Volume, activation molar

The dependence of the rate constant on pressure provides another activation parameter of mechanistic utility. From thermodynamics we have (dGldP)T = V, where V is the molar volume (partial molar volume in solutions). We define the free energy of activation by AG = G — SGr. where SGr is the sum of the molar free energies of the reactants. Thus, we obtain... 

Using PCA, Cramer found that more than 95% of the variances in six physical properties (activity coefficient, partition coefficient, boiling point, molar refractivity, molar volume, and molar vaporization enthalpy) of 114 pure liquids can be explained in terms of only two parameters which are characteristic of the solvent molecule (Cramer 111, 1980). These two factors are correlated to the molecular bulk and cohesiveness of the individual solvent molecules, the interaction of which depends mainly upon nonspecific, weak intermolecular forces. 

Measurements of sedimentation behaviour of polymer molecule in solution can provide a consideratble amount of information, e.g., hydrodynamic volume, average molar masses and even some indication of molar mass distribution. Such measurements have been extensively used to characterise biologically-active polymers which often exist in solution as compact spheroids or rigid rods. However, sedimentation methods are rarely used to study synthetic polymers and so will be given only brief non-theoretical consideration here. 

Therefore, it is important to have a theoretical tool which allows one to examine (or even predict) the thickness of the LC region and the value of the LC on the basis of more easily available experimental information regarding liquid mixtures. A powerful and most promising method for this purpose is the fluctuation theory of Kirkwood and Buff (KB). " The KB theory of solutions allows one to extract information about the excess (or deficit) number of molecules, of the same or different kind, around a given molecule, from macroscopic thermodynamic properties, such as the composition dependence of the activity coefficients, molar volume, partial molar volumes and isothermal compressibilities. This theory was developed for both binary and multicomponent solutions and is applicable to any conditions including the critical and supercritical mixtures. 

Tab. 8.6. Activation volumes, partial molar volumes of the reactant (Vsubstrate). the ortho product (Vo-Produci), and of the ortho transition state (Vts) and reaction volume (AVr = V substrate Ws) for the intramolecular cycloaddition of 48.

Tabulated are the equivalent conductances, transference numbers, activity coefficients, densities and partial molar volumes, apparent molar compressibilities, heats of solution and dilution for the rare earth salts in aqueous solution at 25 "C. 

Since we make the simplifying assumption that the partial molar volumes are functions only of temperature, we assume that, for our purposes, pressure has no effect on liquid-liquid equilibria. Therefore, in Equation (23), pressure is not a variable. The activity coefficients depend only on temperature and composition. As for vapor-liquid equilibria, the activity coefficients used here are given by the UNIQUAC equation. Equation (15). ... 

An equation algebraically equivalent to Eq. XI-4 results if instead of site adsorption the surface region is regarded as an interfacial solution phase, much as in the treatment in Section III-7C. The condition is now that the (constant) volume of the interfacial solution is i = V + JV2V2, where V and Vi are the molar volumes of the solvent and solute, respectively. If the activities of the two components in the interfacial phase are replaced by the volume fractions, the result is... 

The intrinsic volume of activation was estimated to correspond to the molar volume difference between cyclohexene and cyclohexane, adding the molar volume difference between ethane and ethene to account for... 

A more active product is obtained by the following slight modification of the above procedure. Dissolve the succinimide in a slight molar excess of sodium hydroxide solution and add the bromine dissolved in an equal volume of carbon tetrachloride rapidly and with vigorous stirring. A finely crystalline white product is obtained. Filter with suction and dry thoroughly the crude product can be used directly. It may be recrystallised from acetic acid. 

Remember that Vj is the partial molar volume of the solvent. Therefore a completely general relationship between n and the solvent activity is given by... 

An overview of some basic mathematical techniques for data correlation is to be found herein together with background on several types of physical property correlating techniques and a road map for the use of selected methods. Methods are presented for the correlation of observed experimental data to physical properties such as critical properties, normal boiling point, molar volume, vapor pressure, heats of vaporization and fusion, heat capacity, surface tension, viscosity, thermal conductivity, acentric factor, flammability limits, enthalpy of formation, Gibbs energy, entropy, activity coefficients, Henry s constant, octanol—water partition coefficients, diffusion coefficients, virial coefficients, chemical reactivity, and toxicological parameters. 

