Debye-Hiickel relationship

In order to solve the equations completely, the mean activity coefficients f of the ions are required. If one asstunes the extended Debye-Hiickel relationship for the concentration-dependence of the mean activity coefficients... 

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 Gibbs phase rule is the basis for organizing the models. In general, the number of independent variables (degrees of freedom) is equal to the number of variables minus the number of independent relationships. For each unique phase equilibria, we may write one independent relationship. In addition to this (with no other special stipulations), we may write one additional independent relationship to maintain electroneutrality. Table I summarizes the chemical constituents considered as variables in this study and by means of chemical reactions depicts independent relationships. (Throughout the paper, activity coefficients are calculated by the Debye-Hiickel relationship). Since there are no data available on pressure dependence, pressure is considered a constant at 1 atm. Sulfate and chloride are not considered variables because little specific data concerning their equilibria are available. Sulfate may be involved in a redox reaction with iron sulfides (e.g., hydrotroilite), and/or it may be in equilibrium with barite (BaS04) or some solid solution combinations. Chloride may reach no simple chemical equilibrium with respect to a phase. Therefore, these two ions are considered only to the... 

The standard state is defined as the hypothetical state that would exist if the solutewere at a concentration oTTMTbut with the molecules experiencing the environment of an extremely dilute solution with this standard state, activity coefficients approach unity with increasing dilution. For electrolytes in dilute solution in water, the departure of the coefficients from unity can be calculated from the Debye-Hiickel relationship.8... 

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) ... 

However even monoanions differ in their inhibitory effects, as shown in Figure 16. Thus in Figure 16 data obtained for six inorganic anions are plotted according to the Debye-Hiickel relationship which, incidentally, does not seem to pertain except perhaps in the cases of phosphate and fluoride. However reactivities decline with higher concentrations of... 

Based on agreement between calculated and experimental values of mean ionic activity coefficients, we can infer that the Debye-Hiickel relationship and the data in Table 10-2 give satisfactory activity coefficients for ionic strengths up to about 0.1 M. Beyond this value, the equation fails, and we must determine mean activity coefficients experimentally. 

Formal molal-scale values are approximated here by molar-scale values calculated by using molarities in place of molalities, m, in the Debye-Hiickel relationship, 0.5 z w. 

The experimental data are interpreted through the average number of fluorides per thorium atom. The concentration-dependent stability constants thus calculated were converted to constants at zero ionic strength through the modified Debye-Hiickel relationship for the experimental temperatures. These constants were then used to calculate other thermodynamic parameters through the use of van t Hoff relationship assuming that Aj/f ° (A.60) is independent of temperature. The log, (A.60) values for ... 

As explained in section 3.6.1, many modifications have been proposed for the Debye-Hiickel relationship for estimating the mean ionic activity coefficient 7 of an electrolyte in solution and the Davies equation (equation 3.35) was identified as one of the most reliable for concentrations up to about 0.2 molar. More complex modifications of the Debye-Huckel equation (Robinson and Stokes, 1970) can greatly extend the range of 7 estimation, and the Bromley (1973) equation appears to be effective up to about 6 molar. The difficulty with all these extended equations, however, is the need for a large number of interacting parameters to be taken into account for which reliable data are not always available. 

Chapters 7 to 9 apply the thermodynamic relationships to mixtures, to phase equilibria, and to chemical equilibrium. In Chapter 7, both nonelectrolyte and electrolyte solutions are described, including the properties of ideal mixtures. The Debye-Hiickel theory is developed and applied to the electrolyte solutions. Thermal properties and osmotic pressure are also described. In Chapter 8, the principles of phase equilibria of pure substances and of mixtures are presented. The phase rule, Clapeyron equation, and phase diagrams are used extensively in the description of representative systems. Chapter 9 uses thermodynamics to describe chemical equilibrium. The equilibrium constant and its relationship to pressure, temperature, and activity is developed, as are the basic equations that apply to electrochemical cells. Examples are given that demonstrate the use of thermodynamics in predicting equilibrium conditions and cell voltages. 

