Hildebrand’s equation

Ertl and DuUien [ibid.] found that Hildebrand s equation could not fit their data with B as a constant. They modified it by applying an empirical exponent n (a constant greater than unity) to the volumetric ratio. The new equation is not generally useful, however, since there is no means for predicting /i. The theory does identify the free volume as an important physical variable, since n > for most hquids implies that diffusion is more stronglv dependent on free volume than is viscosity. 

As early as 1926, Hildebrand showed a relationship between solubility and the internal pressure of the solvent, and in 1931 Scatchard incorporated the CED concept into Hildebrand s equation. This led to the concept of a solubility parameter, S, which is the square root of CED. Thus, as shown below, the solubility parameter S for nonpolar solvents is equal to the square root of the heat of vaporization per unit volume ... 

The Chao-Seader method uses the Redlich-Kwong equation of state for the calculation of Hildebrand s equation for the calculation of the liquid activity coefficient yf, and an extension of Pitzer s modified form of the principle of corresponding states for the calculation of the liquid fugacity ratio [. 

The activity coefficients yf were predicted by use of Hildebrand s equation... 

Interaction parameters between components of the polymer-solvent system are estimated on the base of a simplified Hildebrand s equation at 25°C... 

Thus where Hildebrand s equation is valid, if a polymer and a solvent have similar solubility parameters then solution will occur. Such validity covers amorphous hydrocarbon rubbers and can also be used qualitatively with caution with the more polar rubbers. The solubility parameter of natural rubber, in (MJ/m ) is 16-5 which suggests that a not too entangled mass of rubber molecules will dissolve in turpentine (16-5), carbon tetrachloride (17-6) and toluene (18-2) but not in acetone (20-4) or ethanol (26-0). (All figures are expressed in units of (MJ/m )". ) This expectation is realized in practice. 

Ertl, H. Dullien, F.A.L. (1973). Hildebrand s equations for viscosity and diffusivity. 

To illustrate the effectiveness of Hildebrand s equation in predicting solubility data at low temperatures, the calculated solubility values for eight hydrocarbons and for carbon dioxide are presented in Table II, together with the corresponding experimental solubility values. Hydrocarbons, such as ethane, propane, and propylene, do not freeze at -297 F and therefore, ideally, should be miscible in all proportions with liquid oxygen at that temperature. However, as can be seen in Table II, three solubility values exist for each of the above three hydrocarbons. The use of Hilde-brand s equation for predicting the solubilities of these three hydrocarbons in liquid oxygen is not applicable. 

For the monomers in the polymerization under consideration the fugacity coefficients were estimated by Redlich-Kwong equation of state and were found to be close to unity. The activity coefficients (8) for the monomers were estimated by Scatchard-Hildebrand s method (5) for the most volatile monomer there was a temperature dependence but none for the other monomer. These were later confirmed by applying the UNIFAC method (6). The saturation vapor pressures were calculated by Antoine coefficients (5). 

A good fit was obtained for data obtained with carbon tetrachloride. Figure 3.8a shows a plot of Hildebrand s data.10 A good linear fit was obtained over the temperature range 278 K to 338 K note however that the exponential fit e.g. Equation (3.34)] is poor over this range when plotted as fluidity. A good fit is found when the data is plotted as the... 

Substitutional Disorder In Regular Solid Solutions. Most simple ionic solutions in which substitution occurs in one sublattice only are not ideal, but regular 2, J3) Most complex ionic solid solutions in which substitution occurs in more than one sublattice are not only regular in the sense of Hildebrand s definition (15) but also exhibit substitutional disorder. The Equations describing the activities of the components as a function of the composition of their solid solutions are rather complex ( 7, V7, 1 ), and these can be evaluated best for each individual case. Both type II and type III distributions can result from these conditions. 

For liquids that do not have a reported molar enthalpy of vaporization, a convenient method of approximation is Hildebrand s empirical equation, based on the boiling plQjnb Kelvin units ... 

It is possible to determine C quantitatively using Hildebrand s theory of microsolutes. An example of the accuracy that can be achieved is provided by the calculation of the solubilities of a series of p-aminobenzoate esters in hexane (17,18). Michaels, et al. (19) used this approach to estimate the solubility of steroids in various polymers. The solubilities of seven steroids in six polymers were calculated from the steroid melting points, heats of fusion, and solubility parameters. Equation 8 was derived, where Jjj is the maximum steady state flux, h is the membrane thickness, x is the product of V, the molar volume of the liquid drug, and the square of the difference in the solubility parameters of the drug and polymer, p is the steroid density, T is melting point (°K), T is the temperature of the environment, R is the gas constant, and AH and ASf are the enthalpy and entropy of fusion, respectively. 

