Partial molar volum

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). ... 

VSTR is useful for estimating partial molar volumes at infinite dilution but is not used here because of Equation (4-17)... 

We define a partial molar volume Vi such that V = riiVi -I- U2V2... 

We may define, say, partial molar volume, enthalpy, or entropy by analogy with Eq. (8.5) ... 

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... 

As in osmotic pressure experiments, polymer concentations are usually expressed in mass volume units rather than in the volume fraction units indicated by the Einstein equation. For dilute solutions, however, Eq. (8.100) shows that

partial molar volume of the polymer in solution, and M is the molecular weight of the polymer. Substituting this relationship for (pin Eq. (9.9)gives... 

P rtl IMol r Properties. The properties of individual components in a mixture or solution play an important role in solution thermodynamics. These properties, which represent molar derivatives of such extensive quantities as Gibbs free energy and entropy, are called partial molar properties. For example, in a Hquid mixture of ethanol and water, the partial molar volume of ethanol and the partial molar volume of water have values that are, in general, quite different from the volumes of pure ethanol and pure water at the same temperature and pressure (21). If the mixture is an ideal solution, the partial molar volume of a component in solution is the same as the molar volume of the pure material at the same temperature and pressure. 

Vi Partial molar volume of i mVkmol or cmVmol ftVlbmol... 

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... 

Beck, et al. have used the permeation technique to study the effect of uniaxial tensile stresses in the elastic region on hydrogen permeation through pure iron, and have shown that it increases with increase in stress. The partial molar volume of hydrogen (cubic centimetres of hydrogen per mole of iron) in ferrous alloys can be evaluated from the variation of permeation with applied stress, and from the relationship... 

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

B. Partial Molar Volumes in Saturated Liquids from an Equation of State. 162... 

Thermodynamics gives limited information on each of the three coefficients which appear on the right-hand side of Eq. (1). The first term can be related to the partial molar enthalpy and the second to the partial molar volume the third term cannot be expressed in terms of any fundamental thermodynamic property, but it can be conveniently related to the excess Gibbs energy which, in turn, can be described by a solution model. For a complete description of phase behavior we must say something about each of these three coefficients for each component, in every phase. In high-pressure work, it is important to give particular attention to the second coefficient, which tells us how phase behavior is affected by pressure. 

In order to simplify Eq. (5), it was suggested many years ago by Lewis (L3) that Amagat s law be used, viz., to assume that the partial molar volume of component i at any temperature and pressure is equal to the molar volume... 

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... 

For dilute solutions, the technical literature contains some direct (dilato-metric) measurements of v2, the partial molar volume of the more volatile component, but the accuracy of these measurements is usually not high. A survey was made by Lyckman and Eckert (L6) and they established the rough correlation shown in Fig. 5. On the ordinate, the partial molar volume is... 

Flo. 5. Partial molar volumes of gases in dilute liquid solutions. 

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... 

FIg. 6. Partial molar volumes of gaseous solutes at infinite dilution in expanded solvents. 

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. 

Fig. 7. Partial molar volumes in the saturated liquid phase of the n-butane-carbon...

Fig. 8. Partial molar volumes in the saturated liquid phase of the propane-methane system at IOO°F. (O) (0) Data of B. H. Sage and W. N. Lacey, Some Properties of the Lighter Hydrocarbons, Hydrogen Sulfide, and Carbon Dioxide. American Petroleum...

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. 

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