Entropies of Ions

If the energy to put an isolated ion and its electrons into the lattice is denoted s, then the chemical potential is 

This equation ignores interactions between the ions. The simplest way to treat the interactions is just to add them to this equation, and assume that the ions remain randomly arranged. Suppose U is the total interaction energy that a given ion would feel if all the other sites were full. When only a fraction x of the sites is occupied, it costs an extra energy Ux to add another ion to the lattice, so fi becomes 

This is often called a mean-field expression, because each ion feels the mean interaction or field from its neighbours. It is exact if the interaction between ions extends over a long range, because then the interaction will not cause the ions to rearrange from a random distribution. If the ions do rearrange, the entropy will change and better approximations are needed. 

As a function of x, these expressions lead to s-shaped curves like those shown in Fig. 7.10. The scale of the variation is set by kT/e, which is 25 mV at room temperature. The partial entropy diverges near x = 1 and X = 0, reflecting the impossibility of filling every last site or removing every last ion. 

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Table 25. Conventional Paivtial Molal Entropies of Ions in Aqueous Solution...

Accordingly, the conventional partial molal entropies of ions in solution are often said to refer to the ions in a one-molal solution (m = 1) hot in a real one-molal solution, but in a hypothetical ideal one-molal solution, where the contribution from the interionic forces is taken to be zero, and the cratic term replaces the communal term. 

Table 45. Conventional Partial Molal Entropies of Ions in Water at 25°C...

The entropies of individual ions in solution are determined by setting the entropy <>1 II in water equal to 0 and then defining the entropies of all other ions relative to this value hence a negative entropy is one that is lower than the entropy of H in water. All absolute entropies are positive, and no sign need he given all entropies of ions are relative to that ot H+ and are listed here with a sign (either + or —). 

Water solubility of ionic substances is dependent on a fine balance between lattice energy, hydration energy and entropy of ions. The scheme shown hereunder as ... 

Recently, the Pitzer equation has been applied to model weak electrolyte systems by Beutier and Renon ( ) and Edwards, et al. (10). Beutier and Renon used a simplified Pitzer equation for the ion-ion interaction contribution, applied Debye-McAulay s electrostatic theory (Harned and Owen, (14)) for the ion-molecule interaction contribution, and adoptee) Margules type terms for molecule-molecule interactions between the same molecular solutes. Edwards, et al. applied the Pitzer equation directly, without defining any new terms, for all interactions (ion-ion, ion-molecule, and molecule-molecule) while neglecting all ternary parameters. Bromley s (1) ideas on additivity of interaction parameters of individual ions and correlation between individual ion and partial molar entropy of ions at infinite dilution were adopted in both studies. In addition, they both neglected contributions from interactions among ions of the same sign. 

The (iibbs energies of formation, the enthalpies of formation and the standard entropies of ions were described. 

Once the system of Eqs. (A.9) and (A. 13) is solved for the boundaty conditions ((A.14), (A.15), (A.18) and (A.19)), the total free energy of the system can be calculated by adding various contributions, due to the electric field, entropy of ions, chemical energy and interactions between ions and surfaces. 

The double-layer contribution Fa contains the entropic Fcni of the ions and the electrostatic contributions Feie- The entropic contribution of the electrolyte to the free energy of the system is due to the change in the entropy of ions between the plates compared to that in the reservoir. 

Partial molar entropies of ions can, for example, be calculated assuming S (H+) = 0. Alternatively, because K+ and Cl ions are isoelectronic and have similar radii, the ionic properties of these ions in solution can be equated, e.g. analysis of B-viscosity coefficients (Gurney, 1953). In other cases, a particular theoretical treatment which relates solvation parameters to ionic radii indicates how the subdivision could be made. For example, the Bom equation requires that AGf (ion) be proportional to the reciprocal of the ionic radius (Friedman and Krishnan, 1973b). However, this approach involves new problems associated with the definition of ionic radius (Stem and Amis, 1959). In another approach to this problem, the properties of a series of salts in solution are plotted in such a way that the value for a common ion is obtained as the intercept. For example, when the partial molar volumes of some alkylammonium iodides, V (R4N+I ) in water (Millero, 1971) are plotted against the relative molecular mass of the cation, M+, the intercept at M + = 0 is equated to Ve (I-) (Conway et al., 1966). This procedure has been used to... 

