Solute partial molar volume

Figure 5 Calculated and experimental (1) solute partial molar volume ys. density at T = 1.37.

When a component A is added to a solvent, there is a volume change usually the volume increases but occasionally it decreases. The volume change on adding 1 mol of A is termed the partial molar volume, and it is denoted by It is dependent on all the thermodynamic conditions the temperature, the pressure, the nature of the solvent, the concentration of A, the concentration of all the other solutes, and any other pertinent thermodynamic variables. Measuring is equivalent to measuring the solution density at specified conditions, a measurement that can be made quite accurately. For stable solutes, partial molar volumes are frequently quoted to a precision of 0.01 cm mol It often suffices to consider only the partial molar volume at... 

Liu, H. and J. P. O Connell. 1998. On the measurement of solute partial molar volumes in near-critical fluids with supercritical fluid chromatography. Industrial and Engineering Chemistry Research. 37, 3323. 

The solute partial molar volume V can be estimated via solute-solvent site correlation functions using the following 3D RISM theory expression [105,106] ... 

Table 9.6 Expressions for the dependence of pressure factors of nonelectrolytes on pressure. The approximate expressions assume the phase is incompressible, or the solute partial molar volume is independent of pressure.

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. 

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... 

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. 

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. 

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

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. 

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... 

In their correlation, Chao and Seader use the original Redlich-Kwong equation of state for vapor-phase fugacities. For the liquid phase, they use the symmetric convention of normalization for y and partial molar volumes which are independent of composition, depending only on temperature. For the variation of y with temperature and composition, Chao and Seader use the equation of Scatchard and Hildebrand for a multicomponent solution ... 

In Eq. (128), the superscript V stands for the vapor phase v2 is the partial molar volume of component 2 in the liquid phase y is the (unsym-metric) activity coefficient and Hffl is Henry s constant for solute 2 in solvent 1 at the (arbitrary) reference pressure Pr, all at the system temperature T. Simultaneous solution of Eqs. (126) and (128) gives the solubility (x2) of the gaseous component as a function of pressure P and solvent composition... 

Volume is an extensive property. Usually, we will be working with Vm, the molar volume. In solution, we will work with the partial molar volume V, which is the contribution per mole of component i in the mixture to the total volume. We will give the mathematical definition of partial molar quantities later when we describe how to measure them and use them. Volume is a property of the state of the system, and hence is a state function.1 That is... 

The volume of a solution is sometimes expressed as a function of composition and the partial molar volume is then obtained by differentiation. For example, Klotz and Rosenburg2 have expressed the volume of aqueous sodium chloride solutions at 298.15 K and ambient pressure as a function of the molality m of the solution by the equation ... 

Figure 5.3 shows V and V2 for the (benzene + cyclohexane) system as a function of mole fraction, obtained in this manner.3 Shown on the graph are Fm, i and F, 2, the partial molar volumes (which are the molar volumes) of the pure benzene and pure cyclohexane. The opposite ends of the curves gives Vf and Vf, the partial molar volumes in an infinitely dilute solution. We note that... 

In the case of solutions, lnF] would be determined from equation (6.85) using the partial molar volume of the solvent, V, in the solution, rather than the molar volume of the pure solvent. 

In solution studies, the modification of the equilibrium nuclear configuration appears primarily as a change of the partial molar volume A V° of the two spin states. The presently available values of AV° for spin conversions in solution are collected in Table 16. There is no apparent difference between the values for iron (II) and iron (III) spin transition compounds, the variation being... 

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. 

The effect of pressure on chemical equilibria and rates of reactions can be described by the well-known equations resulting from the pressure dependence of the Gibbs enthalpy of reaction and activation, respectively, shown in Scheme 1. The volume of reaction (AV) corresponds to the difference between the partial molar volumes of reactants and products. Within the scope of transition state theory the volume of activation can be, accordingly, considered to be a measure of the partial molar volume of the transition state (TS) with respect to the partial molar volumes of the reactants. Volumes of reaction can be determined in three ways (a) from the pressure dependence of the equilibrium constant (from the plot of In K vs p) (b) from the measurement of partial molar volumes of all reactants and products derived from the densities, d, of the solution of each individual component measured at various concentrations, c, and extrapolation of the apparent molar volume 4>... 

Figure 3.1 Mixing of nA moles of A and wg moles of B at constant p and T. The molar volumes of pure A and B are Fa and Fg. The partial molar volumes of A and B in the solution are VA and FB, respectively.

The coordination numbers of the Ln3+ ions in water are now well established from different experimental techniques (214-221). The lighter La3+-Nd3+ ions are predominantly nine-coordinate, Pm3+ Eu3+ exist in equilibria between nine- and eight-coordinate states and the heavier Gd3+-Lu3+ are predominantly eight-coordinate. The change in coordination number is also reflected in the absolute partial molar volumes, U°bs, of several Ln3+ ions determined in aqueous solutions (222,223). 

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