Ion association in the MSA

And it is derived from the same variational principle explained above. 

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Abstract Analytical solution of the associative mean spherical approximation (AMSA) and the modified version of the mean spherical approximation - the mass action law (MSA-MAL) approach for ion and ion-dipole models are used to revise the concept of ion association in the theory of electrolyte solutions. In the considered approach in contrast to the traditional one both free and associated ion electrostatic contributions are taken into account and therefore the revised version of ion association concept is correct for weak and strong regimes of ion association. It is shown that AMSA theory is more preferable for the description of thermodynamic properties while the modified version of the MSA-MAL theory is more useful for the description of electrical properties. The capabilities of the developed approaches are illustrated by the description of thermodynamic and transport properties of electrolyte solutions in weakly polar solvents. The proposed theory is applied to explain the anomalous properties of electrical double layer in a low temperature region and for the treatment of the effect of electrolyte on the rate of intramolecular electron transfer. The revised concept of ion association is also used to describe the concentration dependence of dielectric constant in electrolyte solutions. 

In this chapter some aspects of the present state of the concept of ion association in the theory of electrolyte solutions will be reviewed. For simplification our consideration will be restricted to a symmetrical electrolyte. It will be demonstrated that the concept of ion association is useful not only to describe such properties as osmotic and activity coefficients, electroconductivity and dielectric constant of nonaqueous electrolyte solutions, which traditionally are explained using the ion association ideas, but also for the treatment of electrolyte contributions to the intramolecular electron transfer in weakly polar solvents [21, 22] and for the interpretation of specific anomalous properties of electrical double layer in low temperature region [23, 24], The majority of these properties can be described within the McMillan-Mayer or ion approach when the solvent is considered as a dielectric continuum and only ions are treated explicitly. However, the description of dielectric properties also requires the solvent molecules being explicitly taken into account which can be done at the Born-Oppenheimer or ion-molecular approach. This approach also leads to the correct description of different solvation effects. We should also note that effects of ion association require a different treatment of the thermodynamic and electrical properties. For the thermodynamic properties such as the osmotic and activity coefficients or the adsorption coefficient of electrical double layer, the ion pairs give a direct contribution and these properties are described correctly in the framework of AMSA theory. Since the ion pairs have no free electric charges, they give polarization effects only for such electrical properties as electroconductivity, dielectric constant or capacitance of electrical double layer. Hence, to describe the electrical properties, it is more convenient to modify MSA-MAL approach by including the ion pairs as new polar entities. 

Modern theory of associative fluids is based on the combination of the activity and density expansions for the description of the equilibrium properties. The activity expansions are used to describe the clusterization effects caused by the strongly attractive part of the interparticle interactions. The density expansions are used to treat the contributions of the conventional nonassociative part of interactions. The diagram analysis of these expansions for pair distribution functions leads to the so-called multidensity integral equation approach in the theory of associative fluids. The AMSA theory represents the two-density version of the traditional MSA theory [4, 5] and will be used here for the treatment of ion association in the ionic fluids. 

In the mean spherical approximation (MSA) treatment of the ion association in aqueous solutions, the linearity of the relative permittivity and of the hydrated cation diameters with the electrolyte concentration was taken into account and a good fit of the experimental activity and osmotic coefficient was obtained [72-75]. The MSA model was elaborated on the basis of cluster expansion considerations involving the direct correlation function the treatment can deal with the many-body interaction term and with a screening parameter and proved expedient for the interpretation of experimental results concerning inorganic electrolyte solutions [67,75-77]. 

Taking the chosen set of radii, very good agreement between theory and experiment was obtained without the need of introducing the concept of ion association for the description of the variation of the conductivity with concentration of an electrolyte solution in the case of 3 different simple ionic species (strong electrolytes). This model provides analytical expressions which are easy to use. However, above the limit of 1 mol/L in total concentration, its validity becomes questionable. A further extension of the theory should involve a modification in the equilibrium model. One possibility would be the use of the HNC model or of other improvements of MSA (softs- MSA, exp- MSA, The problem is then the connection to the low concentration (limiting laws) and the increase in adjustable parameters. Moreover,... 

The DH and MSA theory, that are linear in charge can be considered in the framework of linearized Poisson-Boltzmann (PB) equation. The concept of ion association entails nonlinearity in the treatment of electrostatic interactions by the formulation of appropriate thermodynamic equilibrium constants between free ions and ion clusters [14], In general, this formulation can be considered as the division of ion-ion interaction potentials into an associative part responsible for the ion association, and nonassociative part which is more or less arbitrary. In order to optimize this division in the framework of associative hypernetted chain approximation (AHNC), the division of energy and distance were considered [17] with the parameters calculated from the condition of sta-... 

The results obtained also are useful for the calculation of the ionic conductivity of nonaqueous electrolyte solutions. Several attempts exist for the calculation of the molar conductivity of associating electrolytes beyond the limiting law at the level of the MSA [3, 32, 33], where, however, only ion pairs were taken into account. Ion pairs and tetramers are electrically neutral, nonconducting species in the solution, by contrast to the ion trimers. The total concentration of charged particles is given by,... 

In summary, the models discussed in this chapter focus on the physical aspects of electrolyte solutions but they ignore the chemical aspects. This is especially apparent in the treatment of ion solvation where an empirical correction to the MSA model was applied to treat the differences in behavior seen for cations and anions in water. The same problem arises in using classical electrostatics to describe ion pairing. In spite of the fact that the Bjerrum and Fuoss models give a good qualitative description of an ion association, this phenomenon can only be understood in detail by using quantum-mechanical methods. Needless to say, such calculations in condensed media are much more difficult to carry out. 

Figure 6 shows pair correlation functions from HNC calculations of an aqueous tetraalkyl-ammonium salt solution. In contrast to the MSA calculations, the HNC calculations yield g/y (r) functions that give a realistic picture of the solution structure. It can be inferred that no noticeable cation-anion association takes place, in agreement with chemical model calculations at lower concentrations. Also, no cation-cation association exists, which indicates that no hydrophobic interaction occurs between the large organic ions. Interpenetration of the cations is found, which can be reproduced because of the soft-sphere COR++(r) potential in Eq. (93). 

Subsequently, the model has been extended [37, 38] to the case of associated electrolytes by using a recent model for associating electrolytes[39]. Unlike the classic chemical model of the ion pair the effect of the pairing association is included in the computation of the MSA screening parameter F. Simple formulas for the thermodynamic excess properties have been obtained in terms of this parameter when a new EXP approximation is used. The new formalism based on closures of the Wertheim-Ornstein-Zernike equation (WOZ)[40, 41 does accommodate all association mechanisms (coulombic, covalent and solvation) in one single association parameter, the association constant. The treatment now includes the fraction of particles that are bonded, which is obtained by imposing the chemical equilibrium mass action law. This formalism was shown to be very successful for ionic systems, both in the HNC approximation and MSA [42, 43, 44, 45, 46, 47]. 

Here, we describe the application and typical modelling results for a G model (MSA-NRTL) as well as for an EOS (ePC-SAFT). In addition to strong electrolytes which are almost fully dissociated, we also consider some weak electrolytes (acids like HE or ion-paired electrolytes) that do only partially dissociate in aqueous solution. Here, ion pairing is accounted for by an association/dissociation equilibrium between the ion pair and the respective free ions in solution. 

In this section the results of the associative MSA theory for the ion-dipole model will be reviewed. Moreover, the modified MSA-MAL version for the ion-dipole system will be applied to describe the dielectric constant of electrolyte solutions. 

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