Table 13-1, based on the binary-system activity-coefficient-eqnation forms given in Table 13-3. Consistent Antoine vapor-pressure constants and liquid molar volumes are listed in Table 13-4. The Wilson equation is particularly useful for systems that are highly nonideal but do not undergo phase splitting, as exemplified by the ethanol-hexane system, whose activity coefficients are snown in Fig. 13-20. For systems such as this, in which activity coefficients in dilute regions may...

It is conventional to take as the activation volume the value of AV when P = 0, namely —bRT. (This is essentially equal to the value at atmospheric pressure.) Pressure has usually been measured in kilobars (kbar), or 10 dyn cm 1 kbar = 986.92 atm. The currently preferred unit is the pascal (Pa), which is 1 N m 1 kbar = 0.1 GPa. Measurements of AV usually require pressures in the range 0-10 kbar. The units of AV are cubic centimeters per mole most AV values are in the range —30 to +30 cm moP, and the typical uncertainty is 1 cm moP. Rate constant measurements should be in pressure-independent units (mole fraction or molality), not molarity. ... 

General considerations. Conventional d.c. polarographic analysis is most conveniently carried out if the concentration of the electro-active substance is 10-4 — 10 3 molar and the volume of the solution is between 2 and 25 mL. It is, however, possible to deal with concentrations as high as 10 2 molar or as low as 10 5 molar and to employ volumes appreciably less than 1 mL. 

IV. Effect of Pressure on Activity Coefficients Partial Molar Volumes. 160... 

For a dilute solution at high pressure, the variation of activity coefficient with pressure cannot be neglected. But when x2 is small, it is often a good approximation to assume, as above, that the activity coefficient is not significantly affected by composition. If we also assume that v2 the partial molar volume of the solute, is independent of both pressure and composition... 

Since Eqs. (5) and (6) are not restricted to the vapor phase, they can, in principle, be used to calculate fugacities of components in the liquid phase as well. Such calculations can be performed provided we assume the validity of an equation of state for a density range starting at zero density and terminating at the liquid density of interest. That is, if we have a pressure-explicit equation of state which holds for mixtures in both vapor and liquid phases, then we can use Eq. (6) to solve completely the equations of equilibrium without explicitly resorting to the auxiliary-functions activity, standard-state fugacity, and partial molar volume. Such a procedure was discussed many years ago by van der Waals and, more recently, it has been reduced to practice by Benedict and co-workers (B4). 

In addition to deciding on the method of normalization of activity coefficients, it is necessary to undertake two additional tasks first, a method is required for estimating partial molar volumes in the liquid phase, and second, a model must be chosen for the liquid mixture in order to relate y to x. Partial molar volumes were discussed in Section IV. This section gives brief attention to two models which give the effect of composition on liquid-phase thermodynamic properties. 

The difficulties encountered in the Chao-Seader correlation can, at least in part, be overcome by the somewhat different formulation recently developed by Chueh (C2, C3). In Chueh s equations, the partial molar volumes in the liquid phase are functions of composition and temperature, as indicated in Section IV further, the unsymmetric convention is used for the normalization of activity coefficients, thereby avoiding all arbitrary extrapolations to find the properties of hypothetical states finally, a flexible two-parameter model is used for describing the effect of composition and temperature on liquid-phase activity coefficients. The flexibility of the model necessarily requires some binary data over a range of composition and temperature to obtain the desired accuracy, especially in the critical region, more binary data are required for Chueh s method than for that of Chao and Seader (Cl). Fortunately, reliable data for high-pressure equilibria are now available for a variety of binary mixtures of nonpolar fluids, mostly hydrocarbons. Chueh s method, therefore, is primarily applicable to equilibrium problems encountered in the petroleum, natural-gas, and related industries. 

While the dilated van Laar model gives a reliable representation of constant-pressure activity coefficients for nonpolar systems, the good agreement between calculated and experimental high-pressure phase behavior shown in Fig. 14 is primarily a result of good representation of the partial molar volumes, as discussed in Section IV. The essential part of any thermodynamic description of high-pressure vapor-liquid equilibria must depend,... 

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