Hildebrand theory

Regular solutions, the solubility parameter and Scatchard-Hildebrand theory... 

A theory of regular solutions leading to predictions of solution thermodynamic behavior entirely in terms of pure component properties was developed first by van Laar and later greatly improved by Scatchard [109] and Hildebrand [110,1 11 ]. It is Scatchard-Hildebrand theory that will be briefly outlined here. Its point of departure is the statement that It is next assumed that the volume... 

Another formal shortcoming of Scatchard-Hildebrand theory, or any of its variants, is its requirement that the interacting molecules be of essentially the same size and shape. This disallows an accounting for any non-randomness of the pack-... 

The Scatchard-Hildebrand theory of regular solutions is most attractive because of its simplicity, and it is of special interest here because it has been applied to hydrocarbon mixtures at high pressures (PI 3), leading to the correlation of Chao and Seader (Cl). 

Because the entropy of formation in Hildebrand theory is ideal, this approach should be restricted to those systems in which there are no structure effects due to solute-solvent and solvent-solvent interactions. The implication of this is that the solute should be non-ionic and not have functional groups which can interact with the solvent. According to Equation (4.8), the maximum solubility occurs when the Hildebrand parameter of the solvent is equal to the Hildebrand parameter of the solute. That is, when plotting the solubility versus the Hildebrand parameter, the solubility exhibits a maximum when the solubility parameter of the solvent is equal to the solubility parameter of the solute. 

None of the material presented in the last section was restricted to a specific solvent system. In fact, the use of the Hildebrand theory should be limited to those systems... 

Even though Hildebrand theory should not apply to solvent systems having considerable solvent-solvent or solute-solvent interactions, the solubility of compounds in co-solvent systems have been found to correlate with the Hildebrand parameter and dielectric constant of the solvent mixture. Often the solubility exhibits a maximum when plotting the solubility versus either the mixed solvent Hildebrand parameter or the solvent dielectric constant. When comparing different solvent systems of similar solvents, such as a series of alcohols and water, the maximum solubility occurs at approximately the same dielectric constant or Hildebrand parameter. This does not mean that the solubilities exhibit the same maximum solubility. 

This sample shows a slightly broadened peak similar to PPO 1000/50. As described in the introduction, the driving force for phase separation comes from the large negative enthalpy of the phase-separated state relative to the homogeneous mixture. The enthalpy (AH) can be quantified in terms of the Scott-Hildebrand theory for these multiblocks (10,12,16) ... 

Experience has shown that the RED number combined with a solvent molecular size parameter, such as the molar volume, frequently provides all the technical information required to evaluate solvent quality. The original Hildebrand theory predicted that larger solvent molecules are inherently poorer solvents than smaller counterparts with the same cohesion energy parameter. Molecular size... 

TABLE 1.9 Solubility of Napthalene in Various Solvents by UNIFAC and Scatchard-Hildebrand Theory... 

The groups contribution methods can also be used to calculate solubility in binary (solute-solvent) systems. A comparison of solubilities calculated employing the UNIFAC method with experimental values and values obtained from the Scatchard-Hildebrand theory is given in Table 1.9. 

Finding an appropriate mixed solvent system should not be done on a strictly trial and error basis. It should be examined systematically based on the binary solubility behavior of the solute in solvents of interest. It is important to remember that the mixed solvent system with the solute present must be miscible at the conditions of interest. The observed maximum in the solubility of solutes in mixtures is predicted by Scatchard-Hildebrand theory. Looking at Eq. (1.50) we see that when the solubility parameter of the solvent is the same as that of the subcooled liquid solute, the activity coefficient will be 1. This is the minimum value of the activity coefficient possible employing this relation. When the activity coefficient is equal to 1, the solubility of the solute is at a maximum. This then tells us that by picking two solvents with solubility parameters that are greater than and less than the solubility parameter of the solute, we can prepare a solvent mixture in which the solubility will be a maximum. As an example, let us look at the solute anthracene. Its solubility parameter is 9.9 (cal/cm ). Looking at Table 1.8, which lists solubility parameters for a number of common solvents, we see that ethanol and toluene have solubility parameters that bracket the value of anthracene. If we define a mean solubility parameter by the relation... 

On the other hand, the quality of the solvent or the solubility of the polymer in a solvent is determined by the solubility parameter ( ) and the Flory-Huggins polymer-solvent interaction parameter (j). Solvating potential of a solvent can be written by using Hildebrand theory [34, 63, 64]. 

The theory implies that the entropy of mixing is combinatorial, i.e., it is stipulated by permutations of molecules into solution in which the molecules of mixed components differ greatly in size. The next assumption is that AV ,j,j = 0 and that the enthalpy of mixing does not influence the value of AS i. The last assumptions are the same as in the Hildebrand theory of regular solutions. The expression for the Gibbs energy of mixing is... 

Equation (79) is directly analogous to the Berthelot relationship, while Eq. (80) has some relationship to similar equations in the Hildebrand theory of solutions. The attraction of Eq. (79) is that it allows 712 to be determined from parameters that can be measured or approximated, provided that 4>o can be determined. Good and coworkers suggested a method involving Eq. (80) and following the procedure described in Section III.B. of summing over all infinitesimal volumes in two dissimilar bodies separated by a distance Zq. The result of this treatment is analogous to that determined in Section III.B. for y and is shown in Eq. (81). 

Browarzik et al calculated asphaltenes flocculation at high pressures for methane + crude oil - - 2,2,4-trimethylpentane [i-octane] using continuous thermodynamics where 2,2,4-trimethylpentane acts as a precipitant. The asphaltene flocculation was considered to be a liquid -b liquid equilibrium. Browarzik et al applied the van der Waals equation of state. The polydispersity of the crude oil was considered to be described by the solubility parameter of the Scatchard-Hildebrand theory. Within this distribution the asphaltenes represent the species with the highest solubility parameters. The calculated results were compared to experimental data. For oils with a very low content of asphaltenes the model describes the experimental flocculation data reasonably well. However, on contrary to the experimental results, the model predicts the asphaltenes to show a higher flocculation tendency with increasing asphaltenes content of the crude oil. Based on these comparisons further work was undertaken by Browarzik et al and the associates formed... 

A second part of the study of Singh and Schweizer" described in Section V.C was to investigate the validity of a solubility parameter approach to polymer blend miscibility. Solubility parameter, or Hildebrand, theory is potentially extremely useful from a practical viewpoint... 

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