Derivation of the Debye-Hiickel Equation

Fig. 59 this is particularly true if both the saturating salt and the added electrolyte are of high valence types. The deviations are often due to incomplete dissociation, and also to the approximations made in the derivation of the Debye-Hiickel equations as already seen, both these factors become of importance with ions of high valence. 

Mathematical Proof of the Caratheodory Theorem and Resulting Interpretations Derivation of the Debye-Hiickel Equation... 

We present here a simplified derivation of the Debye-Hiickel equation as specified by Eq. (4.2.2a). The presentation is adapted from Ref. 9.5.1 for more advanced treatments that provide better insights on the various approximations inherent in the Debye-Hiickel theory the reader may consult standard references. The present exposition is also of intrinsic interest, as will be detailed below. 

The present volume involves several alterations in the presentation of thermodynamic topics covered in the previous editions. Obviously, it is not a trivial exercise to present in a novel fashion any material that covers a period of more than 160 years. However, as best as I can determine the treatment of irreversible phenomena in Sections 1.13, 1.14, and 1.20 appears not to be widely known. Following much indecision, and with encouragement by the editors, I have dropped the various exercises requiring numerical evaluation of formulae developed in the text. After much thought I have also relegated the Caratheodory formulation of the Second Law of Thermodynamics (and a derivation of the Debye-Hiickel equation) as a separate chapter to the end of the book. This permitted me to concentrate on a simpler exposition that directly links entropy to the reversible transfer of heat. It also provides a neat parallelism with the First Law that directly connects energy to work performance in an adiabatic process. A more careful discussion of the basic mechanism that forces electrochemical phenomena has been provided. I have also added material on the effects of curved interfaces and self assembly, and presented a more systematic formulation of the basics of irreversible processes. A discussion of critical phenomena is now included as a separate chapter. Lastly, the treatment of binary solutions has been expanded to deal with asymmetric properties of such systems. 

The derivation of the Debye-Hiickel equations is not included here, but only the end results. A complete discussion is given by Bull (1964), Chapter 3. There are, however, several basic phenomena that we need to examine. There are three mechanisms which produce ions in water solutions. These are (1) solution of an ionic crystal, (2) oxidation of a metal or reduction of a nonmetal, and (3) ionization of a neutral molecule. Most metals when they ionize give up electrons to an electronegative element so that both acquire the electronic structure of a rare gas. Exceptions of interest in electrode work are iron, copper, silver, mercury, and zinc. In the ionized state, these metals do not acquire the completed outer shell structure of a rare gas and do have residual valences. They are then somewhat unstable and complex with various molecules more easily than do the stable ionized metals with completed shells. This accounts for the poisoning of silver-silver chloride electrodes and p02-measuring electrodes when used in high-protein environments such as blood. 

In this introductory foray into the subject, we present a simplified derivation of the Debye-Hiickel equation, guided by the exposition of Ref. 1, which is also of intrinsic interest. References to a more complete exposition are listed in Ref. 2. 

The derivation of the Debye— Hiickel equation for the activity coefficient is based on the linearized Boltzmann equation for electrostatic charge distribution around an ion. This limits the applicability of Eq. (57) to solutes with low surface potentials, which occurs for solution concentrations of monovalent ions of < 0.01 M. However, it is important to note that die method used for deriving activity coefficient equation (25) is based on rigorous thermodynamics and is not limited by the Debye—Huckel theory. If, for example, the Gouy—Chapman equation [22] was... 

A more rigid but laborious method, for deriving transference numbers from E.M.p. data, makes use of the fact that the activity coefficient of an electrolyte can be expressed, by means of an extended form of the Debye-Hiickel equation, as a function of the concentration and of two empirical constants.When applied to the same data, however, this procedure gives results which are somewhat different from those obtained by the method just described. Since the values are in better agreement with the transference data derived frorq moving boundary and other measurements, they are probably more reliable. 

Equation 6-33 suggests that extrapolation of equilibrium constants to infinite dilution is done appropriately by plotting log vs Yh- For example. Fig. 6-1 shows plots of pfC a for dissociation of H2PO4, AMP , and ADP , and ATP vs Yh- The variation of pK with yjl at low concentrations (Eq. 6-35) is derived by application of the Debye-Hiickel equation (Eq. 6-33) ... 

In actual experiments, as indicated above, ionization quotients Q are usually measured in a solution at finite ionic strength made up by the addition of supporting electrolytes such as NaCl, KCl, or NaCFsSOs. Therefore, activity coefficient models are needed to extrapolate the Q values to infinite dilution for such equilibria. All of these models are based on some version of the Debye-Hiickel equation, which determines the initial slope the logio0 versus ionic strength dependence, with additional empirical ionic strength terms which are typically derived from those used in the Pitzer ion interaction model (Pitzer, 1991). An example of this empirical approach is given in Equation (3.29). 

Equation (7.44) is known as the third approximation of the Debye-Hiickel theory. Numerous attempts have been made to interpret it theoretically, hi these attempts, either individual simplifying assumptions that had been made in deriving the equations are dropped or additional factors are included. The inclusion of ionic solvation proved to be the most important point. In concentrated solutions, solvation leads to binding of a significant fraction of the solvent molecules. Hence, certain parameters may change when solvation is taken into account since solvation diminishes the number of free solvent molecules (not bonded to the ions). The influence of these and some other factors was analyzed in 1948 by Robert A. Robinson and Robert H. Stokes. 

The activity a2 of an electrolyte can be derived from the difference in behavior of real solutions and ideal solutions. For this purpose measurements are made of electromotive forces of cells, depression of freezing points, elevation of boiling points, solubility of electrolytes in mixed solutions and other characteristic properties of solutions. From the value of a2 thus determined the mean activity a+ is calculated using the equation (V-38) whereupon by application of the analytical concentration the activity coefficient is finally determined. The activity coefficients for sufficiently diluted solutions can also be calculated directly on the basis of the Debye-Hiickel theory, which will bo explained later on. 

