Extended Debye-Hiickel theory

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 ... 

Thermodynamic properties in dilute aqueous solutions are taken to be functions of ionic strength so that concentrations of reactants, rather than their activities can be used. This also means that pHc = — log[H+] has to be used in calculations, rather than pHa = — log a(H + ). When the ionic strength is different from zero, this means that pH values obtained in the laboratory using a glass electrode need to be adjusted for the ionic strength and temperature to obtain the pH that is used to discuss the thermodynamics of dilute aqueous solutions. Since pHa = — log-/(H + ) + pHc, the use of the extended Debye-Hiickel theory yields... 

The thermodynamic model of Krissmann [53] was used in the calculations of these experiments, though this was limited by the phase equilibrium (Eq. (3)) and the reaction equilibrium (Eq. (4)). Calculation of the activity coefficients of the H+ ions and HSOj was performed according to the extended Debye-Hiickel theory, using the approximation of Pitzer... 

The values of /3 for each C, were obtained by iteration from the total thermodynamic equilibrium constants based on conductivity measurements. The activity coefficients f2- and / were evaluated from the extended Debye-Hiickel theory. 

Use the extended Debye-Hiickel theory to estimate the mean ionic activity coefficient for Na2S04 at concentrations of 0.01 and 0.1 M and 25°C assuming an ion size parameter of 400 pm. Also calculate the mean electrolyte activity and the electrolyte activity. 

The diffusion coefficient of Na+ is 1.34 x 10 m s at infinite dilution in water at 25°C. Estimate its value for a Na+ concentration of 0.1 M in a solution of the same ionic strength. Use the extended Debye-Hiickel theory to estimate the concentration dependence of the activity coefficient assuming that the ion size parameter is 400 pm. 

The Extended Debye-Hiickel Theory. The Debye-Hitckel theory is successful in accounting for the experimental results when its application is limited to solutions in which the ratio of the electrical to the thermal energy of the ions is very smalt, i.e.,... 

When applicable, the extended Debye-Hiickel theory of Gronwail, LaMer and Sandved,21r although rather laborious, furnishes a means for the determination of Eo values. Its use may be illustrated by the work of Cowperthwaite and LaMer 22 on zinc sulphate, using a cell of type (25). Their data are given in Table II. Due to the fact that this... 

The following relation between the activity coefficient and ionic strength J is derived from the extended Debye-Hiickel theory for dissolved ions. 

Bromley, L.A., "Approximate individual ion values of 3 (or B) in extended Debye-Hiickel theory for uni-univalent aqueous solutions at 298.15K", J. Chem. Thermo., v4. pp669-673 (1972)... 

Fig. B.2 The variation of the mean activity coefficient with ionic strength according to the extended Debye-Hiickel theory, (a) The limiting law for a 1,1-electrolyte, (b) The extended law with B = 0.5. (c) The extended law, extended further by the addition of a term CI-, in this case with C = 0.02. The last form of the law reproduces the observed behavior reasonably well.

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... 

Normally, the validity of the Debye—Hiickel theory extends little further than kR <1. At room temperature, this requires ionic concentrations < 0.1 mol dm-3 for univalent ions in water, 0.03moldm-3 for univalent ions in ethanol or <0.01 mol dm-3 for univalent ion in ethers. In these cases, ions may be regarded as point particles and the strong repulsive core potential ignored. Furthermore, the time taken for non-reactive ions to diffuse far enough to establish an ionic-atmosphere around an ion, which was suddenly formed in solution containing only univalent ions, is... 

Issue is taken here, not with the mathematical treatment of the Debye-Hiickel model but rather with the underlying assumptions on which it is based. Friedman (58) has been concerned with extending the primitive model of electrolytes, and recently Wu and Friedman (159) have shown that not only are there theoretical objections to the Debye-Hiickel theory, but present experimental evidence also points to shortcomings in the theory. Thus, Wu and Friedman emphasize that since the dielectric constant and relative temperature coefficient of the dielectric constant differ by only 0.4 and 0.8% respectively for D O and H20, the thermodynamic results based on the Debye-Hiickel theory should be similar for salt solutions in these two solvents. Experimentally, the excess entropies in D >0 are far greater than in ordinary water and indeed are approximately linearly proportional to the aquamolality of the salts. In this connection, see also Ref. 129. 

In concluding this section, we note that the physical arguments here are involved for a subject (the Debye-Hiickel theory) that is at once so basic, so firmly established, and so limited in physical scope to molecular science. The traditional presentation (e.g. Hill, 1986 Lewis etal., 1961, see Section 23) is fine as far as it goes but gives little support for extensions of the theory, and little perspective on the basic issues of the theory of solutions. The physical discussion here is different from the most conventional presentations, does give further perspective on the role of the PDT, but is too extended without other pieces of the theory of solutions in place. All these points surely mean that this is one area where the beautiful but more esoteric theoretical tools (Lebowitz et al, 1965) of professional theory of liquids are relevant to a simple view of the problem. This topic is taken up again after the developments of Section 6.1 see Eq. (6.28), p. 132. 

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. 

In summary, Onsager s extension of the Debye-Hiickel theory to the nonequilibrium properties of electrolyte solutions provides a valuable tool for deriving single ion properties in electrolyte solutions. Examination of the large body of experimental data for aqueous electrolyte solutions helped confirm the model for a strong electrolyte. In more recent years, these studies have been extended to non-aqueous solutions. Results in these systems are discussed in the following section. 

A modification of the Debye-Hiickel theory to include the possible formation of ion pairs was suggested as early as 1926 by Bjerrum.22 As has been shown in Chapter 7, the unmodified form of the Debye-Hiickel theory, leads, even with water solutions, to absurdities, for electrolytes with small ions, or salts of the higher valence types and the extended theory was necessary to account for these cases. 

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]... 

Upon comparing Pitzer-theory calculations for typical scrubber and model solutions with the association-equilibrium, extended Debye-Hiickel code in current use for FGD systems, one sees differences which reflect the differences in concentration range and applicability to mixtures of the two approaches. 

His treatment extended the range of the theory to ionic strengths of 0.1 molal, and showed quite unambiguously that values predicted by the numerical integration for 1-1 electrolytes fit almost exactly those predicted by the Debye-Hiickel theory. It follows that the Debye-Hiickel expression is an excellent base-line for describing the properties of 1-1 electrolyte solutions. 

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] ... 

Nonelectrolyte G mcxlels only account for the short-range interaction among non-charged molecules (—One widely used G model is the Non-Random-Two-Liquid (NRTL) theory developed in 1968. To extend this to electrolyte solutions, it was combined with either the DH or the MSA theory to explicitly account for the Coulomb forces among the ions. Examples for electrolyte models are the electrolyte NRTL (eNRTL) [4] or the Pitzer model [5] which both include the Debye-Hiickel theory. Nasirzadeh et al. [6] used a MSA-NRTL model [7] (combination of NRTL with MSA) as well as an extended Pitzer model of Archer [8] which are excellent models for the description of activity coefficients in electrolyte solutions. Examples for electrolyte G models which were applied to solutions with more than one solvent or more than one solute are a modified Pitzer approach by Ye et al. [9] or the MSA-NRTL by Papaiconomou et al. [7]. However, both groups applied ternary mixture parameters to correlate activity coefficients. Salimi et al. [10] defined concentration-dependent and salt-dependent ion parameters which allows for correlations only but not for predictions or extrapolations. 

Instead of taking the ions as point charges, as done above, the Debye-Hiickel theory can be extended to ions with finite sizes. Let each of the ions (cations and anions) be modeled as a hard sphere of diameter a with its charge located... 

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