Standard partial molar volume

As seen from Eq. (130) an activity coefficient may deviate significantly from unity at higher salt concentrations. The activity coefficient can therefore also be used as a measure of the deviation of the salt solution from a thermodynamically ideal solution. If the chemical potential of a solute in a (pressure-dependent) standard state of infinite dilution is /x°, we find the standard partial molar volume from... 

Thus, the partial molar volume is constant along the Henry s-law line and equal to the standard partial molar volume. The only real solution along the Henry s-law line is the infinitely dilute solution, so... 

Ion pairing is also predicted to produce a positive volume change since the sum of the standard partial molar volumes of the free ions is lower than the standard partial molar volumes of the ion-pair. The difference between these two qnantities can be obtained by the pressure derivative of (corrected for isothermal compressibility if... 

Fig. 17.10. The standard partial molar volume and standard partial molar heat capacity of aqueous NaCl as represented by the HKF model, showing the characteristic inverted-U shape and steep negative slopes at high and low temperatures.

Fig. 17.11. Solvation versus non-solvation contribution to the standard partial molar volume of aqueous Na at saturation pressure, according to the HKF model. Labels are T°C.

Fig. 17.13. The standard partial molar volume of aqueous NaCl as a function of temperature. Squares are experimental data, and the solvation and non-solvation contributions are from the HKF model for Cr.

Usually the second term of Eq. (2.14) is replaced by wie (9 Ve/3wie )7-, because of the square root dependence of Ve on wie in dilute solutions according to the Debye-Hiickel theory. On extrapolation to infinite dilution Ve becomes equal to the standard partial molar volume of the electrolyte Ve" = Ve". ... 

The standard partial molar volume of an ion in aqueous solution, Vf, is the actual volume to be assigned to the ion in the solution (at infinite dilution). It is the sum of its intrinsic volume, Tnntr", and the electrostriction that the ion has caused in the water around it, yieiec , the latter being a negative quantity. The volume of a bare unhydrated ion, A%Np,l i)r, cannot represent its intrinsic volume and must be enlarged to account for the void spaces between the water molecules and the ion and among themselves in order to represent the intrinsic volume of the ion in the solution. A factor of A = 1.213 was proposed by Mukerjee (1961) for the alkali metal and the halide ions, producing ... 

Marcus Y (2008) On the relation between thermodynamic, transport and structural properties of electrolyte solutions Russ. J Electrochem 44 16-27 Marcus Y (2008a) Properties of individual ions in solution. In Bostrelli DV (ed) Solution chemistry research progress. Nova Science pubhshers, Inc., Hauppauge, 51-68 Marcus Y (2009) The standard partial molar volumes of ions in solution, part 4. Ionic volumes in water at 0-100 °C. J Phys Chem B 113 10285-10291 Marcus Y (2009a) The effects of ions on the structure of water structure-breaking and—making. Chem Rev 109 1346-1370... 

Marcus Y (2012a) The standard partial molar volumes of ions in solution. Part 5. Ionic volumes in water at 125-200 °C. J. Phys. Chem. B 117 http //dx.doi/10.1021/jp212518t Marcus Y (2012b) Are ionic Stokes radii of any use J. Solution Chem in the press Marcus Y, Hefter G (1999) On the pressure and electric field dependencies of the relative permittivity of liquids. J Sol Chem 28 575-591 Marcus Y, Hefter G (2006) Ion pairing. Chem Rev 106 4585-4621... 

In the unsymmetric standard state convention, where the solvent is referenced to its pure state (Raoult s law reference state) and the solutes to the infinite dilution state (Henry s law reference state), the standard partial molar volume is equal to the pressure derivative of the chemical potential at... 

K° being the isothermal compressibility of the pure solvent. This equation will be used later to explain the observed behavior of the standard partial molar volume of solutes near the solvent critical point. 

For aqueous electrolytes the ionic association become important when b is higher than 5, a value typical of a 2 2 electrolyte at room temperature or a 1 1 electrolyte above 300 °C. Thus, the extrapolation of the apparent partial volume of these electrolytes at infinite dilution to obtain the standard partial molar volume is uncertain, because the free ions concentration depends on the stoichiometric electrolyte concentration. For a 2 2 electrolyte, as MgS04, at 25 °C the apparent partial molar volume approach the DHLL value at concentrations bellow 0.01 mol kg (Franks and Smith, 1967) and, at least the density could be measured with a precision of 1 ppm, V° for MgS04 can not be obtained by extrapolation. In this case one can calculate the standard partial molar volume from the known values of standard partial volume of 1 2 and 1 1 electrolytes by using the additivity rule (Lo Surdo et al, 1982) ... 

Since the amount of HCl is small as compared with the organic component, its apparent molar volume is approximated by the standard partial molar volume. The excess mixing term, 5, is ignored in the calculation of the apparent partial molar volume of the major component having a common anion. 

In this section we will briefly summarize the models proposed to assess the standard partial molar volume of solute at high temperature and pressure. 

