Approaches to Ionic Solvation

In considering various computational approaches to solvation, it must first be understood that the ion-water association alone offers a great range of behavior as far as the residence time of water in a hydration shell is concerned. Certain ions form hydrates with lifetimes of months. However, for the ions that are nearly always the goal of computation (ions of groups lA and HA in the Periodic Table and halide ions), the lifetime may be fractions of a nanosecond. 

As indicaled earlier in this chapter (see Section 2.3), there are three approaches to calculating solvation-related phenomena in solution Quantum mechanic, Monte Carlo, and molecular dynamics. 

The quantum mechanical approach, which at first seems the most fundamental, has major difficulties. It is basically a 0 K approach, neglecting aspects of ordering and entropy. It is suited to dealing with the formation of molecular bonds and reactivity by the formation in terms of electron density maps. However, ionic solutions are system in which order and entropy, its converse, are paramount considerations. 

The most fruitful of the three approaches, and the one likely to grow most in the future, is the molecular dynamics approach (Section 2.3.2). Here, a limited system of ions and molecules is considered and the Newtonian mechanics of the movement of 

There is a more fundamental difficulty the great time such calculations take. If they have to deal with more than ten electrons, ab initio calculations in quantum mechanics may not be practical. 

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A straightforward approach to ionic solvation, particularly for monatomic ions, is by means of the Bom equation,17 Eq. (32), which was introduced in Section III.2.i ... 

COMPUTER-SIMULATION APPROACHES TO IONIC SOLVATION 2.17.1. General... 

In summary, the empirical approach to ionic solvation based on the MSA is quite successful for monoatomic ions of the main group elements. It helps one to understand the important differences between the way cations and anions are solvated in water. It can also be applied to other ions, including polyatomic ions, provided the solvation is essentially electrostatic in character. Thus, one may estimate effective radii for anions such as nitrate and perchlorate from the Gibbs solvation energy using the value of 8s calculated for the halide ions. Considering the simplicity of the model, it provides an useful means of understanding the thermodynamics of solvation. 

From the experimental results and theoretical approaches we learn that even the simplest interface investigated in electrochemistry is still a very complicated system. To describe the structure of this interface we have to tackle several difficulties. It is a many-component system. Between the components there are different kinds of interactions. Some of them have a long range while others are short ranged but very strong. In addition, if the solution side can be treated by using classical statistical mechanics the description of the metal side requires the use of quantum methods. The main feature of the experimental quantities, e.g., differential capacitance, is their nonlinear dependence on the polarization of the electrode. There are such sophisticated phenomena as ionic solvation and electrostriction invoked in the attempts of interpretation of this nonlinear behavior [2]. 

Theoretical considerations leading to a density functional theory (DFT) formulation of the reaction field (RF) approach to solvent effects are discussed. The first model is based upon isolelectronic processes that take place at the nucleus of the host system. The energy variations are derived from the nuclear transition state (ZTS) model. The solvation energy is expressed in terms of the electrostatic potential at the nucleus of a pseudo atom having a fractional nuclear charge. This procedure avoids the introduction of arbitrary ionic radii in the calculation of insertion energy, since all integrations involved are performed over [O.ooJ The quality of the approximations made are discussed within the frame of the Kohn-Sham formulation of density functional theory. 

In the Bom like approaches to solvation energy, the electrostatic potential of the ion appears as the basic variable of the theory. From Eq (1), it may be seen that if we have accurate electron densities at hand, the electrostatic potential strongly depends on the ionic radius r. The choice of suitable ionic radii usually introduces some arbitrariness in the calculation of AESolv there is no a physical criterium to justify the use of empirical rA values coming from different sources [15-16],... 

In chapter 2, Profs. Contreras, Perez and Aizman present the density functional (DF) theory in the framework of the reaction field (RF) approach to solvent effects. In spite of the fact that the electrostatic potentials for cations and anions display quite a different functional dependence with the radial variable, they show that it is possible in both cases to build up an unified procedure consistent with the Bom model of ion solvation. The proposed procedure avoids the introduction of arbitrary ionic radii in the calculation of insertion energy. Especially interesting is the introduction of local indices in the solvation energy expression, the effect of the polarizable medium is directly expressed in terms of the natural reactivity indices of DF theory. The paper provides the theoretical basis for the treatment of chemical reactivity in solution. 

