Debye-Hiickel methods

In 1923, Debye and Hiickel published their famous papers describing a method for calculating activity coefficients in electrolyte solutions. They assumed that ions behave as spheres with charges located at their center points. The ions interact with each other by coulombic forces. Robinson and Stokes (1968) present their derivation, and the papers are available (Interscience Publishers, 1954) in English translation. 

The result of their analysis, known as the Debye-Huckel equation, 

Variable di in Equation 8.2 is the ion size parameter. In practice, this value is determined by fitting the Debye-Huckel equation to experimental data. Variables A and B are functions of temperature, and I is the solution ionic strength. At 25 °C, given I in molal units and taking a, in A, the value of A is 0.5092, and B is 0.3283. 

Equation 8.2 is notable in that it predicts a species activity coefficient using only two numbers (z/ and di) to account for the species properties and a single value 

I to represent the solution. As such, it can be applied readily to study a variety of geochemical systems, simple and complex. 

1 Activity coefficients y predicted at 25°C for a singly charged ion with size a of 4 A, according to the Debye-Hiickel (Eqn. 7.2), Davies (Eqn. 7.4), and B-dot (Eqn. 7.5) equations. Dotted line shows the Davies equation evaluated with a coefficient of 0.2 instead of 0.3. 

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Can the species activity coefficients be calculated accurately An activity coefficient relates each dissolved species concentration to its activity. Most commonly, a modeler uses an extended form of the Debye-Hiickel equation to estimate values for the coefficients. Helgeson (1969) correlated the activity coefficients to this equation for dominantly NaCl solutions having concentrations up to 3 molal. The resulting equations are probably reliable for electrolyte solutions of general composition (i.e., those dominated by salts other than NaCl) where ionic strength is less than about 1 molal (Wolery, 1983 see Chapter 8). Calculated activity coefficients are less reliable in more concentrated solutions. As an alternative to the Debye-Hiickel method, the modeler can use virial equations (the Pitzer equations ) designed to predict activity coefficients for electrolyte brines. These equations have their own limitations, however, as discussed in Chapter 8. 

The Debye-Hiickel methods work poorly, however, when carried to moderate or high ionic strength, especially when salts other than NaCl dominate the solute. In the theory, the ionic strength represents all the properties of a solution. For this reason, a Debye-Hiickel method applied to any solution of a certain ionic strength (whether dominated by NaCl, KC1, HC1, H2SO4, or any salt or salt mixture) gives the same set of activity coefficients, irregardless of the solution s composition. This result, except for dilute solutions, is, of course, incorrect. Clearly, we cannot rely... 

Unlike the Debye-Hiickel equations, the virial methods provide little or no information about the distribution of species in solution. Geochemists like to identify the dominant species in solution in order to write the reactions that control a system s behavior. In the virial methods, this information is hidden within the complexities of the virial equations and coefficients. Many geochemists, therefore, find the virial methods to be less satisfying than methods that predict the species distribution. The information given by Debye-Hiickel methods about species distributions in concentrated solutions, however, is not necessarily reliable and should be used with caution. 

Stochastic aggregation does not emerge for oppositely charged particles, when electroneutrality holds due to conditions nk(t) = nB(f) = n(f), particle charge ea = — eB = e. Let us introduce, following the Debye-Hiickel method, the self-consistent potential (J> through Poisson equation... 

Figure 1.2. The change in the value of the ion activity coefficient as a function of ionic strength for -1 and +2 ions calculated from the Davies equation, and HC03 and Ca2+ calculated using the Debye-Hiickel method.

Since the Debye-Hiickel method is limited to very dilute solutions, experimental methods must be invoked to find F for m > 10 molal. The derivations are largely patterned after Section 2.8. The use of emf methods for the same purpose is briefly dealt with below Eq. (4.7.2). Once again, the choice of P = 1 bar serves as the standard pressure in this case the activity coefficient F, introduced in Section 3.3 is appropriate. When molalities are used as concentration units the expression f serves the purpose. 

In the non-linear differential equation Eq. (43), k is related to the inverse Debye-Hiickel length. The method briefly outlined above is implemented, e.g., in the pro-... 

We have problems when we attempt to repeat this calculation in 0.010/77 KC1 and 0.010m KNO, since 7- is a function of the total concentration of ions, and, as we saw in Chapter 7, at m = 0.010 mol-kg-1, 7 differs significantly from one. Debye-Hiickel theory provides a method for calculating 7 for dilute solutions. 

For concentrated solutions, there are approaches that are more sophisticated than that of Debye Hiickel. A particularly successful method of describing such solutions is that due to McMillan Mayer (1945) which has subsequently been developed by Ramanathan ... 

