Debye-Huckel relationship

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). 

Taking ionic strength into consideration, the modified dissociation constant pK may be calculated by the Debye-Huckel relationship ... 

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

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. 

Debye-Huckel effects are significant in the dilute range and are not considered, and (2) the usual composition scale for the solute standard state is molality rather than mole fraction. Both of these problems have been overcome, and the more complex relationships are being presented elsewhere (17). However, for most purposes, the virial coefficient equations for electrolytes are more convenient and have been widely used. Hence our primary presentation will be in those terms. 

The Debye-Huckel theory was developed to extend the capacitor model and is based on a simplified solution of the Poisson equation. It assumes that the double layer is really a diffuse cloud in which the potential is not a discontinuous function. Again, the interest is in deriving an expression for the electrical potential function. This model states that there is an exponential relationship between the charge and the potential. The distribution of the potential is ... 

The relationship between y and / for the extended Debye-Huckel and the Davis equation for selected ions is shown in Fig. 5.1. 

Abstract, The solution of the Debye-Huckel equation for a system of spheres with arbitrary radii and surface charge in electrolyte solutions is described. The general theoretical approach to describe such systems is elaborated. The practically important case of two spheres is considered in detail. Finite closed formulae to calculate the interaction energy of two spherical particles with constant surface charges are obtained from general expressions in zero approximation. Known relationships follow from our formulae in limiting cases. 

As the dependency does not include any specific property of the ion (in particular its chemical identity) but only its charge the explanation of this dependency invokes properties of the ionic cloud around the ion. In a similar approach the Debye-Huckel-Onsager theory attempts to explain the observed relationship of the conductivity on c1/2. It takes into account the - electrophoretic effect (interactions between ionic clouds of the oppositely moving ions) and the relaxation effect (the displacement of the central ion with respect to the center of the ionic cloud because of the slightly faster field-induced movement of the central ion, - Debye-Falkenhagen effect). The obtained equation gives the Kohlrausch constant ... 

The simplest theoretical relationship between I and yi is expressed in the Debye Huckel (DH) Equation (3.3). The DH model assumes that ions can be represented as point charges, i.e. of infinitely small radius, and that long-range coulombic forces between ions of opposite charge are responsible for the differences between the observed chemical behaviour, i.e. activity, and the predicted behaviour on the basis of solute concentration. 

Electromotive force measurements of the cell Pt, H2 HBr(m), X% alcohol, Y% water AgBr-Ag were made at 25°, 35°, and 45°C in the following solvent systems (1) water, (2) water-ethanol (30%, 60%, 90%, 99% ethanol), (3) anhydrous ethanol, (4) water-tert-butanol (30%, 60%, 91% and 99% tert-butanol), and (5) anhydrous tert-butanol. Calculations of standard cell potential were made using the Debye-Huckel theory as extended by Gronwall, LaMer, and Sandved. Gibbs free energy, enthalpy, entropy changes, and mean ionic activity coefficients were calculated for each solvent mixture and temperature. Relationships of the stand-ard potentials and thermodynamic functons with respect to solvent compositions in the two mixed-solvent systems and the pure solvents were discussed. 

In view of these uncertainties, it may prove more advantageous provisionally to work with an empirical relationship between activity coefficients and ionic strength of more concentrated solutions, instead of the Debye-Huckel equation. Bjerrum has found from experience that the following equation holds within wide limits ... 

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. 

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-Huckel 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. 

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. 

According to the Smoluchowski theory (Hunter 1981,1993), there is a linear relationship between the electrophoretic mobility and the potential U =At, where A is a constant for a thin EDL at Kfl 1 (where a denotes the particle radius and k is the Debye-Huckel parameter). For a thick EDL (Kfl< 1), e.g., at pH close to the isoelectric point (lEP), the equation with the Henry correction factor is more appropriate ... 

All the assumptions involved in the Debye-Huckel approximation (DH) are incorporated in this model because approximations have been used to establish the relationship between the electrostatic transfer constant Kj and (Tb/Tf)j (where Tj is the activity coefficient of the j-th species) in terms of measurable parameters. 

P. Debye and E. Huckel succeeded, in 1923, in deriving the relationship between the different coefficients, on the one hand, and the concentration, valence, and specific nature (diameter) of the ions on the other. At present, this very complicated problem is not yet completely solved. Debye and Huckel found it necessary to make certain approximations in order to simplify their complicated mathematical expressions. These simplifications were shown by Gronwall, La Mer and Sandved to introduce rather large deviations, especially for polyvalent ions. The equations of Debye and... 

It is interesting to note that the simple relationship used by Kohlrausch to obtain Aq values has been subjected to considerable theoretical study and correction, notably by Peter Debye (1884-1966) and Erich HUckel (1896-1980) and by Lars Onsager (1903-1976). 

T is the temperature in degrees Kelvin. After some rather simple algebra, Debye and Huckel could prove that the logarithm of the activity coefficients (T ) should vary linearly with the square root of the ionic strength (I) in the very dilute regime, and in general, the relationship between ionic strength would be... 

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