Molar volume development

Figure 3. Molar volume development in the crystalline, smectic B, smectic A, and isotropic phases within the homologous series of 4-bromo-iV-(4-n-alkyloxybenzyli-dene)anilines. Adapted from Seurin et al. [94],...

This approach to solution chemistry was largely developed by Hildebrand in his regular solution theory. A regular solution is one whose entropy of mixing is ideal and whose enthalpy of mixing is nonideal. Consider a binary solvent of components 1 and 2. Let i and 2 be numbers of moles of 1 and 2, 4>, and 4>2 their volume fractions in the mixture, and Vi, V2 their molar volumes. This treatment follows Shinoda. ... 

In this approach, connectivity indices were used as the principle descriptor of the topology of the repeat unit of a polymer. The connectivity indices of various polymers were first correlated directly with the experimental data for six different physical properties. The six properties were Van der Waals volume (Vw), molar volume (V), heat capacity (Cp), solubility parameter (5), glass transition temperature Tfj, and cohesive energies ( coh) for the 45 different polymers. Available data were used to establish the dependence of these properties on the topological indices. All the experimental data for these properties were trained simultaneously in the proposed neural network model in order to develop an overall cause-effect relationship for all six 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. 

In subsequent works, Marcus developed his theory further in a series of papers providing expressions for the work terms, the reorganization energy and the macroscopic ET rate constants [3 6]. Assuming a sharp liquid-liquid boundary, the solution of the mean molar volume of reactants yields an expression for of the form... 

From the viewpoint of developing quantitative correlations it is desirable to seek a linear relationship between descriptor and property, but a nonlinear or curvilinear relationship is adequate for illustrating relationships and interpolating purposes. In this handbook we have elected to use the simple descriptor of molar volume at the normal boiling point as estimated by the Le Bas method (Reid et al. 1987). This parameter is very easily calculated and proves to be adequate for the present purposes of plotting property versus relationship without seeking linearity. 

First-order estimates of entropy are often based on the observation that heat capacities and thereby entropies of complex compounds often are well represented by summing in stoichiometric proportions the heat capacities or entropies of simpler chemical entities. Latimer [12] used entropies of elements and molecular groups to estimate the entropy of more complex compounds see Spencer for revised tabulated values [13]. Fyfe et al. [14] pointed out a correlation between entropy and molar volume and introduced a simple volume correction factor in their scheme for estimation of the entropy of complex oxides based on the entropy of binary oxides. The latter approach was further developed by Holland [15], who looked into the effect of volume on the vibrational entropy derived from the Einstein and Debye models. 

Several methods have been developed for calculating fugacities from measurements of pressures and molar volumes of real gases. 

Replacing V by the mole-weighted sum of the partial molar volumes, as was done in developing Equation (31), we write... 

Other researchers examined the deviations of the Benedict-Webb-Rubin constants with pressure, temperature, and molar volume.12 Equations with as many as 20 empirical constants were developed. Although... 

LSER Model of Leahy In the LSER model of Leahy [22], the cavity term is substituted by the molar volume, Vm, at 25°C in g cm-3 or by the intrinsic molecular volume, V), in mLmoL1. The dipolar term and the hydrogen-bonding terms are represented by the dipole moment, n, and the HBA basicity, (3, respectively. Group contribution schemes have been developed to calculate the solvatochromic parameters from molecular structure input [23]. Leahy [22] gives the following equation derived with a diverse set of monofunctional liquids ... 

In the event that the molar volumes of solute and solvent are not comparable, and the thermal agitation is not adequate to achieve maximum entropy of mixing, a nonideal entropy of mixing exists (Bustamante et al., 1989). Two equations that account for the nonideal entropy of mixing have been derived by considering the partial molal volurfiA,and the volume fraction, occupied by each solution component. The L rst was developed by Flory and Huggins (Hildebrand, 1949 Kertes, 1965) ... 

In many instances it is possible to determine the partial molar volumes of binary systems by graphical means. Some cases simply require the application of calculus to the equations developed. As an example we consider Equations (6.57) and (6.58). We plot 4>V against m, determine the slope of the curve at a given m, and then determine Vx and V2 by means of these equations. 

Partial molar volumes of electrolytes may present an additional difficulty if the partial molar volumes of individual constituent ions are needed. The subject has been dealt with as part of a valuable discourse on metal ion solvation.158 For species in water and some other solvents, the reference point for calculation of other partial molar volumes of single ions is the partial molar volume of the solvated proton, although there have been different views on what this value should be.159 However, the variation in values is not large and the consequence of choice of value does not have a huge impact particularly for large ionic species. Theoretical models of partial molar volumes of hydrated ions have been developed in order that estimates of them can be calculated. 160-162... 

Careful consideration was taken in the parameterization process to insure that the parameters were deemed reasonable for the atom types, using the OPLS-AA force field atom types as a comparison. As one of the goals of this project was to ensure that robustness was achieved in many different calculated properties of the newly developed model, several sets of simulations were also performed to ensure that the parameters could achieve a reasonable agreement with experiment. Some of the properties calculated included the gas phase density, the partial molar volume in aqueous solution, and the bulk solvent structure as well. The calculation of the solubility was discussed in the previous section for the parameterization process and the viewing of these results, the solubility will be reported in log S values, as many of the literature values are reported as log S values, and therefore, the comparison would not lose any sensitivity due to rounding error from the log value. 

This sensor responds to octane. This compound is significantly more volatile than the other test componds, proving to be problematic in the development of a significant QSAR. The compound 3-methylheptane clearly does not fall into the same group as the n-alkanes. The value predicted for induction (111%) using Equation 17.5 is different from the actual value of 36%. Further testing with a wider range of branched alkanes would be necessary to determine if, like -alkanes, they formed a QSAR that was dependent on molar volume. 

All current activity coefficient estimation models are by necessity semi-empirical in nature, because too little is known about solution theory for outright estimation. Chemical modeling is not readily available and is not far enough developed to make this type of calculation. The constants required by these models must be estimated using either experimental data (e.g. an infinite dilution activity coefficient or a molar volume) or group contributions derived from experimental data (e.g. interaction constants, molecular volumes and surface areas). 

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