Partial molal volumes, determination

Interesting is a comparison of the volumes occupied by individual complexes in solution and in the solid state. The partial molal volumes can be obtained from precise measurements of the solution densities of the complexes as a function of concentration [177]. These values may be subsequently compared with the unit cell volumes per complex molecule derived from the crystal structure. For Fe[HB(pz)3]2, the apparent molal volume in tetrahydrofuran solution was determined as 340.9 em mol Taking into account that the complex in solution forms an equilibrium between 86% LS and 14% HS isomers and employing the volume difference between the two spin states AF° = 23.6 cm mol S the volume of the LS isomer was calculated as 337.6 cm mol This value agrees closely with the volume of 337.3 cm mol for the completely LS complex in solid Fe[HB(pz)3]2 [105]. 

The pressure-volume-temperature (PVT) properties of aqueous electrolyte and mixed electrolyte solutions are frequently needed to make practical engineering calculations. For example precise PVT properties of natural waters like seawater are required to determine the vertical stability, the circulation, and the mixing of waters in the oceans. Besides the practical interest, the PVT properties of aqueous electrolyte solutions can also yield information on the structure of solutions and the ionic interactions that occur in solution. The derived partial molal volumes of electrolytes yield information on ion-water and ion-ion interactions (1,2 ). The effect of pressure on chemical equilibria can also be derived from partial molal volume data (3). 

In kinetics, similar relationships apply, but the volume of activation AV can be determined only from the pressure dependence of the rate coefficient k, since the partial molal volumes V of transition states are not directly measurable. Conversely, however, equation 4 can yield values of V. ... 

Richardson, P. J., D. F. McCafferty, and A. D. V foolfson. 1992. Determination of three-component partial solubility parameters for temazepam and the effects of change in partial molal volume on the thermodynamics of drug solubility J. Pharm. 78 189-198. 

FIGURE 1.22.1 Plot illustrating how partial molal volumes may be determined. 

Using Equation IV.2 we can readily determine the pressure dependence of the Gibbs free energy as needed in the last bracket of Equation IV. 11—namely, dG/dP)TEhn.n is equal to Fby Equation IV.12. Next, we have to consider the partial derivative of this V with respect to rij (see the last equality of Eq. IV. 11). Equation2.6 indicates that dV/dnj)TPEkn. is Vj, the partial molal volume of species j. Substituting these partial derivatives into Equation IV. 11 leads to the following useful expression ... 

Hawley and Pytkowicz (36) determined the partial molal volume change for aragonite in seawater at 22 and 2°C. At 22 C... 

There are several studies that have been successful in determining the dissolution rate at conditions near seawater saturation. Acker et al. (1987) was able to employ very precise determinations of pH to measure the rate of dissolution of a single pteropod shell at different pressures from 15 atm to 300 atm. Because his measurements were at different pressures and is a function of pressure, he was able to determine whether the rate constant is indeed a function of K p. He found that Equation (9) fit his data better than (10), suggesting that the constant is not pressure dependent and the former is a more accurate universal rate law. An exponent oin= 1.9 was obtained for this surface-controlled dissolution reaction and a partial molal volume. Ay, of —39 cm mol (very close to the mean of the values determined in laboratory experiments for calcite) best fit the data. 

Here V, is known as the partial molal volume of component i. It is the effective volume of one mole of that component in an infinite copy of the solution. Alternatively, this quantity may be regarded as the incremental change in volume of the solution in which temperature, pressure, and all other components are held at fixed values. In general, the volume of the mixture is not equal to the sum of volumes of the individual components. Thus, for Eq. (1.19.2) to be useful we need to find out how the individual V, are to be determined experimentally. We shall address that problem below. 

Fig. 1.19.1. Plot illustrating how the partial molal volumes of components in a binary mixture may be determined by extrapolation.

Alternative Method for Determining Partial Molal Volumes... 

Since G is not as readily measured as V, methods other than those discussed above for specifying partial molal volumes must be introduced to determine the chemical potentials. These procedures will be taken up at a later stage. 

To determine the in situ properties of the carbonate system in the ocean, it is necessaiy to determine the effect of pressure on the thermodynamic constants. This correction can be made in two ways (1) using direct measurements of the constants and (2) using partial molal volume and compressibility data (Mil-lero, 1979). The two methods are in good agreement (Millero, 1979) when comparisons are made for the carbonate system. The effect of pressure on the dissociation constants of acids (A ,) can be made from (Millero, 1979) equations... 

Because of the great depth of the ocean, the most important physical property determining the solubility of carbonate minerals in the sea is pressure. The pressure dependence of the equilibrium constants is related to the difference in volume, AV, occupied by the ions of Ca + and CO3 in solution versus in the solid phase. The volume difference between the dissolved and solid phases is called the partial molal volume change, AV (see also Appendix A4.1) ... 

