Theory of partitions

The Theory of Partitions, Encyclopedia of Mathematics and its Applications, Addison-Wesley, Reading, Massachusetts 1976. 

The term plate height comes from distillation theory. Some high-performance distillation columns contain discrete units called plates, in which liquid and vapor equilibrate with each other. As a teenager, A. J. R Martin, coinventor of partition chromatography, built distillation columns in discrete sections from coffee cans. (We don t know what he was distilling ) When he formulated the theory of partition chromatography, he adopted terms from distillation theory. 

The theory of partitioning for bodies of complex geometry in convoluted pore spaces with different-sized passages is obviously quite complicated, but... 

The treatment of partition equilibrium was further generalized to the cases in the presence of ion-pair formation [19] and ion-ionophore complex formation [21]. An important corollary of this theory of partition equilibrium based on standard ion transfer potentials of single ions is to give a new interpretation to liquid extraction processes. Kakutani et al. analyzed the extraction of anions with tris(l,10-phenan-throline) iron(II) cation from the aqueous phase to nitrobenzene [22], which demonstrated the effectiveness of the theory and gave a theoretical backbone for ion-pair extraction from an electrochemical point of view. 

V 5. Vink, H. Theory of partition chromatography. II. Numerical calculations. 

Hardy, G.H. (1940). Ramanujan, Chapter VIII Asymptotic theory of partitions. Chelsea Publ. Co., New York. 

Also see Andrews, G.E. (1976). The theory of partitions. Addison-Wesley, Londoa... 

Pankow, J. F. (1987) Review and comparative analysis of the theories of partitioning between the gas and aerosol particulate phases in the atmosphere, Atmos. Environ. 21, 2275-2283. 

McMillan-Mayer theory of solutions [1,2], which essentially seeks to partition the interaction potential into tln-ee parts that due to the interaction between the solvent molecules themselves, that due to die interaction between the solvent and the solute and that due to the interaction between the solute molecules dispersed within the solvent. The main difference from the dilute fluid results presented above is that the potential energy u(r.p is replaced by the potential of mean force W(rp for two particles and, for particles of solute in the solvent, by the expression... 

There is an inunediate coimection to the collision theory of bimolecular reactions. Introducing internal partition functions excluding the (separable) degrees of freedom for overall translation. 

We now consider the connection between the preceding equations and the theory of Aharonov et al. [18] [see Eqs. (51)-(60)]. The tempting similarity between the structures of Eqs. (56) and (90), hides a fundamental difference in the roles of the vector operator A in Eq. (56) and the vector potential a in Eq. (90). The fomrer is defined, in the adiabatic partitioning scheme, as a stiictly off-diagonal operator, with elements (m A n) = (m P n), thereby ensuring that (P — A) is diagonal. By contiast, the Mead-Truhlar vector potential a arises from the influence of nonzero diagonal elements, (n P /i) on the nuclear equation for v), an aspect of the problem not addressed by Arahonov et al. [18]. Suppose, however, that Eq. (56) was contracted between (n and n) v) in order to handle the adiabatic nuclear dynamics within the Aharonov scheme. The result becomes... 

Partitioning Electron Density The Theory of Atoms in Molecules... 

R F W Bader s theory of atoms in molecules [Bader 1985] provides an alternative way to partition the electrons between the atoms in a molecule. Bader s theory has been applied to many different problems, but for the purposes of our present discussion we will concentrate on its use in partitioning electron density. The Bader approach is based upon the concept of a gradient vector path, which is a cuiwe around the molecule such that it is always perpendicular to the electron density contours. A set of gradient paths is drawn in Figure 2.14 for formamide. As can be seen, some of the gradient paths terminate at the atomic nuclei. Other gradient paths are attracted to points (called critical points) that are... 

The theory of the process can best be illustrated by considering the operation, frequently carried out in the laboratory, of extracting an orgaiuc compound from its aqueous solution with an immiscible solvent. We are concerned here with the distribution law or partition law which, states that if to a system of two liquid layers, made up of two immiscible or slightly miscible components, is added a quantity of a third substance soluble in both layers, then the substance distributes itself between the two layers so that the ratio of the concentration in one solvent to the concentration in the second solvent remains constant at constant temperature. It is assumed that the molecular state of the substance is the same in both solvents. If and Cg are the concentrations in the layers A and B, then, at constant temperature ... 

Now we make the usual assumption in nonadiabatic transition theory that non-adiabaticity is essential only in the vicinity of the crossing point where e(Qc) = 0- Therefore, if the trajectory does not cross the dividing surface Q = Qc, its contribution to the path integral is to a good accuracy described by adiabatic approximation, i.e., e = ad Hence the real part of partition function, Zq is the same as in the adiabatic approximation. Then the rate constant may be written as... 

The reader who is less familiar with the theory of grand partition functions may directly proceed to Eqs. 12a and 13. The physical basis of these formulas and the significance of the quantities CK% will then become apparent in the subsequent paragraph is the vapor pressure (or fugacity) of solute K and y i is the probability of finding a K molecule in a cavity of type i. 

Again, therefore, all thermodynamic properties of a system in quantum statistics can be derived from a knowledge of the partition function, and since this is the trace of an operator, we can choose any convenient representation in which to compute it. The most fruitful application of this method is probably to the theory of imperfect gases, and is well covered in the standard reference works.23... 

There is, however, a fatal objection to the theory of Boltzmann. At very low temperatures the oscillations will be small, and should conform to the theory. But the atomic heats, instead of approaching the limit 5 955 at low temperatures, diminish very rapidly, and in the case of diamond the specific heat is already inappreciable at the temperature of liquid air. A new point of view is therefore called for, and it is a priori very probable that this will consist of a replacement of the hypothesis of Equipartition of Energy adopted by Boltzmann. This supposition has been verified, and the new law of partition of energy derived... 

In the following, the MO applications will be demonstrated with two selected equilibrium reactions, most important in radical chemistry disproportionation and dimerization. The examples presented will concern MO approaches of different levels of sophistication ab initio calculations with the evaluation of partition functions, semiempirical treatments, and simple procedures employing the HMO method or perturbation theory. 

In the framework of the mobile order and disorder (MOD) theory five components contribute most to the Gibbs free energy of partitioning of a solute in a biphasic system of two essentially immiscible solvents [23] ... 

A relationship correlating the weak acid uncouplers activity with their A %-, pAi°, and has been presented on the basis of protonophoric theory of uncoupling activity, in which the concentration of anionic ionophore (A ) within a biomembrane is supposed to be controlled by the ionic partition of A at the biomembrane solution interface according to Eq. (28) [19]. The biomembrane solution interface could be polarized or electrogenic [37]. Experimental results on the activities of uncouplers on rat liver mitochondria [30] have been explained reasonably [19,24]. 

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