Electrolyte solutions, statistical mechanics

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

It seems appropriate to assume the applicability of equation (A2.1.63) to sufficiently dilute solutions of nonvolatile solutes and, indeed, to electrolyte species. This assumption can be validated by other experimental methods (e.g. by electrochemical measurements) and by statistical mechanical theory. 

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)... 

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 UNIFAC (Unified quasi chemical theory of liquid mixtures Functional-group Activity Coefficients) group-contribution method for the prediction of activity coefficients in non-electrolyte liquid mixtures was first introduced by Fredenslund et al. (1975). It is based on the Unified Quasi Chemical theory of liquid mixtures (UNIQUAC) (Abrams and Prausnitz, 1975), which is a statistical mechanical treatment derived from the quasi chemical lattice model (Guggenheim, 1952). UNIFAC has been extended to polymer solutions by Oishi and Prausnitz (1978) who added a free volume contribution term (UNIFAC-FV) taken from the polymer equation-of-state of Flory (1970). 

Not to be forgotten is the assumption that neither the presence of the electrolyte nor the interface itself changes the dielectric medium properties of the aqueous phase. It is assumed to behave as a dielectric continuum with a constant relative dielectric permittivity equal to the value of the bulk phase. The electrolyte is presumed to be made up of point charges, i.e. ions with no size, and responds to the presence of the charged interface in a competitive way described by statistical mechanics. Counterions are drawn to the surface by electrostatic attraction while thermal fluctuations tend to disperse them into solution, surface co-ions are repelled electrostatically and also tend to be dispersed by thermal motion, but are attracted to the accumulated cluster of counterions found near the surface. The end result of this electrical-thermodynamic conflict is an ion distribution which can be represented (approximately) by a Boltzmann distribution dependent on the average electrostatic potential at an arbitrary point multiplied by the valency of individual species, v/. 

The upper sign corresponds to a water-dielectric , and the lower one to a water-conductor type of interface. Equation (7) shows that a charge located next to a conductor will be attracted by its own image, and dielectrics in aqueous solutions will repel it. For a review of statistical-mechanical models of the double layer near a single interface we refer to [7], and here we would like only to illustrate how the image forces will alter the ion concentration and the electrostatic potential distribution next to a single wall. At a low electrolyte concentration the self-image forces will mostly dominate, and the ion-surface interaction will only be affected by the polarization due... 

The following corrections can be obtained in a similar way, however, one could see that every following step becomes more and more complicated. Moreover, now more advanced computational method are developed, which can utilize a better statistical mechanics while describing correlations in such a system [6,27,28]. The present treatment can be considered as an introduction into the subject, but it gives a better insight of the problem, because of its simplicity and possibility of an analytical solution. One may still use the analysis presented in a system with low electrolyte concentrations. To illustrate results obtained, we will proceed with elaboration based on the zero approximations for the one-body and pair potentials. 

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. 

Arrhenius theory applies well to solutions of weak acids and bases in water, but fails in the case of strong electrolytes such as ordinary salts. Debye and Hiickel [26] solved this problem assuming complete dissociation, but considering the Coulomb interactions between the ions by a patchwork theory based on both macroscopic electrostatics and statistical mechanics. 

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. 

Equation 3.108 predicts a higher local concentration of cations near a negatively charged clay surface than in bulk solution, and a lower concentration of anions near the surface than in solution. Figure 3.24 shows this predicted distribution of monovalent cations and anions near the clay surface for two different concentrations of electrolyte in solution. More modem statistical mechanical models of this clay interfacial region have predicted that ion-ion correlation (electrostatic) effects should cause deviations from this classical picture, such as the positive adsorption of anions at intermediate distances from the surface when the cation is divalent or multivalent. 

Considerable progress has been made in the solution theory of poly-electrolytes. However, for the condensed-phase analogs of polyelectrolytes, ionomers, this is not the case. Eisenberg (1) has put forth an initial theory of ionomer structure that contains conceptual formalisms of general use. His theory has been consulted extensively in the work reported here. Ponomarev and Ionova (2) have attempted to construct a sophisticated statistical mechanical model to describe the thermodynamics of ionomers. Recently, Gierke (3) has described a theory of ion transport in the Nafion ionomer based on a specific molecular organization. 

Modem theories of electrolyte solutions using statistical mechanical ideas are now able to take cognisance of the shapes of ions (see Sections 10.17.3 and 10.19). 

In the Debye-Hilckel model the interactions contributing to the potential energy, 0, are long range coulombic interactions between the ions. However, because of the versatility of the computer simulation calculations, the statistical mechanical description of an electrolyte solution could include all conceivable electrostatic interactions such as the ion-ion, ion-dipole and dipole-dipole, dipole-quadrupole and quadrupole-quadrupole as weU as induced dipole interactions between the ions, and between the ions and the solvent and between solvent molecules. The total potential energy, 0, fed into the calculations which ultimately lead to g( (ri2) could be made up of contributions such as these and those given in Sections... 

D) As indicated in Section 10.17.3, the statistical mechanical approach can be used to describe the behaviour of a solution containing an electrolyte made up of at least one grossly non-spherically symmetrical ion. All shapes of ions can be considered and this represents a vast improvement on the Debye-Hiickel theory. 

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