TABLE 4.1. Crystallographic Radii and Heats and Entropies of Ion Hydration at 25°C... 

Statistical mechanical approaches apply mainly to deductions about structure and are the basis of interpretations of the entropy of ions in solution and the solution s heat capacity. The entropy of a system can be calculated if the partition functions of the ions and the water molecules surrounding them are known." The partition functions (translatory, rotational, and vibrational) can be obtained from textbook material by assuming a structure of the ion-solvent complex. By comparing calculations based on various assumptions about stmcture with the values obtained from experiments, certain stmctures can be shown to be more likely (those giving rise to the calculations that match the experiment), others less probable, and some so far from the experimental values that they may be regarded as impossible. 

Association Constants, Gibbs Energies, Enthalpies, and Entropies of Ion-Pair Formation of Alkaii Metai and Tetraaikylammonium Iodides in 1-Propanol from Temperature-Dependent Conductivity Measurements... 

In Table 4.27, one can see the free energies, enthalpies, and entropies of ion-pair formation. The enthalpies and entropies of the alkali metal salts are on the order of 16 kJ mor and 100 J K" mol , respectively. In contrast, the small enthalpies and entropies of the tetraaikylammonium ions reflect lesser solvation of the cations in the protic solvents. 

But take care these will also be dependent on the convention used in the determination of the individual partial molar entropies of ions in aqueous solution. 

Table 3.2. Crystallographic radii and heats and entropies of Ion hydration at 25° C...

Table 7.3 Conventional Standard Entropies of Ions, and Conventional Entropies jof Hydration at 25 Ct...

The Born equation also leads to a simple interpretation of entropies of hydration and of absolute entropies of ions. The thermodynamic relationship between entropy and Gibbs energy (see equation (5.153)) is... 

The last term on the right-hand side is conventionally taken to be zero, so the conventional standard entropies of ions are given by the... 

Conventional and absolute v alues lor the entropies of ions were delined and derived. 

The data analysis of dilution measurements with the set of Eqs. (25) yields the basic quantities and R of the chemical model and AH (heat of ion-pair formation). The entropy of ion-pair formation, Al, can be calculated from and AH in the usual way. Table III shows examples of a simultaneous determination of Ka. R., andAH . 

Criss and Cobble have also used a correspondence method of dividing entropies into ionic components. Their division was made by plotting entropies of ions at one temperature against the entropies of the corresponding ions at another temperature. Linear results could be obtained only when the entropies at 25°C were based on an assignment of —5.0 cal mol" K" for the hydrogen ion. 

Franks and Reid, using data from the literature, have evaluated partial molal entropies of ions in aqueous methanol and dioxan. The data are listed in Appendix 2.4.42. In their division they made the assumption that -----for each solvent system, which appears reason-... 

That there is a direct connection between the entropy of certain single systems and the amount of order or disorder in the system has led many workers to interpret entropy changes as reflecting the degree of structure in liquid systems. In particular the standard entropies of ions in solution are often used for assessing the effect of the ion on the structure of the solvent. 

In the framework of LEA, the electrostatic interactions manifest themselves through the entropy of ions disproportionated between the interior of a strongly branched macromolecule and bulk solution. Therefore, the driving force for swelling of a branched polyion can be formulated in terms of the differential osmotic pressure of small ions inside and outside of the macromolecule. 

Marcus Y (1986) The hydration entropies of ions and their effects on the structure of water. J Chem Soc, Faraday Trans l(82) 233-240... 

Criss and Cobble noted that the entropies of ions can be expressed as functions of mass, charge and ionic size, and that the functional dependences appear to be the same at 25°C and higher temperatures. This suggested that it would not be necessary to know the complete functional dependences and the entropies at elevated temperatures could be determined from 25°C entropies. They suggested that the following equation could be used to determine the standard state entropy at tj having an experimental standard state entropy at temperature ti ... 

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