Hildebrand model

Following the introduction of the Hildebrand model, the topic of solubility parameters has been extensively discussed (Hildebrand and Scott, 1962 Hildebrand et al., 1970 Kumar and Prausnitz, 1975 Barton, 1983), and values of S can be found in these reference works. As a general rule, compounds having stronger London forces will be characterized by larger solubility parameters values. 

Scatchard-Hildebrand model / xs(Com) Expansion of ( Au / RT) limited to the S" term... 

Statistical thermodynamics can be used to calculate both the enthalpic and the entropic contributions to the free energy of mixing by means of statistical calculations. Attempts were first made to apply a theory appropriate to simple regular solutions, the Hildebrand model, to the case of macromolecular solutions. This model does not adequately account for the specific behavior of polymer solutions primarily because the entropy of mixing in a polymer solution is strongly affected by the connectivity of the polymer—that is, by the existence of covalent bonds between the repetitive units. 

Some attempts were made to improve the concept of the solubility volume. Jac-quel et al. found relationship between radius of this value for three approaches (Hildebrand model, Hansen model, and Beeibower model) and some conditions (temperature, ciystallinity, and molecular weight)." Hughes et al. tried to find a more deep theoretical base for the Hansen s solubility volume essentially in case of dipolar interactions." This last attempt included taking into account more reasonable partial parameters. But it was not able to overcome limitations of the initial approach. The reason can be referred to contradicted basic postulates. 

Hildebrand T., Ruegsegger P. Quantifieation of bone microarchitecture with the structure model index. CMBBE, v.l, 15-23, 1997. 

Margules, and Scatchard-Hildebrand) are particular mathematical solutions to Eq. (48) these models do not satisfy Eqs. (45) and (46), except in the limiting case where the right-hand sides of these equations vanish. This limiting case provides a good approximation for mixtures at low pressures but introduces serious error for mixtures at high pressures, especially near critical conditions. 

The solubility parameter 5 of a pure solvent defined initially by Hildebrand and Scott based on a thermodynamic model of regular solution theory is given by Equation 4.4 [13] ... 

Various models of SFE have been published, which aim at understanding the kinetics of the processes. For many dynamic extractions of compounds from solid matrices, e.g. for additives in polymers, the analytes are present in small amounts in the matrix and during extraction their concentration in the SCF is well below the solubility limit. The rate of extraction is then not determined principally by solubility, but by the rate of mass transfer out of the matrix. Supercritical gas extraction usually falls very clearly into the class of purely diffusional operations. Gere et al. [285] have reported the physico-chemical principles that are the foundation of theory and practice of SCF analytical techniques. The authors stress in particular the use of intrinsic solubility parameters (such as the Hildebrand solubility parameter 5), in relation to the solubility of analytes in SCFs and optimisation of SFE conditions. 

The regular solution model, originally introduced by Hildebrand [2] and further developed by Guggenheim [3], is the most used physical model beside the ideal... 

Finally, it should be noted that although we have used a model in which the liquid solution conceptually has been divided into a lattice, Hildebrand [12] has shown that a similar expression may be derived without resorting to a hypothetical lattice. 

The extension of the cell model to multicomponent systems of spherical molecules of similar size, carried out initially by Prigogine and Garikian1 in 1950 and subsequently continued by several authors,2-5 was an important step in the development of the statistical theory of mixtures. Not only could the excess free energy be calculated from this model in terms of molecular interactions, but also all other excess properties such as enthalpy, entropy, and volume could be calculated, a goal which had not been reached before by the theories of regular solutions developed by Hildebrand and Scott8 and Guggenheim.7... 

First a database of solute-solvent properties are created in SoluCalc. The database needs the melting point, the enthalpy of fusion and the Hildebrand solubility parameter of the solute (Cimetidine) and the solvents for which solubility data is available. Using the available data, SoluCalc first prepares a list of the most sensitive group interactions and fits sequentially, the solubility data for the minimum set of group interaction parameters that best represent the total data set. For a small set of solvents, the fitted values from SoluCalc are shown in Table 9. It can be noted that while the correlation is very good, the local model is more like a UNIQUAC model than a group contribution model... 

