Geometrical partitioning factor

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... 

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)),... 

The gj are degeneration factors, Fgeo a geometrical correction factor. The partition function of a classical gas is QcUss = 1 /N (V[/A)NQ. ... 

The equilibrium partition factor is defined as the ratio of the concentration of species A inside and outside the pores. This concept was first introduced by Ferry in terms of a geometric exclusion effect [25]. Since the center of mass of the molecule, assumed to be a hard sphere, cannot be closer to the pore wall than the distance of the molecular radius, Ferry obtained... 

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... 

In the following discussion we will consider an ensemble of molecules that represents a liquid system. Assuming there is no correlation between molecular interactions and geometrical restraints, the partition sum Z of an ensemble can be factorized into three contributions ... 

The reduced partition functions of isotopic molecules determine the isotope separation factors in all equilibrium and many non-equilibrium processes. Power series expansion of the function in terms of even powers of the molecular vibrations has given explicit relationships between the separation factor and molecular structure and molecular forces. A significant extension to the Bernoulli expansion, developed previously, which has the restriction u = hv/kT < 2n, is developed through truncated series, derived from the hyper-geometric function. The finite expansion can be written in the Bernoulli form with determinable modulating coefficients for each term. They are convergent for all values of u and yield better approximations to the reduced partition function than the Bernoulli expansion. The utility of the present method is illustrated through calcidations on numerous molecular systems. 

Another approach consists of expressing the (3 parameter as a sum over the partial waves of the ejected electron, of products of an energy-independent geometrical factor and an energy-dependent dynamical factor. This is the partitioning scheme of Thiel (1982 and 1983). This scheme is based on the expectation that, in the case of a resonance (shape or autoionization), only one contribution is dominant. In such a case, f3 reduces to a purely geometrical factor. The geometrical factors, G , are listed in Table 8.4 for the case where the ionic state is E and the partial wave is la. 

Clearly, a number of structural factors (geometric faaors and molecular states) will influence intramolecular interactions, hence solubility and partitioning. This set of effects is schematized in Figure 2, part of which is briefly discussed below. 

Experimental evidence on the relationship between structure and reactivity in Norrish type II photochemistry of spirobenzoyladamantanes in the solid state has been produced. Geometric factors clearly determine partitioning between cleavage, cychzation, and hydrogen transfer reactions of the 1,4-biradical intermediate Cleavage becomes predominant when the singly occupied orbitals overlap poorly with one another but well with the central CC bond, which, as earlier sections indicate, has been presumed for years based on ketone structure. 

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