Hildebrand-Scott equation

Figure III-9u shows some data for fairly ideal solutions [81] where the solid lines 2, 3, and 6 show the attempt to fit the data with Eq. III-53 line 4 by taking ff as a purely empirical constant and line 5, by the use of the Hildebrand-Scott equation [81]. As a further example of solution behavior, Fig. III-9b shows some data on fused-salt mixtures [83] the dotted lines show the fit to Eq. III-SS.

Charles M. Hansen (4) was working in the area of paint technology. He was aware oT the Hildebrand/Scott solubility parameter, and explored the use of the solubility parameter in polymer-solvent interactions. He began his research with the consideration of the thermodynamic equation for the energy of mixing... 

Based on the pioneering work of van der Waals [16] and van Laar [17], the Hildebrand-Scatchard equation was derived on semitheoretical grounds by Scatchard [18,19] and Hildebrand [20-24] and popularized by Hildebrand, Scott, Prausnitz and others [25-37, 10-12]. For a binary mixture, the Hildebrand-Scatchard equation is expressed as... 

Unfortunately values of A// at sueh low temperatures are not readily available and they have to be eomputed by means of the Clausius-Clapeyron equation or from the equation given by Hildebrand and Scott" ... 

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] ... 

Molecular Orientation and Surface Tension. One disconcerting fact is the molecular orientation of nonsymmetrical and polar liquids. With orientation, the surface of these liquids may represent only a part of the molecules. For this type of liquid, a correlation between the surface and the bulk properties could present a problem. The original assumption made by Hildebrand and Scott (9) was that nonpolar liquid molecules assume a spherical symmetry. The general application of their equations to liquids without a spherical symmetry is really remarkable. However, we noted (17) that deviations were found for nearly all oxygenated liquids. In the case of polymers, the molecular orientation is expected to play a more important role. This could be one of the reasons that the relationship between surface and bulk properties of polymers is rather sporadic. In the Experimental section, we shall explain the importance of the induced orientation (3,26) with respect to obtaining a surface tension reading for a polar polymer. 

The original Hildebrand and Scott equation was obtained on the premise that nonpolar liquids possessed a spherical symmetry. For liquids without symmetry, the molecules tend to orient. The result of orientation could contribute to the departure of the surface properties from the bulk properties of polar and hydrogen-bonded liquids. This departure should be enhanced in the case of a long-chain polymer. 

Under a separate study (17), we found that oxygenated liquids generally do not follow the Hildebrand and Scott equation. For the simplest oxygenated liquids—e.g., ethers and esters—the proper equation should be ... 

Hee any text on the liquid state, c.g., J. H. Hildebrand and R. L. Scott, Solubility of Non-electrolytes, Reinhold Publishing Corporation, New York, 1960. From an operational point of view we can use this equation to define the free volume of a liquid. 

The emphasis on RaOult s law and the liquid state raises difficulties for solutes which are above their critical temperatures and for which A// , p , and F appear to have little meaning. From an empirical point of view it appears that the extrapolation of the Clausius-Clapeyron equation to super-critical temperatures gives usable values of p (see Hildebrand and Scott, loc. ciL, chap. 14). 

Before presenting examples, we should remark that the subject of solubility of nonelectrolytes has been treated in detail by Hildebrand and Scott (921). They discuss (in their Chapter XI) the various chemicaT and physical theories of the interactions, such as H bonding, that are responsible for extreme deviations from regular behavior. Both approaches can provide equations to fit experimental data. The first does so by relating equilibrium constants and activity coefficients for assumed reactions, and the second by the use of varying values of the energy of interaction and empirical factors for the effective volume of solute and solvent molecules. They conclude with the observation, still valid, that no satisfactory theoretical treatment is available. 

More commonly used descriptors of polymer solubility are the solubility parameters introduced by Hildebrand and Scott for dispersive interaction forces, and extended by Hansen " for dispersive (8 ), polar (8d), and hydrogen bonding contributions (8 ) to interaction energies. An equation sometimes used to estimate the solubility range of Polymer 2 in a solvent (subscript 1) is ... 