Geochemical modelers currently employ two types of methods to estimate activity coefficients (Plummer, 1992 Wolery, 1992b). The first type consists of applying variants of the Debye-Hiickel equation, a simple relationship that treats a species activity coefficient as a function of the species size and the solution s ionic strength. Methods of this type take into account the distribution of species in solution and are easy to use, but can be applied with accuracy to modeling only relatively dilute fluids. 

It is interesting to compare the Debye-Hiickel and virial methods, since each has its own advantages and limitations. The Debye-Hiickel equations are simple to apply and readily extensible to include new species in solution, since they require few coefficients specific to either species or solution. The method can be applied as well over the range of temperatures most important to an aqueous geochemist. There is an extensive literature on ion association reactions, so there are few limits to the complexity of the solutions that can be modeled. 

Figure 8.8 shows the resulting saturation indices for halite and anhydrite, calculated for the first four samples in Table 8.8. The Debye-Hiickel (B-dot) method, which of course is not intended to be used to model saline fluids, predicts that the minerals are significantly undersaturated in the brine samples. The Harvie-Mpller-Weare model, on the other hand, predicts that halite and anhydrite are near equilibrium with the brine, as we would expect. As usual, we cannot determine whether the remaining discrepancies result from the analytical error, error in the activity model, or error from other sources. 

At least a partial solution to this problem is attained by the conventional activity scale method [5, 6, 7, 9, 10, 11]. This procedure was first used by Bates and Guggenheim [8] when formulating the operational definition of pH (see [86a], chapter 1), on the basis of which the National Bureau of Standards in the USA developed a method for determining conventional hydrogen ion activities. The basic assumption is the use of the Debye-Hiickel relationship for the individual activity of chloride ions ... 

The Debye-Hiickel term, which is the dominant term in the expression for the activity coefficients in dilute solution, accounts for electrostatic, nonspecific long-range interactions. At higher concentrations, short-range, nonelectrostatic interactions have to be taken into account. This is usually done by adding ionic strength dependent terms to the Debye-Hiickel expression. This method was first outlined by Bronsted [5,6], and elaborated... 

The other method uses an extended Debye-Hiickel expression where the activity coefficients of reactants and products depend only on the... 

Figure 8JO Comparison between individual ionic activity coefficients obtained with Debye-Hiickel equation and with mean salt method for various ionic strength values. Reprinted from Garrels and Christ (1965), with kind permission from Jones and Bartlett Publishers Inc., copyright 1990.

For studying the stability of colloidal particles in suspension (Chapter 13) or for determining the potential at the surface of particles (Chapter 12), one often needs expressions for potential distributions around small particles that have curved surfaces. Solving the Poisson-Boltzmann equation for curved geometries is not a simple matter, and one often needs elaborate numerical methods. The linearized Poisson-Boltzmann equation (i.e., the Poisson-Boltzmann equation in the Debye-Hiickel approximation) can, however, be solved for spherical electrical double layers relatively easily (see Section 12.3a), and one obtains, in place of Equation (37),... 

The solution of the linearized Poisson-Boltzmann equation around cylinders also requires numerical methods, although when cylindrical symmetry and the Debye-Hiickel approximation are assumed the equation can be solved. The solution, however, requires advanced mathematical techniques and we will not discuss it here. It is nevertheless useful to note the form of the solution. The potential for symmetrical electrolytes has been given by Dube (1943) and is written in terms of the charge density a as... 

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. 

Chapter 18 describes electrolyte solutions that are too concentrated for the Debye-Hiickel theory to apply. Gugenheim s equations are presented and the Pitzer and Brewer tabulations, as a method for obtaining the thermodynamic properties of electrolyte solutions, are described. Next, the complete set of Pitzer s equations from which all the thermodynamic properties can be calculated, are presented. This discussion ends with an example of the extension of Pitzer s equations to high temperatures and high pressures. Three-dimensional figures show the change in the thermo-... 

Although there is no straightforward and convenient method for evaluating activity coefficients for individual ions, the Debye-Hiickel relationship permits an evaluation of the mean activity coefficient (y+), for ions at low concentrations (usually <0.01 moll-1) ... 

In the following we outline the method of Ref. [18] which attempts to retain the simplicity of the PB theory but also accommodates correlation effects within a local density approximation (LDA) where all the relevant interactions are included at the level of the free energy density. One starts out with the free energy density of the PB approach and adds an appropriate correlational correction to the mean-field free energy density. One attempt at the level of the Debye-Hiickel theory (DH) is called DH plus Hole (DHH)... 

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