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. 

When one takes into account the effects of interaction between the polar head groups using similar degree of the approximation as in the Debye-Hiickel theory, the following relationship results [38] ... 

It is shown that the properties of fully ionized aqueous electrolyte systems can be represented by relatively simple equations over wide ranges of composition. There are only a few systems for which data are available over the full range to fused salt. A simple equation commonly used for nonelectrolytes fits the measured vapor pressure of water reasonably well and further refinements are clearly possible. Over the somewhat more limited composition range up to saturation of typical salts such as NaCl, the equations representing thermodynamic properties with a Debye-Hiickel term plus second and third virial coefficients are very successful and these coefficients are known for nearly 300 electrolytes at room temperature. These same equations effectively predict the properties of mixed electrolytes. A stringent test is offered by the calculation of the solubility relationships of the system Na-K-Mg-Ca-Cl-SO - O and the calculated results of Harvie and Weare show excellent agreement with experiment. 

The Debye Hiickel Limiting Law. Although beyond the scope of this Handbook, the derivation of this quantitative relationship rests on the following simplifying assumptions (a) Electrolytes are assumed to be completely... 

In solution thermodynamics, the concentration (C) of ions is replaced by their activity, a, where a = Cy and y is the activity coefficient that takes into account nonideal behavior due to ion-solvent and ion-ion interactions. The Debye-Hiickel limiting law predicts the relationship between the ionic strength of a solution and y for an ion of charge Z in dilute solutions ... 

Using the Debye-Hiickel limiting law for the relationship between the activity coefficients y and the ionic strength of the solution, one finds... 

This same relationship is the starting point of the Debye-Hiickel theory of electrolyte nonideality, except that the Debye-Hiickel theory uses the value of V2 p required for spherical symmetry. It is interesting to note that Gouy (in 1910) and Chapman (in 1913) applied this relationship to the diffuse double layer a decade before the Debye-Hiickel theory appeared. 

Equation (45) provides a relationship between the surface charge density and the slope of the potential at the surface. Next, we turn to Equation (37) —the Debye-Hiickel approximation for p — to evaluate (dip/dx)0. Differentiation leads to the value... 

The Debye-Hiickel approximation to the diffuse double-layer problem produces a number of relatively simple equations that introduce a variety of double-layer topics as well as a number of qualitative generalizations. In order to extend the range of the quantitative relationships, however, it is necessary to return to the Poisson-Boltzmann equation and the unrestricted Gouy-Chapman theory, which we do in Section 11.6. 

Equation (62) describes the variation in potential with distance from the surface for a diffuse double layer without the simplifying assumption of low potentials. It is obviously far less easy to gain a feeling for this relationship than for the low-potential case. Anticipation of this fact is why so much attention was devoted to the Debye-Hiickel approximation in the first place. Note that Equation (62) may be written... 

Remember that Table 11.3 contains some useful numerical values of k at different concentrations of various electrolytes. Equation (10) is the basic relationship of the Debye-Hiickel theory and may be integrated as follows. The variable x is introduced with the following definition ... 

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. 

A theoretical approach for explaining the relationship between S and the characteristics of the electrolyte was provided by Onsager on the basis of the model of ions plus ionic cloud developed in the Debye-Hiickel theory, obtaining [4]... 

Correlations for the determination of the dissociation equilibrium constants and solubility values for SO2 and CO2 as functions of temperature as well as the equations for activity coefficients are given in Ref. [70], Thermodynamic non-idealities are taken into account depending on whether species are charged, or not. For uncharged species, a simple relationship from Ref. [102] is applied, whereas for individual ions, the extended Debye-Hiickel model is used according to Ref. [103]. 

The correlation between activity coefficient and ionic strength can be deduced from the quantitative relationships of the Debye-Hiickel-Onsager theory. Without giving details of this deduction it is interesting to quote the final result ... 

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