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... 

Some other theoretical aspects of ionic solvation have been reviewed in the last few years. The interested reader is referred to them ionic radii and enthalpies of hydration 20>, a phenomenological approach to cation-solvent interactions mainly based on thermodynamic data 21>, relationship between hydration energies and electrode potentials 22>, dynamic structure of solvation shells 23>. Brief reviews, monographs, and surveys on this subject from a more or less different point of view have also been published 24—28) ... 

Theoretical considerations based upon a molecular approach to solvation are not yet very sophisticated. As in the case of ionic solvation, but even more markedly, the connection between properties of liquid mixtures and models on the level of molecular colculations is, despite all the progress made, an essentially unsolved problem. Even very crude approximative approaches utilizing for example the concept of pairwise additivity of intermolecular forces are not yet tractable, simply because extended potential hypersurfaces of dimeric molecular associations are lacking. A complete hypersurface describing the potential of two diatomics has already a dimensionality of six In this light, it is clear that advanced calculations are limited to very basic aspects of intermolecular interactions,... 

In none of these examples has the potential for removal of an electron approached the ionization potentials of the metals. Although traditional treatments attribute the potentials of Eqs. (10.1), (10.15), and (10-17)—(10-21) to the removal of electrons from the metals, coupled with large ionic solvation energies, this requires a pathway with the ionization potential as a kinetic barrier. Furthermore, the spontaneous reaction of iron with acidified water is driven by the formation of Fe—OH2+ and H—H covalent bonds that facilitate hydrogen-atom transfer from water (rather than electron transfer from iron) ... 

Another viable method to compare experiments and theories are simulations of either the cell model with one or more infinite rods present or to take a solution of finite semi-flexible polyelectrolytes. These will of course capture all correlations and ionic finite size effects on the basis of the RPM, and are therefore a good method to check how far simple potentials will suffice to reproduce experimental results. In Sect. 4.2, we shall in particular compare simulations and results obtained with the DHHC local density functional theory to osmotic pressure data. This comparison will demonstrate to what extent the PB cell model, and furthermore the whole coarse grained RPM approach can be expected to hold, and on which level one starts to see solvation effects and other molecular details present under experimental conditions. 

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... 

Finally, there is the theoretical method of approaching ionic solvation including the molecular dynamics simulations. These have become increasingly used because they are cheap and quick. However, MD methods use two-body interaction equations and the parameters used here need experimental data to act as a guide for the determination of parameters that fit. 

Apart from neutron diffraction, what other method distinguishes between the static or equilibrium coordination number and the dynamic solvation number, the number of solvent molecules that travel with an ion when it moves One method is to obtain the sum of the solvation numbers for both cation and anion by using a compressibility approach, assuming that the compressibility of the primary solvation shell is small or negligible, then using the vibration potential approach of Debye to obtain the difference in mass of the two solvated ions. From these two measurements it is possible to get the individual ionic solvation numbers with some degree of reliability. 

From (4-26), for a 1 1 electrolyte in water the Bjerrum critical distance is 3.6 X 10 cm. When it is realized that ions in water are normally highly solvated and that the sum of ionic crystal radii for typical anions and cations often approaches or exceeds 3.6 x 10 cm, it is reasonable to find that dissociation constants for ion pairs in water are large thus for sodium hydroxide the dissociation constant is about 5. On the other hand, for a 2 2 electrolyte in water the critical distance is 14.3 x 10 cm, and for a 1 1 electrolyte in ethanol, 11.5 x 10 cm. In these cases, even highly solvated ions can readily approach to the distance necessary to form an ion pair. For magnesium sulfate the dissociation constant in water is 6 x 10 , and for sodium sulfate, 0.2. 

Cryoscopic measurements in sodium nitrate indicate that this is an ionic melt. The extension of studies with organic solutes to this solvent or (Na, K)N03 eutectic (mp 233°) could give information about the solvated species, role of the solvent, and at higher temperatures the mechanism of oxidation. Based on inorganic reactions, currently there is disagreement about the existence in fused nitrates of oxide ion and nitronium ion. A fresh approach to the question with some organic reactions might help clarify the mechanism. 

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