Also, the original Hildebrand approach has been refined to take into account the contribution of polar groups and hydrogen bsolubility parameters. These mndifications of the Flory-Huggins theory and of the solubility parameter concept have made these methods an even more useful tool in the description of solutions, especially of mixtures containing polymer compounds. A comprehensive treatment of these extensions of Flory-Huggins and Hildebrand s theories, as well as the new equation of state approach of Flory (1965), bns re ntly been published (Shinoda, 1978 Olahisi et al 1979). 

The extraction (and hence the transport) efficiency depends on several diluent factors such as Schmidt empirical diluent parameter [124,125], the Swain s acity and basity parameters along with the Dimroth and Reichardt polarity indices [126], dielectric constant [127], refractive index [127] and viscosity [127], and the Hildebrand s solubility parameter [128]. The permeability coefficients (Paio) were computed from the Wlke-Chang, Scheibel, and Ratcliff [129,130] equations, which compared reasonably well with the experimentally determined values as shown in Table 31.10. Efiiassadi and Do [131] have, on the other hand, taken into account only the viscosity and solubility effect of the diluent and the carrier immobilized in SLM. They have reported that these two factors influenced the transport rates significantly. 

The second approach proposed by Thorlaksen et al. is based on a combination of the Entropic-FV term with Hildebrand s regular solution theory and developed a model for estimating gas solubilities in elastomers. The so-called Hildebrand-Entropic-FV model is given by the equation ... 

If this process is favorable in the direction of the arrow in Equation (15.18) and Figure 15.6, K xch > 1. In that case, xab, which has units of energy divided by kT, is negative. According to Hildebrand s principle, for most systems, the AB affinity is weaker than the AA and BB affinities, so usually xab > 0. The quantity xah also contributes to the interfacial free energy between two materials, which we take up in the next section. 

These equations are identical with those given by Van Laar, but the assumptions are somewhat different. In the case of the Van Laar, Scatchard, and Hildebrand derivations, both A and B should be positive, while in Cooper s equation B could be either positive or negative. 

This dimensionless interaction parameter characterizes the interaction energy per solvent molecule normalized in terms of kT. Assuming that the molar volume of the solvent is almost independent of temperature, like the Hildebrand s parameters of solubility for both solvent and polymer. Equation (5.5) can be used to take into account the influence of the temperature of membrane fabrication. 

Equation (3), Hildebrand s basic equation for the theory of regular solutions 0, is useful in estimating solubility. 

The association constant (K ) of the FLZ P-CD complex were determined by using well known Scott s method [9] which is a modification of Benesi-Hildebrand equation [10]. Equation (1) refers the Scott s equation ... 

They calculated y,-, by a slightly modified version of Equation 9.24, based on Scatchard and Hildebrand s regular solution theory, and found (< i)pure liquid t by a Pitzer-type equation... 

Based on his results for carbon dioxide and nitrous oxide in a number of organic liquids, Kunerth (1922) ° concluded that there was no correlation between solubility and the internal pressures of solvent and solute he expressed disagreement with Hildebrand s views. In his reply, Hildebrand (1923) emphasized that he required the condition of nonpolarity. Of the 21 liquids S used by Kunerth, only 8 had dielectric constants as low as 5. In the many references to the solubility parameter and the parameter equation since that time, there is the inherent difficulty over the terms polar and nonpolar and what constitutes a chemical reaction. 

The parameters a and p indicate the capacity of a solvent to donate or accept a hydrogen bond from a solute, i.e., the solvent s hydrogen bond acidity or basicity. % is intended to reflect van der Waals-type solute-solvent interactions (dipolar, dispersion, exchange-repulsion, etc.). Equation (43) was subsequently expanded to include a term representing the need to create a cavity for the solute (and thus to interrupt solvent-solvent interactions).188 For this purpose was used the Hildebrand solubility parameter, 5, which is defined as the square root of the solvent s energy of vaporization per unit volume.189 Thus Eq. (43) becomes,190... 

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