The Hildebrand-Scatchard equation (see Hildebrand and Scott, 1951), which is derived on semiempirical grounds, is... 

Hildebrand and Scott used the geometric mean rule to describe the interaction between molecules of two unlike species to arrive at the total solubility parameter. Strictly speaking, this was valid for nonpolar type molecules only. The fact that Equation 10.4 has produced hundreds of satisfactory correlations of solubility, swelling, permeation, surface phenomena, etc., confirms that the geometric mean rule is likewise applicable to molecules engaging in permanent dipole-permanent dipole and hydrogen bonding interactions as well. 

With these comments in mind, the assumptions involved in the use of Equation 1 as the criterion for compatibility can be examined. Many of the points to be raised have been considered in more detail by Hildebrand and Scott (J[, 26) and by Hildebrand, Prausnitz, and Scott (27). but more recent experimental evidence bearing on the subject justifies a reiteration of this earlier material with an emphasis on what now appear to be the critical problems. 

Benesi and Hildebrand derived the equation for the calculation of association constants of iodine with aromatic hydrocarbons by spectrophotometry in 1949 [47]. At present, many techniques used for the determination of binding constants represent the various modifications of the Benesi-Hildebrand equation [47]. That commonly used in spectrometric techniques is Scott s modification of this technique [48]. 

The solubility parameter, as defined by Hildebrand and Scott [43], is the square root of the cohesive energy density. The equation is... 

A plot of the left-hand sides against 1 /[i)]o (Benesi-Hildebrand equation) or against [D]o (Scott equation) yields equilibrium constants and molar linear absorption coefficients from the slope and ordinate intercepts of the straight lines. The equilibrium constants obey the van t Hoff relationship as long as only one CT complex occurs. 

A simple, convenient method of estimating the vaporization enthalpy is the application of the Hildebrand and Scott equation [12,24,35], which expresses AH at 25 °C (A//°) in terms of the normal boiling point T/, ... 

Scott, R.L. (1956) Some comments on the Benesi-Hildebrand equation. Rec. Trav. Chim., 75,787-789. 

Hildebrand and Scott [12] first correlated the enthalpy of mixing with the cohesive properties of the permeant, and for a binary mixture of a polymer and a solvent, the Hildebrand-Scatchard equation is expressed as ... 

Equation (3.121) is identical with that obtained by Prigogine [4] in cell theory. Prigogine has further employed the cell model to calculate Eg he evaluted the potential field of a representative molecule of the system as a function of its position in the cell, defined by the geometry of the nearest neighbours whose positions were assumed to be fixed. Flory assumed that Eg depends only on the volume of the system according to Hildebrand and Scott s relation ... 

In some cases, in order to apply the theory, it is necessary to extrapolate the vapour pressure of the liquefied gas beyond the critical point. For example, suppose that it is desired to estimate the ideal solubility of methane at a temperature of 25 C, which is far above critical. If the observed vapour pressures are extrapolated by means of the Clausius-Clapeyron equation, the estimated value of p at 25 is found to be 289 atm— but of course this does not correspond to a stable state of gas-liquid ecjuUibrium. The ideal solubility of methane at 25 is therefore 1/289=0.0035. Some of the observed solubilities, as quoted by Hildebrand and Scott, are given in the table. 

The molar volumes of the reactants, V, V, are presumably known. The ds are known for many substances or can be calculated from known quantities (Hildebrand and Scott, 1962 Marcus, 1985). The molar volume of the activated complex is not usually known may be obtained through the effect of pressure on the reaction rate (see earlier. Section 1.10). Finally, may be obtained from the best fit of data for the rate constant of the reaction in a variety of solvents to Equation 2.14. 

The next step is treatment of data collected to obtain the K values after titration experiments. Some data treatment methods are general, some rely on approximations and thus are subject to some cautions, and some are only regression methods. Typical examples of the approximate methods are the Benesi-Hildebrand, Ketelaar, Nagakura-Baba, Scott, and Hammond methods that approximate [G] by [G]q. From Equations (15.8) and (15.10) and a = b = 1, we derive... 

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 ... 

Scott R. L. Some comments on the Benesi-Hildebrand equation. 

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