Hildebrand dependence

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. 

In their correlation, Chao and Seader use the original Redlich-Kwong equation of state for vapor-phase fugacities. For the liquid phase, they use the symmetric convention of normalization for y and partial molar volumes which are independent of composition, depending only on temperature. For the variation of y with temperature and composition, Chao and Seader use the equation of Scatchard and Hildebrand for a multicomponent solution ... 

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). 

With traditional solvents, the solvent power of a fluid phase is often related to its polarity. Compressed C02 has a fairly low dielectric constant under all conditions (e = 1.2-1.6), but this measure has increasingly been shown to be insufficiently accurate to define solvent effects in many cases [13], Based on this value however, there is a widespread (yet incorrect ) belief that scC02 behaves just like hexane . The Hildebrand solubility parameter (5) of C02 has been determined as a function of pressure, as demonstrated in Figure 8.3. It has been found that the solvent properties of a supercritical fluid depend most importantly on its bulk density, which depends in turn on the pressure and temperature. In general higher density of the SCF corresponds to stronger solvation power, whereas lower density results in a weaker solvent. 

There have been numerous approaches to describing the temperature dependence of the properties. For aqueous solubility, the most common expression is the van t Hoff equation of the form (Hildebrand et al. 1970) ... 

The properties of organic liquids relevant to their use as solvating agents have also been reviewed [76]. The ability of liquids to solvate a solute species depends mainly on their polarity and polarizability properties, ability to hydrogen bond, and cohesive electron density. These molecular properties are best measured by the Kamlet-Taft solvatochromic parameters, and the square of Hildebrand s solubility parameter. 

The exponential dependence on temperature was taken by Hildebrand to be due to the variation of the free volume ratio with temperature. 

Many solvent properties are related to density and vary with pressure in a SCF. These include the dielectric constant (er), the Hildebrand parameter (S) and n [5], The amount a parameter varies with pressure is different for each substance. So, for example, for scC02, which is very nonpolar, there is very little variation in the dielectric constant with pressure. However, the dielectric constants of both water and fluoroform vary considerably with pressure (Figure 6.3). This variation leads to the concept of tunable solvent parameters. If a property shows a strong pressure dependence, then it is possible to tune the parameter to that required for a particular process simply by altering the pressure [6], This may be useful in selectively extracting natural products or even in varying the chemical potential of reactants and catalysts in a reaction to alter the rate or product distributions of the reaction. 

Heimer L, Zahm DS, ChurchUl L, Kalivas PW, Wohltman C (1991) Specificity in the projection patterns of accumbal core and medial shell in the rat. Neuroscience 41 89-125 Hemby SE, No C, Koves TR, Smith JE, Dworkin SI (1997) Differences in extracellular dopamine concentration in the nucleus accumbens during response-dependent and response-independent cocaine administration in the rat. Psychopharmacology 133 7-16 Hildebrand BE, Nomikos GG, Hertel P, Sclrilstrom B, Svensson TH (1998) Reduced dopamine output in the nucleus accumbens but not the prefrontal cortex in rats displaying mecamylamine-precipitated nicotine withdrawal syndrome. Brain Res 779 214-225 Ikemoto S (2003) Involvement of the olfactory tubercle in cocaine reward intracranial selfadministration studies. J Neurosd 23 9305-9311... 

Mecamylamine injected into the VTA of nicotine-dependent rats induced decreased DA output in the NAcc (Hildebrand et al. 1999), while injection directly... 

Ab initio SCRF/MO methods have been applied to the hydrolysis and methanol-ysis of methanesulfonyl chloride (334). ° The aminolysis by aromatic amines of sulfonyl and acyl chlorides has been examined in terms of solvent parameters, the former being the more solvent-dependent process.Solvent effects on the reactions of dansyl chloride (335) with substituted pyridines in MeOH-MeCN were studied using two parameters of Taft s solvatochromatic correlation and four parameters of the Kirkwood-Onsager, Parker, Marcus and Hildebrand equations. MeCN solvent molecules accelerate charge separation of the reactants and stabilize the transition... 

As already noted, equation 1.102 is valid when vibrational energy does not depend explicitly on volume. In a more refined treatment of the problem, the vibrational energy of the crystal can be treated as purely dependent on T. This assumption brings us to the Hildebrand equation of state and to its derivative on V ... 

