Hildebrand solution model

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

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... 

Experimental vapor-liquid-equilibrium data for benzene(l)/n-heptane(2) system at 80°C (176°F) are given in Table 1.8. Calculate the vapor compositions in equilibrium with the corresponding liquid compositions, using the Scatchard-Hildebrand regular-solution model for the liquid-phase activity coefficient, and compare the calculated results with the experimentally determined composition. Ignore the nonideality in the vapor phase. Also calculate the solubility parameters for benzene and n-heptane using heat-of-vaporization data. 

The Scatchard-Hildebrand regular-solution model expresses the liquid activity coefficients y in a binary mixture as... 

Related Calculations. The regular-solution model of Scatchard and Hildebrand gives a fair representation of activity coefficients for many solutions containing nonpolar components. This procedure is suggested for estimating vapor-liquid equilibria if experimental data are not available. The solubility parameters and liquid molar volumes used as characteristic constants may be obtained from Table 1.10. For substances not listed there, the solubility parameters may be calculated from heat of vaporization and liquid molar volume data as shown in step 4. 

The regular solution model, introduced by the American UC Berkeley chemist Joel Henry Hildebrand (1881-1983) is the simplest way to consider these other contributions to the Gibbs energy (Hildebrand, 1929). In this case, Gex is of the form ... 

Occasionally the basic Scott-Hildebrand regular solution model might be useful for non-hydrocarbon mixtures. 

This is the case for the C6H6 - CC14 system, but not for the C6H6 - CS2 system. Therefore, regular solution theory is not applicable to the C6H6-CS2 system. To test the Hildebrand-Scatchard model we use... 

J. Sci.. in press). Indeed, in the case of a solid-solution with a small difference in the size of the substituting ions (relative to the size of the non-substituting ion), the first parameter, ao, is usually sufficient (8). Equations 5 and 6 then become identical to those of the "regular" solid-solution model of Hildebrand (9). For the case where both ao and ai parameters are needed, equations 5 and 6 become equivalent to those of the "subregular" solid-solution model of Thompson and Waldbaum (10). a model much used in high-temperature work. Equations 5 and 6 can also be shown equivalent to Margules activity coefficient series (11). 

In ideal solutions, the pure solute and solvent mix with no heat of mixing, AH"" = 0, and the heat of dissolution is numerically equal to the heat of fusion. However, only a limited number of systems form ideal solutions. A less restrictive assumption is that the solution is represented by a regular solution model. This model assumes the heat of mixing is nonzero, but independent of solution composition and temperature (i.e., AH= constant) (Hildebrand and Scott 1950). For a regular solution the differential entropy of mixing is also assumed ideal (i.e., AS = —R In x). 

The sodium-lithium phase system has been studied by thermal analysis in the liquid and solid regions to temperatures in excess of 400°C. Two liquid phases separate at 170.6°C. with compositions of 3.4 and 91.6 atom % sodium. The critical solution temperature is 442° zt 10°C. at a composition of 40.3 atom % sodium. The freezing point of pure lithium is depressed from 180.5°C. to 170.6°C. by the addition of 3.4 atom % sodium, and the freezing point of pure sodium is depressed from 97.8° to 92.2°C. by the addition of 3.8 atom % lithium. From 170.6° to 92.2°C. one liquid phase exists in equilibrium with pure lithium. Regardless of the similarity in the properties of the pure liquid metals, the binary system deviates markedly from simple nonideal behavior even in the very dilute solutions. Correlation of the experimentally observed data with the Scatchard-Hildebrand regular solution model using the Flory-Huggins entropy correction is discussed. 

This model was first described by JH Hildebrand in 1929, and is called the regular solution model [4]. Solutions with free energies of mixing that are well described by Equation (15.14) are called regular solutions. Here s an application of Equation (15.14). 

JH Hildebrand and RL Scott, Regular Solutions, Prentice-Hall, Englewood Cliffs, 1962. The regular solution model is described in detail. 

We can see that the regular solution model does not give us the variations of the activity coefficient with the amounts of substance. In fact, it is a family of models which includes several possible subsets - each one defined by a relation 9tj(x). We shall come across two of these subsets in our study the Van Laar equation (see Table 3.3) and the Hildebrand-strictly-regular solution model, which we shall touch on. 

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] ... 

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... 

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... 

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. 

Other molecular properties have been also proposed to model the hydrophobic interactions. The parachor, which is related to the surface tension of a compound (139, 140) represents mainly the intermolecular interactions in a liquid. The Hildebrand-Scott solubility parameter, 6, (141) is related to intermolecular van der Waals forces and the closely related molar attraction constant, F, is obtained by multiplying 6 by the molar volume (142). The partition coefficient between two solvents can be obtained from the solubility parameters and the molar volumes of the solute and the solvents (193). This relationship is based on regular solution theory (194) and the assumption that the partial molar volumes of the solute is not different from its molar volume. Recently this has been criticized and a new derivation was proposed (195) in which the partial molar volumes are taken into account. The molar refractivity, MR, is related to dispersion forces and can be obtained as a sum of the partial molar refractivi-ties assigned to atoms and bonds (140, 143). These parameters have been compared (144) to establish their relative applicability to correlations with biological activity. The conclusion was that logP and molecular refractivity were the best parameters. Parameters obtained from high pressure liquid chromatography (144,... 

A number of methods based on regular solution theory also are available. Only pure-component parameters are needed to make estimates, so they may be applied when UNIFAC group-interaction parameters are not available. The Hansen solubility parameter model divides the Hildebrand solubility parameter into three parts to obtain parameters 8d, 5p, and 5 accounting for nonpolar (dispersion), polar, and hydrogenbonding effects [Hansen,/. Paint Technot, 39, pp. 104-117 (1967)]) An activity coefficient may be estimated by using an equation of the form... 

Solute-solvent interactions were largely studied and modeled by Linear Solvation Energy Relationships and the —> Hildebrand solubility parameter. 

The entropy of mixing of van der Waals-type equations, which is derived from the repulsive term, bears a remarkable functional similarity with that of explicit activity coefficient models, which are successfully applied to polymer solutions. Below are shown these equations for the van der Waals-type equation of state, the EH model and the Entropic-EV model (Elbro-Hildebrand term) ... 

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