The revised HKF equation for the noneleclrostatic part of the ionic standard partial molar volume is ... 

The values of the parameters for several aqueous ions (Shock and Helgeson, 1988) are summarized in Table 2.1 and according to the authors allow calculating partial molar volume of ions up to 723 K and 500 MPa. For ionic species linear correlations were found between oi and the nonelec-trostatic part, V, of the standard partial molar volume, and between 02 and the nonelectrostatic part, sr", of the compressibility. On the other hand, a linear correlation was observed between 04 and 02, namely 04 = —4.134 02 - 27790, and as could be calculated from the measured standard partial molar volume using Equation (2.80). 

Cooney and O Connell (1987) found a correlation between A and the reduced density for electrolytes which allow them to estimate the standard partial molar volumes of aqueous salts. 

Table 2.3 Test of the different equations for the standard partial molar volume of aqueous ions (Reproduced from Chemical Geology, A new equation of state for correlation and prediction of standard molal thermodynamic properties of aqueous species at high temperatures and pressures with permission from Elsevier)...

The failure of the revised HFK model to predict the standard partial molar volume of electrolytes and nonelectrolytes in the near critical conditions is not unexpected taking into account the effect of the solvent compressibility on V2 near the critical point, as mentioned before, and the limited data set of high temperature data considered in the fit for aqueous nonelectrolytes (Shock et ah, 1989). 

Because the compressibility of pure water has a singularity at the critical point. Equation (2.11) predicts that the standard partial molar volume should diverge in the critical point. The sign of the critical divergence, determined by the spatial integral of the direct correlation function, C 2, depends on the nature of the solute-solvent interaction. 

Figure 2.17 shows that the standard partial molar volume is negative for nonvolatile strong electrolytes and becomes increasingly positive for volatile nonelectrolytes with decreasing polarity. 

The generalized Krichevskii parameter or its equivalent, the direct correlation function integral, Cu, is well behaved in the critical region, as shown in Figure 2.18, and on this is based Equation (2.85) (O Connell et al., 1996), which was used to fit the standard partial molar volume of nonelectrolytes all over the density range. 

Similarly, the entropy of ionization, AS°, the standard partial molar heat capacity of ionization, AjC , and the standard partial molar volume of ionization, AV°, can be derived from AG using standard thermodynamic identities (Mesmer et al., 1988) ... 

Values of the ionic standard partial molar volumes of the hydrogen, alkali metal, alkaline earth metal, and anunonium cations and hydroxide, hahde, nitrate, perchlorate, and sulfate anions from 0 to 200°C at 25°C intervals have been reported by Marcus in Refs. 79-81. Conventional values V " for these ions (except CIO ) and also for HCOj" and HS are reported by Tanger and Helgeson [84] from 0 to 350°C at 25°C intervals. 

As mentioned earlier, the actual volume to be assigned to an ion in the solution at infinite dilution is its standard partial molar volume V". It may be negative, in particular for small highly charged ions, because the electrostriction (volume diminution), 1/,", such ions cause in the water surrounding the ion may be numerically larger than the intrinsic volume of the ion, [80], Ionic intrinsic volumes that are independent of the concentration are discussed in Section 2.2 and are shown for... 

Thermodynamic properties of ions in nonaqueous solvents are described in terms of the transfer from water as the source solvent to nonaqueous solvents as the targets of this transfer. These properties include the standard molar Gibbs energies of transfer (Table 4.2), enthalpies of transfer (Table 4.3), entropies of transfer (Table 4.4) and heat capacities of transfer (Table 4.5) as well as the standard partial molar volumes (Table 4.6) and the solvation numbers of the ions in non-aqueous solvents (Table 4.10). The transfer properties together with the properties of the aqueous ions yield the corresponding properties of ions in the nonaqueous solvents. 

The standard partial molar volumes of electrolytes in mixed solvents can be modeled, as can those in neat solvents, in terms of the sum of the intrinsic volumes of the ions and their electrostriction. It is assumed that the intrinsic volumes, that is, the volumes of the ions proper and including the voids between ions and solvent molecules, are solvent independent, so that they do not depend on the natures of the solvents near the ions. Then, if no preferential solvation of the ions by the components of the solvent mixture takes place, the electrostriction can be calculated according to Marcus [32] as for neat solvents (Section 4.3.2.5), with the relevant properties of the solvents prorated according to the composition of the mixture. This appeared to be the case for the ions Li+, Na", K+, CIO ", AsE , and CFjSOj in mixtures of PC with MeCN, in which V (P,PC+MeCN) is linear with the composition over nearly the entire composition range. This is the case also for Me NBr in W+DMSO, as shown in Figure 6.1. Similarly, in aqueous methanol mixtures, smooth curves result for the ions Li", Na ", K+, Cs", CF, Br", and I" like those shown in Figure 6.1 for NaBr and KBr. However, when preferential solvation occurs, the... 

---