There have been numerous attempts to determine HLB numbers from other fundamental properties of surfactants, e.g., from cloud points of nonionics (Schott, 1969), from CMCs (Lin, 1973), from gas chromatography retention times (Becher, 1964 Petrowski, 1973), from NMR spectra of nonionics (Ben-et, 1972), from partial molal volumes (Marszall, 1973), and from solubility parameters (Hayashi, 1967 McDonald, 1970 Beerbower, 1971). Although relations have been developed between many of these quantities and HLB values calculated from structural groups in the molecule, particularly in the case of nonionic surfactants, there are few or no data showing that the HLB values calculated in these fashions are indicative of actual emulsion behavior. 

By studying the pressure dependence of the hydrogen evolution reaction on Hg at pH 1 they obtained a negative volume of activation (—3.4 ml. mole"1), which, they argued, was not in accord with any H-atom process but could result from the emission and hydration of metallic electrons as the rate-determining step. This interpretation thus invokes the formation of e aq even at pH 1. (This value of —3.4 agrees well with the partial molal volume of e m (—5.5 to —1.1 mole 1) obtained indirectly in a radiation chemical system (45). It is interesting to note the difference in V for e aq and e ( ammonia >, where the value is —60 ml. mole 1 (49)). 

Of course the partial molal volume of Haq is not directly measurable. This parameter could be measured for the solutes Ho and He by determining densities of solutions, but no record of such observations has been found. The entry for V° of Haq in Table I is simply the volume of a sphere of radius 1.0 A. (the volume containing 99% of the electronic charge distribution). The ideas of Nemethy and Scheraga (6) on hydro-phobic bonding suggest that even this small estimate may be somewhat too large. 

T is temperature, Ak is the change in compressibility caused by the reaction, V indicates partial molal volume, and sub- scripts refer to pressure in atmospheres. Determination of AV would allow calculation of stCOYOACaa) quantity necessary for predicting the direction of tne pressure effect for several geochemical reactions which are difficult to investigate experimentally, for exaniple, mineral formation at sea floor conditions. However, this calculation demands accuracy in knowledge of the atmospheric pressure solubility, and confidence in the reproducibility of the pressure versus solubility plots. 

Evaluation of partial molal volumes of electrolytes, and in particular the standard partial molal volume, Fg, is similar to the determination of partial molal heat capacities. As in the case of heat capacities, the apparent molal quantity is of interest. Corresponding to eqns. 2.3.52, 2.3.54 and 2.3.55 we have... 

In an attempt to correlate gross structure of solute-water with packing and stereochemistry, the partial molal volumes and isentropic partial molal compressibilities were measured for sugars, uronic acids, and some di- and trisaccharides in water at 25 C. Attempts to systematize the results were unsuccessful. In a different approach, the structural transitions at saturation temperature using Arrhenius plots of results obtained by conductance measurements on electrolyte-sucrose-water solution were determined. These transitions were postulated to be due to the ability of the sucrose to form intermolecular hydrogen bonds with the solvent, so that at saturation temperature it was considered to have entered the total structure of the solution. ... 

This method involves the accurate measurement of the density of the solution and the determination of the surfactant concentration. Any change in the partial molal volume of surfactant will be reflected in the density of the solution. As with all of the other methods, very careful measurement procedures and precise temperature control are required in order to obtain reliable data. This method can be used for any type of surfactant in aqueous, as well as non-aqueous, systems. 

The determination of partial molal volumes shown below provides another illustration of thermodynamic methodology. We begin by defining the molar volume of a mixture as V = VIn terms of mole fractions, rewrite Eq. (1.14.2) as... 

P is the experimentally-determined pressure of the equilibrium end member reaction (Hariya and Kennedy, I968) and the assumptions are made of independence of the partial molal volume of CaTs of composition and independence of AV of pressure. The reference state of CaTs is the pure phase at the T and P of solid solution equilibration. 

A related matter is the electrostriction suffered by these water molecules in the first solvation sphere. One of the ways to approach it is through determination of the partial molal volumes of the ions (with various assumptions we will not go into here). These ionic partial molal volumes can be split, using the semi-empirical Hepler equation (eq. 1), into a geometric... 

The partial molal volume (F ) at zero concentration was determined by extrapolation of The F was found to increase with rising temperature, and the temperature coefficients for tetrabutylammonium salts of polyacrylic add, polyethylenesulfonic add, and polystyrenesulfonic acid were in the range of 0.2 0.3 ml/monomole-deg 80), which are in a rough agreement with those found for tetrabutylammonium bromide [0.2 0.5 ml/mole deg] (S5). The values were discu d in terms of the partial molal voluine of macroions (Fj ) and g enions (F, )... 

Both these considerations would be taken into accoimt if the activation process were assumed to occur at a constant pressure, p, such that the partial molar volume of the solvent is independent of the temperature, though this possibility does not appear to have been considered. A full discussion is beyond the scope of this chapter, but the resulting heat capacities of activation are unlikely to differ greatly from those determined at a constant pressme of, say, 1 atm. (see p. 137). Unfortunately, this approach requires the definition of rather clumsy standard states for solutes, e.g., hypothetically ideal, 1 molal, under a pressure such that a given mass of the pure solvent occupies a particular volume. 

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