Debye-Huckel activity coefficients

Single-ion activity coefficients, DEBYE-HUCKEL TREATMENT... 

This standard textbook on chemical thermodynamics contains an Appendix (no. 4) of selected data for aqueous electrolyte solutions. Compiled are activity coefficients, Debye-Huckel parameters, relative partial molar enthalpies, and relative partial molar heat capacities for about 70 of the most common electrolytes in aqueous solution at 25 °C. More recent Debye-HUckel parameters are to be found 1n the pages of Pitaer, Pelper, and Busey and of Bradley and Pitaer (see item [121]) and the paper of Clarke and Glew, Hem [223. 

E. 7.2 Electrostatic Field due to Distributed Ions and Activity Coefficient Debye-HUckel Theory... 

As a result of these electrostatic effects aqueous solutions of electrolytes behave in a way that is non-ideal. This non-ideality has been accounted for successfully in dilute solutions by application of the Debye-Huckel theory, which introduces the concept of ionic activity. The Debye-Huckel Umiting law states that the mean ionic activity coefficient y+ can be related to the charges on the ions, and z, by the equation... 

If the activity coefficients are estimated from the Debye-Huckel theory in dilute regions of simple electrolyte systems, we have for aqueous solutions at 25 °C,... 

Figure 9.6 Mean ionic activity coefficients for HCl(aq) at T = 298.15 K obtained from the emf results of G. A. Linhart, J. Am. Chem. Soc.. 41, 1175-1180 (1919). The dashed line is the Debye-Huckel limiting law prediction.

Provided the ionic strength is not too high, this equation is obeyed as well as (but no better than) the Debye-Huckel equation for activity coefficients. One can expect deviations at higher ionic strength, and they are in general more serious the higher... 

Changes in activity coefficients (and hence the relationship between concentration and chemical activity) due to the increased electrostatic interaction between ions in solution can be nicely modeled with well-known theoretical approaches such as the Debye-Huckel equation ... 

The question of the relationship between activity and concentration arises. Here the Debye-Huckel theory of activity coefficients, although valid only below 0.01 M, has proved to be most helpful, either for establishing an acid concentration from its H+ activity or for calculating H+ activity from its previously known acid concentration. 

Although it is not possible to measure an individual ionic activity coefficient,, it may be estimated from the following equation of the Debye-Huckel theory ... 

At moderate ionic strengths a considerable improvement is effected by subtracting a term bl from the Debye-Huckel expression b is an adjustable parameter which is 0.2 for water at 25°C. Table 8.4 gives the values of the ionic activity coefficients (for z, from 1 to 6) with a taken to be 4.6A. 

Fig. 4.5. Convergence of the iteration to very small residuals in the reduced basis method, for systems contrived to have large positive and negative residuals at the start of the iteration. The tests assume Debye-Huckel activity coefficients ( ) and an ideal solution in which the activity coefficients are unity (o).

Fig. 8.1. Activity coefficients y, predicted at 25 °C for a singly charged ion with size a of 4 A, according to the Debye-Huckel (Eqn. 8.2), Davies (Eqn. 8.4), and B-dot (Eqn. 8.5) equations. Dotted line shows the Davies equation evaluated with a coefficient of 0.2 instead...

And (b) the extended Debye-Huckel equation for the approximation of the activity coefficient yj of the j-th ion. It needs the charge Zi and the ionic radius ay... 

The local compostion model is developed as a symmetric model, based on pure solvent and hypothetical pure completely-dissociated liquid electrolyte. This model is then normalized by infinite dilution activity coefficients in order to obtain an unsymmetric local composition model. Finally the unsymmetric Debye-Huckel and local composition expressions are added to yield the excess Gibbs energy expression proposed in this study. 

The original Debye-Huckel expression for the calculation of the activity coefficients of ions is... 

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... 

The statistical thermodynamic approach of Pitzer (14), involving specific interaction terms on the basis of the kinetic core effect, has provided coefficients which are a function of the ionic strength. The coefficients, as the stoichiometric association constants in our ion-pairing model, are obtained empirically in simple solutions and are then used to predict the activity coefficients in complex solutions. The Pitzer approach uses, however, a first term akin to the Debye-Huckel one to represent nonspecific effects at all concentrations. This weakens somewhat its theoretical foundation. 

Very few generalized computer-based techniques for calculating chemical equilibria in electrolyte systems have been reported. Crerar (47) describes a method for calculating multicomponent equilibria based on equilibrium constants and activity coefficients estimated from the Debye Huckel equation. It is not clear, however, if this technique has beep applied in general to the solubility of minerals and solids. A second generalized approach has been developed by OIL Systems, Inc. (48). It also operates on specified equilibrium constants and incorporates activity coefficient corrections for ions, non-electrolytes and water. This technique has been applied to a variety of electrolyte equilibrium problems including vapor-liquid equilibria and solubility of solids. 

With the above considerations in mind, Debye and HUckel devised a means of calculating the ionic strength I and the mean ionic activity coefficient y . Before we can progress, however, we must define the extent to which a solute promotes association, and thus screening. The parameter of choice is the ionic strength /, which is defined formally as follows ... 

Knowing (Zn ) from the Nemst equation and the activity coefficient y for the zinc cation from the Debye-Huckel law, we can therefore calculate the concentration, [ZnS04], from equation (3.10). 

Sulfuric acid is a 2 1 electrolyte, and so (by using the data in Table 3.1) the ionic strengthlis three times the concentration, i.e.l = 0.03 moldm f Next, from the Debye-Huckel extended law equation (3.15), we can obtain the mean ionic activity coefficient y as follows ... 

While this relationship is simple, it introduces more errors because the activity coefficient (or more normally, the mean ionic activity coefficient y ) is wholly unknown. While y can sometimes be calculated (e.g. via the Debye-Huckel relationships described in Section 3.4), such calculated values often differ quite significantly from experimental values, particularly when working at higher ionic strengths. In addition, ionic strength adjusters and TISABs are recommended in conjunction with calibration curves. 

The activity a and concentration c are related by a = (c/c ) x y (equation (3.12)), where y is the mean ionic activity coefficient, itself a function of the ionic strength /. Approximate values of y can be calculated for solution-phase analytes by using the Debye-Huckel relationships (equations (3.14) and (3.15)). The change of y with ionic strength can be a major cause of error in electroanalytical measurements, so it is advisable to buffer the ionic strength (preferably at a high value), e.g. with a total ionic strength adjustment buffer (TISAB). 

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... 

The Pitzer virial coefficient method, see section 6.2.2. Methods 1 and 2 are equivalent and differ only in the form of the denominator in the Debye-Huckel term. Method 3 requires more parameters for the description of the activity factors. These parameters are not available in many cases. This is generally the case for complex formation reactions. 

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

I. 46. The magnitude of the coefficient reflects the electric charge distribution of the ionic species. A 0.1 molal solution of Al2(S04)3 has an activity coefficient of only 0.035. It should also be noted that, in dilute solutions, activity coefficients of electrolytes decrease in magnitude with increasing concentration. A minimum is reached and the coefficient then increases with concentration. See Activity Debye-Huckel Law Biomineralization... 

An equation for estimating how the activity coefficient of an ion in dilute solutions is influenced by ionic strength. See Debye-Huckel treatment... 

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