Debye-Hiickel theory statistical mechanical

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. 

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. 

The Debye-Hiickel theory is a study of the equilibrium properties of electrolyte solutions, where departures from ideal behaviour are considered to be a result of coulombic interactions between ions in an equilibrium situation. It is for this reason that equilibrium statistical mechanics can be used to calculate an equilibrium Maxwell-Boltzmann distribution of ions. 

Debye and Hiickel derived Eq. 10.4.1 using a combination of electrostatic theory, statistical mechanical theory, and thermodynamics. This section gives a brief outline of their derivation. 

For some reactions use can also be made of statistical-mechanical equations either (rarely) alone or in combination with some of the quantities alluded to above. These are reactions taking place in systems for which we have a model which is at once realistic enough and mathematically tractable enough to be useful. An example is the calculation of the standard equilibrium constant of a gas reaction (and thence of the yield, but only if the gas mixture is nearly enough perfect) from spectroscopically determined molecular properties. Another example is the use of the Debye-Hiickel theory or its extensions to improve the calculation of the yield of a reaction in a dilute electrolyte solution from the standard equilibrium constant of the reaction when, as is usually so, it is not accurate enough to assume that the solution is ideal-dilute. 

Kirkwood, J. G. Poirier, J. C. (1954) The Statistical Mechanical Basis of the Debye-Hiickel Theory of Strong Electrolytes. J. Phys. Chem. 58, 8, 591-5%, ISSN 0022-3654 Kjellander, R Mitchell, D.J. (1992) An exact but linear and Poisson—Boltzmann-like theory for electrolytes and coUoid dispersions in the primitive model. Chem. Phys. Lett. 200,1-2, 76-82, ISSN 0009-2614... 

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... 

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. 

Another approach to the conductance of electrolytes, which is less complex than that of Lee and Wheaton, is due to Blum and his co-workers. This theory goes back to the original Debye-Hiickel-Onsager concepts, for it does not embrace the ideas of Lee and Wheaton about the detailed structure around the ion. Instead, it uses the concept of mean spherical approximation of statistical mechanics. This is the rather portentous phrase used for a simple idea, which was fully described in Section 3.12. It is easy to see that this is an approximation because in reality an ionic collision with another ion will be softer than the brick-wall sort of idea used in an MSA approach. However, using MSA, the resulting mathematical treatment turns out to be relatively simple. The principal equation from the theory of Blumet al. is correspondingly simple and can be quoted. It runs... 

In discussing the theory of Debye and Hiickel (1923) we shall defer a rather lengthy derivation of the final results to the end of Chapter 9, so as not to interrupt the flow of the underlying concepts. One may even adopt the view that the results listed below represent excellent limiting laws that are known to represent a large body of experimental data. However, this obscures the fact that an elementary exposition of the theory is presented in Section 9.5 without the use of statistical mechanics and electrodynamics, though a more detailed derivation is needed for a proper understanding of the model. We now proceed without proof. 

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. 

The first chapter of the book sets the stage for many of the topics dealt with later, and, in particular, is a prelude to the development of the two major theoretical topics described in the book, namely the theory of non-ideality and conductance theory. The conventional giants of these fields are Debye and Hiickel with their theory of non-ideality and Debye, Huckel, Fuoss and Onsager with their various conductance equations. These topics are dealt with in Chapters 10 and 12. In addition, the author has included for both topics a qualitative account of modern work in these fields. There is much exciting work being done at present in these fields, especially in the use of statistical mechanics and computer simulations for the theory of nonideality. Likewise some of the advances in conductance theory are indicated. 

A theory of ionic solutions developed by Peter Debye and Erich Hiickel in 1923 (which is based on statistical mechanics and beyond the scope of this text) provides an expression for the activity. We shall only state the main result of this theory, which works well for dilute electrolytes. The activity depends on a quantity called the ionic strength /, defined by... 

The replacement of the potential of mean force with the mean electrostatic potential by Debye and Hiickel (and implicit in the Gouy-Chapman approach) has caused the greatest amount of concern for those applying the PB equation. Fowler severely criticized use of the PB equation on this basis, but his investigation was soon shown to be overly restrictive.Still, the effect of neglecting ion-ion correlation, which this mean-field approximation implies, is a continual source of study. Hence there have been published numerous comparisons between PB theory and more detailed statistical-mechanical theories or calculations that do include correlation. While the size of the effect depends on the particular system studied, calculations on the cylindrical and all-atom models of DNA show that PB calculations tend to underestimate ion concentrations at the surface by 15-25% for mono- or divalent ions, respec-tively. " "- ... 

Integral Equation and Eield-Theoretic Approaches In addition to theories based on the direct analytical extension of the PB or DH equation, PB results are often compared with statistical-mechanical approaches based on integral equation or density functional methods. We mention only a few of the most recent theoretical developments. Among the more popular are the mean spherical approximation (MSA) and the hyper-netted chain (HNC) equation. Kjellander and Marcelja have developed an anisotropic HNC approximation that treats the double layer near a flat charged surface as a series of discrete layers.Attard, Mitchell and Ninham have used a Debye-Hiickel closure for the direct correlation function to obtain an analytical extension (in terms of elliptic integrals) to the PB equation for the planar double layer. Both of these approaches, which do not include finite volume corrections, treat the fluctuation potential in a manner similar to the MPB theory of Outhwaite. 

Recently, Manning [19] has developed a theory in which the real polyelectrolyte chain is replaced by an infinite line charge. Statistical mechanical considerations lead to the conclusion that sufficiently many counter-ions have to condense on the polyion to lower the charge-density parameter to unity. The non-condensed counterions may by treated by the Debye-Hiickel approximation. Thus, a limiting law has been developed which is supposed to be valid at high dilution. For the osmotic coefficient this limiting law is... 

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