Partial molar volume calculation

Here, is the number density of solvent sites a. Unfortunately, HFEs calculated using the GF free energy functional have only a qualitative agreement with experiment. The error in hydration free energies calculated by the GF functional in 3D RISM is strongly correlated with the partial molar volume calculated by 3D RISM [67, 94, 104]. The 3D RISM/UC free energy functional developed from this observation is a linear combination of the 07 , the dimensionless partial molar contribution, pV, and a bias correction, b (intercept) [104] ... 

It is difficult to measure partial molar volumes, and, unfortunately, many experimental studies of high-pressure vapor-liquid equilibria report no volumetric data at all more often than not, experimental measurements are confined to total pressure, temperature, and phase compositions. Even in those cases where liquid densities are measured along the saturation curve, there is a fundamental difficulty in calculating partial molar volumes as indicated by... 

If we can write an equation of state for liquid mixtures, we can then calculate partial molar volumes directly by differentiation. For a pressure-explicit equation, the most convenient procedure is to use the exact relation... 

By adopting mixing rules similar to those given in Section II, Chueh showed that Eq. (55) can be used for calculating partial molar volumes in saturated liquid mixtures containing any number of components. Some results for binary systems are given in Figs. 7 and 8, which compare calculated partial molar volumes with those obtained from experimental data. 

Chueh s method for calculating partial molar volumes is readily generalized to liquid mixtures containing more than two components. Required parameters are and flb (see Table II), the acentric factor, the critical temperature and critical pressure for each component, and a characteristic binary constant ktj (see Table I) for each possible unlike pair in the mixture. At present, this method is restricted to saturated liquid solutions for very precise work in high-pressure thermodynamics, it is also necessary to know how partial molar volumes vary with pressure at constant temperature and composition. An extension of Chueh s treatment may eventually provide estimates of partial compressibilities, but in view of the many uncertainties in our present knowledge of high-pressure phase equilibria, such an extension is not likely to be of major importance for some time. 

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). 

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,... 

From the Debye-Hiickel expressions for lny , one can derive equations to calculate other thermodynamic properties. For example L2, the relative partial molar enthalpy,q and V2, the partial molar volume are related to j by the equations... 

Fig. 13. Absolute partial molar volumes, Vab8°, of [Ln(H20) P in aqueous LnCl3 solutions (301) (closed circles), compared with the calculated Vabs° values (4, 42) for [Ln(H20)8]3+ and [Ln(H20)9]3 indicated by the upper and lower solid curves, respectively. Interchange rate constants, kj (298 K) (310), for the substitution of S042 on [Ln(H20) ]3+ are shown as open squares, and water exchange rate constants, fcn2o (298 K) (311, 312), for [Ln(H20)8]3+ are shown as open circles. 

All volumes are given in cm3 mol-1. The structural parameters necessary for the calculation of the van der Waals volume for the transition state (TS) were taken from ab initio calculations159,160. The partial molar volume for the TS was calculated from the equation ... 

For the mechanistic interpretation of activation volume data for nonsymmetrical electron-transfer reactions, it is essential to have information on the overall volume change that can occur during such a process. This can be calculated from the partial molar volumes of reactant and product species, when these are available, or can be determined from density measurements. Efforts have in recent years focused on the electrochemical determination of reaction volume data from the pressure dependence of the redox potential. Tregloan and coworkers (139, 140) have demonstrated how such techniques can reveal information on the magnitude of intrinsic and solvational volume changes associated with electron-transfer reactions of transition... 

The partial molar volume data of Table I on ammonia was calculated from density data in Landolt-Bornstein(57). The values of partial molal volume at infinite dilution can be expressed as ... 

The partial molar volumes for water in the ethanol solutions can be calculated by analogous procedures. An interesting alternative method is the tangent method [2]. 

Table 18.6 show the experimental results of Gucker et al. [10] for the molar concentration of urea, the density p of the solutions in grams per liter, and the partial molar volume of urea in cm calculated from the density data. 

Bottinga Y. and Weill D. F. (1970). Densities of liquid silicates systems calculated from partial molar volumes of oxide components. Amer. Jour. Scl, 269 169-182. 

For Eq. (2) it is assumed that the volume of the micellar phase is proportional to the tenside concentration and that the partial molar volume v remains constant. (See Chapter 2.) A further prerequisite for the application of Eq. (2) is a constant ionic mobility of the micellar phase independent of the uptake of a solute (/x, . = const.). In contrast to HPLC, substances that have an infinitely high kP value, i.e., that are completely dissolved in the micellar phase, can be detected. In this case the sample molecule migrates with the mobility of the micelle. In the presence of several different micellar phases (coexistence of simple and mixed micelles), the calculation of kP is possible only when partial capacity factors are known (20). The determination of kP is then considerably more complicated. 

KP and v can, in contrast to kp, not be determined via the concentration gradient for binary and ternary mixed micelles, because for the calculation of the Nemstian distribution a constant CMC and an almost constant partial molar volume must be assumed. The calculation of aggregation constants of simple bile salt systems based on Eq. (4) yields similar results (Fig. 8b). Assuming the formation of several concurrent complexes, a brutto stability constant can be calculated. For each application of any tenside, suitable markers have to be found. The completeness of dissolution in the micellar phase is, among other parameters, dependent on the pH value and the ionic strength of the counterions. Therefore, the displacement method should be used, which is not dependent on the chemical solubilization properties of markers. For electrophoretic MACE studies, it is advantageous for the micellar constitution (structure of micelle, type of phase micellar or lamellar) to be known for the relevant range of concentrations (surfactant, lipids). 

Special classes of apparatus are used for the determination of particular thermodynamic properties, such as activity coefficients at infinite dilution, Henry s constants, or partial molar volumes at infinite dilution [105,106]. These data, together with a thermodynamic model, can be used for the calculation of the compositions of the coexisting phases at equilibrium, and for that reason - in this context - these methods are considered as indirect methods of measurement. 

Fig. 13. Partial molar volume of alcohols Fj as a function of the alcohol mol fraction 2 in water-alcohol mixtures at 15 °C. V02 mol volume of the pure alcohols. (Calculated by densities values of D Ans-Lax85))...

Partial molar volumes and the isothermal compressibility can be calculated from an equation of state. Unfortunately, these equations require properties of the components, such as critical temperature, critical pressure and the acentric factor. These properties are not known for the benzophenone triplet and the transition state. However, they can be estimated very roughly using standard techniques such as Joback s modification of Lyderson s method for Tc and Pc and the standard method for the acentric factor (Reid et al., 1987). We calculated the values for the benzophenone triplet assuming a structure similar to ground state benzophenone. The transition state was considered to be a benzophenone/isopropanol complex. The values used are shown in Table 1. 

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