Electrolyte solutions, statistical

Outhwaite, C.W., 1975, Equilibrium theory of electrolyte solutions "Statistical Mechanics Vol. 2", Specialist Periodical Reports, Chemical Society, London. 

Friedman H L and Dale W T 1977 Electrolyte solutions at equilibrium Statistical Mechanics part A, Equilibrium Techniques ed B J Berne (New York Plenum)... 

According to the statistical thermodynamics of solutions the logarithm of the activity coefficient for an electrolyte solution can be expanded in the series... 

The contribution of short-range forces to the activity coefficient can be described much better and in greater detail by the methods of the statistical thermodynamics of liquids, which has already created several models of electrolyte solutions. However, the procedures employed in the statistical... 

Historically, one of the central research areas in physical chemistry has been the study of transport phenomena in electrolyte solutions. A triumph of nonequilibrium statistical mechanics has been the Debye—Hiickel—Onsager—Falkenhagen theory, where ions are treated as Brownian particles in a continuum dielectric solvent interacting through Cou-lombic forces. Because the ions are under continuous motion, the frictional force on a given ion is proportional to its velocity. The proportionality constant is the friction coefficient and has been intensely studied, both experimentally and theoretically, for almost 100... 

Electrostatic and statistical mechanics theories were used by Debye and Hiickel to deduce an expression for the mean ionic activity (and osmotic) coefficient of a dilute electrolyte solution. Empirical extensions have subsequently been applied to the Debye-Huckel approximation so that the expression remains approximately valid up to molal concentrations of 0.5 m (actually, to ionic strengths of about 0.5 mol L ). The expression that is often used for a solution of a single aqueous 1 1, 2 1, or 1 2 electrolyte is... 

Fig. 6.77. Calculations done using the statistical mechanical theory of electrolyte solutions. Probability density p(6,r) for molecular orientations of water molecules (tetrahedral symmetry) as a function of distance rfrom a neutral surface (distances are given in units of solvent diameter d = 0.28 nm) (a) 60H OH bond orientation and (b) dipolar orientation, (c) Ice-like arrangement found to dominate the liquid structure of water models at uncharged surfaces. The arrows point from oxygen to hydrogen of the same molecule. The peaks at 180 and 70° in p(0OH,r) for the contact layer correspond to the one hydrogen bond directed into the surface and the three directed to the adjacent solvent layer, respectively, in (c). (Reprinted from G. M. Tome and G. N. Patey, ElectrocNm. Acta 36 1677, copyright 1991, Figs. 1 and 2, with permission from Elsevier Science.

During the last two decades, studies on ion solvation and electrolyte solutions have made remarkable progress by the interplay of experiments and theories. Experimentally, X-ray and neutron diffraction methods and sophisticated EXAFS, IR, Raman, NMR and dielectric relaxation spectroscopies have been used successfully to obtain structural and/or dynamic information about ion-solvent and ion-ion interactions. Theoretically, microscopic or molecular approaches to the study of ion solvation and electrolyte solutions were made by Monte Carlo and molecular dynamics calculations/simulations, as well as by improved statistical mechanics treatments. Some topics that are essential to this book, are included in this chapter. For more details of recent progress, see Ref. [1]. 

In 1923, Peter Debye and Erich Hiickel developed a classical electrostatic theory of ionic distributions in dilute electrolyte solutions [P. Debye and E. Hiickel. Phys. Z 24, 185 (1923)] that seems to account satisfactorily for the qualitative low-ra nonideality shown in Fig. 8.3. Although this theory involves some background in statistical mechanics and electrostatics that is not assumed elsewhere in this book, we briefly sketch the physical assumptions and mathematical techniques leading to the Debye-Hiickel equation (8.69) to illustrate such molecular-level description of thermodynamic relationships. 

The three appendices in this volume give selected sets of thermodynamic data (Appendix 5), review the statistical calculations covered in Principles and Applications (Appendix 6), and summarize the equations and parameters required to calculate the properties of electrolyte solutions, principally from Pitzer s equations (Appendix 7). 

