Hildebrand equation

Real solutions are rarely completely athermal, even when there is considerable similarity between the nature of the molecules. For cases in which some energy effects must be taken into account, Flory introduced an additional term into the expression for excess Gibbs free energy. Adapting the format of the Scatchard-Hildebrand equation, the additional contribution to the excess Gibbs free energy is assumed to be of the form ... 

Scatchard-Hildebrand equation, 173-175 Separation-process, 203 Solubility parameters, 174 Sorbital, 76 State, equation of empirical... 

It is advantageous with a new drug substance to be able to estimate what its solubility will be prior to carrying out dissolution experiments. There are several systems of solubility prediction, most notably those published by Amidon and Yalkowsky [14-16] in the 1970s. Their equation for solubility of p-aminobenzo-ates in polar and mixed solvents is a simplified two-dimensional analog of the Scatchard-Hildebrand equation and is based on the product of the interfacial tension and the molecular surface area of the hydrocarbon portion of a molecule. 

To determine the constant (Kc) of complex formation between TBSM and MA from the shifts of protons of anhydride, CH2 =, and CH2—Sn-groups, the modified Beneshi-Hildebrand equation 26) was used ... 

Ideas very similar to those presented here were developed by Litt and Wellinghoff [19] with regard to the formation of D—A complexes and the involvement of competition of donors (D) and solvents in the solvation shell of the acceptors (A) and of the complexes. Their theory was used successfully to account for deviations from the Benesi-Hildebrand equation [20]. 

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 ... 

The solubility parameter 6 is a measure of the cohesion of a material, or of the strength of molecular attractive forces between like molecules. The relationship between the solubility parameter and enthalpy of mixing is given by the Hildebrand equation (1) ... 

By simple algebra, a double-reciprocal plot form, often called the Bensi-Hildebrand equation, can be derived ... 

Because of the need for better predictability in more aqueous cosolvent systems, the Scatchard-Hildebrand equation was modiLed into the Extended Hildebrand equation [40,41] ... 

Acree and Rytting [54,55] developed the nearly ideal binary solvent (NIBS) approach to predict solubilities. NIBS did show an increased ability to predict solubilities versus the Scatchard-Hildebrand equation. However, it is limited to use in systems only involving nonspeciLc interactions. 

The Xh term quantifies the nonideal heat of mixing between the HOC and DOM phase and is typically quantified using the one-solubility parameter Scatchard-Hildebrand equation... 

ACTIVITY COEFFICIENTS FROM THE SCATCHARD-HILDEBRAND EQUATION 1.40... 

The premise of ideal solubility is that the system must be considered as a mixture of components that do not form mixed crystals (Wesdorp et al. 2005). If the assumption of ideality holds true, then the Hildebrand equation may predict the solubility behavior of a binary TAG mixture (Knoester et al. 1972) ... 

Major limitations of the Hildebrand equation occur when co-crystallization occurs between the fractions of TAGs (Humphrey et al. 2003). Also, deviations from ideality occur at a high solvent/solute ratios. Inclusion of solvent in the crystal may also cause deviation from ideality as imperfect crystals have a higher solubility than perfect crystals. These major limitations, which have been observed in our SSS/OOO ideal system, have lead authors to explore other methods to quantify and predict solubility. 

Scatchard-Hildebrand equation, 374 Schacham equation, friction factor, 94 Scheibel extractor, 483,487 Schildknecht column, 543,544,546 Screens, separating, 335, 337 capacity, 335, 340 flat, 335... 

In order to account for polar and hydrogenbonding interactions, the Extended Hildebrand equation was introduced by Martin, Newberger, and Adjei P2-2 l... 

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] ... 

With the assumption that ACp is independent of temperature, integration of equation (16) and the replacement of A by A /, yields the Hildebrand equation... 

For a mixture of components that behaves ideally, it can be derived that there is no change in enthalpy when the components are mixed, i.e., no heat is released nor consumed. The decrease in free energy due to mixing then is purely due to an increase in entropy. Such a situation may occur for two components of very similar properties, for instance for a mixture of closely related triglycerides. However, if one of the components is a solid at the temperature of mixing, it has to melt, and this means an increase in enthalpy, equal to the enthalpy of fusion AH( (the enthalpy of mixing is still assumed to be zero). This implies that there is a limited solubility (xs), given by the Hildebrand equation,... 

When crystallization from solution is desired, the first thing one wants to know is the solubility of the solute as a function of temperature. For ideal solutions, this relation can be calculated by the Hildebrand equation (2.9) from the heat of fusion and the fusion temperature of the solute. However, sugar solutions are strongly nonideal, and solubilities have to be determined experimentally. 

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