Other thermodynamic quantities of solvation

In this section, we derive some more relations between the solvation quantities and standard thermodynamic quantities of solution. 

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Having defined the Gibbs energy of solvation, we can derive all the other thermodynamic quantities of solvation by using standard thermodynamic relationships. The most important quantities are the first derivatives of the Gibbs energy, i.e.,... 

Having obtained the Gibbs energy of solvation through one of the relations cited above, it is a straightforward matter to calculate other thermodynamic quantities of solvation using the standard relations. For instance, the solvation entropy can be calculated from the temperature dependence of the AG /, i.e.,... 

When evaluating other thermodynamic quantities of solvation from data along the equilibrium line, care must be exercised to distinguish between derivatives at constant pressure and derivatives along the equilibrium line. The connection between the two is... 

Before considering different theoretical approaches to determining the free energies and other thermodynamic properties of ionic solvation, it is important to be aware of a problem on the experimental level. There are several methods available for obtaining these quantities for electrolyte solutions, both aqueous and nonaqueous some of these have been described by Conway and Bockris162 and by Padova.163 For example, enthalpies of solvation can be found via thermodynamic cycles, free energies from solubilities or galvanic cell potentials. However the results... 

According to the Lorentz-Lorenz equation (4.3.21) for the molar refraction at optical frequencies, Y is directly proportional to the molecular polarizability p. The Koppel-Palm equation has also been applied to the analysis of solvent effects on thermodynamic quantities related to the solvation of electrolytes [48, 49]. In the case of the systems considered in table 4.11, addition of the parameter X to the linear equation describing the solvent effect improves the quality of the fit to the experimental data, especially in the case of alkali metal halide electrolytes involving larger ions. The parameter Y is not important for these systems but does assist in the interpretation of other thermodynamic quantities which are solvent dependent [48, 49]. Addition of these parameters to the analysis is only possible when the solvent-dependent phenomenon has been studied in a large number of solvents. 

Third, the equality of the Gibbs energies for the p-process and the solvation process do not imply equality between any other thermodynamic quantities pertaining to these two processes. One must exercise extreme care in deriving the relations between, say, the standard entropy of the p-process and the entropy of solvation these cannot be obtained by taking the temperature derivative of equation (7.54). As we shall see in the next section, this is a tricky point which has been overlooked even by experts working in this field. 

Other quantities of solvation and their relations with the solvation thermodynamic quantities were discussed at great length in Ben-Naim (2006). 

Equation (6.13.27) is the required connection between the Gibbs energy of the x-process and the solvation Gibbs energy. Again, we note that this equality holds only for the ideal-gas and ideal solutions. To obtain the corresponding relations for the other thermodynamic quantities associated with the processes defined above, we start from the general equation, Eq. (6.13.1), for the chemical potential. The partial molar (or molecular) entropy of s is obtained from... 

Fajans (1962) was the first scientist to put these thoughts into practice. One finds that the determined heats of solvation are reiativeiy small and endothermic (+) for some salts but exothermic (-) for others. However, lattice energies are known to be in the region of several hundreds of kilojoules mor, so that, in rough terms [Eq. (2.3)], heats of solvation should not be more than a dozen kilojoules moF (numerically) different from lattice energies. In Table 2.4 a compilation is given of the quantities mentioned earlier in the case of the alkali haUdes. Now, the method described here gives the sum of the heat of hydration of the ions of a salt. The question of how to divide this sum up into individual contributions from each of the ions of a salt requires more than the thermodynamic approach that has been used so far. The way this is done is described in later sections (e.g., in Section 2.6.2 or 2.15.9). 

Consider a system of Nw solvent molecules, say water, and ND solute molecules contained in a volume V at temperature T. By solute, we mean those molecules D for which we wish to evaluate the solvation thermodynamic quantities. The solvent, which is usually water, may be any liquid or any mixture of liquids and could contain any number of other solutes besides D. In the most general case, Nw will be the total number of molecules in the system except those that are counted as solutes in ND. However, for notational simplicity, we shall treat only two-component systems, W and D. 

The excess chemical potential of solute, or the solvation free energy , at infinite dilution is of particular interest, because it is the quantity which measures the stability of solute in solvent, and because all other excess thermodynamic quantities are derived from the free energy. The excess chemical potential, which is defined as an excess from the ideal gas, can be expressed in terms of the so called Kirkwood coupling parameter. The excess chemical potential is defined as the free energy change associated with a process in which a solute molecule is coupled into solvent [41]. The coupling procedure can be expressed by. 

It should be noted that the SPT is not a pure molecular theory in the following sense. A molecular theory is supposed to provide, say, the Gibbs free energy as a function of T, P, N as well as of the molecular parameters of the system. Once this function is available, the density of the system can be computed from the relation p = (9/x/9 )t (with pi = G/N). The SPT utilizes the effective diameter of the solvent molecules as the only molecular parameter (which is the case for a hard-sphere fluid) and, in addition to the specification of T and P, the solvent density Pw is also used as input in the theory. The latter being a measurable quantity carries with it implicitly any other molecular properties of the system. The first application of the SPT to calculate the thermodynamics of solvation in liquids was carried out by Pierotti (1963, 1965). 

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