It will be seen later (p. 230) that there does not appear to be any experimental method of evaluating the activity coefficient of a single ionic species, so that the Debye-Hiickel equations cannot be tested in the forms given above. It is possible, however, to derive very readily an expression for the mean activity coefficient, this being the quantity that is obtained experimentally. The mean activity coefficient f of an electrolyte is defined by an equation analogous to (30), and... 

The product of the activity coefficients can be estimated from the Debye-Hiickel equations, and mcr and E are known hence nin in the given solution can be derived from the measured e.m.f. of the cell. 

Derivation of the Limiting Form for the Debye-Hiickel Equation 9.5.1 Fundamentals... 

Attention may be called to the fact that equation (44.41) can be derived from the Debye-Hiickel treatment in an alternative manner, which is based on the same fundamental principles as that just described. In 40a the deviation from ideal behavior, as represented by the activity coefficient, was attributed to the interaction of the ions, and the heat of dilution to infinite dilution may be ascribed to the same cause. The quantity AHc o can thus be identified with — He, corresponding to Foi. evaluated in 40c these quantities are then related by a form of the Qibbs-Helmholtz equation [cf. (25.28)3,... 

In 1923, P. Debye and E. Hiickel used tbe ionic atmosphere model, described in Section lOA-3, to derive an equation that permits the calculation of activity coefficients of ions from their charge and their average size. This equation, which has become known as the Debye-Hiickel equation, takes the form... 

If the surface potential is small, we can expand in series the logarithm in the right-hand side of Equation 5.343 and derive the Debye-Hiickel equation ... 

There is no detail of the derivation of the equations of the Debye-Hiickel theory that has not been criticized. Its incompleteness mathematically is evident, since only the first term of the expansion of equations (3) and (29) is used. The extensions of the theory to overcome this deficiency are, however, briefly considered below. A possibly more serious deficiency of the theory as given is that it does not take account of "fluctuation terms. This amounts to the statement that the Boltzmann equation does not yield a correct average potential this being subject to wide variations for which allowance should he made in the theory. This part of the criticism is still in active progress, and cannot be briefly summarized. In Chapters 8 and 12 an attempt will be made to show both the successes and the limitations of the theory in its present state from the experimental point of view. 

When the Debye-Hiickel equation is tested against experimental results it is very successful in accounting for behaviour at low concentrations, and it is believed that the theory is basically correct for low concentrations (see Section 10.10). Having to test the theory rigorously at very low concentrations proved a great stimulus in developing precision techniques for deriving experimental values of y. . 

In the extension of the Debye-Hiickel theory by Onsager an equation similar to Kohlrausch s law was derived... 

Coimterion condensation has detractors (28-34), who point to flaws in the concept s derivation, such as artificial subdivision of the counterions into two populations, inappropriate extrapolation of the Debye-Hiickel approximation to regions of high electrostatic potential, and inconsistent treatment of counterions. The full nonlinear Poisson-Boltzmann equation offers a more rigorous way to interpret electrostatic phenomena in electrolyte solutions, but the physical picture obtained through this equation is different in some ways from the one suggested by condensation (21,34,35). In particular, a Poisson-Boltzmann analysis does not readily identify distinct populations of condensed and free counterions but rather a smoothly varying Gouy-Chapman layer. Nevertheless, Poisson-Boltzmann-based... 

The above analysis provides a simple route to the measurement of transference numbers from readily measurable quantities. The method may be applied to a wide range of electrolyte solutions (hence a large working range of electrolyte concentrations). The drawback is the rather severe set of approximations made in the derivation of equation (20.1.2-10) which limits the accuracy of the transference numbers derived. More accurate results can be obtained by considering the non-ideality of the electrolytes, to a first approximation by use of the Debye-Hiickel limiting law. The fact that the potentials respond to the logarithm of the concentration ratio also reduces the accuracy of this method. 

The magnitude of the kinetic salt effect for a bimolecular reaction depends strongly on the charges on the reacting species. Equation 2.30, derived from the Debye-Hiickel theory, has been shown to represent most experimental data well (Atkins, 1998, p. 836). 

The ideal partial molar enthalpy equation (47) was derived from Eq. (45) under the constraint that in an ideal solution, k = 0. The interpretation of this Ar con-straint is that the thickness of the Debye—Hiickel diffuse ion atmosphere, has become infinite, indicative of the absence of electrostatic screening among solute ions. While K is required to be zero for ideal solutions and real solutions as they approach... 

The Debye-Hiickel equation derives from a combination of the Poisson equation and a statistical-mechanical distribution formula (Debye and Hiickel, 1923). The Poisson equation is a general expression of the Coulomb law of force between charged bodies and can be written as... 

APPENDIX A Derivation of the Main Equation of Debye-Hiickel Theory... 

A new theory of electrolyte solutions is described. This theory is based on a Debye-Hiickel model and modified to allow for the mutual polarization of ions. From a general solution of the linearized Poisson-Boltzmann equation, an expression is derived for the activity coefficient of a central polarized ion in an ionic atmosphere of non-spherical symmetry that reduces to the Debye-Hiickel limiting laws at infinite dilution. A method for the simultaneous charging of an ion and its ionic cloud is developed to allow for ionic polarization. Comparison of the calculated activity coefficients with experimental values shows that the characteristic shapes of the log y vs. concentration curves are well represented by the theory up to moderately high concentrations. Some consequences in relation to the structure of electrolyte solutions are discussed. 

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