions are larger and there are no specific solute-pore wall interactions, the partitioning effect is indicated by a geometrical partitioning factor for a cylindrical pore as... 

Figure 3.3.5A. Geometrical partitioning effect only a distance of — r,- from the pore center is available for locating the center of the molecules of the solute.

The cases considered above were such that r,- < Tp and there was no pore-solute interaction. If the solute dimension is no longer orders of magnitude smaller than the pore dimensions and there are no specific solute-pore waU interactions, we may employ flux expression (3.1.112Q and integrate (Lane and Higgle, 1959) under the conditions of hindered diffusion and geometrical partitioning ... 

Case (1) The solvent present outside the membrane is identical to the solvent/liquid inside the membrane pores. Since there are no solute-pore interactions and T < Tp (excludes the geometrical partitioning effect (3.3.88a)), the solute concentration in the solvent in the pores is identical to that immediately outside the pores ... 

We can use Faxen s expression (3.1.112e) for GDr Ti,Tp) as long as (ri/fp) < 0.5. Further, Kt may be obtained from relation (3.3.88a). The membrane mass-transfer coefficient and the permeability coefficient in the case of hindered ditfiision and geometrical partitioning are defined as follows ... 

Derive the expression given below for the geometrical partitioning factor Kim for a spherical rigid solute i... 

Derive the expression for the geometrical partitioning factor ki for a spherical rigid solute of radius r,-between an external solution and a pore in the shape of a rectangle of sides bi and b2. The pore is infinitely long. Rewrite this result in terms of a hydraulic radius of the pore. (Ans.Xjm = (1 — 2ri/b2)) (1 - 2n/bi)y,)... 

The quantity (C /Ca) is the geometrical partitioning factor defined earlier (see (3.3.88a) and (3.3.89a)) for the partitioning of a solute between a solution and a porous membrane. Here the porous membrane is replaced by the microporous adsorbent. The quantity k, , is less than 1 unless there are specific or nonspecific interactions (electrostatic or van der Waals interactions) between the solute and the pores it can be quite small if the radius r, of the solute molecules is of the order of the pore radius tp. For cylindrical pores (see (3.3.88a)),... 

It is of particular interest to be able to correlate solubility and partitioning with the molecular stmcture of the surfactant and solute. Likes dissolve like is a well-wom plirase that appears applicable, as we see in microemulsion fonnation where reverse micelles solubilize water and nonnal micelles solubilize hydrocarbons. Surfactant interactions, geometrical factors and solute loading produce limitations, however. There appear to be no universal models for solubilization that are readily available and that rest on molecular stmcture. Correlations of homologous solutes in various micellar solutions have been reviewed by Nagarajan [52]. Some examples of solubilization, such as for polycyclic aromatics in dodecyl sulphonate micelles, are driven by hydrophobic... 

Theoretical efforts a step beyond simply fitting standard statistical curves to fragment size distribution data have involved applications of geometric statistical concepts, i.e., the random partitioning of lines, areas, or volumes into the most probable distribution of sizes. The one-dimensional problem is reasonably straightforward and has been discussed by numerous authors... 

Squire [364] and Porath [300,301] developed geometrical pore models for gel chromatography media. Squire considered a gel with a set of conical, cylindrical, and rectangular crevices, and found the pore volume, assumed equal to the partition coefficient K y, to vary as... 

Waldmann-Meyer, H, Structure Parameters of Molecules and Media Evalnated hy Chromatographic Partition II. Geometrical Exclusion in Gels, Journal of Chromatography 410, 233, 1987. 

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