A second example is provided by a semiempirical correlation for multi-component activity coefficients in aqueous electrolyte solutions shown in Fig. 2. This correlation, developed by Fritz Meissner at MIT [3], presents a method for scale-up activity-coefficient data for single-salt solutions, which are plentiful, are used to predict activity coefficients for multisalt solutions for which experimental data are rare. The scale-up is guided by an extended Debye-Hilckel theory, but essentially it is based on enlightened empiricism. Meissner s method provides useful estimates of thermodynamic properties needed for process design of multieffect evaporators to produce salts from multicomponent brines. It will be many years before sophisticated statistical mechanical techniques can perform a similar scale-up calculation. Until then, correlations such as Meissner s will be required in a conventional industry that produces vast amounts of inexpensive commodity chemicals. 

When treating the association of counterions one may also apply the association statistics (AS) model which is equivalent to the Bjerrum theory for ion pairing in an electrolyte solution [29,30]. However, in the case of surface association spaee is available only on the side of the liquid. Another difference is due to the critical distance which depends on direction and is a function of the surface potential. This theory explains why two ionic species may associate at the surface despite the fact that they do not undergo ion pairing in the bulk of solution. According to the Bjerrum theory, ions of large effective size cannot approach the critical distance and such an electrolyte is completely dissociated. At the surface the critical distance extends by increasing surface potential and once the surface potential exceeds the critical value, at which the critical distance matches the minimum separation, association at the interface proceeds. 

In the following chapters, some of the theories introduced here are used to discuss other systems, including polar solvents and electrolyte solutions. The statistical mechanical tools introduced here are important because they help one to develop an understanding of the way that molecular properties of a given system influence its macroscopic properties. 

In many cases, it is more convenient to use activity coefficients on the molarity scale. Not only is molarity more commonly used as a concentration unit in chemistry but values of are more directly related to the results of statistical mechanical theories of electrolyte solutions discussed later in this chapter. For a given molality, m, one must calculate the corresponding molarity, c, using the relationship... 

The theory of electrolyte solutions developed in this chapter relies heavily on the classical laws of electrostatics within the context of modern statistical mechanical methods. On the basis of Debye-Hiickel theory one understands how ion-ion interactions lead to the non-ideality of electrolyte solutions. Moreover, one is able to account quantitatively for the non-ideality when the solution is sufficiently dilute. This is precisely because ion-ion interactions are long range, and the ions can be treated as classical point charges when they are far apart. As the concentration of ions increases, their finite size becomes important and they are then described as point charges within hard spheres. It is only when ions come into contact that the problems with this picture become apparent. At this point one needs to add quantum-mechanical details to the description of the solution so that phenomena such as ion pairing can be understood in detail. 

In summary, diffraction techniques provide a powerful means of investigating the structure of electrolyte solutions. They give information about the pair correlation functions which can be directly related to modern theoretical techniques such as molecular dynamics calculations. This information can also be used to improve the statistical thermodynamic models of electrolyte solutions discussed in chapter 3. 

The first section of this book covers liquids and. solutions at equilibrium. I he subjects discussed Include the thcrmodvnamics of solutions, the structure of liquids, electrolyte solutions, polar solvents, and the spectroscopy of solvation. The next section deals with non-equilibrium properties of solutions and the kinetics of reactions in solutions. In the final section emphasis is placed on fast reactions in solution and femtochemistry. The final three chapters involve important aspects of solutions at interfaces. Fhese include liquids and solutions at interfaces, electrochemical equilibria, and the electrical double layer. Author W. Ronald Fawcett offers sample problems at the end of every chapter. The book contains introductions to thermodynamics, statistical thermodynamics, and chemical kinetics, and the material is arranged in such a way that It may be presented at different levels. Liquids, Solutions, and Interfaces is suitable for senior undergr.iduates and graduate students and will be of interest to analytical chemists, physical chemists, biochemists, and chemical environmental engineers. 

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