Heindel, J.J. Powell, C.J. (1992) Phthalate ester effects on rat Sertoli cell function in vitro effects of phthalate side chain and age of animal. Toxicol appl. Pharmacol, 115, 116-123 Hellwig, J., Freudenberger, H. Jackh, R. (1997) Differential prenatal toxicity of branched phthalate esters in rats. Food Cosmel Toxicol, 35, 501-512 Hildebrand, H., Schmidt, U., Kempka, G, Jacob, R., Ahr, H.J., Ebener, C., Goretzki, RE. Bader, A. (1999) An in-vitro model for peroxisome proliferation utilizing primary hepatocytes in sandwich culture. Toxicol. In Vitro, 13, 265-273... 

Hildebrand, E.E. and Schack-Kirchner, H. 2000. Initial effects of lime and rock powder application on soil solution chemistry in a dystric cambisol - results of model experiments. Nutrient Cycling in Agroecosystems 56 69-78. 

The solubility of a gas is an integral part for the prediction of the permeation properties. Various models for the prediction of the solubility of gases in elastomeric polymers have been evaluated (57). Only a few models have been found to be suitable for predictive calculations. For this reason, a new model has been developed. This model is based on the entropic free volume activity coefficient model in combination with Hildebrand solubility parameters, which is commonly used for the theory of regular solutions. It has been demonstrated that mostly good results are obtained. An exception... 

A particularly simple approximation known as regular-solution theory was developed by Hildebrand and co-workers [J. H. Hildebrand. /. Am. Chem. Soc. 51, 66-80 (1929)]. The regular-solution model assumes that the excess enthalpy of mixing can be represented as a simple one-parameter correction... 

In a recent study by Lee (1996), LSER were combined with an extended Hildebrand solubility equation. From a comparison between LSER and the extended solubility parameter model, it has... 

The species concentrations are formulated in activities using the Pitzer model (207) for the aqueous phase and the Hildebrand-Scott solubility parameter (208) for the organic phase. 

Considerable evidence concerning the central olfactory pathways of insects is available and may be used to draw a reasonable hypothesis as to the organization and physiology (Hansson and Anton, 2000). In these model dipteran and lepidopteran systems, electrical activity in the form of action potentials that arise in response to the signal transduction events initiated by the olfactory stimulus is directed along ORN axons to the AL where synaptic connections are made that mediate signaling to higher centers within the insect CNS (Hildebrand and... 

When gas solubility data are lacking or are unavailable at the desired temperature, they can be estimated using available models. The method of Prausnitz and Shair (1961), which is based on regular solution theory and thus has the limitations of that theory. The applicability of regular solution theory is covered in detail by Hildebrand et al. (1970). A more recent model, now widely used, is UNIFAC, which is based on structural contributions of the solute and solvent molecular species. This model is described by Fredenslund et al. (1977) and extensive tabulations of equilibrium data, based on UNIFAC, have been published by Hwang et al. (1992) for aqueous systems where the solute concentrations are low and the solutions depart markedly from thermodynamic equilibrium. 

The Chao-Seader and the Grayson-Streed methods are very similar in that they both use the same mathematical models for each phase. For the vapor, the Redlich-Kwong equation of state is used. This two-parameter generalized pressure-volume-temperature (P-V-T) expression is very convenient because only the critical constants of the mixture components are required for applications. For the liquid phase, both methods used the regular solution theory of Scatchard and Hildebrand (26) for the activity coefficient plus an empirical relationship for the reference liquid fugacity coefficient. Chao-Seader and Grayson-Streed derived different constants for these two liquid equations, however. 

Hildebrand has defined a regular solution13 as one in which deviations from ideality are attributed only to the enthalpy of mixing the intermo-lecular forces are limited to dispersion forces. The equation that defines his model is... 

As a conclusion from the Hildebrand/Trouton Rule, the definition of a standard vapor phase in a standard state with a well known amount of disorder can be made. This definition can be used as a starting point for modeling diffusion coefficients of gases and liquids. 

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