Debye-Huckel model

The mean activity coefficient is the standard form of expressing electrolyte data either in compilations of evaluated experimental data such as Hamer and Wu (2) or in predictions based on extensions to the Debye-Huckel model of electrolyte behavior. Recently several advances in the prediction and correlation of mean activity coefficients have been presented in a series of papers starting in 1972 by Pitzer (3, Meissner 04), and Bromley (5) among others. 

As the basis for predicting ionic activity coefficients we chose to adopt an. empirical modification of Bromley s ( 5) extension of the Debye-Huckel model. The mean activity coefficient of a pure salt in water is given by... 

These equations do not reduce to the Debye-Huckel model for dilute solutions and are thus only justified for the treatment of very concentrated solutions. 

The discussion above is a description of problem that requires answers to the following (1) the determination of the distribution of ions around a reference ion, and (2) the determination of the thickness (radius) of the ionic atmosphere. Obviously this is a complex problem. To solve this problem Debye and Huckel used a rather general approach they suggested an oversimplified model in order to obtain an approximate solutions. The Debye-Huckel model has two basic assumptions. The first is continuous dielectric assumption. In this assumption water (or the solvent) is a continuous dielectric and is not considered to be composed of molecular species. The second, is a continuous charge distribution in the ionic atmosphere. Put differently, charges of the ions in the ionic surrounding atmosphere are smoothened out (continuously distributed). 

The Debye-Huckel model for ion-ion interaction yields the following equation for the activity coefficient of species i ... 

DCI. see Desorption chemical ionization Debye-Huckel model in interface studies, 625-626... 

Table 5 Comparison of the critical radius Rc (in au), n (/cm3) and critical pressure Pc (in atm) between the results obtained by using a Debye-Huckel model and an Ion-Sphere (IS) model. Reprinted with permission from [203] Copyright 2006, John Wiley Sons, Inc.

Fig. 3.7. The Debye-Huckel model is based upon selecting one ion as a reference ion, replacing the solvent molecules by a continuous medium of dielectric constant e and the remaining ions by an excess charge density (the shading usually used in this book to represent the charge density is not indicated in this figure).

Q.l5.5 List three assumptions of the Debye-Huckel model. 

A plot of the data according to equation (9.5.6) is shown in fig. 9.3. It is apparent that it would be very difficult to determine E° on the basis of this plot, which shows curvature even in the lowest concentration range. Under these circumstances it is better to use the extended Debye Huckel model to perform the necessary extrapolation. Then equation (9.5.5) becomes... 

DERIVAnON OF THE DEBYE-HUCKEL THEORY FROM THE SIMPLE DEBYE-HUCKEL MODEL 363... 

Meanwhile, it is constmctive to look again at the physical basis of the simple Debye-Hiickel model and its mathematical development to see where both could be modified, and to consider whether this would be mathematically possible. What has been written in Chapter 1 on ions and solvent structure shows that the Debye-Huckel model is painfiiUy naive and cannot even approach physical reality. A brief reassessment of the features 1-7 of the simple Debye-Hiickel model is given below, along with indications as to how these problems have been tackled. 

Some other kinds of models have shown parameters that seem to follow useful correlation relationships. Among these are the virial coefficient model of Bums (2), the interaction coefficient model of Helgeson, Kirkham, and Flowers (4), and the hydration theory model of Stokes and Robinson (1). The problem shared by all three of these models is that they employ individual ion size parameters in the Debye-Hiickel submodel. This led to restricted applicability to solutions of pure aqueous electrolytes, or thermodynamic inconsistencies in applications to electrolyte mixtures. Wolery and Jackson (in prep.) discuss empirical modification of the Debye-Huckel model to allow ion-size mixing without introducing thermodynamic inconsistencies. It appears worthwhile to examine what might be gained by modifying these other models. This paper looks at the hydration tlieory approach. 

The excess Gibbs Energy for a pure Debye-Huckel model is given by (see Wolery and Jackson, in prep.)... 

Note that the excess Gibbs energy carries an asterisk to denote that it pertains to the set of components with formally hydrated solutes. Thus, w w is the number of kilograms of "free" water (w w = WwD), and I is the molal ionic strength, calculated in the usual way but with the nominal molalities replaced by molalities corrected for hydration (thus I = I/D). If it were desired to place less reliance on the Debye-Huckel model, other terms could be added to the right hand side in the usual way. We will presently ignore this as we are immediately interested in the hydration correction itself. 

In the simplest Debye-Huckel model, the activity coefficient y, is given by... 

Bearing in mind these limitations on the Debye—HUckel model of electrolytes, the influence of ionic concentration on the rate coefficient for reaction of ions was solved numerically by Logan [54, 93] who evaluated the integral of eqn. (56) with the potential of eqn. (55). He compared these numerical values with the predictions of the Bronsted— Bjerrum correction to the rate of a reaction occurring between ions surrounded by equilibrated ionic atmospheres, where the reaction of encounter pairs is rate-limiting... 

Based on the simplified Debye-Huckel model, if you take the mean activity coefficient... 

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