Debye-Hiickel theory extended equation

Experience shows that solutions of other electrolytes behave in a manner similar to the examples we have used. The conclusion we reach is that the Debye-Hiickel equation, even in the extended form, can be applied only at very low concentrations, especially for multivalent electrolytes. However, the behavior of the Debye-Hiickel equation as we approach the limit of zero ionic strength appears to give the correct limiting law behavior. As we have said earlier, one of the most useful applications of Debye-Hiickel theory is to... 

Edwards et al. (6) made the assumption that was equal to 4>pure a at the same pressure and temperature. Further theyused the virial equation, truncated after the second term to estimate pUre a These assumptions are satisfactory when the total pressure is low or when the mole fraction of the solute in the vapor phase is near unity. For the water, the assumption was made that <(>w, , aw and the exponential term were unity. These assumptions are valid when the solution consists mostly of water and the total pressure is low. The activity coefficient of the electrolyte was calculated using the extended Debye-Hiickel theory ... 

The electrostatic methods just discussed suitable for nonelectrolytic solvent. However, both the GB and Poisson approaches may be extended to salt solutions, the former by introducing a Debye-Huckel parameter and the latter by generalizing the Poisson equation to the Poisson-Boltzmann equation. The Debye-Huckel modification of the GB model is valid to much higher salt concentrations than the original Debye-Hiickel theory because the model includes the finite size of the solute molecules. 

Equation (26.41) predicts to within approximately 10% mean molal activity coefficients for salt concentrations up to 0.1 molal. The more accurate form of the activity coefficient equation [Equation (26.40)] allows the model to be extended to salt concentrations up to 0.5 molal. To expand the applicability of the Debye-Hiickel theory to higher concentrations, additional terms are added to Equation (26.40), such as [4]... 

With increasing electrolyte concentration, the short-range interactions become more and more dominating. Therefore, in activity coefficient models the Debye-Hiickel term, which describes the long-range interactions, has to be extended by a term describing the short-range interactions. A well-known empirical extension of the Debye-Hiickel theory is the Bromley equation [5] ... 

Solutions to the Poisson—Boltzmann equation in which the exponential charge distribution around a solute ion is not linearized [15] have shown additional terms, some of which are positive in value, not present in the linear Poisson—Boltzmann equation [28, 29]. From the form of Eq. (62) one can see that whenever the work, q yfy - yfy), of creating the electrostatic screening potential around an ion becomes positive, values in excess of unity are possible for the activity coefficient. Other methods that have been developed to extend the applicable concentration range of the Debye—Hiickel theory include mathematical modifications of the Debye—Hiickel equation [15, 26, 28, 29] and treating solution complexities such as (1) ionic association as proposed by Bjerrum [15,25], and(2) quadrupole and second-order dipole effects estimated by Onsager and Samaras [30], etc. 

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. 

The need for an analytical expressions for the equation of state have led to a revival of the macroscopic electrostatic theory due to Debye, Hiickel and Bjerrum. DH theory becomes exact for large particles. In pilot work by Fisher and Levin (FL) [31], DH-Bj theory is extended by considering the interactions of the pairs with the free ions. Weiss and Schroer (WS) [32] have supplemented this theory accounting for dipole-dipole interactions between pairs and the e-dependence of the association constant. 

General theory devoted to calculation of pH and distribution of species in particular buffer mixtures can be foimd in [20, 67-70]. Chemical formirlation includes a number of the mass-action equations linked with the mass and charge conservation equations. Limiting or extended forms of the Debye-Hiickel expressions for activity... 

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