Hildebrand approximations

Figure 2 shows that with octadecane the theoretical curves fit the experimental values well if Xo = 0.8. If X2 = 0, the swelling of the seed would be lower than that for all cases where x = 0.8. The interaction parameter, Xo> was expressed by Huggins (2CQ using the van Laar -Scatchard - Hildebrand approximations as... 

Solubility Parameter. CompatibiHty between hydrocarbon resins and other components in an appHcation can be estimated by the Hildebrand solubiHty parameter (2). In order for materials to be mutually soluble, the free energy of mixing must be negative (3). The solubiHty of a hydrocarbon resin with other polymers or components in a system can be approximated by the similarities in the solubiHty parameters of the resin and the other materials. Tme solubiHty parameters are only available for simple compounds and solvents. However, parameters for more complex materials can be approximated by relative solubiHty comparisons with substances of known solubiHty parameter. 

Matthews-Akgerman The free-volume approach of Hildebrand was shown to be valid for binary, dilute liquid paraffin mixtures (as well as self-diffusion), consisting of solutes from Cg to Cig and solvents of Cg and C o- The term they referred to as the diffusion volume was simply correlated with the critical volume, as = 0.308 V. We can infer from Table 5-15 that this is approximately related to the volume at the melting point as = 0.945 V, . Their correlation was vahd for diffusion of linear alkanes at temperatures up to 300°C and pressures up to 3.45 MPa. Matthews et al. and Erkey and Akger-man completea similar studies of diffusion of alkanes, restricted to /1-hexadecane and /i-octane, respectively, as the solvents. 

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. 

Lohse et al. have summarized the results of recent work in this area [21]. The focus of the work is obtaining the interaction parameter x of the Hory-Huggins-Stavermann equation for the free energy of mixing per unit volume for a polymer blend. For two polymers to be miscible, the interaction parameter has to be very small, of the order of 0.01. The interaction density coefficient X = ( y/y)R7 , a more relevant term, is directly measured by SANS using random phase approximation study. It may be related to the square of the Hildebrand solubility parameter (d) difference which is an established criterion for polymer-polymer miscibility ... 

Being able to change the density, via either changes in pressure or temperature, is the key difference in SFC over GC and LC separations. Typical density ranges are from 0.3 to 0.8g/ml for pure carbon dioxide. Table 16.2 shows data obtained from ISCO s SF-Solver Program for the calculation of density (g/ml), Hildebrand Solubility Parameter and a relative equivalent solvent for pure carbon dioxide at a constant pressure of 6000 psi, approximately 408 atm. 

The three kinds of forces described above, collectively known as the cohesive forces that keep the molecules of liquids together, are responsible for various properties of the liquids. In particular, they are responsible for the work that has to be invested to remove molecules from the liquid, that is, to vaporize it. The energy of vaporization of a mole of liquid equals its molar heat of vaporization, Ay//, minus the pressure-volume work involved, which can be approximated well by Rr, where R is the gas constant [8.3143 J K" mol" ] and T is the absolute temperamre. The ratio of this quantity to the molar volume of the liquid is its cohesive energy density. The square root of the cohesive energy density is called the (Hildebrand) solubility parameter of the liquid, 8 ... 

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. 

Table 3-12, which is similar to a table presented by Pitzer and Hildebrand, gives information about compounds of sulfur and halogens with atoms that are known to form colorless ions or to form with fluorine analogous compounds that are colorless. The numbers in the table are the enthalpies of formation per M—X bond in kcal/mole, these being also, as mentioned earlier in this chapter, approximately equal to the percentages of ionic character of the bonds. There is seen... 

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... 

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 ... 

Typically, the error in equation (7.1) is called truncation error. This is a hypothetical definition of the error, because it excludes gross errors, which are caused by unpredictable mistakes (for example, human or mechanical mistakes in an experiment), and round off errors, which are a result of having only a finite number of digits. The truncation error is the error that is implied in an approximation. More in-depth definitions on error can be found in the literature, as presented by Achieser [2], Hildebrand [10] or Davis [8],... 

INTRODUCT ION TO NUMERICAL ANALYSIS (2nd Edition). F.B. Hildebrand. Classic, fundamental treatment covers computation, approximation, interpolation, numerical differentiation and integration, other topics. 150 new problems. 669pp. 55 x 85. 65363-3 Pa. 13.95... 

He noted that the solubility increased for the interaction of two materials as the heat of mixing decreased. He, therefore, took note of the Hildebrand expression for the approximation of the heat of mixing... 

The solvency of hydrocarbon solvents used in paint and lacquer formulations is empirically described by their kauri butanol numbers, i.e. the volume in milHliters at 25 °C of the solvent required to produce a defined degree of turbidity when added to 20 g of a standard solution of kauri resin in 1-butanol [120]. Standard values are KB = 105 for toluene and KB = 40 for -heptane/toluene (75 25 cL/L). A high KB number corresponds to high solvent power. An approximately linear relationship exists between Hildebrand s 8 values and KB numbers for hydrocarbons with KB >35 8 = Q.Q6-KB+U.9 [99, 177]. 

Hildebrand proposed to compare the values of Ml T at temperatures at which the vapour concentrations are equal for 0-00507 mol/lit. the value is about 27-5 for normal liquids, 32-4 for 1 3, and 32-0 for H2O. He found that the plots of log p (vapour pressure) against log T gave curves having the same gradients at the same vapour concentrations. The approximate Clapeyron-Clausius equation (11a), 7.VIIIL ... 

In addition, it is worthwhile noting that the use of the Benesi-Hildebrand equation provides only approximate Kf values, because more emphasis is placed on the lower concentration values than on the higher ones, and the data are not weighted properly [85,103,104], Therefore, a better estimation of Kf can be made by using Eq. 4 based on a non-linear regression (NLR) analysis [83,86] ... 

Some data treatment methods are general, some rely on approximations and thus are subject to some premises, and some are just regression methods. Typical examples of the approximate methods are the Benesi-Hildebrand [12], Ketelaar [13], Nagakura-Baba [14], Scott [15], Scatchard, and Hammond [16] methods which approximate [G] by [G]o. 

This quantity was investigated extensively by Hildebrand [1], who showed that the internal pressure is approximately equal to the enthalpy of vaporization divided by the molar volume. Thus,... 

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