Homomorph concept

The component of solubility parameter due to dispersion or van der Waals forces is usually approximated in the literature through the homomorph concept. The hydrocarbon homomorph of... 

The homomorph concept may also be used here in the case of hydrogen-bonded liquids. Equation 2.39a provides the component In principle, then, we may obtain the polar component of the solnbility parameter as follows ... 

As already mentioned, in evaluating the dispersive component of the solubility parameter in the literature, reference is usually made to the homomorph concept. The homomorph is typically the hydrocarbon with the structure closest to the studied substance. The homomorph of n-pentanol, for example, is n-hexane and of isopropanol is isobutane. The homomorph concept may be used with the present approach as well. Equation 2.36 indicates how to use it The homomorph and the studied substance should be brought at the same reduced density — a type of corresponding states. Table 2.5 also includes the calculated homomorph component, 8, of the solubility parameter. 

The model used may evaluate the hydrogen bonding component of the solubility parameter directly and with no recourse to ambiguous concepts such as the homomorph concept. Apart from this, the model does not distinguish between the remaining polar and nonpolar contributions to 8. 

The separation of 62 into the three components employs the homomorph concept to determine 6j as described by Blanks and Prausnitz (20) followed by a subsequent semiempirical estimate of 6p2 and 6, 2 separately. 

Summaries of three-dimensional solubility parameters are given by Hansen and Beerbower (21) and by Hoy (22). A problem that now arises, however, and that is of real significance, is the fact that the three-dimensional parameters tabulated by Hansen and Beerbower and by Hoy do not always agree. This disagreement is illustrated in Table I for three common solvents, chosen at random from the source tables. An additional problem, noted by Hansen, is that the homomorph concept for estimating 6j fails in cases of solvents containing chlorine or sulfur atoms, and, in addition, homomorphs for cyclic compounds are hard to find. 

Hansen firstly determined d for a solvent using the homomorph concept. The energy of vaporisation of a hydrocarbon molecule of the same size and shape as the solvent molecule in question at the same reduced temperature (absolute temperature divided by the critical temperature) is assumed to be that due to dispersion forces existing in the solvent. The difference between the energy of vaporisation of the solvent, AE, and that calculated as the contribution due to dispersion forces, A d, is taken as that due to both polar and hydrogen bonding forces, i.e. ... 

The prochirality concept is useful if it is applied to factored structures within a molecule rather than to the whole, because chiral compounds may also contain centers of prostereoisomerism that would become chiral if their homomorphic ligands were made distinct. The methylene carbons of cholesterol or C(3) of chiral trihydroxyglutaric acid (20b) are appropriate examples. 

Readers should take a Utde time to familiarize themselves with this homomorphism by concrete calculations such as those in Exercise 4.38. Readers who wish to check by brute calculation that 4> is indeed a homomorphism should consult Exercise 4.32. We will take another approach, one that is more appealing geometrically (because we will see how an element of 50(2) can rotate an actual geometric object) and theoretically (because it uses concepts that generalize to other Lie groups). 

Clearly we can define these concepts using just the T-action. Let T be any group acting as automorphisms of a group F. The maps / T - F satisfying f at) =/(crossed homomorphisms (they are homomorphisms when the action is trivial). The ones cohomologous to/are those of the form cf (o)[a(c)Y1 for some fixed c in F this is the definition that matches up with ours for G(S), and one can easily check that it is an equivalence relation in general. The set of equivalence classes is denoted // (T, F). 

Parameters involved in this equation may be estimated using the concept of the homomorph. The homomorph of a polar molecule is a non-polar molecule with nearly the same size and shape as its polar counterpart. The cohesion energy of the homomorph is assumed to be the measure of the effect of the dispersion forces. The polar contribution to the cohesion energy is the difference between the total cohesion energy and the cohesion energy of the homomorph. 

Originally intended for application to substances whose cohesion arose from dispersion forces, the parameter seem to be of limited use with polymers, which generally decompose before vaporization enthalpies can be determined. The concept now has been greatly expanded. The overall 6 can be divided into dispersion and polar contributions. Often non-polar homomorphs of polar molecules can provide values of 5, and polar contributions, can then be obtained from differences between 6 and Further refinements due to Hansen " have introduced a three-component solubility parameter, which separates non-dispersive contributions into polar and hydrogen bond components. This has been applied to organic liquids, and to some polymers. Calculations of b for macromolecules also can be made from tabulated values of molar attraction constants, and extensive summaries of b and of other cohesion parameters are readily available to the potential user. Ultimately, however, the application of b to polymer systems is impeded for the following reasons ... 

A. Kerber and R. Laue. Group actions, double cosets, and homomorphisms unifying concepts for the constructive theory of discrete structures. ActaAppl. Math., 52 63-90,1998. 

For the estimation of nonpolar component of 6, Brown et al proposed the homo-morph concept. The homomorph of a polar molecule is the nonpolar molecule most closely resembling it in the size and the stmctuie (e.g., n-butane is the homomorph of n-butyl alcohol). The nonpolar component of the cohesion eneigy of a polar solvent is taken as the experimentally determined total vaporization eneigy of the corresponding homomorph at the same reduced temperature (the actual temperature divided by the critical temperature in Kelvin s scale). For this comparison the molar volumes must also be equal. Blanks and Prausnitz proposed plots of dependencies of dispersion energy density on a molar volume for straight-chain, alicyclic and aromatic hydrocarbons. If the vaporization... 

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