Dispersion polymerizations of methyl methacrylate ntUizing poly(l,l,-dihydroper-fluorooctyl acrylate) as a steric stabilizer in snpercritical CO2 were carried out in the presence of helium. Particle size and particle size distribution were found to be dependent on the amonnt of inert helium present. Particle sizes ranging from 1.64 to 2.66 pm were obtained with varions amounts of helium. Solvatochromic investigations using 9-(a-perflnoroheptyl-p,p-dicyanovinyl)julolidine indicated that the solvent strength of CO2 decreases with increasing helium concentration. This effect was confirmed by calcnlations of Hildebrand solubility parameters (Hsiao and DeSimone, 1997). 

Table I Phase transitions, Flory interaction parameters (/), free energies (AG) and differences of Hildebrand solubility parameters (A<5) depending on the molecular structure (fluorination of the alkyl chain) [25]...

It is, therefore, not surprising that there exists a definite relationship between Aand the enthalpy of vaporization, Av H, the former constituting a fraction between 0.2 and 0.3 of the latter, as is readily obtained from the data in Tables 3.1 and 3.9. The pressure dependence of the viscosity is also closely related to the free volume of the solvent. The fluidity (O = l/r ) is proportional to the ratio between the free and the occupied volume, the former, as mentioned above, being the difference between the actual molar volume and the intrinsic molar volume (Tables 3.1 and 3. 4) (Hildebrand 1978). In fact, the logarithm of the viscosity of liquids was found (Marcus 1998) to be described well for some 300 liquids by the empirical relationship ... 

The latter can be taken as either Vx, or Evdw, or VL, related by Eqs. (3.19), (3.20), and (3.21), respectively, to the constitution of the solvents. It is obvious that the free volumes defined according to these choices of the intrinsic volume are not the same, and caution must be exercised when this notion is applied to concrete problems. The fluidity O = l/r of solvents depends on the free volume O = B[(V - V0)/V0J, according to (Hildebrand 1978), where B is a temperature-independent constant and V0 is the occupied volume, that may be equated with the intrinsic volume, see also Eq. (3.33). As mentioned in Chapter 3, the compressibilities of solvents appear to depend mainly on their free volumes, according to Eq. (3.8), so that there exists a relationship between the compressibilities of solvents and their fluidities (Marcus 1998). Two non-linear curves result from plots of log O v s kt, one for non-associated liquids and the... 

Christensen T. A., Pawlowski V. M., Lei H. and Hildebrand J. G. (2000) Multi-unit recordings reveal context-dependent modulation of synchrony in odor-specific neural ensembles. Nat. Neurosci. 3(9), 927-931. 

In the derivation of Eq. (7.3) by Hildebrand only dispersion forces between structural units have been taken into account. For many liquids and amorphous polymers, however, the cohesive energy is also dependent on the interaction between polar groups and on hydrogen bonding. In these cases the solubility parameter as defined corresponds with the total cohesive energy. 

The above formula for the solubility, in which also the corrections for the temperature dependence of the latent heat of fusion AH were taken into account, has been very extensively tested by Hildebrand on the solubility of iodine in a very large number of solvents (violet solution, p. 342). Since the specific cohesion (p. 359) of iodine is very high (y/UjV — 14.1), the solubility line only approaches the ideal line for solvents with similarly large specific cohesion, such as Snl4 (11.7) and Sg (12.7), molten sulphur (Table 33). 

Although the kinetics of liquid uptake to attain gel-saturation is history-dependent, the composition at the true end-state (i.e. thermodynamic equilibrium in excess liquid) is not therefore the observed end-state is usually reproducible [19]. Gel-saturation is attained when the restraining force (per unit area) of the polymeric crosslinked network becomes equal and opposite to the osmotic pressure that causes the system to swell [20], In other words saturation is achieved when the chemical potential of swelling liquid, p1 in the swollen network is equal to the chemical potential of the excess pure liquid, p , outside the network. It was logical to anticipate that the volume of liquid sorbed per gram of polymer, at this state of thermodynamic equilibrium with excess liquid, would correlate with the molecular structure of the liquid. In fact two parameters already exist which relate the sorption affinity to the molecular structure, namely the solubility parameter, 8, first proposed by Hildebrand [21], and the interaction parameter, %, introduced by Flory [22] and Huggins [23-26],... 

In mixtures which are regular solutions the mutual solubility of the components depends on the cohesive pressure, hence Hildebrand termed the square root of c the solubility parameter 8, according to Eq. (5-77) cf. also Eq. (2-1) in Section 2.1 [228, 229, 231, 238]. 

Sometimes, Hildebrand s solubility parameter 8 has been incorrectly used in linear Gibbs energy relationships cf. for example [96, 97, 226, 255]. Since in linear Gibbs energy relationships the correlated solvent-dependent solute properties e.g. Ig K, Ig k, h v) are proportional to Gibbs energy changes of reaction or activation (AG, AG ) and... 

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