Hildebrand’s theory

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. 

As early as 1916 Hildebrand pointed out that the order of solubility of a given solute in a series of solvents is determined by the internal pressures of the solvents. Later Scatchard (1931) introduced the concept of "cohesive energy density" into Hildebrand s theories, identifying this quantity with the cohesive energy per unit volume. Finally Hildebrand (1936) gave a comprehensive treatment of this concept and proposed the square root of the cohesive energy density as a parameter identifying the behaviour of specific solvents. In 1949 he proposed the term solubility parameter and the symbol S. 

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). 

How does Hildebrand s theory of regular solutions apply to the role of the organic solvent in extraction processes ... 

The H-F Eq. 18.6 has two parts the (j)-dependent configurational entropy derived from the lattice model without free volume and the enthalpic part taken from the Hildebrand s theory of regular solutions (Shinoda 1978 Reichart et al. 1997 Maranas et al. 1998). More recent version of Eq. 18.6 was used for the interpretation of SANS data, and it will be discussed in reference to the lattice cluster theory (LCT) (Freed and Dudowicz 2005). 

The concept of solubility parameters as it relates to the internal energy of a solution and solute was first introduced by Hildebrand (1916, 1970). Internal pressure is defined as the energy needed to vaporize one cubic centimeter of a substance. In Hildebrand s theory molecules with similar internal pressures will attract and interact with each other. Internal energy is described by the cohesive energy density (CED). The theory predicts that dissolution of a solute will occur in a solvent or solvent blend having similar CED value. Solubility parameters in the originally stated theory have the units of (cal/cm ). In modern SI convention the units are as follows ... 

Others have attempted to extend Hildebrand s theory to nonpolar gases in polar liquids, and there appears to be a propensity to write about polar associated liquids, and polar nonassociated liquids. ... 

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. 

A better estimate of all attractive forces surrounding a molecule was found in the use of the solubility parameter [32,33], Hancock et al. [34] has reviewed the use of solubility parameters in pharmaceutical dosage form design. The solubility parameter is used as a measure ofthe internal pressures ofthe solvent and solute in nonideal solutions. Cosolvents that are more polar have larger solubility parameters. The square root ofthe cohesive energy density, that is, the square root of the energy of vaporization per unit volume of substance, is known as the solubility parameter and was developed from Hildebrand s Regular Solution Theory in the Scatchard-Hildebrand... 

Guillet and coworkers (8-10) have determined the solubility parameter of polymers from the probe-polymer interaction coefficients. They separated the interaction parameter into entropic and enthalpic contributions, such that Xi2=X h+Xs to yield, in combination with Hildebrand s solution theory, the following expression ... 

Hoff. He then returned to the University of Pennsylvania, where he held the position of instructor in chemistry until 1913. Then, at the invitation of G. N. Lewis, he joined the faculty of the University of California at Berkeley, where he remained for the rest of his scientific career. Hildebrand s main scientific research was in the area of the physical chemistry of liquids and non-electrolyte solutions. He was a major contributor to the theory of regular solutions. Much of his work in this field is summarized in his monograph with Robert Scott,... 

More recently a number of systems using three solubility parameters has therefore been suggested in order to predict the solubility characteristics of polymers more precisely. Hildebrand s original theory was intended only for application to mixtures of non-polar liquids and for such liquids the intermolecular forces are only one type, namely dispersion forces. For other liquids, as discussed, also polar and hydrogen bonding forces play a role, and it has been argued by several workers that a solvent should be described by a parameter representing each of these three types of intermolecular forces [10]. 

Another useful tool is the Hildebrand solubility theory, which is applicable to apolar and moderately polar systems. For strongly polar systems, it is unable to correctly qualify the compatibility between components. However, the massive amount of interaction parameters data obtained in recent decades, and mainly Small s method, allowing to assess them, make this method quite efficient and readily applicable. The Hildebrand solubility parameter, 5, can be defined as the square root of the cohesive energy density (CED) and it is measured in (MJ m )° . This parameter indicates the polarity level of the component and goes from 12 (MJ m )° for nonpolar components to 23 (MJ m )° for water. The larger the difference... 

The energy of interaction between molecules was calculated by Hildebrand and Scatchard [22—25] according to van laiar s theory [26] for a Lennard-Jones potential. Let denote the interaction energy between a pair of i molecules then Et, the cohesive energy (potential energy) of one mole of substance becomes ... 

Martire [29] utilized van Arkel s modification of the Hildebrand — Scatchard theory [30] to extend eqn (3.40) so that it takes accoimt of dipole interactions the following relation was obtained for the activity coefficient ... 

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 ... 

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

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... 

According to Hansen s theory [13], 5, the Hildebrand parameter, can be calculated using the three components 5, which represents the energy from dispersion bonds 5i which represents the energy from hydrogen bonds between molecules and 8p, which represents the energy from dipolar intermolecular forces ... 

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